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Optics part i

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The document outlines key concepts in optics, including Maxwell's equations, boundary conditions, optical aberrations, and various optical theories. It covers wave equations, geometrical and wave optics, diffraction, and image quality metrics, providing derivations for fundamental principles in optics. The document serves as a comprehensive reference for understanding the fundamentals of optical science and electromagnetic theory.

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1 OPTICS Part I SOLO HERMELIN Updated: 16.01.10http://www.solohermelin.com
2 Table of Content SOLO OPTICS Maxwell’s Equations Boundary Conditions Electromagnatic Wave Equations Monochromatic Planar Wave Equations Spherical Waveforms Cylindrical Waveforms Energy and Momentum Electrical Dipole (Hertzian Dipole) Radiation Reflections and Refractions Laws Development Using the Electromagnetic Approach IR Radiometric Quantities Physical Laws of Radiometry Geometrical Optics Foundation of Geometrical Optics – Derivation of Eikonal Equation The Light Rays and the Intensity Law of Geometrical Optics The Three Laws of Geometrical Optics Fermat’s Principle (1657)
3 Table of Content (continue) SOLO OPTICS Plane-Parallel Plate Prisms Lens Definitions Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law Derivation of Lens Makers’ Formula First Order, Paraxial or Gaussian Optics Ray Tracing Matrix Formulation
4 Table of Content (continue) SOLO OPTICS Optical Diffraction Fresnel – Huygens’ Diffraction Theory Complementary Apertures. Babinet Principle Rayleigh-Sommerfeld Diffraction Formula Extensions of Fresnel-Kirchhoff Diffraction Theory Phase Approximations – Fresnel (Near-Field) Approximation Phase Approximations – Fraunhofer (Near-Field) Approximation Fresnel and Fraunhofer Diffraction Approximations Fraunhofer Diffraction and the Fourier Transform Fraunhofer Diffraction Approximations Examples Resolution of Optical Systems Optical Transfer Function (OTF) Point Spread Function (PSF) Modulation Transfer Function (MTF) Phase Transfer Function (PTF) Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function Other Metrics that define Image Quality – Srahl Ratio Other Metrics that define Image Quality - Pickering Scale Other Metrics that define Image Quality – Atmospheric Turbulence Fresnel Diffraction Approximations Examples O P T I C S P a r t I I
5 Table of Content (continue) SOLO OPTICS References Optical Aberration Monochromatic Seidel Aberrations Chromatic Aberration Interference O p t i c s P a r t I I
6 OpticsSOLO Hierarchy of Optical Theories • Quantum Light as particle (photon) Emission, absorption, interaction of light and matter • Electromagnetic Maxwell’s Equations Reflection/Transmission, polarization • Scalar Wave Light as wave Interference and Diffraction • Geometrical Light as ray Image-forming optical systems λ → 0
7 OpticsSOLO Hierarchy of Optical Theories
8 MAXWELL’s EQUATIONSSOLO SYMMETRIC MAXWELL’s EQUATIONS Magnetic Field IntensityH   1  mA Electric DisplacementD   2  msA Electric Field IntensityE   1  mV Magnetic InductionB   2  msV Electric Current DensityeJ   2 mA Free Electric Charge Distributione  3  msA Fictious Magnetic Current DensitymJ   2 mV Fictious Free Magnetic Charge Distributionm  3  msV 1. AMPÈRE’S CIRCUIT LW (A) eJ t D H        2. FARADAY’S INDUCTION LAW (F) mJ t B E        3. GAUSS’ LAW – ELECTRIC (GE) eD   4. GAUSS’ LAW – MAGNETIC (GM) mB   Although magnetic sources are not physical they are often introduced as electrical equivalents to facilitate solutions of physical boundary-value problems. André-Marie Ampère 1775-1836 Michael Faraday 1791-1867 Karl Friederich Gauss 1777-1855 James Clerk Maxwell (1831-1879)
9 SOLO The Electromagnetic Spectrum
10 SOLO Visible Spectrum
11 SOLO The Infrared (IR) Spectrum of Interest Return to TOC
12 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions 2 ˆt 1 ˆt h 2H  1H  1 2 C CS1P 2P 3P 4P bˆ 21 ˆ n ek      ldtHtHhldtHldtHldH h C 2211 0 2211 ˆˆˆˆ     where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ  nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba          ldbkldbHHnldnbHHldtHH e ˆˆˆˆˆˆ 21212121121    Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ   ekHHn   2121 ˆ              S e C Sd t D JdlH      dlbkbdlh t D JSd t D J e h e S e ˆˆ 0                             AMPÈRE’S LAW  1 0 lim:              mAh t D Jk e h e  
13 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 1) 2 ˆt 1 ˆt h 2E  1E  1 2 C CS1P 2P 3P 4P bˆ 21 ˆ n mk      ldtEtEhldtEldtEldE h C 2211 0 2211 ˆˆˆˆ     where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ  nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba          ldbkldbEEnldnbEEldtEE m ˆˆˆˆˆˆ 21212121121    Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ   mkEEn   2121 ˆ              S m C Sd t B JdlE      dlbkbdlh t B JSd t B J m h m S m ˆˆ 0                             FARADAY’S LAW  1 0 lim:              mVh t B Jk m h m  
14 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 2) h 2D  1D  1 2 21 ˆ n dS 1 ˆn 2 ˆn e     SdnDnDhSdnDSdnDSdD h S 2211 0 2211 ˆˆˆˆ     where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and 21 ˆ,ˆ nn 2121 ˆˆˆ  nnn - a unit vector normal to the boundary between region (1) and (2)21 ˆ n     SdSdnDDSdnDD e 2121121 ˆˆ  Since this must be true for any dS on the boundary between regions (1) and (2) we must have:   eDDn  2121 ˆ    dSdShdv e h e V e  0  GAUSS’ LAW - ELECTRIC  1 0 lim:    msAhe h e    V e S dvSdD  
15 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 3) h 2B  1B  1 2 21 ˆ n dS 1 ˆn 2 ˆn m     SdnBnBhSdnBSdnBSdB h S 2211 0 2211 ˆˆˆˆ     where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and 21 ˆ,ˆ nn 2121 ˆˆˆ  nnn - a unit vector normal to the boundary between region (1) and (2)21 ˆ n     SdSdnBBSdnBB m 2121121 ˆˆ  Since this must be true for any dS on the boundary between regions (1) and (2) we must have:   mBBn  2121 ˆ    dSdShdv m h m V m  0  GAUSS’ LAW – MAGNETIC  1 0 lim:    msVhm h m    V m S dvSdB  
16 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (summary) 2 ˆt 1 ˆt h 22 ,HE  11,HE  1 2 C CS1P 2P 3P 4P bˆ 21 ˆ n me kk  , 21 ˆ n dS 11,BD  22,BD  me ,   mkEEn   2121 ˆ FARADAY’S LAW   ekHHn   2121 ˆ AMPÈRE’S LAW  1 0 lim:              mAh t D Jk e h e    1 0 lim:              mVh t B Jk m h m     eDDn  2121 ˆ  GAUSS’ LAW ELECTRIC  1 0 lim:    msAhe h e    mBBn  2121 ˆ  GAUSS’ LAW MAGNETIC  1 0 lim:    msVhm h m  Return to TOC
17 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS For Homogeneous, Linear and Isotropic Medium ED   HB   where are constant scalars, we have, J t E J t D H t t H t B E ED HB                               Since we have also tt            t J t E E DED EEE t J t E E                                               2 2 22 2 2 &
18 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS (continue 1) Define meme KK c KK v   00 11  where   smc /103 10 36 1 104 11 8 9700               is the velocity of light in free space. The absolute index of refraction n is me KK v c n     0 The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is t J t E v E              2 2 2 2 1
19 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS (continue 2) In the same way The Inhomogeneous Wave (Helmholtz) Differential Equation for the Magnetic Field Intensity is J t E J t D H t H t B E t ED HB                               Since are constant and tt      ,       J t H H HHB HHH J t H H                                     2 2 22 2 2 0&    J t H v H        2 2 2 2 1 Return to TOC
20 ELECTROMAGNETICSSOLO Monochromatic Planar Wave Equations Let assume that can be written as:   trHtrE ,,,             tjrHtrHtjrEtrE 00 exp,,exp,    where are phasor (complex) vectors.                rHjrHrHrEjrErE  ImRe,ImRe  We have          tjrEjtj t rEtrE t 00 expexp,         Hence                                               m e m e j t m e m e B D JBjE JDjH BGM DGE J t B EF J t D HA                )(
21 ELECTROMAGNETICSSOLO Fourier Transform The Fourier transform of can be written as:   trHtrE ,,,                                     dttjtrHrHdtjrHtrH dttjtrErEdtjrEtrE     exp,,&exp, 2 1 , exp,,&exp, 2 1 ,   This is possible if:                         drHdttrH drEdttrE 22 22 , 2 1 , , 2 1 ,   JEAN FOURIER 1768-1830
22 ELECTROMAGNETICSSOLO Note The assumption that can be written as:   trHtrE ,,,             tjrHtrHtjrEtrE 00 exp,,exp,    is equivalent to saying that has a Fourier transform; i.e.:   trHtrE ,,,                                           dtjrHtrHdttjtrHrH dtjrEtrEdttjtrErE exp, 2 1 ,&exp,, exp, 2 1 ,&exp,,   This is possible if:                         drHdttrH drEdttrE 22 22 , 2 1 , , 2 1 ,                                    00 0 exp expexpexp,, rEdttjrE dttjtjrEdttjtrErE   End Note
23 ELECTROMAGNETICSSOLO                        m e m e ED HB m e JHjE JEjH JHjE JEjH JBjE JDjH           me JJjEkE   2   em JJjHkH   2     22 f c c f k    Using the vector identity      AAA   For a Homogeneous, Linear and Isotropic Media:                         m e ED HB m e H E B D    e me JJjEkE   22    m em JJjHkH   22 and we obtain Monochromatic Planar Wave Equations (continue)
24 ELECTROMAGNETICSSOLO Assume no sources: we have Monochromatic Planar Wave Equations (continue) 0,0,0,0  meme JJ   022  EkE 022  HkH nkk n k 0 00 00 0                                    rktjtj rktjtj eHerHtrH eEerEtrE         0 0 ,, ,,   022     rkj rkjrkjrkjrkj ek ekkeekje    Helmholtz Wave Equations satisfy the Helmholtz wave equations    ,,, rHrE             rkj rkj eHrH eErE     0 0 , ,   Assume a progressive wave of phase  rkt   (a regressive wave has the phase ) rkt   For a Homogeneous, Linear and Isotropic Media k  0E 0H r  t k  Planes for which constrkt   
25 ELECTROMAGNETICSSOLO To satisfy the Maxwell equations for a source free media we must have: Monochromatic Planar Wave Equations (continue) we haveUsing: 1ˆˆ&ˆˆ  sss c n sk              0 0 H E HjE EjH                 0ˆ 0ˆ ˆ ˆ 0 0 00 00 Hs Es HEs EHs     sˆ Planar Wave 0E 0H r                    0 0 0 0 00 00 rkj rkj rkjrkj rkjrkj ekje eHkj eEkj eHjeEkj eEjeHkj rkjrkj                        0 0 0 0 00 00 Hk Ek HEk EHk       For a Homogeneous, Linear and Isotropic Media: Return to TOC
26 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS Spherical Waveforms z x y  r cosr  ,,rP   sinsinr  cossinr The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is t J t E v E              2 2 2 2 1 In spherical coordinates:    cos sinsin cossin rz ry rx    2 2 222 2 2 2 sin 1 sin sin 11                           rrr r rr For a spherical symmetric wave:    rErE  ,,  Er rrr E rr E r E r rr E    2 2 2 2 2 2 2 121                     
27 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS SourceSourceSource Spherical Waveforms z x y  r cosr  ,,rP   sinsinr  cossinr The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is assuming no sources   0 11 2 2 22 2          t E v Er rr In spherical coordinates:    cos sinsin cossin rz ry rx        0 1 2 2 22 2        Er tv Er r or: A general solution is:        wave regressive wave eprogressiv tvrFtvrFEr  21 0,0,0,0  meme JJ         r e EerEtrE rktj tj        0,, Assume a progressive monochromatic wave of phase  rkt   (a regressive wave has the phase ) rkt     r e ErE rkj     0, Return to TOC
28 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS Cylindrical Waveforms z x y r  zrP ,,  sinr cosr In cylindrical coordinates: zz ry rx      sin cos 2 2 2 2 2 2 11 zrr r rr                  For a cylindrical symmetric wave:    rEzrE  ,,              r E r rr E   12 The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is assuming no sources 0 11 2 2 2                  t E vr E r rr 0,0,0,0  meme JJ  
29 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS Source Cylindrical Waveforms z x y r  zrP ,,  sinr cosr SourceSource In cylindrical coordinates: zz ry rx      sin cos The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is assuming no sources 0 11 2 2 2                  t E vr E r rr 0,0,0,0  meme JJ   Assume a progressive monochromatic wave of phase  rkt   (a regressive wave has the phase ) rkt       tj erEtrE  ,,    0 1 2 2 2              E vr E rr E k  The solutions are Bessel functions which for large r approach asymptotically to:   rkj e r E rE   0 ,  Return to TOC
30 SOLO Energy and Momentum Let start from Ampère and Faraday Laws                      t B EH J t D HE e      EJ t D E t B HHEEH e              HEHEEH  But Therefore we obtain   EJ t D E t B HHE e             First way This theorem was discovered by Poynting in 1884 and later in the same year by Heaviside. ELECTROMAGNETICS John Henry Poynting 1852-1914 Oliver Heaviside 1850-1925
31 SOLO Energy and Momentum (continue -1) We identify the following quantities - Power density of the current densityEJe    HEDE t BH t EJe                     2 1 2 1           BH t pBHw mm  2 1 , 2 1           DE t pDEw ee  2 1 , 2 1  HEpR   eJ  - Magnetic energy and power densities, respectively - Electric energy and power densities, respectively - Radiation power density For linear, isotropic electro-magnetic materials we can write HBED  00 ,    DE tt D E ED             2 10  BH tt B H HB             2 10 ELECTROMAGNETICS
32 SOLO Energy and Momentum (continue – 3) Let start from the Lorentz Force Equation (1892) on the free charge  BvEF e    Free Electric Chargee  3  msA Velocity of the chargev   1  sm Electric Field IntensityE   1  mV Magnetic InductionB   2  msV Hendrik Antoon Lorentz 1853-1928 e Force on the free chargeF   Ne Second way ELECTROMAGNETICS
33 SOLO Energy and Momentum (continue – 4) The power density of the Lorentz Force the charge     EJBvEvp e Bvv Jv e ee       0   or             HE t B HE t D E t D HEEH E t D HEJp t B E HEHEEH J t D H e e                                                      e ELECTROMAGNETICS
34 SOLO Energy and Momentum (continue – 5)  HEDE t BH t EJe                     2 1 2 1 dve E  B  eJv  , V  FdF  Fd  Let integrate this equation over a constant volume V               VVVV e dvSdvDE td d dvBH td d dvEJ  2 1 2 1 If we have sources in V then instead of we must use E  source EE   Use Ohm Law (1826)  source ee EEJ               VV td d t Georg Simon Ohm 1789-1854 source e e EJE    1 For linear, isotropic electro-magnetic materials  HBED  00 ,   ELECTROMAGNETICS
35 SOLO Energy and Momentum (continue – 6)               VVVR n V source e dvSdvDE td d dvBH td d dRIdvEJ  2 1 2 12         V FieldMagnetic dvBH td d P  2 1         V FieldElectric dvDE td d P  2 1   SV Radiation SdSdvSP    V source eSource dvEJP          V source e R n V source e L S e ee V source e L S e ee V e dvEJdRI dvEJ dS dl dSJdSJdvEJldSdJJdvEJ   2 11   R nJoule dRIP 2 RadiationFieldMagneticFieldElectricJouleSource PPPPP  For linear, isotropic electro-magnetic materials  HBED  00 ,   R – Electric Resistance Define the Umov-Poynting vector:  2 / mwattHES   The Umov-Poynting vector was discovered by Umov in 1873, and rediscovered by Poynting in 1884 and later in the same year by Heaviside. ELECTROMAGNETICS John Henry Poynting 1852-1914
36 ElectromagnetismSOLO EM People John Henry Poynting 1852-1914 Oliver Heaviside 1850-1925 Nikolay Umov 1846-1915 1873 “Theory of interaction on final distances and its exhibit to conclusion of electrostatic and electrodynamic laws” 1884 1884 Umov-Poynting vector HES   The Umov-Poynting vector was discovered by Umov in 1873, and rediscovered by Poynting in 1884 and later in the same year by Heaviside. 1873 - 1884 Return to TOC
37 Note: Since there are not magnetic sources the Magnetic Hertz’s Vector Potential is : 0  m Electrical Dipole (Hertzian Dipole) RadiationSOLO Given a dipole monochromatic of electric charges defined by the Polarization Vector Intensity  tq  tq d  r  dqP   dr       tdqdeqaltP tj e  cosRe 00   we want to find the radiation properties. We start with the Helmholtz Non-homogeneous Differential Equation of the Electric Hertz’s Vector Potential : te       trPtr tc tr eee , 1 , 1 , 0 2 2 2 2       Heinrich Rudolf Hertz 1857-1894 - speed of propagation of the EM wave [m/s] 00 1  c - Polarization Vector IntensityeP   2  msA - Permitivity of space  2122   mNsA - Electric Hertz’s Vector Potential (1888)e   NsA   11 t A e      000  eV   0 Using the Electric Hertz’s Vector Potential we obtain : The field vectors are given by  e e tc V t A E           2 2 20 0 1 t AH e      00 0 1  
38 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1  Compute (continue-3)  e e tc E        2 2 2 1 We have                  32 0 4 0 2 5 0 2 2 2 2 44 3 4 31 rc rpr rc rprpr r rprrp tc E ee e                230 44 rc rp r rp t H e               r p tre 0 4 ,         krtjkrtj epedqp    00  Let use spherical coordinates         zyxr rrr 1111 cossinsincossin                     111 sincos00 rz krtjkrtj epepp                 krtj ep rccr j r rc rprrp rc rprrp r rprrp E rr                                 02 0 2 2 0 3 0 32 0 2 4 0 2 5 0 2 4 sin 4 sincos2 4 sincos2 44 3 4 3 11111  r1  1  1             pckpp pckjpjp 222      
39 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1  Using we can write              11 0 2 0 2 2 sin1 4 sin 44                  krtjkrtj ep rk j r kc ep rcr j H  krtj ep r k r kj r rccr j r E r rr                                                 0 2 23 0 2 0 2 2 0 3 0 111 11111 sinsincos2 1 4 1 4 sin 4 sincos2 4 sincos2 We can divide the zones around the source, as function of the relation between dipole size d and wavelength λ, in three zones: Near, Intermediate and Far Fields   22 :  c f c k The Magnetic Field Intensity is transverse to the propagation direction at all ranges, but the Electric Field Intensity has components parallel and perpendicular to .r1  r1  E  However and are perpendicular to each other.H  • Near (static) zone: rd • Intermediate (induction) zone: ~rd  • Far (radiation) zone: rd 
40 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1     102 sin 4   tj FieldNear ep r kc jH tj FieldNear ep r E r    03 0 11 sincos2 4 1         Near, Intermediate and Far Fields (continue – 1) • Near (static) zone: rd In the near zone the fields have the character of the static fields. The near fields are quasi-stationary, oscillating harmonically as , but otherwise static in character.tj e  0 2    r rk
41 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1       102 sin 4    krtj FieldteIntermedia ep r kc jH  krtj FieldteIntermedia ep r kj r E r                   023 0 11 sincos2 1 4 1 Near, Intermediate and Far Fields • Intermediate (induction) zone: ~rd  • Far (radiation) zone: rd       10 2 sin 4    krtj FieldFar ep r kc H       10 0 2 sin 4    krtj FieldFar ep r k E r1  FieldFarE FieldFarH At Far ranges are orthogonal; i.e. we have a transversal wave. rHE 1,,  In the Radiation Zone the Field Intensities behave like a spherical wave (amplitude falls off as r-1) 1 2    r rk              120 10 36 1 1041 : 9 7 0 0 1 0 00    c FieldFar FieldFar cH E Z
42 SOLO Electric Dipole Radiation http://dept.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_apr_07_2003.shtml#tth_sEc12.1 http://www.falstad.com/mathphysics.html Electric Field Lines of Force
43 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1  The phasors of the Magnetic and Electric Field Intensities are:       10 2 sin 4 1          krtj ep cr j r H  krtj ep crc j rc j rrr E r                            02 2 2 0 11 sin 11 cos 12 4 1 Poynting Vector of the Electric Dipole Field The Poynting Vector of the Electric Dipole Field is The Magnetic and Electric Field Intensities are:            1sincossin 4 2 0         krt c krt rr p HrealH                                                  11 sinsin 1 cos 1 cossincos 12 4 2 2 2 0 0 krt rc krt cr krt c krt rrr p ErealE r                                       1 1 cossincossinsincos 1 4 2 sincossinsincos 1 4 2 0 32 2 0 2 2 2 2 2 0 22 2 0                                   krt c krt r krt c krt rr p krt c krt r krt rc krt crr p HES r  The Poynting Vector of the Electric Dipole Field is given by:
44 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1  Let compute the time average < > of the Poynting vector: Poynting Vector of the Electric Dipole Field Using the fact that:                                      1 1 cossincossinsincos 1 4 2 sincossinsincos 1 4 2 0 32 2 0 2 2 2 2 2 0 22 2 0                                   krt c krt r krt c krt rr p krt c krt r krt rc krt crr p HES r     T T dttS T S 0 1 lim        2 1 2cos 1 lim 2 11 lim 2 1 cos 1 limcos 0 0 1 00 22       T T T T T T dtrkt T dt T dtrkt T rkt        2 1 2cos 1 lim 2 11 lim 2 1 sin 1 limsin 0 0 1 00 22       T T T T T T dtrkt T dt T dtrkt T rkt            02sin 1 lim 2 1 cossin 1 limcossin 0 00       T T T T dtrkt T dtrktrkt T rktrkt    r rc p S 1 2 23 0 2 42 0 sin 42                11 cossin 4 sin 1 42 22 0 32 2 02 2 2 2 2 2 2 0 22 2 0                     rcrcr p rccrcr p S r  we obtain: or: Radar Equation Irradiance
45 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1  Poynting Vector of the Electric Dipole Field   r rc p S 1 2 23 0 2 42 0 sin 42      Radar Equation 45 90 135 1800 0 5 10 15 20 25 30  0  45  90  135  180  225  270  315 z y 5.0 0.1 Polar Angle , in degrees RelativePower,indb The Total Average Radiant Power is:                0 22 23 0 2 42 0 sin2sin 42 dr rc p dSSP A rad  22 0 22 120 123 0 42 0 3/4 0 3 23 0 42 0 40 12 sin 16 0              p rc p d rc p P c c rad                3 4 3 2 3 2 cos 3 1 coscoscos1sin 0 3 0 2 0 3                  dd Return to TOC
46 ELECTROMAGNETICSSOLO To satisfy the Maxwell equations for a source free media we must have: Monochromatic Planar Wave Equations we haveUsing: 1ˆˆ&ˆˆ 0  kkknkkk              0 0 H E HjE EjH                  0ˆ 0ˆ ˆ ˆ 0 0 00 00 Hk Ek HEk EHk     kˆ Planar Wave 0E 0H r                    0 0 0 0 00 00 rkj rkj rkjrkj rkjrkj ekje eHkj eEkj eHjeEkj eEjeHkj rkjrkj               22 22 & 2 ˆ 2 ˆ HwEwwcn k wwcn k S meme    Time Average Poynting Vector of the Planar Wave Reflections and Refractions Laws Development Using the Electromagnetic Approach
47 SOLO REFLECTION & REFRACTION iE  iH  rE  rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t Plan of incidence Consider an incident monochromatic planar wave     c n k eEkH eEE iiii rktj iii rktj ii ii ii 1 00 11 0011 0 0                  The monochromatic planar reflected wave from the boundary is     1 1 1 1 0 0 & n c v vc n k eEkH eEE r rr rktj rrr rktj rr rr rr                The monochromatic planar refracted wave from the boundary is     2 2 2 2 0 0 & n c v vc n k eEkH eEE t tt rktj ttt rktj tt tt tt                Reflections and Refractions Laws Development Using the Electromagnetic Approach
48 SOLO REFLECTION & REFRACTION The Boundary Conditions at z=0 must be satisfied at all points on the plane at all times, implies that the spatial and time variations of This implies that iE  iH  rE  rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t Plan of incidence Phase-Matching Conditions       yxteEeEeE z rktj t z rktj r z rktj i ttrrii ,,,, 0 0 0 0 0 0                 yxtrktrktrkt z tt z rr z ii ,, 000     ttri         yxrkrkrk z t z r z i , 000    must be the same Reflections and Refractions Laws Development Using the Electromagnetic Approach
49 SOLO REFLECTION & REFRACTION tri nnn  sinsinsin 211  iE  iH  rE  rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t Plan of incidence Phase-Matching Conditions            zyx c n k zyx c n k ttttttt irirrrr ˆcossinˆsinsinˆcos ˆcossinˆsinsinˆcos 2 1                       yyx c n rk yx c n rk y c n rk ttt z t irr z r i z i ˆsinsincos sinsincos sin 2 0 1 0 1 0             yxrkrkrk z t z r z i , 000    2    tr ttri   x y Coplanar Snell’s Law         zzyyxxr zy c n k iiii ˆˆˆ ˆcosˆsin1    Given: Let find: Reflections and Refractions Laws Development Using the Electromagnetic Approach
50 SOLO REFLECTION & REFRACTION Second way of writing phase-matching equations ri   11 22 2 1 1 2 sin sin      v v n n t iRefraction Law Reflection Law Phase-Matching Conditions         zzyyxxr zy c n k iiii ˆˆˆ ˆcosˆsin1               zyx c n k zyx c n k ttttttt irirrrr ˆcossinˆsinsinˆcos ˆcossinˆsinsinˆcos 2 1                      ynnyn c kkz ynnyn c kkz ittrti irrrri ˆsinsinsinˆcosˆ ˆsinsinsinˆcosˆ 122 111       ttri   We can see that                tri tiri kkzkkz 0ˆˆ                  tri tri tr nnn sinsinsin 2/ 211 iE  iH  rE  rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t Plan of incidence Reflections and Refractions Laws Development Using the Electromagnetic Approach
51 SOLO REFLECTION & REFRACTION ri   11 22 2 1 1 2 sin sin      v v n n t iRefraction Law Reflection Law Phase-Matching Conditions (Summary) ttri                  tri tiri kkzkkz 0ˆˆ                  tri tri tr nnn sinsinsin 2/ 211 iE  iH  rE  rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t Plan of incidence       yxrkrkrk z t z r z i , 000          yxtrktrktrkt z tt z rr z ii ,, 000     Vector Notation Scalar Notation Reflections and Refractions Laws Development Using the Electromagnetic Approach
52 SOLO REFLECTION & REFRACTION iE  iH  rE rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t i r t tH  tE  tk  rH  rk  rE  iH  iE  ik  21 ˆ n Boundary Plan of incidence ti ti i r nn nn E E r         coscos coscos 2 2 1 1 2 2 1 1 0 0            ti i i t nn n E E t       coscos cos2 2 2 1 1 1 1 0 0           For most of media μ1= μ2 , and using refraction law: 1 2 sin sin n n t i       ti ti i r E E r               sin sin21 0 0  ti it i t E E t              sin cossin221 0 0 Assume is normal to plan of incidence (normal polarization) E  xEExEExEE ttrrii ˆ&ˆ&ˆ 000000    Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations See full development in P.P. “Reflection & Refractions”
53 SOLO REFLECTION & REFRACTION iE  iH  rE  rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t i r t tH  tE  tk  rH  rk  rE  iH  iE  ik  21 ˆ n Boundary Plan of incidence Assume is parallel to plan of incidence (parallel polarization) E       zyEE zyEE zyEE tttt rrrr iiii ˆsinˆcos ˆsinˆcos ˆsinˆcos 0||0 0||0 0||0          ti ti i r nn nn E E r         coscos coscos 1 1 2 2 1 1 2 2 ||0 0 ||          ti i i t nn n E E t       coscos cos2 1 1 2 2 1 1 ||0 0 ||         For most of media μ1= μ2 , and using refraction law: 1 2 sin sin n n t i       ti ti i r E E r             tan tan21 ||0 0 ||    titi it i t E E t            cossin cossin221 ||0 0 || Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations See full development in P.P. “Reflection & Refractions”
54 SOLO REFLECTION & REFRACTION ti ti i r nn nn E E r         coscos coscos 1 1 2 2 1 1 2 2 ||0 0 ||          ti i i t nn n E E t       coscos cos2 1 1 2 2 1 1 ||0 0 ||         ti ti i r nn nn E E r         coscos coscos 2 2 1 1 2 2 1 1 0 0            ti i i t nn n E E t       coscos cos2 2 2 1 1 1 1 0 0           The equations of reflection and refraction ratio are called Fresnel Equations, that first developed them in a slightly less general form in 1823, using the elastic theory of light. Augustin Jean Fresnel 1788-1827 The use of electromagnetic approach to prove those relations, as described above, is due to H.A. Lorentz (1875) Reflections and Refractions Laws Development Using the Electromagnetic Approach Hendrik Antoon Lorentz 1853-1928 See full development in P.P. “Reflection & Refractions” Return to TOC
55 IR Radiometric QuantitiesSOLO  RTA DA  2 cm  2 cm TARGET SOURCE DETECTOR RECEIVER Radiation Flux Power  W Spectral Radial Power          m W   Irradiance          2 mc W A E Spectral Radiant Emittance          mmc WM M   2 Radiant Intensity          str W I Spectral Radiant Intensity          mstr WI I   Radiance          strmc W A I L 2 cos Spectral Radiance          mstrmc WL L   2 Radiant Emittance          2 mc W A M Spectral Irradiance          mmc WE E   2    T T dttS T S 0 1 lim  Irradiance is the time- average of the Poynting vector Return to TOC
56 Physical Laws of RadiometrySOLO Plank’s Law   1/exp 1 2 5 1   Tc c M BB   Plank 1900 Plank’s Law applies to blackbodies; i.e. perfect radiators. The spectral radial emittance of a blackbody is given by:   KT KWk Wh kmc Kmkhcc mcmWchc    ineTemperaturAbsolute- constantBoltzmannsec/103806.1 constantPlanksec106260.6 lightofspeedsec/458.299792 10439.1/ 107418.32 23 234 4 2 4242 1           MAX PLANCK (1858 - 1947) Plank’s Law
57 Physical Laws of RadiometrySOLO Plank’s Law   1/exp 1 2 5 1   Tc c M BB   Plank 1900 Plank’s Law applies to blackbodies; i.e. perfect radiators. The spectral radial emittance of a blackbody is given by: MAX PLANCK (1858 - 1947) Plank’s Law
58 Physical Laws of Radiometry (Continue -1)SOLO Wien’s Displacement Law 0   d Md BB Wien 1893 from which: The wavelength for which the spectral emittance of a blackbody reaches the maximum is given by: m KmTm    2898 Wien’s Displacement Law Stefan-Boltzmann Law Stefan – 1879 Empirical - fourth power law Boltzmann – 1884 Theoretical - fourth power law For a blackbody:                             42 12 32 45 2 4 0 2 5 1 0 10670.5 15 2 : 1/exp 1 Kcm W hc k cm W Td Tc c dMM BBBB       LUDWIG BOLTZMANN (1844 - 1906) WILHELM WIEN (1864 - 1928) Stefan-Boltzmann Law JOSEF STEFAN (1835 – 1893)
59 Physical Laws of Radiometry (Continue -1a)SOLO Black Body Emittance M [W/m2] M (300ºK) 5.86 121 M (301ºK) - M (300ºK) 0.22 2 M (600ºK) 1,719 1,555 M (601ºK) - M (600ºK) 17 7 3 – 5 µm 8 - 12 µm
60 Physical Laws of Radiometry (Continue -2)SOLO Emittance of Real Bodies (Gray Bodies) For real (gray) bodies: BB MM   - Directional spectral emissivity is a measure of how closely the flux radiated from a given temperature radiator approaches that from a blackbody at the same temperature   ,   BB M M
61 Physical Laws of Radiometry (Continue -3)SOLO Kirchhoff’s Law rM iE aE tM Gustav Robert Kirchhoff 1824-1887 - Incident IrradianceiE - Absorbed IrradianceaE - Reflected Radiant ExcitancerM - Transmitted Radiant ExcitancetM Law of Conservation of Energy: trai MMEE      i t i r i a E M E M E E 11  i a E E : - fraction of absorbed energy (absorptivity) i r E M : - fraction of reflected energy (reflectivity) i t E M : - fraction of transmitted energy (transmissivity) Opaque body (no transmission): 01   Blackbody (no reflection or transmission): 0&01   Sharp boundary (no absorption): 01  
62 Physical Laws of Radiometry (Continue -4)SOLO Kirchhoff’s Law (Continue – 1) Gustav Robert Kirchhoff 1824-1887 Kirchhoff’s Law (1860) states that, for any temperature and any wavelength, the emissivity of an opaque body in an isothermal enclosure is equal to it’s absorptivity. This is because if the body will radiate to the surrounding less than it absorbs it’s temperature will rise above the surrounding and will be a transfer of energy from a cold surrounding to a hot body contradicting the second law of thermodynamics.    TT    222 ,, T 2A 111 ,, T 1A
63 Physical Laws of Radiometry (Continue -5)SOLO Lambert’s Law Johann Heinrich Lambert 1728 - 1777 http://www-groups.dcs.st-andrews.ac.uk/~history/Biographies/Lambert.html A Lambertian Surface is defined as a surface from which the radiance L [W/(cm2 str)] is independent of the direction of radiation.  dr    d rsin    2 sin r drdr d   A cosAAn   z x y   0 2 cos , L A L          coscos, 00 IALI     Lambert’s Law 0 2 0 2/ 0 00 sincoscos LddLdL A M         The Radiant Intensity from a Lambertian Surface is The Radiant Emittance (Exitance) from a Lambertian Surface is
64 Physical Laws of Radiometry (Continue -6)SOLO Transfer of Radiant Energy We have two bodies 1 and 2. The radiant power (radiance) transmitted from 1 to 2 is: 2 12 22 22 211 12 2 1 cos & cos R Ad d strcm W A L            1A 1dA 1 12R Radiating (Source) Surface 2A 2dA Receiving Surface 2 2d    2 12 22111 12 coscos R AdAdL d   The total radiant power (radiance) received at surface A2 from A1 is:   2 1 212 12 211 12 coscos A A AdAd R L 
65 Physical Laws of Radiometry (Continue -7)SOLO Transfer of Radiant Energy (Continue – 1) Define the projected areas: and the solid angles: 222111 cos&cos AdAdAdAd nn   1A 1dA 1 12R Radiating (Source) Surface 2A 2dA Receiving Surface 2 2d 2 12 22 22 12 11 1 cos & cos R Ad d R Ad d   1A 1dA 1 12R Radiating (Source) Surface 2A 2dA Receiving Surface 2 1d then:    2 12 211 2 12 22111 12 coscos R AdAdL R AdAdL d nn   12121112  dAdLdAdLd nn The Power is the product of the Radiance, the projected Area, and the Solid Angle using the other area.
66 Physical Laws of Radiometry (Continue -8)SOLO Transfer of Radiant Energy (Continue – 2) Optics R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, For an Optical System define: ATARGET – Target Area ADETECTOR – Detector Area AOPTICS – Optics Area R – Range from Target to Optics f – Focal Length (from Optics to Detector) ΩO,T – solid angle of Optics as seen from the Target 2, R AOPTICS TO  ωT,O – solid angle of Target as seen from the Optics 2, R ATARGET OT  ΩD,O – solid angle of Detector as seen from the Optics 2, f ADETECTOR OD  ωO,D – solid angle of Optics as seen from the Detector 2, f ADETECTOR DO 
67 Physical Laws of Radiometry (Continue -9)SOLO Transfer of Radiant Energy (Continue – 3) Optics (continue – 1) R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, For the Figure we can see that: ODOT ,,  22 f A R A DETECTORTARGET  Also we found that: DODETECTOROTOPTICS ODOPTICSTOTARGET ALAL ALAL OTOD ,, ,, ,,     
68 Physical Laws of Radiometry (Continue -10)SOLO Transfer of Radiant Energy (Continue – 4) Optics (continue – 2) R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, TOTARGETAL , ODOPTICSAL , OTOPTICSAL , DODETECTORAL ,
69 Physical Laws of Radiometry (Continue -11)SOLO Transfer of Radiant Energy (Continue – 5) Optics (continue – 3)   2 , R A AL AL TARGET DETECTOR DTDETECTOROpticsNo    2 , f A AL AL OPTICS DETECTOR DODETECTOROpticsWith    R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, R ATARGET ADETECTOR TD, DT , • IR Detector without Optics • IR Detector with Optics     2 # / 2 4 0 4 4 0# f AL f D AL DETECTOR Dff DETECTOR     The Optics increases the energy collected by the Detector since DTDO ,,   22 # 2 4 R A ff A TARGETOPTICS   OpticsNoOpticsWith 
70 Physical Laws of Radiometry (Continue -12)SOLO Targets The parts of the aircraft that are especially hot are: • The exhaust nozzle of the jet engine • The hot exhaust gas area, or the plume • The areas in which aerodynamic heating is the highest
71 Physical Laws of Radiometry (Continue -13)SOLO Targets
72 Physical Laws of Radiometry (Continue -14)SOLO Targets
73 Physical Laws of Radiometry (Continue -15)SOLO Targets
74 Physical Laws of Radiometry (Continue -16)
75 Physical Laws of Radiometry (Continue -17)SOLO Targets (continue – 1) • The exhaust nozzle of the jet engine The exhaust nozzle can be regarded as a gray body with ε = 0.9. Example: Turbojet Engine 4-P&W JT4A-9 2 3660 cmANOZZLE  rafterburnewithCT  538   24124 207.22735381067.59.0   cmWTM  We are interested only in the band 3 μm ≤ λ ≤ 5 μm. By numerically integration or using infrared radiation calculators we obtain: 397.0 811 4 5 3    KT BB T dM    Hence:   2 876.0207.2397.053   cmWmmM  In a tail-on situation the radiant intensity is:   1 1020 3660876.0 53           strWA M mmI NOZZLE Lambertian  
76 Physical Laws of Radiometry (Continue -18)SOLO Targets (continue – 2) • The plume The plume is characterized by the radiant emittance of the hot gases that are expanding into the atmosphere after passing through the exhaust nozzle. The products of combustion are H2O, CO2, some times CO (incomplete combustion), OH, HF, HCl. The infrared emission is produced by changes in the energy contained in the molecular vibrations and rotations, only at certain frequencies..
77 Physical Laws of Radiometry (Continue -19)SOLO Targets (continue – 3) • The plume (continue – 1)
78 Physical Laws of Radiometry (Continue -20)SOLO Targets (continue – 4) • The plume (continue – 2) Breathing engines have exhaust plume temperatures of K 600450  Cruise flight K 800600  Maximum Un-augmented Thrust K 15001000  Augmented (After burner) Thrust Rockets have exhaust plume temperatures of K 75002500  Liquid propellant K 35001700  Solid propellant Example Assume:   mm  55.433.45.0  KCCTPLUME  643273370  then:   2222 55.4 33.4 1075.1105.35.0      cmWcmWdMM     For a plume surface of APLUME = 10000 cm2 = 1 m2 the Radiant Intensity is:   1 42 8.27 101075.1 55.433.4            strWA M mmI PLUME Lambertian  
79 Physical Laws of Radiometry (Continue -21)SOLO Targets (continue – 5) • Aerodynamic Heating The Target body is heated by the compression and friction of the air against it’s surface and by friction. Assuming a negligible friction effect and an adiabatic compression the Target skin temperature is given by:         2 0 2 1 1, MachrMachHTT  - air temperature at altitude HTARGET and mach number Mach MachHT ,0 - recovery factorr vp CC / - specific heat ratio = 1.4 for air Example Mach = 2.0, HTARGET = 5000 m   27.0,250.2,50000  KMachmHT  then KT  4142 2 14.1 82.01250 2             23 5 3 1066.1414      cmWdKTMM      assume 2 15mATARGET    1 43 3.79 10151066.1 53            strWA M mmI TARGET Lambertian  
80 Physical Laws of Radiometry (Continue -22)
81 Physical Laws of Radiometry (Continue -23)
82 Physical Laws of Radiometry (Continue -24)SOLO Targets (continue – 6) • Aerodynamic Heating (continue – 1) EmissivityReflectanceAbsorptanceMaterial .04.81.19Polished Aluminium .04.63.37Unpolished Aluminium .18.43.57Titanium .05.60.40Polished Stainless Steel .88.79.21White Paint .92.05.95Black Paint .27.71.29Aluminum Paint
83 Physical Laws of Radiometry (Continue -25)SOLO Sun, Background and Atmosphere
84 Physical Laws of Radiometry (Continue -26)SOLO Sun, Background and Atmosphere (continue – 1) The spectrum distribution of the sun radiation is like a black body with a temperature of T = 5900 °K From Wien’s Law the maximum of Mλ is at m T m  49.0 5900 28982898  This is almost at the middle of the visible spectrum mm  75.040.0  Loss by Scattering
85 Physical Laws of Radiometry (Continue -27)SOLO Sun, Background and Atmosphere (continue – 2) Atmosphere Atmosphere affects electromagnetic radiation by     3.2 1 1        R kmRR  • Absorption • Scattering • Emission • Turbulence Atmospheric Windows: Window # 2: 1.5 μm ≤ λ < 1.8 μm Window # 4 (MWIR): 3 μm ≤ λ < 5 μm Window # 5 (LWIR): 8 μm ≤ λ < 14 μm For fast computations we may use the transmittance equation: R in kilometers. Window # 1: 0.2 μm ≤ λ < 1.4 μm includes VIS: 0.4 μm ≤ λ < 0.7 μm Window # 3 (SWIR): 2.0 μm ≤ λ < 2.5 μm
86 Physical Laws of Radiometry (Continue -28)SOLO Sun, Background and Atmosphere (continue – 3) Atmosphere Absorption over Electromagnetic Spectrum
87 Physical Laws of Radiometry (Continue -29)SOLO Sun, Background and Atmosphere (continue – 4) Rain Attenuation over Electromagnetic Spectrum FREQUENCY GHz ONE-WAYATTENUATION-Db/KILOMETER WAVELENGTH
88 Physical Laws of Radiometry (Continue -30)SOLO Sun, Background and Atmosphere (continue – 3) Add scanned Figure from McKenzie Atmosphere (continue – 1)
89 GEOMETRICAL OPTICSSOLO http://en.wikipedia.org/wiki/Optics From “Cyclopaedia” or “An Universal Dictionary of Art and Science” Published by Ephraim Chambers In London in 1728 Return to TOC
90 SOLO DERIVATION OF EIKONAL EQUATION Foundation of Geometrical Optics Derivation from Maxwell Equations Consider a general time-harmonic field:                                tjrHtjrHtjrHaltrH tjrEtjrEtjrEaltrE     exp,exp, 2 1 exp,Re, exp,exp, 2 1 exp,Re, * *   in a non-conducting, far-away from the sources  0,0  eeJ   No assumption of isotropy of the medium are made; i.e.:     rr   , Far from sources, in the High Frequencies we can write using the phasor notation:             00000 &,&, 00     kerHrHerErE rSjkrSjk Note The minus sign was chosen to get a progressive wave: End Note                SktjSktj erHaltrHerEaltrE 00 00 Re,&Re,     James Clerk Maxwell (1831-1879) See full development in P.P. “Foundation of Geometrical Optics”
91 SOLO From those equations we have Foundation of Geometrical Optics           Sjktj SjkSjktjSjktjtj eeESjkE EeeEeeEeerE 0 000 000 000,               Sjk SjktjSjktjtj eHjk eHejeHejerH t 0 00 0 00 0 0 00 000 1 1 ,           from which   0 00 0000 HjkESjkEF    and 0 1 0 0 0 0 00 0   k E jk HES   DERIVATION OF EIKONAL EQUATION (continue – 2) Derivation from Maxwell Equations (continue – 2)
92 SOLO From Maxwell equations we also have Foundation of Geometrical Optics from which and DERIVATION OF EIKONAL EQUATION (continue – 3) Derivation from Maxwell Equations (continue – 3)           Sjktj SjkSjktjSjktjtj eeHSjkH HeeHeeHeerH 0 000 000 000,               Sjk SjktjSjktjtj eEjk eEejeEejerE t 0 00 0 00 0 0 00 000 1 1 ,             0 00 0000 EjkHSjkHA    0 1 0 0 0 0 00 0   k H jk EHS  
93 SOLO DERIVATION OF EIKONAL EQUATION (continue – 4) Foundation of Geometrical Optics Derivation from Maxwell Equations (continue – 4) We have Faradey (F), Ampére (A), Gauss Electric (GE), Gauss Magnetic (GM) equations:                       0 0 HGM EGE EjHA HjEF                                                         0&0 2 0 0 0 ee e e J c k j t HB ED BGM DGE J t D HA t B EF                  André-Marie Ampère 1775-1836 Michael Faraday 1791-1867 Karl Friederich Gauss 1777-1855
94 SOLO From Maxwell equations we also have Foundation of Geometrical Optics from which and DERIVATION OF EIKONAL EQUATION (continue – 4) Derivation from Maxwell Equations (continue – 4)             0 , 0 000 0000 000     Sjktj SjkSjktjSjktjtj eeESjkEE EeeEeeEeerE       00000  ESjkEEGE  0 1 0 00 0 0           k EE jk ES   We also have from which and             0 , 0 000 0000 000     Sjktj SjkSjktjSjktjtj eeHSjkHH HeeHeeHeerH       00000  HSjkHHGM  0 1 0 00 0 0           k HH jk HS  
95 SOLO To summarize, from k0 → ∞ we have Foundation of Geometrical Optics DERIVATION OF EIKONAL EQUATION (continue – 5) Derivation from Maxwell Equations (continue – 5)   00 00 0  HESF     00 00 0  EHSA     00  ESGE   00  HSGM We will use only the first two equations, because the last two may be obtained from the previous two by multiplying them (scalar product) by .S
96 SOLO Foundation of Geometrical Optics DERIVATION OF EIKONAL EQUATION (continue – 6) Derivation from Maxwell Equations (continue – 6)   00 00 0  HESF     00 00 0  EHSA   From the second equation we obtain 0 00 0 HSE    And by substituting this in the first equation   00 0 00 00 00 0 00           HHSSHHSS       But         2 00 0 2 0 0 00 n HSHSSSHSHSS     
97 SOLO Foundation of Geometrical Optics DERIVATION OF EIKONAL EQUATION (continue – 7) Derivation from Maxwell Equations (continue – 7) Finally we obtain    00 22  HnS or    zyxn z S y S x S ornS ,,0 2 222 22                          S is called the eikonal (from Greek έίκων = eikon → image) and the equation is called Eikonal Equation. Return to TOC
98 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS From Max Born & Emil Wolf, “Principles of Optics”, 6th Ed., Ch. 3             00000 &,&, 00     kerHrHerErE rSjkrSjk We found the following relations   00 00 0  HESF     00 00 0  EHSA     00  ESGE   00  HSGM We can see that the vectors are perpendicular in the same way as the vectors for the planar waves (where is the Poynting vector). SHE ,, 00 SHE  ,, 00 00 HES   S 0E 0H
99 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 1)                                                     T T T TT e dttjrErErEtjrE T dttjrEtjrEtjrEtjrE T dttjrEal T dttrEtrE T dttrDtrE T w 0 2**2 0 ** 0 2 00 2exp,,,22exp, 4 1 exp,exp,exp,exp, 4 1 exp,Re 1 ,, 1 ,, 1           But               0 2 2exp 2exp 2 1 2exp 1 0 2 2exp 2exp 2 1 2exp 1 0 0 0 0         T T T T T T Tj Tj tj Tj dttj T Tj Tj tj Tj dttj T           Therefore                rErEerEerEdt T rErEw rSjkrSjk T e  * 00 * 00 0 * 22 1 ,, 2 00      Let compute the time averages of the electric and magnetic energy densities
100 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 2) In the same way                rErEerEerEdt T rErEw rSjkrSjk T e  * 00 * 00 0 * 22 1 ,, 2 00                 rHrHdttrHtrH T dttrBtrH T w TT m * 00 00 2 ,, 1 ,, 1      Using the relations   0 00 0 HSEA      0 00 0 ESHF    since and are real values , where * is the complex conjugate, we obtain S )**,( SS                                              e m e wrHSrErHSrErHSrE rESrHrESrHrHrHw rHSrErHSrErErEw    * 00 * 0 * 00 ** 0 * 00 * 0 00 0 * 00 * 00 * 0 00 0 * 00 2 1 2 1 2 1 2 1 22 2 1 22        S 0E 0H
101 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 3) Therefore     * 00 2 1 rHSrEww me   Within the accuracy of Geometrical Optics, the time-averaged electric and magnetic energy densities are equal.             * 0000 * 00 22 rHSrErHrHrErEwww me    The total energy will be: The Poynting vector is defined as:      trHtrEtrS ,,:,                               T tjtjtjtj T tjtj T dterHerHerEerE T dterHerEal T dttrHtrE T trHtrES 0 ** 00 ,, 2 1 ,, 2 11 ,,Re 1 ,, 1 ,,                                  ,,,, 4 1 ,,,,,,,, 4 11 ** 0 2****2 rHrErHrE dterHrErHrErHrEerHrE T T tjtj                      rHrErHrE erHerEerHerE rSjkrSjkrSjkrSjk 0 * 0 * 00 )( 0 )(* 0 )(* 0 )( 0 4 1 4 1 0000    The time average of the Poynting vector is: John Henry Poynting 1852-1914
102 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 3) Using the relations   0 00 0 HSEA      0 00 0 ESHF                           rHrHSrESrErHrErHrES 0 * 0 * 00 00 0 * 0 * 00 222 1 4 1    we obtain                  * 00 0 0 * 0 0 0 * 0 * 00 00 22222 1 HHSHSHESEEES                 * 0000 * 00 22 rHSrErHrHrErEwww me    we obtain Using   wS n c wwSS me  2 00 00 22 1     00 2 00 & 1     nc
103 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 4) Using   22 nS  Eikonal Equation we obtain nS  Define snS n S S S s ˆ:ˆ       We have swvwS n c S n c v ˆ 2 1 2 2    sˆ constS  constdSS  sˆ r  0 ˆs 0r  A Bundle of Light Rays
104 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 5) swvwS n c S n c v ˆ 2 1 2 2    sˆ constS  constdSS  sˆ r  0 ˆs 0r  From this equation we can see that average Poynting vector is the direction of the normal to the geometrical wave-front , and its magnitude is proportional to the product of light velocity v and the average energy density, therefore we say that defines the direction of the light ray. S sˆ sˆ Suppose that the vector describes the light path, then the unit vector is given by r  sˆ sd rd rd rd s ray ray ray    ˆ where is the differential of an arc length along the ray pathrayrdsd  
105 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 6) Let substitute in and differentiate it with respect to s. sd rd rd rd s ray ray ray    ˆ rayrdsd    S sd d sd rd n sd d       ray   S sd rd  ray    sd rd f sd zd zd fd sd yd yd fd sd xd xd fd sd zyxfd   ,,   SS n  1 S sd rd n  ray           ABBAABBABA   AB        AAAAAA   2 1 SA           SSSSSSSS    0 2 1 SS n  2 1 2 nSS  2 2 1 n n  n
106 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 7) Therefore we obtained   nS sd d  and n sd rd n sd d         ray  We obtained a ordinary differential equation of 2nd order that enables to find the trajectory of an optical ray , giving the relative index and the initial position and direction of the desired ray.  srray   zyxn ,,   00 rrray   0 ˆs sˆ constS  constdSS  sˆ r  0 ˆs 0r  We can transform the 2nd order differential equation in two 1st order differential equations by the following procedure. Define Ssn sd rd np  ˆ: ray   We obtain   0 ˆ0 snpnp sd d     0 ˆ0 snpnp sd d   Return to TOC
107 SOLO The Three Laws of Geometrical Optics 1. Law of Rectilinear Propagation In an uniform homogeneous medium the propagation of an optical disturbance is in straight lines. 2. Law of Reflection An optical disturbance reflected by a surface has the property that the incident ray, the surface normal, and the reflected ray all lie in a plane, and the angle between the incident ray and the surface normal is equal to the angle between the reflected ray and the surface normal: 2v 1v Refracted Ray 21 ˆ n 2n 1n i t Reflected Ray 21 ˆ n 2n 1n i r 3. Law of Refraction An optical disturbance moving from a medium of refractive index n1 into a medium of refractive index n2 will have its incident ray, the surface normal between the media , and the reflected ray in a plane, and the relationship between angle between the incident ray and the surface normal θi and the angle between the reflected ray and the surface normal θt given by Snell’s Law: ti nn  sinsin 21  ri   “The branch of optics that addresses the limiting case λ0 → 0, is known as Geometrical Optics, since in this approximation the optical laws may be formulated in the language of geometry.” Max Born & Emil Wolf, “Principles of Optics”, 6th Ed., Ch. 3 Foundation of Geometrical Optics Return to TOC
108 SOLO Foundation of Geometrical Optics Fermat’s Principle (1657) 1Q 1P 2P 2Q 1Q 2Q 1S SdSS  12  2PS  1PS 2'Q rd  sˆ sˆ The Principle of Fermat (principle of the shortest optical path) asserts that the optical length of an actual ray between any two points is shorter than the optical ray of any other curve that joints these two points and which is in a certai neighborhood of it. An other formulation of the Fermat’s Principle requires only Stationarity (instead of minimal length).  2 1 P P dsn An other form of the Fermat’s Principle is: Princple of Least Time The path following by a ray in going from one point in space to another is the path that makes the time of transit of the associated wave stationary (usually a minimum). The idea that the light travels in the shortest path was first put forward by Hero of Alexandria in his work “Catoptrics”, cc 100B.C.-150 A.C. Hero showed by a geometrical method that the actual path taken by a ray of light reflected from plane mirror is shorter than any other reflected path that might be drawn between the source and point of observation.
109 SOLO 1. The optical path is reflected at the boundary between two regions     0 21 21          rd sd rd n sd rd n rayray   In this case we have and21 nn        0ˆˆ 21 21          rdssrd sd rd sd rd rayray   We can write the previous equation as: i.e. is normal to , i.e. to the boundary where the reflection occurs. 21 ˆˆ ss  rd    0ˆˆˆ 2121  ssn 11 ˆsn 21 ˆsn 1121 ˆˆˆ snsn  rd    0ˆˆ 121  rdssn  Reflected Ray 21 ˆ n 1n i r REFLECTION & REFRACTION Reflection Laws Development Using Fermat Principle ri   Incident ray and Reflected ray are in the same plane normal to the boundary. This is equivalent with: &
110 SOLO 2. The optical path passes between two regions with different refractive indexes n1 to n2. (continue – 1)     0 21 21          rd sd rd n sd rd n rayray   where is on the boundary between the two regions andrd      sd rd s sd rd s rayray 2 :ˆ, 1 :ˆ 21   rd  22 ˆsn 11 ˆsn 1122 ˆˆˆ snsn    0ˆˆˆ 1122  rdsnsn  Refracted Ray 21 ˆ n 2n 1n i t Therefore is normal to .2211 ˆˆ snsn  rd  Since can be in any direction on the boundary between the two regions is parallel to the unit vector normal to the boundary surface, and we have rd  2211 ˆˆ snsn  21 ˆ n   0ˆˆˆ 221121  snsnn We recovered the Snell’s Law from Geometrical Optics REFLECTION & REFRACTION Refraction Laws Development Using Fermat Principle ti nn  sinsin 21  Incident ray and Refracted ray are in the same plane normal to the boundary. & Return to TOC
111 SOLO Plane-Parallel Plate i r ri   r t l d i A C B E 2n 1n A single ray traverses a glass plate with parallel surfaces and emerges parallel to its original direction but with a lateral displacement d. Optics    irriri lld  cossincossinsin  r t l cos         r i ritd    cos cos sinsin ir nn  sinsin 0Snell’s Law        n n td r i i 0 cos cos 1sin    For small anglesi        n n td i 0 1
112 SOLO Plane-Parallel Plate (continue – 1) t        r i i n n td    cos cos 1sin 1 2 1n  2n i r        r i i n n t d l    cos cos 1 sin 1 2 l Two rays traverse a glass plate with parallel surfaces and emerge parallel to their original direction but with a lateral displacement l. Optics    irriri lld  cossincossinsin  r t l cos         r i ritd    cos cos sinsin ir nn  sinsin 0Snell’s Law        n n td r i i 0 cos cos 1sin           r i i n n t d l    cos cos 1 sin 0 For small anglesi        n n tl 0 1 Return to TOC
113 SOLO Prisms   2i1i 1t    11 ti   2t  22 it   Type of prisms: A prism is an optical device that refract, reflect or disperse light into its spectral components. They are also used to polarize light by prisms from birefringent media. Optics - Prisms 2. Reflective 1. Dispersive 3. Polarizing
114 OpticsSOLO Dispersive Prisms   2i1i 1t    11 ti   2t  22 it      2211 itti   21 it     21 ti 202 sinsin ti nn  Snell’s Law 10 n     1 1 2 1 2 sinsinsinsin tit nn         11 21 11 1 2 sincossin1sinsinsincoscossinsin ttttt nn    Snell’s Law 110 sinsin ti nn   11 sin 1 sin it n     1 2/1 1 221 2 sincossinsinsin iit n          1 2/1 1 221 1 sincossinsinsin iii n The ray deviation angle is 10 n
115 OpticsSOLO Prisms   2i1i 1t    11 ti   2t  22 it         1 2/1 1 221 1 sincossinsinsin iii n
116 OpticsSOLO Prisms   2i1i 1t    11 ti   2t  22 it         1 2/1 1 221 1 sincossinsinsin iii n   21 ti Let find the angle θi1 for which the deviation angle δ is minimal; i.e. δm. This happens when  01 0 11 2 1  ii t i d d d d d d       Taking the differentials of Snell’s Law equations 22 sinsin tin   11 sinsin ti n   2222 coscos iitt dnd   1111 coscos ttii dnd   Dividing the equations  1 2 1 2 1 1 2 1 2 1 cos cos cos cos   i t i t t i t i d d d d         2 22 1 22 2 2 2 2 1 2 2 2 1 2 2 2 1 2 sin sin /sin1 /sin1 sin1 sin1 sin1 sin1 t i t i i t t i n n n n                    1 1 2  i t d d   21 it   1 2 1  i t d d   2 2 1 2 2 2 1 2 cos cos cos cos i t t i      21 ti   1n
117 OpticsSOLO Prisms   2i1i 1t    11 ti   2t  22 it         1 2/1 1 221 1 sincossinsinsin iii n We found that if the angle θi1 = θt2 the deviation angle δ is minimal; i.e. δm. Using the Snell’s Law equations 22 sinsin tin   11 sinsin ti n   21 ti   21 it   This means that the ray for which the deviation angle δ is minimum passes through the prism parallel to it’s base.   2i 1i 1t  m  11 ti   2t  22 it   21 ti   21 it   Find the angle θi1 for which the deviation angle δ is minimal; i.e. δm (continue – 1).
118 OpticsSOLO Prisms       1 2/1 1 221 1 sincossinsinsin iii n Using the Snell’s Law 11 sinsin ti n   21 it   This equation is used for determining the refractive index of transparent substances.   2i 1i 1t  m  11 ti   2t  22 it   21 ti   21 it   21 it     21 ti 21 ti   m  2/1  t   12 im   2/1   mi    2/sin 2/sin     m n Find the angle θi1 for which the deviation angle δ is minimal; i.e. δm (continue – 2).
119 OpticsSOLO Prisms The refractive index of transparent substances varies with the wavelength λ.        1 2/1 1 221 1 sincossinsinsin iii n   2i1i 1t    11 ti   2t  22 it  
120 OpticsSOLO http://physics.nad.ru/Physics/English/index.htm Prisms υ [THz]λ0 (nm)Color 384 – 482 482 – 503 503 – 520 520 – 610 610 – 659 659 - 769 780 - 622 622 - 597 597 - 577 577 - 492 492 - 455 455 - 390 Red Orange Yellow Green Blue Violet 1 nm = 10-9m, 1 THz = 1012 Hz        1 2/1 1 221 1 sincossinsinsin iii n In 1672 Newton wrote “A New Theory about Light and Colors” in which he said that the white light consisted of a mixture of various colors and the diffraction was color dependent. Isaac Newton 1542 - 1727
121 SOLO Dispersing Prisms Pellin-Broca Prism Abbe Prism Ernst Karl Abbe 1840-1905 At Pellin-Broca Prism an incident ray of wavelength λ passes the prism at a dispersing angle of 90°. Because the dispersing angle is a function of wavelength the ray at other wavelengths exit at different angles. By rotating the prism around an axis normal to the page different rays will exit at the 90°. At Abbe Prism the dispersing angle is 60°. Optics - Prisms
122 SOLO Dispersing Prisms (continue – 1) Amici Prism Optics - Prisms
123 SOLO Reflecting Prisms  2i 1i 1t  2t E B D G A F C BED  180  360 ABEBEDADE 1 90 i ABE   2 90 t ADE    3609090 12  it BED  12 180 it BED      21 180 ti BED The bottom of the prism is a reflecting mirror Since the ray BC is reflected to CD DCGBCF  Also CGDBFC  CDGFBC  FBCt   901  CDGi   902 21 it   202 sinsin ti nn  Snell’s Law Snell’s Law 110 sinsin ti nn   21 ti     12 i CDGFBC  ~ Optics - Prisms
124 SOLO Reflecting Prisms Porro Prism Porro-Abbe Prism Schmidt-Pechan Prism Penta Prism Optics - Prisms Roof Penta Prism
125 SOLO Reflecting Prisms Abbe-Koenig Prism Dove Prism Amici-roof Prism Optics - Prisms
126 SOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Nicol Prism The Nicol Prism is made up from two prisms of calcite cemented with Canada balsam. The ordinary ray can be made to totally reflect off the prism boundary, leving only the extraordinary ray.. Polarizing Prisms Optics - Prisms
127 SOLO Polarizing Prisms A Glan-Foucault prism deflects polarized light transmitting the s-polarized component. The optical axis of the prism material is perpendicular to the plane of the diagram. A Glan-Taylor prism reflects polarized light at an internal air-gap, transmitting only the p-polarized component. The optical axes are vertical in the plane of the diagram. A Glan-Thompson prism deflects the p-polarized ordinary ray whilst transmitting the s-polarized extraordinary ray. The two halves of the prism are joined with Optical cement, and the crystal axis are perpendicular to the plane of the diagram. Optics - Prisms Return to TOC
128 OpticsSOLO Lens Definitions Optical Axis: the common axis of symmetry of an optical system; a line that connects all centers of curvature of the optical surfaces. FFL First Focal Point Second Focal Point Principal Planes Second Principal Point First Principal Point Light Rays from Left EFL BFL Optical System Optical Axis Lateral Magnification: the ratio between the size of an image measured perpendicular to the optical axis and the size of the conjugate object. Longitudinal Magnification: the ratio between the lengthof an image measured along the optical axis and the length of the conjugate object. First (Front) Focal Point: the point on the optical axis on the left of the optical system (FFP) to which parallel rays on it’s right converge. Second (Back) Focal Point: the point on the optical axis on the right of the optical system (BFP) to which parallel rays on it’s left converge.
129 OpticsSOLO Definitions (continue – 1) Aperture Stop (AS): the physical diameter which limits the size of the cone of radiation which the optical system will accept from an axial point on the object. Field Stop (FS): the physical diameter which limits the angular field of view of an optical system. The Field Stop limit the size of the object that can be seen by the optical system in order to control the quality of the image. Entrance Pupil: the image of the Aperture Stop as seen from the object through the elements preceding the Aperture Stop. Exit Pupil: the image of the Aperture Stop as seen from an axial point on the image plane. A.S. F.S. I Aperture and Field Stops Entrance pupil Exit pupil A.S. I xpE npE Chief Ray Entrance and Exit pupils Entrance pupilExit pupil A.S. I xpE npE Chief Ray
130 OpticsSOLO Definitions (continue – 2) Aperture Stop (AS): the physical diameter which limits the size of the cone of radiation which the optical system will accept from an axial point on the object. Field Stop (FS): the physical diameter which limits the angular field of view of an optical system. The Field Stop limit the size of the object that can be seen by the optical system in order to control the quality of the image. Entrance Pupil: the image of the Aperture Stop as seen from the object through the elements preceding the Aperture Stop. Exit Pupil: the image of the Aperture Stop as seen from an axial point on the image plane. Entrance pupil Exit pupil A.S. I Chief Ray Marginal Ray Exp Enp
131 OpticsSOLO Definitions (continue – 3) Principal Planes: the two planes defined by the intersection of the parallel incident rays entering an optical system with the rays converging to the focal points after passing through the optical system. FFL First Focal Point Second Focal Point Principal Planes Second Principal Point First Principal Point Light Rays from Left EFL BFL Optical System Optical Axis Principal Points: the intersection of the principal planes with the optical axes. Nodal Points: two axial points of an optical system, so located that an oblique ray directed toward the first appears to emerge from the second, parallel to the original direction. For systems in air, the Nodal Points coincide with the Principal Points. Cardinal Points: the Focal Points, Principal Points and the Nodal Points.
132 OpticsSOLO Definitions (continue – 4) Relative Aperture (f# ): the ratio between the effective focal length (EFL) f to Entrance Pupil diameter D. Numerical Aperture (NA): sine of the half cone angle u of the image forming ray bundles multiplied by the final index n of the optical system. If the object is at infinity and assuming n = 1 (air): Dff /:#  unNA sin:  # 1 2 1 2 1 sin ff D uNA        EFL EFL D u Last Principal Plane of the Optical System (Spherical)
133 OpticsSOLO Perfect Imaging System • All rays originating at one object point reconverge to one image point after passing through the optical system. • All of the objects points lying on one plane normal to the optical axis are imaging onto one plane normal to the axis. • The image is geometrically similar to the object. Object Image SystemOptical Object Image SystemOptical Object Image SystemOptical
134 OpticsSOLO Lens Convention of Signs 1. All Figures are drawn with the light traveling from left to right. 2. All object distances are considered positive when they are measured to the left of the vertex and negative when they are measured to the right. 3. All image distances are considered positive when they are measured to the right of the vertex and negative when they are measured to the left. 4. Both focal length are positive for a converging system and negative for a diverging system. 5. Object and Image dimensions are positive when measured upward from the axis and negative when measured downward. 6. All convex surfaces are taken as having a positive radius, and all concave surfaces are taken as having a negative radius. Return to TOC
135 OpticsSOLO Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle Karl Friederich Gauss 1777-1855  s 's n 'n h l 'l    ' M T CA M’ R The optical path connecting points M, T, M’ is ''lnlnpathOptical  Applying cosine theorem in triangles MTC and M’TC we obtain:      2/122 cos2 RsRRsRl       2/122 cos'2'' RsRRsRl            2/1222/122 cos'2''cos2  RsRRsRnRsRRsRnpathOptical  Therefore According to Fermat’s Principle when the point T moves on the spherical surface we must have   0 d pathOpticald       0 ' sin''sin      l RsRn l RsRn d pathOpticald   from which we obtain           l sn l sn Rl n l n ' ''1 ' ' For small α and β we have ''& slsl  and we obtain R nn s n s n   ' ' ' Gaussian Formula for a Single Spherical Surface Return to TOC
136 OpticsSOLO Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law Apply Snell’s Law: 'sin'sin  nn  If the incident and refracted rays MT and TM’ are paraxial the angles and are small and we can write Snell’s Law:  ' From the Figure    ' '' nn        nnnnnn  ''' For paraxial rays α, β, γ are small angles, therefore '/// shrhsh     r h nn s h n s h n  ' ' ' or   r nn s n s n   ' ' ' Gaussian Formula for a Single Spherical Surface Karl Friederich Gauss 1777-1855 Willebrord van Roijen Snell 1580-1626  s 's n 'n h l 'l    ' M T CA M’ r
137 OpticsSOLO Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law for s → ∞ the incoming rays are parallel to optical axis and they will refract passing trough a common point called the focus F’.   r nn s n s n   ' ' ' s '' fs  n 'n h 'l    ' T C A F’ R  fs  's n 'n h l   F T CA R '   r nn f nn    ' ' ' r nn n f   ' ' ' for s’ → ∞ the refracting rays are parallel to optical axis and therefore the incoming rays passes trough a common point called the focus F.   r nnn f n     '' r nn n f   ' '' n n f f  Return to TOC
138 OpticsSOLO Derivation of Lens Makers’ Formula We have a lens made of two spherical surfaces of radiuses r1 and r2 and a refractive index n’, separating two media having refraction indices n a and n”. Ray MT1 is refracted by the first spherical surface (if no second surface exists) to T1M’.   111 ' ' ' r nn s n s n   11111 ''& sMAsTA  Ray T1T2 is refracted by the second spherical surface to T2M”. 2222 ""&'' sMAsMA    222 '" " " ' ' r nn s n s n   Assuming negligible lens thickness we have , and since M’ is a virtual object for the second surface (negative sign) we have 21 '' ss  21 '' ss    221 '" " " ' ' r nn s n s n   M’ M '1f1f 1s Axis T1 T2 A1 A2 C1 1rC2 F’1F’’2 M’’ F’2F1 ''2f'2f '1s '2s ''2s 2r n 'n ''n
139 OpticsSOLO Derivation of Lens Makers’ Formula (continue – 1) M” M f s Axis A1 A2 C1 1rC2 F” F ''f ''s 2r n 'n ''n   111 ' ' ' r nn s n s n   Add those equations   221 '" " " ' ' r nn s n s n       2121 '"' " " r nn r nn s n s n     M’ M '1f1f 1s Axis T1 T2 A1 A2 C1 1rC2 F’1F’’2 M’’ F’2F1 ''2f'2f '1s '2s ''2s 2r n 'n ''n The focal lengths are defined by tacking s1 → ∞ to obtain f” and s”2 → ∞ to obtain f     f n r nn r nn f n      212 '"' " " Let define s1 as s and s”2 as s” to obtain     21 '"' " " r nn r nn s n s n         f n r nn r nn f n      21 '"' " "
140 OpticsSOLO Derivation of Lens Makers’ Formula (continue – 2) M” M f s Axis A1 A2 C1 1rC2 F” F ''f ''s 2r n 'n n If the media on both sides of the lens is the same n = n”.              21 11 1 ' " 11 rrn n ss              21 11 1 '1 " 1 rrn n ff Therefore " 11 " 11 ffss  Lens Makers’ Formula
141 OpticsSOLO First Order, Paraxial or Gaussian Optics In 1841 Gauss gave an exposition in “Dioptrische Untersuchungen” for thin lenses, for the rays arriving at shallow angles with respect to Optical axis (paraxial). Karl Friederich Gauss 1777-1855 Derivation of Lens Formula 'y s 's M’A F’ M T F 'ffx 'x Q Q’ 'y y S Axis yFrom the similarity of the triangles and using the convention:   '' ' '~' f y s yy TAFTSQ    Lens Formula in Gaussian form     f y s yy FASQTS '' ~       0'  y Sum of the equations:       ' ' ' '' f y f y s yy s yy       since f = f’ fss 1 ' 11  Return to TOC
142 OpticsSOLO First Order, Paraxial or Gaussian Optics (continue – 1) Gauss explanation can be extended to the first order approximation to any optical system. Karl Friederich Gauss 1777-1855 'y s 's M’P1 F’ M T F 'ffx 'x Q Q’ 'y y Axis y P2 First Focal Point First Principal Point Second Focal PointSecond Principal Point Optical System Object Image Lens Formula in Gaussian form 'y s 's M’A F’ M T F 'ffx 'x Q Q’ 'y y S Axis y fss 1 ' 11  s – object distance (from the first principal point to the object). s’ – image distance (from the second principal point to the image). f – EFL (distance between a focal point to the closest principal plane).
143 OpticsSOLO Derivation of Lens Formula (continue) 'y s 's M’A F’ M T F 'ffx 'x Q Q’ 'y y S Axis y From the similarity of the triangles and using the convention:   f y x y FASQMF ' ~   Lens Formula in Newton’s form   f y x y QMFTAF    ' ' '''~'   0'  y Multiplication of the equations:     2 ' ' ' f yy xx yy     or 2 ' fxx  Isaac Newton 1643-1727 First Order, Paraxial or Gaussian Optics (continue – 2) Published by Newton in “Opticks” 1710
144 OpticsSOLO Derivation of Lens Formula (continue) 'h s 's M’A F’ M T F 'ffx 'x Q Q’ 'h h S Axis h First Order, Paraxial or Gaussian Optics (continue – 3) Lateral or Transverse Magnification f x x f s s h h mT '''  (-) sign(+) signQuantity virtual objectreal objects virtual imagereal images’ diverging lensconverging lensf inverted objecterect objecth inverted imageerect imageh’ inverted imageerect imagemT
145 OpticsSOLO Concave Spherical Convex Spherical Paraboloidal Conic Ellipsoidal General Aspherical Plane Converging : General use Diverging : General use Accurately focuses a parallel beam or produces a parallel beam from a point source Refocuses a diverging bundle at another point (P) displaced from the point of origin (O) Change the direction of beam Used mostly in combination systems of two or more components BASIC MIRRORS FORMS
146 OpticsSOLO Convex Plano Convex Meniscus Concave Plano Concave Meniscus Doublet Multi- Element Aspheric Converging: General Use, Magnification Converging: Used often in opposed doubles to reduce spherical aberration Converging: reduced spherical aberration Diverging: General Use, Demagnification Diverging: Used in multi-element combinations Diverging: reduced spherical aberration Corrected for chromatic aberration High order of aberration correction used in complex systems Corrected for spherical aberration used in condenser systems BASIC LENS FORMS Return to TOC
147 OpticsSOLO Ray Tracing F C O I Object Virtual Image Convex Mirror R/2 R/2 R F F’C O I Object Real Image Converging Lens FC O I Object Real Image Concave Mirror F F’C O I Object Virtual Image Diverging Lens Ray Tracing is a graphically implementation of paralax ray analysis. The construction doesn’t take into consideration the nonideal behavior, or aberration of real lens. The image of an off-axis point can be located by the intersection of any two of the following three rays: 1. A ray parallel to the axis that is reflected through F’. 2. A ray through F that is reflected parallel to the axis. 3. A ray through the center C of the lens that remains undeviated and undisplaced (for thin lens).
148 OpticsSOLO Infinity Principal focus SUMMARY OF SIMPLE IMAGING LENSES f f2f2 f 0 's 'ss fs 2 fsf 2' fs 2 fs 2' fsf 2 fs 2' 's 's s s fs  's s s 's fs  fs ' s 's fsf 2 fs ' Real, inverted small Telescope Real, inverted smaller Camera Real, inverted same size Photocopier Real, inverted larger Projector No image Searchlight Virtual, erect larger Microscope Virtual, erect smaller Various Figure Object Location Image Location Image Properties Example L.J. Pinson, “Electro-Optics”, John Wiley & Sons, 1985, pg.54 Return to TOC
149 OpticsSOLO Matrix Formulation The Matrix Formulation of the Ray Tracing method for the paraxial assumption was proposed at the beginning of nineteen-thirties by T.Smith. Assuming a paraxial ray entering at some input plane of an optical system at the distance r1 from the symmetry axis and with a slope r1’ and exiting at some output plane at the distance r2 from the symmetry axis and with a slope r2’, than the following linear (matrix) relation applies: Principal PlanesInput plane Output plane Ray path 1h 2h 1r 2r '1r '2r Symmetry axis                         ''' 1 1 1 1 2 2 r r M r r DC BA r r        DC BA Mwhere ray transfer matrix When the media to the left of the input plane and to the right of the output plane have the same refractive index, we have: 1det  CBDAM
150 OpticsSOLO Matrix Formulation (continue -1) Uniform Optical Medium In an Uniform Optical Medium of length d no change in ray angles occurs: Ray path d 1r 2r '1r '2r Symmetry axis 1 2 '' ' 12 112 rr rdrr          10 1 d M Medium Optical Uniform Planar Interface Between Two Different Media Ray path 1r 2r '1r '2r Symmetry axis 1 2 1n 2n 12 rr  '' 1 2 1 2 12 r n n r rr   Apply Snell’s Law: 2211 sinsin  nn  paraxial assumption:   tan'sin r From Snell’s Law: '' 1 2 1 2 r n n r         21 /0 01 nn M Interface Planar 1det 2 1  n n M Interface Planar 1det  Medium Optical UniformM The focal length of this system is infinite and it has not specific principal planes.
151 OpticsSOLO Matrix Formulation (continue -2) A Parallel-Sided Slab of refractive index n bounded on both sides with media of refractive index n1 = 1 Ray path d 21 rr  43 rr  '1r '4r Symmetry axis '2r '3r nn 211 n 11 n We have three regions: • on the right of the slab (exit of ray):                   '/0 01 ' 3 3 124 4 r r nnr r • in the slab:                   '10 1 ' 2 2 3 3 r rd r r • on the left of the slab (entrance of ray):                   '/0 01 ' 1 1 212 2 r r nnr r Therefore:                               '/0 01 10 1 /0 01 ' 1 1 21124 4 r r nn d nnr r                                21 21 122112 /0 /1 /0 01 /0 01 10 1 /0 01 nn nnd nnnn d nn M mediaentranceslabmediaexit Slab Sided Parallel         10 /1 21 nnd M Slab Sided Parallel 1det  Slab Sided ParallelM
152 OpticsSOLO Matrix Formulation (continue -3) Spherical Interface Between Two Different Media Ray path 21 rr  '1r '2r Symmetry axis 1n 2n i r 1 12 rr  Apply Snell’s Law: rnin sinsin 21  paraxial assumption: rrii  sin&sin From Snell’s Law: rnin 21                           2 1 2 1 2 1 12 21 0101 n n n D n n Rn nnM Interface Spherical 1det 2 1  n n M Interface Spherical 12 11 ' '     rr ri From the Figure:    122111 ''   rnrn 111 / Rr   12 121 2 11 2 ' ' Rn rnn n rn r     1 12 11 112 2 12 ' ' n rn Rn rnn r rr       1 12 1 : R nn D  where: Power of the surface If R1 is given in meters D1 gives diopters
153 OpticsSOLO Matrix Formulation (continue -4) Thick Lens 21 rr  43 rr  '1r i 2 1 '2r '3r r 2R 1R f '4r1C 2F IO 1F 2C Principal planes 2n 1n s 's d We have three regions: • on the right of the slab (exit of ray):                        ' 01 ' 3 3 1 2 1 2 4 4 r r n n n D r r • in the slab:                   '10 1 ' 2 2 3 3 r rd r r • on the left of the slab (entrance of ray):                        ' 01 ' 1 1 2 1 2 1 2 2 r r n n n D r r Therefore:                                                                       ' 101 ' 01 10 1 01 ' 1 1 2 1 2 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 4 4 r r n n n D n n d n D d n n n D r r n n n D d n n n D r r                          2 2 21 21 1 21 2 1 2 1 1 1 n D d nn DD d n DD n n d n D d M Lens Thick   2 21 2 R nn D     1 12 1 : R nn D                            2 1 21 21 1 21 2 1 2 2 1 1 1 n D d nn DD d n DD n n d n D d M Lens Thick 1det  Lens Thick M or 21 DD 
154 OpticsSOLO Matrix Formulation (continue -5) Thick Lens (continue -1) 21 rr  43 rr  '1r i 2 1 '2r '3r r 2R 1R f '4r1C 2F IO 1F 2C Principal planes 2n 1n 2R 1R 2f 1C 2F I O 1F 2C Principal planes 2n 1n 1h 2h s s 's 's d Ray 2 Ray 1 1f Let use the second Figure where Ray 2 is parallel to Symmetry Axis of the Optical System that is refracted trough the Second Focal Point.                                              '1 1 ' 1 1 2 2 21 21 1 21 2 1 2 1 4 4 r r n D d nn DD d n DD n n d n D d r r We found: 2141 /'&0' frrr Ray 2: By substituting Ray2 parameters we obtain: 1 2 1 21 21 1 21 4 1 ' r f r nn DD d n DD r          1 21 21 1 21 2           nn DD d n DD f frrr /'&0' 414 Ray 1: We found:                                             '1 1 ' 4 4 2 1 21 21 1 21 2 1 2 2 1 1 r r n D d nn DD d n DD n n d n D d r r 4 1 4 21 21 1 21 1 1 ' r f r nn DD d n DD r          2 1 21 21 1 21 1 f nn DD d n DD f          
155 OpticsSOLO Matrix Formulation (continue -6) Thin Lens 21 rr  43 rr  '1r i 2 1 '2r '3r r 2R 1R f '4r1C 2F IO 1F 2C Principal planes 2n 1n s 's d For thick lens we found                          2 2 21 21 1 21 2 1 2 1 1 1 n D d nn DD d n DD n n d n D d M Lens Thick          21 21 1 211 nn DD d n DD f For thin lens we can assume d = 0 and obtain             1 1 01 f M Lens Thin 1 211 n DD f     2 21 2 R nn D     1 12 1 : R nn D                  211 2 1 21 11 1 1 RRn n n DD f 21 rr  43 rr  2R 1R f '4r1C 2F IO 1F 2C Principal planes 2n 1n s 's '1r
156 OpticsSOLO Matrix Formulation (continue -7) Thin Lens (continue – 1) For a biconvex lens we have R2 negative                211 2 11 1 1 RRn n f For a biconcave lens we have R1 negative                211 2 11 1 1 RRn n f             1 1 01 f M Lens Thin
157 OpticsSOLO Matrix Formulation (continue -8) A Length of Uniform Medium Plus a Thin Lens                               f d f d d f MMM Medium Uniform Lens Thin Lens Thin Medium Uniform 1 1 1 10 1 1 1 01 21 rr  43 rr  2R 1R f '4r1C 2F IO 1F 2C Principal planes 2n 1n s 's '1r d Combination of Two Thin Lenses 2n 1d 1f 2d 2f 2n                                      21 21 2 2 2 1 1 1 21 2 21 1 21 21 2 2 1 1 1 1 22 2 1 11 1 1 1 1 1 1 1 1122 ff dd f d f d f d ff d ff f dd dd f d f d f d f d f d MMMMM dMedium Uniform fLens Thin dMedium Uniform fLens Thin Lenses Thin Two The Focal Length of the Combination of Two Thin Lenses is: 21 2 21 111 ff d fff 
158 OpticsSOLO Matrix Formulation (continue -9) Mirrors r Spherical Mirror i i ii  i i iy RSpherical Mirror Center of Curvature r   Ryiii /tan  Consider a Spherical Mirror of radius R. From the geometry: For small angles: Ryiii / also: iri  2   2/rii   Ryiri /2 Define by n the index of reflexion of the medium: Rynnn yy iir ir /2                      i i r r n y Rnn y  1/2 01 Therefore:                1/ 01 1/2 01 fnRn M Mirror Spherical
159 OpticsSOLO Matrix Formulation (continue -10) Cavity of two Mirrors d 12M 21M 2MirrorM 1MirrorM Spherical Mirror M1 Radius R1 Spherical Mirror M2 Radius R2 O                             10 1 1/2 01 10 1 1/2 01 12 1221 12 d Rn d Rn MMMMM MMirror Spherical MMirror SphericalCavity Figure shows two spherical mirrors facing each other forming an optical cavity. Light leaves point O, traverse the gap in the positive direction, is reflected by Mirror M1, retraces the gap in the negative direction, and is reflected by Mirror M2. The System Matrix is:                            21221 2 21 1 2 1 1122 /21/21/2/4/2/2 /22/21 /21/2 1 /21/2 1 RdnRdnRdnRRdnRnRn RdndRdn RdnRn d RdnRn d            21 22 2121 2 21 1 2 1 /4/4/21/4/2/2 /22/21 RRdnRdnRdnRRdnRnRn RdndRdn MCavity
160 Go to OPTICS Part II OpticsSOLO
January 5, 2015 161 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA

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Optics part i

  • 1. 1 OPTICS Part I SOLO HERMELIN Updated: 16.01.10http://www.solohermelin.com
  • 2. 2 Table of Content SOLO OPTICS Maxwell’s Equations Boundary Conditions Electromagnatic Wave Equations Monochromatic Planar Wave Equations Spherical Waveforms Cylindrical Waveforms Energy and Momentum Electrical Dipole (Hertzian Dipole) Radiation Reflections and Refractions Laws Development Using the Electromagnetic Approach IR Radiometric Quantities Physical Laws of Radiometry Geometrical Optics Foundation of Geometrical Optics – Derivation of Eikonal Equation The Light Rays and the Intensity Law of Geometrical Optics The Three Laws of Geometrical Optics Fermat’s Principle (1657)
  • 3. 3 Table of Content (continue) SOLO OPTICS Plane-Parallel Plate Prisms Lens Definitions Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law Derivation of Lens Makers’ Formula First Order, Paraxial or Gaussian Optics Ray Tracing Matrix Formulation
  • 4. 4 Table of Content (continue) SOLO OPTICS Optical Diffraction Fresnel – Huygens’ Diffraction Theory Complementary Apertures. Babinet Principle Rayleigh-Sommerfeld Diffraction Formula Extensions of Fresnel-Kirchhoff Diffraction Theory Phase Approximations – Fresnel (Near-Field) Approximation Phase Approximations – Fraunhofer (Near-Field) Approximation Fresnel and Fraunhofer Diffraction Approximations Fraunhofer Diffraction and the Fourier Transform Fraunhofer Diffraction Approximations Examples Resolution of Optical Systems Optical Transfer Function (OTF) Point Spread Function (PSF) Modulation Transfer Function (MTF) Phase Transfer Function (PTF) Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function Other Metrics that define Image Quality – Srahl Ratio Other Metrics that define Image Quality - Pickering Scale Other Metrics that define Image Quality – Atmospheric Turbulence Fresnel Diffraction Approximations Examples O P T I C S P a r t I I
  • 5. 5 Table of Content (continue) SOLO OPTICS References Optical Aberration Monochromatic Seidel Aberrations Chromatic Aberration Interference O p t i c s P a r t I I
  • 6. 6 OpticsSOLO Hierarchy of Optical Theories • Quantum Light as particle (photon) Emission, absorption, interaction of light and matter • Electromagnetic Maxwell’s Equations Reflection/Transmission, polarization • Scalar Wave Light as wave Interference and Diffraction • Geometrical Light as ray Image-forming optical systems λ → 0
  • 8. 8 MAXWELL’s EQUATIONSSOLO SYMMETRIC MAXWELL’s EQUATIONS Magnetic Field IntensityH   1  mA Electric DisplacementD   2  msA Electric Field IntensityE   1  mV Magnetic InductionB   2  msV Electric Current DensityeJ   2 mA Free Electric Charge Distributione  3  msA Fictious Magnetic Current DensitymJ   2 mV Fictious Free Magnetic Charge Distributionm  3  msV 1. AMPÈRE’S CIRCUIT LW (A) eJ t D H        2. FARADAY’S INDUCTION LAW (F) mJ t B E        3. GAUSS’ LAW – ELECTRIC (GE) eD   4. GAUSS’ LAW – MAGNETIC (GM) mB   Although magnetic sources are not physical they are often introduced as electrical equivalents to facilitate solutions of physical boundary-value problems. André-Marie Ampère 1775-1836 Michael Faraday 1791-1867 Karl Friederich Gauss 1777-1855 James Clerk Maxwell (1831-1879)
  • 11. 11 SOLO The Infrared (IR) Spectrum of Interest Return to TOC
  • 12. 12 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions 2 ˆt 1 ˆt h 2H  1H  1 2 C CS1P 2P 3P 4P bˆ 21 ˆ n ek      ldtHtHhldtHldtHldH h C 2211 0 2211 ˆˆˆˆ     where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ  nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba          ldbkldbHHnldnbHHldtHH e ˆˆˆˆˆˆ 21212121121    Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ   ekHHn   2121 ˆ              S e C Sd t D JdlH      dlbkbdlh t D JSd t D J e h e S e ˆˆ 0                             AMPÈRE’S LAW  1 0 lim:              mAh t D Jk e h e  
  • 13. 13 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 1) 2 ˆt 1 ˆt h 2E  1E  1 2 C CS1P 2P 3P 4P bˆ 21 ˆ n mk      ldtEtEhldtEldtEldE h C 2211 0 2211 ˆˆˆˆ     where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ  nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba          ldbkldbEEnldnbEEldtEE m ˆˆˆˆˆˆ 21212121121    Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ   mkEEn   2121 ˆ              S m C Sd t B JdlE      dlbkbdlh t B JSd t B J m h m S m ˆˆ 0                             FARADAY’S LAW  1 0 lim:              mVh t B Jk m h m  
  • 14. 14 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 2) h 2D  1D  1 2 21 ˆ n dS 1 ˆn 2 ˆn e     SdnDnDhSdnDSdnDSdD h S 2211 0 2211 ˆˆˆˆ     where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and 21 ˆ,ˆ nn 2121 ˆˆˆ  nnn - a unit vector normal to the boundary between region (1) and (2)21 ˆ n     SdSdnDDSdnDD e 2121121 ˆˆ  Since this must be true for any dS on the boundary between regions (1) and (2) we must have:   eDDn  2121 ˆ    dSdShdv e h e V e  0  GAUSS’ LAW - ELECTRIC  1 0 lim:    msAhe h e    V e S dvSdD  
  • 15. 15 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 3) h 2B  1B  1 2 21 ˆ n dS 1 ˆn 2 ˆn m     SdnBnBhSdnBSdnBSdB h S 2211 0 2211 ˆˆˆˆ     where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and 21 ˆ,ˆ nn 2121 ˆˆˆ  nnn - a unit vector normal to the boundary between region (1) and (2)21 ˆ n     SdSdnBBSdnBB m 2121121 ˆˆ  Since this must be true for any dS on the boundary between regions (1) and (2) we must have:   mBBn  2121 ˆ    dSdShdv m h m V m  0  GAUSS’ LAW – MAGNETIC  1 0 lim:    msVhm h m    V m S dvSdB  
  • 16. 16 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (summary) 2 ˆt 1 ˆt h 22 ,HE  11,HE  1 2 C CS1P 2P 3P 4P bˆ 21 ˆ n me kk  , 21 ˆ n dS 11,BD  22,BD  me ,   mkEEn   2121 ˆ FARADAY’S LAW   ekHHn   2121 ˆ AMPÈRE’S LAW  1 0 lim:              mAh t D Jk e h e    1 0 lim:              mVh t B Jk m h m     eDDn  2121 ˆ  GAUSS’ LAW ELECTRIC  1 0 lim:    msAhe h e    mBBn  2121 ˆ  GAUSS’ LAW MAGNETIC  1 0 lim:    msVhm h m  Return to TOC
  • 17. 17 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS For Homogeneous, Linear and Isotropic Medium ED   HB   where are constant scalars, we have, J t E J t D H t t H t B E ED HB                               Since we have also tt            t J t E E DED EEE t J t E E                                               2 2 22 2 2 &
  • 18. 18 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS (continue 1) Define meme KK c KK v   00 11  where   smc /103 10 36 1 104 11 8 9700               is the velocity of light in free space. The absolute index of refraction n is me KK v c n     0 The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is t J t E v E              2 2 2 2 1
  • 19. 19 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS (continue 2) In the same way The Inhomogeneous Wave (Helmholtz) Differential Equation for the Magnetic Field Intensity is J t E J t D H t H t B E t ED HB                               Since are constant and tt      ,       J t H H HHB HHH J t H H                                     2 2 22 2 2 0&    J t H v H        2 2 2 2 1 Return to TOC
  • 20. 20 ELECTROMAGNETICSSOLO Monochromatic Planar Wave Equations Let assume that can be written as:   trHtrE ,,,             tjrHtrHtjrEtrE 00 exp,,exp,    where are phasor (complex) vectors.                rHjrHrHrEjrErE  ImRe,ImRe  We have          tjrEjtj t rEtrE t 00 expexp,         Hence                                               m e m e j t m e m e B D JBjE JDjH BGM DGE J t B EF J t D HA                )(
  • 21. 21 ELECTROMAGNETICSSOLO Fourier Transform The Fourier transform of can be written as:   trHtrE ,,,                                     dttjtrHrHdtjrHtrH dttjtrErEdtjrEtrE     exp,,&exp, 2 1 , exp,,&exp, 2 1 ,   This is possible if:                         drHdttrH drEdttrE 22 22 , 2 1 , , 2 1 ,   JEAN FOURIER 1768-1830
  • 22. 22 ELECTROMAGNETICSSOLO Note The assumption that can be written as:   trHtrE ,,,             tjrHtrHtjrEtrE 00 exp,,exp,    is equivalent to saying that has a Fourier transform; i.e.:   trHtrE ,,,                                           dtjrHtrHdttjtrHrH dtjrEtrEdttjtrErE exp, 2 1 ,&exp,, exp, 2 1 ,&exp,,   This is possible if:                         drHdttrH drEdttrE 22 22 , 2 1 , , 2 1 ,                                    00 0 exp expexpexp,, rEdttjrE dttjtjrEdttjtrErE   End Note
  • 23. 23 ELECTROMAGNETICSSOLO                        m e m e ED HB m e JHjE JEjH JHjE JEjH JBjE JDjH           me JJjEkE   2   em JJjHkH   2     22 f c c f k    Using the vector identity      AAA   For a Homogeneous, Linear and Isotropic Media:                         m e ED HB m e H E B D    e me JJjEkE   22    m em JJjHkH   22 and we obtain Monochromatic Planar Wave Equations (continue)
  • 24. 24 ELECTROMAGNETICSSOLO Assume no sources: we have Monochromatic Planar Wave Equations (continue) 0,0,0,0  meme JJ   022  EkE 022  HkH nkk n k 0 00 00 0                                    rktjtj rktjtj eHerHtrH eEerEtrE         0 0 ,, ,,   022     rkj rkjrkjrkjrkj ek ekkeekje    Helmholtz Wave Equations satisfy the Helmholtz wave equations    ,,, rHrE             rkj rkj eHrH eErE     0 0 , ,   Assume a progressive wave of phase  rkt   (a regressive wave has the phase ) rkt   For a Homogeneous, Linear and Isotropic Media k  0E 0H r  t k  Planes for which constrkt   
  • 25. 25 ELECTROMAGNETICSSOLO To satisfy the Maxwell equations for a source free media we must have: Monochromatic Planar Wave Equations (continue) we haveUsing: 1ˆˆ&ˆˆ  sss c n sk              0 0 H E HjE EjH                 0ˆ 0ˆ ˆ ˆ 0 0 00 00 Hs Es HEs EHs     sˆ Planar Wave 0E 0H r                    0 0 0 0 00 00 rkj rkj rkjrkj rkjrkj ekje eHkj eEkj eHjeEkj eEjeHkj rkjrkj                        0 0 0 0 00 00 Hk Ek HEk EHk       For a Homogeneous, Linear and Isotropic Media: Return to TOC
  • 26. 26 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS Spherical Waveforms z x y  r cosr  ,,rP   sinsinr  cossinr The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is t J t E v E              2 2 2 2 1 In spherical coordinates:    cos sinsin cossin rz ry rx    2 2 222 2 2 2 sin 1 sin sin 11                           rrr r rr For a spherical symmetric wave:    rErE  ,,  Er rrr E rr E r E r rr E    2 2 2 2 2 2 2 121                     
  • 27. 27 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS SourceSourceSource Spherical Waveforms z x y  r cosr  ,,rP   sinsinr  cossinr The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is assuming no sources   0 11 2 2 22 2          t E v Er rr In spherical coordinates:    cos sinsin cossin rz ry rx        0 1 2 2 22 2        Er tv Er r or: A general solution is:        wave regressive wave eprogressiv tvrFtvrFEr  21 0,0,0,0  meme JJ         r e EerEtrE rktj tj        0,, Assume a progressive monochromatic wave of phase  rkt   (a regressive wave has the phase ) rkt     r e ErE rkj     0, Return to TOC
  • 28. 28 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS Cylindrical Waveforms z x y r  zrP ,,  sinr cosr In cylindrical coordinates: zz ry rx      sin cos 2 2 2 2 2 2 11 zrr r rr                  For a cylindrical symmetric wave:    rEzrE  ,,              r E r rr E   12 The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is assuming no sources 0 11 2 2 2                  t E vr E r rr 0,0,0,0  meme JJ  
  • 29. 29 ELECTROMAGNETICSSOLO ELECTROMGNETIC WAVE EQUATIONS Source Cylindrical Waveforms z x y r  zrP ,,  sinr cosr SourceSource In cylindrical coordinates: zz ry rx      sin cos The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is assuming no sources 0 11 2 2 2                  t E vr E r rr 0,0,0,0  meme JJ   Assume a progressive monochromatic wave of phase  rkt   (a regressive wave has the phase ) rkt       tj erEtrE  ,,    0 1 2 2 2              E vr E rr E k  The solutions are Bessel functions which for large r approach asymptotically to:   rkj e r E rE   0 ,  Return to TOC
  • 30. 30 SOLO Energy and Momentum Let start from Ampère and Faraday Laws                      t B EH J t D HE e      EJ t D E t B HHEEH e              HEHEEH  But Therefore we obtain   EJ t D E t B HHE e             First way This theorem was discovered by Poynting in 1884 and later in the same year by Heaviside. ELECTROMAGNETICS John Henry Poynting 1852-1914 Oliver Heaviside 1850-1925
  • 31. 31 SOLO Energy and Momentum (continue -1) We identify the following quantities - Power density of the current densityEJe    HEDE t BH t EJe                     2 1 2 1           BH t pBHw mm  2 1 , 2 1           DE t pDEw ee  2 1 , 2 1  HEpR   eJ  - Magnetic energy and power densities, respectively - Electric energy and power densities, respectively - Radiation power density For linear, isotropic electro-magnetic materials we can write HBED  00 ,    DE tt D E ED             2 10  BH tt B H HB             2 10 ELECTROMAGNETICS
  • 32. 32 SOLO Energy and Momentum (continue – 3) Let start from the Lorentz Force Equation (1892) on the free charge  BvEF e    Free Electric Chargee  3  msA Velocity of the chargev   1  sm Electric Field IntensityE   1  mV Magnetic InductionB   2  msV Hendrik Antoon Lorentz 1853-1928 e Force on the free chargeF   Ne Second way ELECTROMAGNETICS
  • 33. 33 SOLO Energy and Momentum (continue – 4) The power density of the Lorentz Force the charge     EJBvEvp e Bvv Jv e ee       0   or             HE t B HE t D E t D HEEH E t D HEJp t B E HEHEEH J t D H e e                                                      e ELECTROMAGNETICS
  • 34. 34 SOLO Energy and Momentum (continue – 5)  HEDE t BH t EJe                     2 1 2 1 dve E  B  eJv  , V  FdF  Fd  Let integrate this equation over a constant volume V               VVVV e dvSdvDE td d dvBH td d dvEJ  2 1 2 1 If we have sources in V then instead of we must use E  source EE   Use Ohm Law (1826)  source ee EEJ               VV td d t Georg Simon Ohm 1789-1854 source e e EJE    1 For linear, isotropic electro-magnetic materials  HBED  00 ,   ELECTROMAGNETICS
  • 35. 35 SOLO Energy and Momentum (continue – 6)               VVVR n V source e dvSdvDE td d dvBH td d dRIdvEJ  2 1 2 12         V FieldMagnetic dvBH td d P  2 1         V FieldElectric dvDE td d P  2 1   SV Radiation SdSdvSP    V source eSource dvEJP          V source e R n V source e L S e ee V source e L S e ee V e dvEJdRI dvEJ dS dl dSJdSJdvEJldSdJJdvEJ   2 11   R nJoule dRIP 2 RadiationFieldMagneticFieldElectricJouleSource PPPPP  For linear, isotropic electro-magnetic materials  HBED  00 ,   R – Electric Resistance Define the Umov-Poynting vector:  2 / mwattHES   The Umov-Poynting vector was discovered by Umov in 1873, and rediscovered by Poynting in 1884 and later in the same year by Heaviside. ELECTROMAGNETICS John Henry Poynting 1852-1914
  • 36. 36 ElectromagnetismSOLO EM People John Henry Poynting 1852-1914 Oliver Heaviside 1850-1925 Nikolay Umov 1846-1915 1873 “Theory of interaction on final distances and its exhibit to conclusion of electrostatic and electrodynamic laws” 1884 1884 Umov-Poynting vector HES   The Umov-Poynting vector was discovered by Umov in 1873, and rediscovered by Poynting in 1884 and later in the same year by Heaviside. 1873 - 1884 Return to TOC
  • 37. 37 Note: Since there are not magnetic sources the Magnetic Hertz’s Vector Potential is : 0  m Electrical Dipole (Hertzian Dipole) RadiationSOLO Given a dipole monochromatic of electric charges defined by the Polarization Vector Intensity  tq  tq d  r  dqP   dr       tdqdeqaltP tj e  cosRe 00   we want to find the radiation properties. We start with the Helmholtz Non-homogeneous Differential Equation of the Electric Hertz’s Vector Potential : te       trPtr tc tr eee , 1 , 1 , 0 2 2 2 2       Heinrich Rudolf Hertz 1857-1894 - speed of propagation of the EM wave [m/s] 00 1  c - Polarization Vector IntensityeP   2  msA - Permitivity of space  2122   mNsA - Electric Hertz’s Vector Potential (1888)e   NsA   11 t A e      000  eV   0 Using the Electric Hertz’s Vector Potential we obtain : The field vectors are given by  e e tc V t A E           2 2 20 0 1 t AH e      00 0 1  
  • 38. 38 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1  Compute (continue-3)  e e tc E        2 2 2 1 We have                  32 0 4 0 2 5 0 2 2 2 2 44 3 4 31 rc rpr rc rprpr r rprrp tc E ee e                230 44 rc rp r rp t H e               r p tre 0 4 ,         krtjkrtj epedqp    00  Let use spherical coordinates         zyxr rrr 1111 cossinsincossin                     111 sincos00 rz krtjkrtj epepp                 krtj ep rccr j r rc rprrp rc rprrp r rprrp E rr                                 02 0 2 2 0 3 0 32 0 2 4 0 2 5 0 2 4 sin 4 sincos2 4 sincos2 44 3 4 3 11111  r1  1  1             pckpp pckjpjp 222      
  • 39. 39 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1  Using we can write              11 0 2 0 2 2 sin1 4 sin 44                  krtjkrtj ep rk j r kc ep rcr j H  krtj ep r k r kj r rccr j r E r rr                                                 0 2 23 0 2 0 2 2 0 3 0 111 11111 sinsincos2 1 4 1 4 sin 4 sincos2 4 sincos2 We can divide the zones around the source, as function of the relation between dipole size d and wavelength λ, in three zones: Near, Intermediate and Far Fields   22 :  c f c k The Magnetic Field Intensity is transverse to the propagation direction at all ranges, but the Electric Field Intensity has components parallel and perpendicular to .r1  r1  E  However and are perpendicular to each other.H  • Near (static) zone: rd • Intermediate (induction) zone: ~rd  • Far (radiation) zone: rd 
  • 40. 40 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1     102 sin 4   tj FieldNear ep r kc jH tj FieldNear ep r E r    03 0 11 sincos2 4 1         Near, Intermediate and Far Fields (continue – 1) • Near (static) zone: rd In the near zone the fields have the character of the static fields. The near fields are quasi-stationary, oscillating harmonically as , but otherwise static in character.tj e  0 2    r rk
  • 41. 41 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1       102 sin 4    krtj FieldteIntermedia ep r kc jH  krtj FieldteIntermedia ep r kj r E r                   023 0 11 sincos2 1 4 1 Near, Intermediate and Far Fields • Intermediate (induction) zone: ~rd  • Far (radiation) zone: rd       10 2 sin 4    krtj FieldFar ep r kc H       10 0 2 sin 4    krtj FieldFar ep r k E r1  FieldFarE FieldFarH At Far ranges are orthogonal; i.e. we have a transversal wave. rHE 1,,  In the Radiation Zone the Field Intensities behave like a spherical wave (amplitude falls off as r-1) 1 2    r rk              120 10 36 1 1041 : 9 7 0 0 1 0 00    c FieldFar FieldFar cH E Z
  • 42. 42 SOLO Electric Dipole Radiation http://dept.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_apr_07_2003.shtml#tth_sEc12.1 http://www.falstad.com/mathphysics.html Electric Field Lines of Force
  • 43. 43 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1  The phasors of the Magnetic and Electric Field Intensities are:       10 2 sin 4 1          krtj ep cr j r H  krtj ep crc j rc j rrr E r                            02 2 2 0 11 sin 11 cos 12 4 1 Poynting Vector of the Electric Dipole Field The Poynting Vector of the Electric Dipole Field is The Magnetic and Electric Field Intensities are:            1sincossin 4 2 0         krt c krt rr p HrealH                                                  11 sinsin 1 cos 1 cossincos 12 4 2 2 2 0 0 krt rc krt cr krt c krt rrr p ErealE r                                       1 1 cossincossinsincos 1 4 2 sincossinsincos 1 4 2 0 32 2 0 2 2 2 2 2 0 22 2 0                                   krt c krt r krt c krt rr p krt c krt r krt rc krt crr p HES r  The Poynting Vector of the Electric Dipole Field is given by:
  • 44. 44 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1  Let compute the time average < > of the Poynting vector: Poynting Vector of the Electric Dipole Field Using the fact that:                                      1 1 cossincossinsincos 1 4 2 sincossinsincos 1 4 2 0 32 2 0 2 2 2 2 2 0 22 2 0                                   krt c krt r krt c krt rr p krt c krt r krt rc krt crr p HES r     T T dttS T S 0 1 lim        2 1 2cos 1 lim 2 11 lim 2 1 cos 1 limcos 0 0 1 00 22       T T T T T T dtrkt T dt T dtrkt T rkt        2 1 2cos 1 lim 2 11 lim 2 1 sin 1 limsin 0 0 1 00 22       T T T T T T dtrkt T dt T dtrkt T rkt            02sin 1 lim 2 1 cossin 1 limcossin 0 00       T T T T dtrkt T dtrktrkt T rktrkt    r rc p S 1 2 23 0 2 42 0 sin 42                11 cossin 4 sin 1 42 22 0 32 2 02 2 2 2 2 2 2 0 22 2 0                     rcrcr p rccrcr p S r  we obtain: or: Radar Equation Irradiance
  • 45. 45 SOLO Electric Dipole Radiation  tq  tq d  r    zSS rrdqP 10     dr    sinr cosr           zyx r r rr 111 1 cossinsincossin   r1  1  1  x1  y1  z1  Poynting Vector of the Electric Dipole Field   r rc p S 1 2 23 0 2 42 0 sin 42      Radar Equation 45 90 135 1800 0 5 10 15 20 25 30  0  45  90  135  180  225  270  315 z y 5.0 0.1 Polar Angle , in degrees RelativePower,indb The Total Average Radiant Power is:                0 22 23 0 2 42 0 sin2sin 42 dr rc p dSSP A rad  22 0 22 120 123 0 42 0 3/4 0 3 23 0 42 0 40 12 sin 16 0              p rc p d rc p P c c rad                3 4 3 2 3 2 cos 3 1 coscoscos1sin 0 3 0 2 0 3                  dd Return to TOC
  • 46. 46 ELECTROMAGNETICSSOLO To satisfy the Maxwell equations for a source free media we must have: Monochromatic Planar Wave Equations we haveUsing: 1ˆˆ&ˆˆ 0  kkknkkk              0 0 H E HjE EjH                  0ˆ 0ˆ ˆ ˆ 0 0 00 00 Hk Ek HEk EHk     kˆ Planar Wave 0E 0H r                    0 0 0 0 00 00 rkj rkj rkjrkj rkjrkj ekje eHkj eEkj eHjeEkj eEjeHkj rkjrkj               22 22 & 2 ˆ 2 ˆ HwEwwcn k wwcn k S meme    Time Average Poynting Vector of the Planar Wave Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 47. 47 SOLO REFLECTION & REFRACTION iE  iH  rE  rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t Plan of incidence Consider an incident monochromatic planar wave     c n k eEkH eEE iiii rktj iii rktj ii ii ii 1 00 11 0011 0 0                  The monochromatic planar reflected wave from the boundary is     1 1 1 1 0 0 & n c v vc n k eEkH eEE r rr rktj rrr rktj rr rr rr                The monochromatic planar refracted wave from the boundary is     2 2 2 2 0 0 & n c v vc n k eEkH eEE t tt rktj ttt rktj tt tt tt                Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 48. 48 SOLO REFLECTION & REFRACTION The Boundary Conditions at z=0 must be satisfied at all points on the plane at all times, implies that the spatial and time variations of This implies that iE  iH  rE  rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t Plan of incidence Phase-Matching Conditions       yxteEeEeE z rktj t z rktj r z rktj i ttrrii ,,,, 0 0 0 0 0 0                 yxtrktrktrkt z tt z rr z ii ,, 000     ttri         yxrkrkrk z t z r z i , 000    must be the same Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 49. 49 SOLO REFLECTION & REFRACTION tri nnn  sinsinsin 211  iE  iH  rE  rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t Plan of incidence Phase-Matching Conditions            zyx c n k zyx c n k ttttttt irirrrr ˆcossinˆsinsinˆcos ˆcossinˆsinsinˆcos 2 1                       yyx c n rk yx c n rk y c n rk ttt z t irr z r i z i ˆsinsincos sinsincos sin 2 0 1 0 1 0             yxrkrkrk z t z r z i , 000    2    tr ttri   x y Coplanar Snell’s Law         zzyyxxr zy c n k iiii ˆˆˆ ˆcosˆsin1    Given: Let find: Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 50. 50 SOLO REFLECTION & REFRACTION Second way of writing phase-matching equations ri   11 22 2 1 1 2 sin sin      v v n n t iRefraction Law Reflection Law Phase-Matching Conditions         zzyyxxr zy c n k iiii ˆˆˆ ˆcosˆsin1               zyx c n k zyx c n k ttttttt irirrrr ˆcossinˆsinsinˆcos ˆcossinˆsinsinˆcos 2 1                      ynnyn c kkz ynnyn c kkz ittrti irrrri ˆsinsinsinˆcosˆ ˆsinsinsinˆcosˆ 122 111       ttri   We can see that                tri tiri kkzkkz 0ˆˆ                  tri tri tr nnn sinsinsin 2/ 211 iE  iH  rE  rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t Plan of incidence Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 51. 51 SOLO REFLECTION & REFRACTION ri   11 22 2 1 1 2 sin sin      v v n n t iRefraction Law Reflection Law Phase-Matching Conditions (Summary) ttri                  tri tiri kkzkkz 0ˆˆ                  tri tri tr nnn sinsinsin 2/ 211 iE  iH  rE  rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t Plan of incidence       yxrkrkrk z t z r z i , 000          yxtrktrktrkt z tt z rr z ii ,, 000     Vector Notation Scalar Notation Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 52. 52 SOLO REFLECTION & REFRACTION iE  iH  rE rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t i r t tH  tE  tk  rH  rk  rE  iH  iE  ik  21 ˆ n Boundary Plan of incidence ti ti i r nn nn E E r         coscos coscos 2 2 1 1 2 2 1 1 0 0            ti i i t nn n E E t       coscos cos2 2 2 1 1 1 1 0 0           For most of media μ1= μ2 , and using refraction law: 1 2 sin sin n n t i       ti ti i r E E r               sin sin21 0 0  ti it i t E E t              sin cossin221 0 0 Assume is normal to plan of incidence (normal polarization) E  xEExEExEE ttrrii ˆ&ˆ&ˆ 000000    Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations See full development in P.P. “Reflection & Refractions”
  • 53. 53 SOLO REFLECTION & REFRACTION iE  iH  rE  rH  ik  rk  tH  tE  tk  Boundary 21 ˆ n z x y i r t i r t tH  tE  tk  rH  rk  rE  iH  iE  ik  21 ˆ n Boundary Plan of incidence Assume is parallel to plan of incidence (parallel polarization) E       zyEE zyEE zyEE tttt rrrr iiii ˆsinˆcos ˆsinˆcos ˆsinˆcos 0||0 0||0 0||0          ti ti i r nn nn E E r         coscos coscos 1 1 2 2 1 1 2 2 ||0 0 ||          ti i i t nn n E E t       coscos cos2 1 1 2 2 1 1 ||0 0 ||         For most of media μ1= μ2 , and using refraction law: 1 2 sin sin n n t i       ti ti i r E E r             tan tan21 ||0 0 ||    titi it i t E E t            cossin cossin221 ||0 0 || Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations See full development in P.P. “Reflection & Refractions”
  • 54. 54 SOLO REFLECTION & REFRACTION ti ti i r nn nn E E r         coscos coscos 1 1 2 2 1 1 2 2 ||0 0 ||          ti i i t nn n E E t       coscos cos2 1 1 2 2 1 1 ||0 0 ||         ti ti i r nn nn E E r         coscos coscos 2 2 1 1 2 2 1 1 0 0            ti i i t nn n E E t       coscos cos2 2 2 1 1 1 1 0 0           The equations of reflection and refraction ratio are called Fresnel Equations, that first developed them in a slightly less general form in 1823, using the elastic theory of light. Augustin Jean Fresnel 1788-1827 The use of electromagnetic approach to prove those relations, as described above, is due to H.A. Lorentz (1875) Reflections and Refractions Laws Development Using the Electromagnetic Approach Hendrik Antoon Lorentz 1853-1928 See full development in P.P. “Reflection & Refractions” Return to TOC
  • 55. 55 IR Radiometric QuantitiesSOLO  RTA DA  2 cm  2 cm TARGET SOURCE DETECTOR RECEIVER Radiation Flux Power  W Spectral Radial Power          m W   Irradiance          2 mc W A E Spectral Radiant Emittance          mmc WM M   2 Radiant Intensity          str W I Spectral Radiant Intensity          mstr WI I   Radiance          strmc W A I L 2 cos Spectral Radiance          mstrmc WL L   2 Radiant Emittance          2 mc W A M Spectral Irradiance          mmc WE E   2    T T dttS T S 0 1 lim  Irradiance is the time- average of the Poynting vector Return to TOC
  • 56. 56 Physical Laws of RadiometrySOLO Plank’s Law   1/exp 1 2 5 1   Tc c M BB   Plank 1900 Plank’s Law applies to blackbodies; i.e. perfect radiators. The spectral radial emittance of a blackbody is given by:   KT KWk Wh kmc Kmkhcc mcmWchc    ineTemperaturAbsolute- constantBoltzmannsec/103806.1 constantPlanksec106260.6 lightofspeedsec/458.299792 10439.1/ 107418.32 23 234 4 2 4242 1           MAX PLANCK (1858 - 1947) Plank’s Law
  • 57. 57 Physical Laws of RadiometrySOLO Plank’s Law   1/exp 1 2 5 1   Tc c M BB   Plank 1900 Plank’s Law applies to blackbodies; i.e. perfect radiators. The spectral radial emittance of a blackbody is given by: MAX PLANCK (1858 - 1947) Plank’s Law
  • 58. 58 Physical Laws of Radiometry (Continue -1)SOLO Wien’s Displacement Law 0   d Md BB Wien 1893 from which: The wavelength for which the spectral emittance of a blackbody reaches the maximum is given by: m KmTm    2898 Wien’s Displacement Law Stefan-Boltzmann Law Stefan – 1879 Empirical - fourth power law Boltzmann – 1884 Theoretical - fourth power law For a blackbody:                             42 12 32 45 2 4 0 2 5 1 0 10670.5 15 2 : 1/exp 1 Kcm W hc k cm W Td Tc c dMM BBBB       LUDWIG BOLTZMANN (1844 - 1906) WILHELM WIEN (1864 - 1928) Stefan-Boltzmann Law JOSEF STEFAN (1835 – 1893)
  • 59. 59 Physical Laws of Radiometry (Continue -1a)SOLO Black Body Emittance M [W/m2] M (300ºK) 5.86 121 M (301ºK) - M (300ºK) 0.22 2 M (600ºK) 1,719 1,555 M (601ºK) - M (600ºK) 17 7 3 – 5 µm 8 - 12 µm
  • 60. 60 Physical Laws of Radiometry (Continue -2)SOLO Emittance of Real Bodies (Gray Bodies) For real (gray) bodies: BB MM   - Directional spectral emissivity is a measure of how closely the flux radiated from a given temperature radiator approaches that from a blackbody at the same temperature   ,   BB M M
  • 61. 61 Physical Laws of Radiometry (Continue -3)SOLO Kirchhoff’s Law rM iE aE tM Gustav Robert Kirchhoff 1824-1887 - Incident IrradianceiE - Absorbed IrradianceaE - Reflected Radiant ExcitancerM - Transmitted Radiant ExcitancetM Law of Conservation of Energy: trai MMEE      i t i r i a E M E M E E 11  i a E E : - fraction of absorbed energy (absorptivity) i r E M : - fraction of reflected energy (reflectivity) i t E M : - fraction of transmitted energy (transmissivity) Opaque body (no transmission): 01   Blackbody (no reflection or transmission): 0&01   Sharp boundary (no absorption): 01  
  • 62. 62 Physical Laws of Radiometry (Continue -4)SOLO Kirchhoff’s Law (Continue – 1) Gustav Robert Kirchhoff 1824-1887 Kirchhoff’s Law (1860) states that, for any temperature and any wavelength, the emissivity of an opaque body in an isothermal enclosure is equal to it’s absorptivity. This is because if the body will radiate to the surrounding less than it absorbs it’s temperature will rise above the surrounding and will be a transfer of energy from a cold surrounding to a hot body contradicting the second law of thermodynamics.    TT    222 ,, T 2A 111 ,, T 1A
  • 63. 63 Physical Laws of Radiometry (Continue -5)SOLO Lambert’s Law Johann Heinrich Lambert 1728 - 1777 http://www-groups.dcs.st-andrews.ac.uk/~history/Biographies/Lambert.html A Lambertian Surface is defined as a surface from which the radiance L [W/(cm2 str)] is independent of the direction of radiation.  dr    d rsin    2 sin r drdr d   A cosAAn   z x y   0 2 cos , L A L          coscos, 00 IALI     Lambert’s Law 0 2 0 2/ 0 00 sincoscos LddLdL A M         The Radiant Intensity from a Lambertian Surface is The Radiant Emittance (Exitance) from a Lambertian Surface is
  • 64. 64 Physical Laws of Radiometry (Continue -6)SOLO Transfer of Radiant Energy We have two bodies 1 and 2. The radiant power (radiance) transmitted from 1 to 2 is: 2 12 22 22 211 12 2 1 cos & cos R Ad d strcm W A L            1A 1dA 1 12R Radiating (Source) Surface 2A 2dA Receiving Surface 2 2d    2 12 22111 12 coscos R AdAdL d   The total radiant power (radiance) received at surface A2 from A1 is:   2 1 212 12 211 12 coscos A A AdAd R L 
  • 65. 65 Physical Laws of Radiometry (Continue -7)SOLO Transfer of Radiant Energy (Continue – 1) Define the projected areas: and the solid angles: 222111 cos&cos AdAdAdAd nn   1A 1dA 1 12R Radiating (Source) Surface 2A 2dA Receiving Surface 2 2d 2 12 22 22 12 11 1 cos & cos R Ad d R Ad d   1A 1dA 1 12R Radiating (Source) Surface 2A 2dA Receiving Surface 2 1d then:    2 12 211 2 12 22111 12 coscos R AdAdL R AdAdL d nn   12121112  dAdLdAdLd nn The Power is the product of the Radiance, the projected Area, and the Solid Angle using the other area.
  • 66. 66 Physical Laws of Radiometry (Continue -8)SOLO Transfer of Radiant Energy (Continue – 2) Optics R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, For an Optical System define: ATARGET – Target Area ADETECTOR – Detector Area AOPTICS – Optics Area R – Range from Target to Optics f – Focal Length (from Optics to Detector) ΩO,T – solid angle of Optics as seen from the Target 2, R AOPTICS TO  ωT,O – solid angle of Target as seen from the Optics 2, R ATARGET OT  ΩD,O – solid angle of Detector as seen from the Optics 2, f ADETECTOR OD  ωO,D – solid angle of Optics as seen from the Detector 2, f ADETECTOR DO 
  • 67. 67 Physical Laws of Radiometry (Continue -9)SOLO Transfer of Radiant Energy (Continue – 3) Optics (continue – 1) R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, For the Figure we can see that: ODOT ,,  22 f A R A DETECTORTARGET  Also we found that: DODETECTOROTOPTICS ODOPTICSTOTARGET ALAL ALAL OTOD ,, ,, ,,     
  • 68. 68 Physical Laws of Radiometry (Continue -10)SOLO Transfer of Radiant Energy (Continue – 4) Optics (continue – 2) R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, TOTARGETAL , ODOPTICSAL , OTOPTICSAL , DODETECTORAL ,
  • 69. 69 Physical Laws of Radiometry (Continue -11)SOLO Transfer of Radiant Energy (Continue – 5) Optics (continue – 3)   2 , R A AL AL TARGET DETECTOR DTDETECTOROpticsNo    2 , f A AL AL OPTICS DETECTOR DODETECTOROpticsWith    R f ATARGET ADETECTOR AOPTICS TO, OD,OT , DO, R ATARGET ADETECTOR TD, DT , • IR Detector without Optics • IR Detector with Optics     2 # / 2 4 0 4 4 0# f AL f D AL DETECTOR Dff DETECTOR     The Optics increases the energy collected by the Detector since DTDO ,,   22 # 2 4 R A ff A TARGETOPTICS   OpticsNoOpticsWith 
  • 70. 70 Physical Laws of Radiometry (Continue -12)SOLO Targets The parts of the aircraft that are especially hot are: • The exhaust nozzle of the jet engine • The hot exhaust gas area, or the plume • The areas in which aerodynamic heating is the highest
  • 71. 71 Physical Laws of Radiometry (Continue -13)SOLO Targets
  • 72. 72 Physical Laws of Radiometry (Continue -14)SOLO Targets
  • 73. 73 Physical Laws of Radiometry (Continue -15)SOLO Targets
  • 74. 74 Physical Laws of Radiometry (Continue -16)
  • 75. 75 Physical Laws of Radiometry (Continue -17)SOLO Targets (continue – 1) • The exhaust nozzle of the jet engine The exhaust nozzle can be regarded as a gray body with ε = 0.9. Example: Turbojet Engine 4-P&W JT4A-9 2 3660 cmANOZZLE  rafterburnewithCT  538   24124 207.22735381067.59.0   cmWTM  We are interested only in the band 3 μm ≤ λ ≤ 5 μm. By numerically integration or using infrared radiation calculators we obtain: 397.0 811 4 5 3    KT BB T dM    Hence:   2 876.0207.2397.053   cmWmmM  In a tail-on situation the radiant intensity is:   1 1020 3660876.0 53           strWA M mmI NOZZLE Lambertian  
  • 76. 76 Physical Laws of Radiometry (Continue -18)SOLO Targets (continue – 2) • The plume The plume is characterized by the radiant emittance of the hot gases that are expanding into the atmosphere after passing through the exhaust nozzle. The products of combustion are H2O, CO2, some times CO (incomplete combustion), OH, HF, HCl. The infrared emission is produced by changes in the energy contained in the molecular vibrations and rotations, only at certain frequencies..
  • 77. 77 Physical Laws of Radiometry (Continue -19)SOLO Targets (continue – 3) • The plume (continue – 1)
  • 78. 78 Physical Laws of Radiometry (Continue -20)SOLO Targets (continue – 4) • The plume (continue – 2) Breathing engines have exhaust plume temperatures of K 600450  Cruise flight K 800600  Maximum Un-augmented Thrust K 15001000  Augmented (After burner) Thrust Rockets have exhaust plume temperatures of K 75002500  Liquid propellant K 35001700  Solid propellant Example Assume:   mm  55.433.45.0  KCCTPLUME  643273370  then:   2222 55.4 33.4 1075.1105.35.0      cmWcmWdMM     For a plume surface of APLUME = 10000 cm2 = 1 m2 the Radiant Intensity is:   1 42 8.27 101075.1 55.433.4            strWA M mmI PLUME Lambertian  
  • 79. 79 Physical Laws of Radiometry (Continue -21)SOLO Targets (continue – 5) • Aerodynamic Heating The Target body is heated by the compression and friction of the air against it’s surface and by friction. Assuming a negligible friction effect and an adiabatic compression the Target skin temperature is given by:         2 0 2 1 1, MachrMachHTT  - air temperature at altitude HTARGET and mach number Mach MachHT ,0 - recovery factorr vp CC / - specific heat ratio = 1.4 for air Example Mach = 2.0, HTARGET = 5000 m   27.0,250.2,50000  KMachmHT  then KT  4142 2 14.1 82.01250 2             23 5 3 1066.1414      cmWdKTMM      assume 2 15mATARGET    1 43 3.79 10151066.1 53            strWA M mmI TARGET Lambertian  
  • 80. 80 Physical Laws of Radiometry (Continue -22)
  • 81. 81 Physical Laws of Radiometry (Continue -23)
  • 82. 82 Physical Laws of Radiometry (Continue -24)SOLO Targets (continue – 6) • Aerodynamic Heating (continue – 1) EmissivityReflectanceAbsorptanceMaterial .04.81.19Polished Aluminium .04.63.37Unpolished Aluminium .18.43.57Titanium .05.60.40Polished Stainless Steel .88.79.21White Paint .92.05.95Black Paint .27.71.29Aluminum Paint
  • 83. 83 Physical Laws of Radiometry (Continue -25)SOLO Sun, Background and Atmosphere
  • 84. 84 Physical Laws of Radiometry (Continue -26)SOLO Sun, Background and Atmosphere (continue – 1) The spectrum distribution of the sun radiation is like a black body with a temperature of T = 5900 °K From Wien’s Law the maximum of Mλ is at m T m  49.0 5900 28982898  This is almost at the middle of the visible spectrum mm  75.040.0  Loss by Scattering
  • 85. 85 Physical Laws of Radiometry (Continue -27)SOLO Sun, Background and Atmosphere (continue – 2) Atmosphere Atmosphere affects electromagnetic radiation by     3.2 1 1        R kmRR  • Absorption • Scattering • Emission • Turbulence Atmospheric Windows: Window # 2: 1.5 μm ≤ λ < 1.8 μm Window # 4 (MWIR): 3 μm ≤ λ < 5 μm Window # 5 (LWIR): 8 μm ≤ λ < 14 μm For fast computations we may use the transmittance equation: R in kilometers. Window # 1: 0.2 μm ≤ λ < 1.4 μm includes VIS: 0.4 μm ≤ λ < 0.7 μm Window # 3 (SWIR): 2.0 μm ≤ λ < 2.5 μm
  • 86. 86 Physical Laws of Radiometry (Continue -28)SOLO Sun, Background and Atmosphere (continue – 3) Atmosphere Absorption over Electromagnetic Spectrum
  • 87. 87 Physical Laws of Radiometry (Continue -29)SOLO Sun, Background and Atmosphere (continue – 4) Rain Attenuation over Electromagnetic Spectrum FREQUENCY GHz ONE-WAYATTENUATION-Db/KILOMETER WAVELENGTH
  • 88. 88 Physical Laws of Radiometry (Continue -30)SOLO Sun, Background and Atmosphere (continue – 3) Add scanned Figure from McKenzie Atmosphere (continue – 1)
  • 89. 89 GEOMETRICAL OPTICSSOLO http://en.wikipedia.org/wiki/Optics From “Cyclopaedia” or “An Universal Dictionary of Art and Science” Published by Ephraim Chambers In London in 1728 Return to TOC
  • 90. 90 SOLO DERIVATION OF EIKONAL EQUATION Foundation of Geometrical Optics Derivation from Maxwell Equations Consider a general time-harmonic field:                                tjrHtjrHtjrHaltrH tjrEtjrEtjrEaltrE     exp,exp, 2 1 exp,Re, exp,exp, 2 1 exp,Re, * *   in a non-conducting, far-away from the sources  0,0  eeJ   No assumption of isotropy of the medium are made; i.e.:     rr   , Far from sources, in the High Frequencies we can write using the phasor notation:             00000 &,&, 00     kerHrHerErE rSjkrSjk Note The minus sign was chosen to get a progressive wave: End Note                SktjSktj erHaltrHerEaltrE 00 00 Re,&Re,     James Clerk Maxwell (1831-1879) See full development in P.P. “Foundation of Geometrical Optics”
  • 91. 91 SOLO From those equations we have Foundation of Geometrical Optics           Sjktj SjkSjktjSjktjtj eeESjkE EeeEeeEeerE 0 000 000 000,               Sjk SjktjSjktjtj eHjk eHejeHejerH t 0 00 0 00 0 0 00 000 1 1 ,           from which   0 00 0000 HjkESjkEF    and 0 1 0 0 0 0 00 0   k E jk HES   DERIVATION OF EIKONAL EQUATION (continue – 2) Derivation from Maxwell Equations (continue – 2)
  • 92. 92 SOLO From Maxwell equations we also have Foundation of Geometrical Optics from which and DERIVATION OF EIKONAL EQUATION (continue – 3) Derivation from Maxwell Equations (continue – 3)           Sjktj SjkSjktjSjktjtj eeHSjkH HeeHeeHeerH 0 000 000 000,               Sjk SjktjSjktjtj eEjk eEejeEejerE t 0 00 0 00 0 0 00 000 1 1 ,             0 00 0000 EjkHSjkHA    0 1 0 0 0 0 00 0   k H jk EHS  
  • 93. 93 SOLO DERIVATION OF EIKONAL EQUATION (continue – 4) Foundation of Geometrical Optics Derivation from Maxwell Equations (continue – 4) We have Faradey (F), Ampére (A), Gauss Electric (GE), Gauss Magnetic (GM) equations:                       0 0 HGM EGE EjHA HjEF                                                         0&0 2 0 0 0 ee e e J c k j t HB ED BGM DGE J t D HA t B EF                  André-Marie Ampère 1775-1836 Michael Faraday 1791-1867 Karl Friederich Gauss 1777-1855
  • 94. 94 SOLO From Maxwell equations we also have Foundation of Geometrical Optics from which and DERIVATION OF EIKONAL EQUATION (continue – 4) Derivation from Maxwell Equations (continue – 4)             0 , 0 000 0000 000     Sjktj SjkSjktjSjktjtj eeESjkEE EeeEeeEeerE       00000  ESjkEEGE  0 1 0 00 0 0           k EE jk ES   We also have from which and             0 , 0 000 0000 000     Sjktj SjkSjktjSjktjtj eeHSjkHH HeeHeeHeerH       00000  HSjkHHGM  0 1 0 00 0 0           k HH jk HS  
  • 95. 95 SOLO To summarize, from k0 → ∞ we have Foundation of Geometrical Optics DERIVATION OF EIKONAL EQUATION (continue – 5) Derivation from Maxwell Equations (continue – 5)   00 00 0  HESF     00 00 0  EHSA     00  ESGE   00  HSGM We will use only the first two equations, because the last two may be obtained from the previous two by multiplying them (scalar product) by .S
  • 96. 96 SOLO Foundation of Geometrical Optics DERIVATION OF EIKONAL EQUATION (continue – 6) Derivation from Maxwell Equations (continue – 6)   00 00 0  HESF     00 00 0  EHSA   From the second equation we obtain 0 00 0 HSE    And by substituting this in the first equation   00 0 00 00 00 0 00           HHSSHHSS       But         2 00 0 2 0 0 00 n HSHSSSHSHSS     
  • 97. 97 SOLO Foundation of Geometrical Optics DERIVATION OF EIKONAL EQUATION (continue – 7) Derivation from Maxwell Equations (continue – 7) Finally we obtain    00 22  HnS or    zyxn z S y S x S ornS ,,0 2 222 22                          S is called the eikonal (from Greek έίκων = eikon → image) and the equation is called Eikonal Equation. Return to TOC
  • 98. 98 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS From Max Born & Emil Wolf, “Principles of Optics”, 6th Ed., Ch. 3             00000 &,&, 00     kerHrHerErE rSjkrSjk We found the following relations   00 00 0  HESF     00 00 0  EHSA     00  ESGE   00  HSGM We can see that the vectors are perpendicular in the same way as the vectors for the planar waves (where is the Poynting vector). SHE ,, 00 SHE  ,, 00 00 HES   S 0E 0H
  • 99. 99 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 1)                                                     T T T TT e dttjrErErEtjrE T dttjrEtjrEtjrEtjrE T dttjrEal T dttrEtrE T dttrDtrE T w 0 2**2 0 ** 0 2 00 2exp,,,22exp, 4 1 exp,exp,exp,exp, 4 1 exp,Re 1 ,, 1 ,, 1           But               0 2 2exp 2exp 2 1 2exp 1 0 2 2exp 2exp 2 1 2exp 1 0 0 0 0         T T T T T T Tj Tj tj Tj dttj T Tj Tj tj Tj dttj T           Therefore                rErEerEerEdt T rErEw rSjkrSjk T e  * 00 * 00 0 * 22 1 ,, 2 00      Let compute the time averages of the electric and magnetic energy densities
  • 100. 100 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 2) In the same way                rErEerEerEdt T rErEw rSjkrSjk T e  * 00 * 00 0 * 22 1 ,, 2 00                 rHrHdttrHtrH T dttrBtrH T w TT m * 00 00 2 ,, 1 ,, 1      Using the relations   0 00 0 HSEA      0 00 0 ESHF    since and are real values , where * is the complex conjugate, we obtain S )**,( SS                                              e m e wrHSrErHSrErHSrE rESrHrESrHrHrHw rHSrErHSrErErEw    * 00 * 0 * 00 ** 0 * 00 * 0 00 0 * 00 * 00 * 0 00 0 * 00 2 1 2 1 2 1 2 1 22 2 1 22        S 0E 0H
  • 101. 101 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 3) Therefore     * 00 2 1 rHSrEww me   Within the accuracy of Geometrical Optics, the time-averaged electric and magnetic energy densities are equal.             * 0000 * 00 22 rHSrErHrHrErEwww me    The total energy will be: The Poynting vector is defined as:      trHtrEtrS ,,:,                               T tjtjtjtj T tjtj T dterHerHerEerE T dterHerEal T dttrHtrE T trHtrES 0 ** 00 ,, 2 1 ,, 2 11 ,,Re 1 ,, 1 ,,                                  ,,,, 4 1 ,,,,,,,, 4 11 ** 0 2****2 rHrErHrE dterHrErHrErHrEerHrE T T tjtj                      rHrErHrE erHerEerHerE rSjkrSjkrSjkrSjk 0 * 0 * 00 )( 0 )(* 0 )(* 0 )( 0 4 1 4 1 0000    The time average of the Poynting vector is: John Henry Poynting 1852-1914
  • 102. 102 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 3) Using the relations   0 00 0 HSEA      0 00 0 ESHF                           rHrHSrESrErHrErHrES 0 * 0 * 00 00 0 * 0 * 00 222 1 4 1    we obtain                  * 00 0 0 * 0 0 0 * 0 * 00 00 22222 1 HHSHSHESEEES                 * 0000 * 00 22 rHSrErHrHrErEwww me    we obtain Using   wS n c wwSS me  2 00 00 22 1     00 2 00 & 1     nc
  • 103. 103 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 4) Using   22 nS  Eikonal Equation we obtain nS  Define snS n S S S s ˆ:ˆ       We have swvwS n c S n c v ˆ 2 1 2 2    sˆ constS  constdSS  sˆ r  0 ˆs 0r  A Bundle of Light Rays
  • 104. 104 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 5) swvwS n c S n c v ˆ 2 1 2 2    sˆ constS  constdSS  sˆ r  0 ˆs 0r  From this equation we can see that average Poynting vector is the direction of the normal to the geometrical wave-front , and its magnitude is proportional to the product of light velocity v and the average energy density, therefore we say that defines the direction of the light ray. S sˆ sˆ Suppose that the vector describes the light path, then the unit vector is given by r  sˆ sd rd rd rd s ray ray ray    ˆ where is the differential of an arc length along the ray pathrayrdsd  
  • 105. 105 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 6) Let substitute in and differentiate it with respect to s. sd rd rd rd s ray ray ray    ˆ rayrdsd    S sd d sd rd n sd d       ray   S sd rd  ray    sd rd f sd zd zd fd sd yd yd fd sd xd xd fd sd zyxfd   ,,   SS n  1 S sd rd n  ray           ABBAABBABA   AB        AAAAAA   2 1 SA           SSSSSSSS    0 2 1 SS n  2 1 2 nSS  2 2 1 n n  n
  • 106. 106 SOLO Foundation of Geometrical Optics THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 7) Therefore we obtained   nS sd d  and n sd rd n sd d         ray  We obtained a ordinary differential equation of 2nd order that enables to find the trajectory of an optical ray , giving the relative index and the initial position and direction of the desired ray.  srray   zyxn ,,   00 rrray   0 ˆs sˆ constS  constdSS  sˆ r  0 ˆs 0r  We can transform the 2nd order differential equation in two 1st order differential equations by the following procedure. Define Ssn sd rd np  ˆ: ray   We obtain   0 ˆ0 snpnp sd d     0 ˆ0 snpnp sd d   Return to TOC
  • 107. 107 SOLO The Three Laws of Geometrical Optics 1. Law of Rectilinear Propagation In an uniform homogeneous medium the propagation of an optical disturbance is in straight lines. 2. Law of Reflection An optical disturbance reflected by a surface has the property that the incident ray, the surface normal, and the reflected ray all lie in a plane, and the angle between the incident ray and the surface normal is equal to the angle between the reflected ray and the surface normal: 2v 1v Refracted Ray 21 ˆ n 2n 1n i t Reflected Ray 21 ˆ n 2n 1n i r 3. Law of Refraction An optical disturbance moving from a medium of refractive index n1 into a medium of refractive index n2 will have its incident ray, the surface normal between the media , and the reflected ray in a plane, and the relationship between angle between the incident ray and the surface normal θi and the angle between the reflected ray and the surface normal θt given by Snell’s Law: ti nn  sinsin 21  ri   “The branch of optics that addresses the limiting case λ0 → 0, is known as Geometrical Optics, since in this approximation the optical laws may be formulated in the language of geometry.” Max Born & Emil Wolf, “Principles of Optics”, 6th Ed., Ch. 3 Foundation of Geometrical Optics Return to TOC
  • 108. 108 SOLO Foundation of Geometrical Optics Fermat’s Principle (1657) 1Q 1P 2P 2Q 1Q 2Q 1S SdSS  12  2PS  1PS 2'Q rd  sˆ sˆ The Principle of Fermat (principle of the shortest optical path) asserts that the optical length of an actual ray between any two points is shorter than the optical ray of any other curve that joints these two points and which is in a certai neighborhood of it. An other formulation of the Fermat’s Principle requires only Stationarity (instead of minimal length).  2 1 P P dsn An other form of the Fermat’s Principle is: Princple of Least Time The path following by a ray in going from one point in space to another is the path that makes the time of transit of the associated wave stationary (usually a minimum). The idea that the light travels in the shortest path was first put forward by Hero of Alexandria in his work “Catoptrics”, cc 100B.C.-150 A.C. Hero showed by a geometrical method that the actual path taken by a ray of light reflected from plane mirror is shorter than any other reflected path that might be drawn between the source and point of observation.
  • 109. 109 SOLO 1. The optical path is reflected at the boundary between two regions     0 21 21          rd sd rd n sd rd n rayray   In this case we have and21 nn        0ˆˆ 21 21          rdssrd sd rd sd rd rayray   We can write the previous equation as: i.e. is normal to , i.e. to the boundary where the reflection occurs. 21 ˆˆ ss  rd    0ˆˆˆ 2121  ssn 11 ˆsn 21 ˆsn 1121 ˆˆˆ snsn  rd    0ˆˆ 121  rdssn  Reflected Ray 21 ˆ n 1n i r REFLECTION & REFRACTION Reflection Laws Development Using Fermat Principle ri   Incident ray and Reflected ray are in the same plane normal to the boundary. This is equivalent with: &
  • 110. 110 SOLO 2. The optical path passes between two regions with different refractive indexes n1 to n2. (continue – 1)     0 21 21          rd sd rd n sd rd n rayray   where is on the boundary between the two regions andrd      sd rd s sd rd s rayray 2 :ˆ, 1 :ˆ 21   rd  22 ˆsn 11 ˆsn 1122 ˆˆˆ snsn    0ˆˆˆ 1122  rdsnsn  Refracted Ray 21 ˆ n 2n 1n i t Therefore is normal to .2211 ˆˆ snsn  rd  Since can be in any direction on the boundary between the two regions is parallel to the unit vector normal to the boundary surface, and we have rd  2211 ˆˆ snsn  21 ˆ n   0ˆˆˆ 221121  snsnn We recovered the Snell’s Law from Geometrical Optics REFLECTION & REFRACTION Refraction Laws Development Using Fermat Principle ti nn  sinsin 21  Incident ray and Refracted ray are in the same plane normal to the boundary. & Return to TOC
  • 111. 111 SOLO Plane-Parallel Plate i r ri   r t l d i A C B E 2n 1n A single ray traverses a glass plate with parallel surfaces and emerges parallel to its original direction but with a lateral displacement d. Optics    irriri lld  cossincossinsin  r t l cos         r i ritd    cos cos sinsin ir nn  sinsin 0Snell’s Law        n n td r i i 0 cos cos 1sin    For small anglesi        n n td i 0 1
  • 112. 112 SOLO Plane-Parallel Plate (continue – 1) t        r i i n n td    cos cos 1sin 1 2 1n  2n i r        r i i n n t d l    cos cos 1 sin 1 2 l Two rays traverse a glass plate with parallel surfaces and emerge parallel to their original direction but with a lateral displacement l. Optics    irriri lld  cossincossinsin  r t l cos         r i ritd    cos cos sinsin ir nn  sinsin 0Snell’s Law        n n td r i i 0 cos cos 1sin           r i i n n t d l    cos cos 1 sin 0 For small anglesi        n n tl 0 1 Return to TOC
  • 113. 113 SOLO Prisms   2i1i 1t    11 ti   2t  22 it   Type of prisms: A prism is an optical device that refract, reflect or disperse light into its spectral components. They are also used to polarize light by prisms from birefringent media. Optics - Prisms 2. Reflective 1. Dispersive 3. Polarizing
  • 114. 114 OpticsSOLO Dispersive Prisms   2i1i 1t    11 ti   2t  22 it      2211 itti   21 it     21 ti 202 sinsin ti nn  Snell’s Law 10 n     1 1 2 1 2 sinsinsinsin tit nn         11 21 11 1 2 sincossin1sinsinsincoscossinsin ttttt nn    Snell’s Law 110 sinsin ti nn   11 sin 1 sin it n     1 2/1 1 221 2 sincossinsinsin iit n          1 2/1 1 221 1 sincossinsinsin iii n The ray deviation angle is 10 n
  • 115. 115 OpticsSOLO Prisms   2i1i 1t    11 ti   2t  22 it         1 2/1 1 221 1 sincossinsinsin iii n
  • 116. 116 OpticsSOLO Prisms   2i1i 1t    11 ti   2t  22 it         1 2/1 1 221 1 sincossinsinsin iii n   21 ti Let find the angle θi1 for which the deviation angle δ is minimal; i.e. δm. This happens when  01 0 11 2 1  ii t i d d d d d d       Taking the differentials of Snell’s Law equations 22 sinsin tin   11 sinsin ti n   2222 coscos iitt dnd   1111 coscos ttii dnd   Dividing the equations  1 2 1 2 1 1 2 1 2 1 cos cos cos cos   i t i t t i t i d d d d         2 22 1 22 2 2 2 2 1 2 2 2 1 2 2 2 1 2 sin sin /sin1 /sin1 sin1 sin1 sin1 sin1 t i t i i t t i n n n n                    1 1 2  i t d d   21 it   1 2 1  i t d d   2 2 1 2 2 2 1 2 cos cos cos cos i t t i      21 ti   1n
  • 117. 117 OpticsSOLO Prisms   2i1i 1t    11 ti   2t  22 it         1 2/1 1 221 1 sincossinsinsin iii n We found that if the angle θi1 = θt2 the deviation angle δ is minimal; i.e. δm. Using the Snell’s Law equations 22 sinsin tin   11 sinsin ti n   21 ti   21 it   This means that the ray for which the deviation angle δ is minimum passes through the prism parallel to it’s base.   2i 1i 1t  m  11 ti   2t  22 it   21 ti   21 it   Find the angle θi1 for which the deviation angle δ is minimal; i.e. δm (continue – 1).
  • 118. 118 OpticsSOLO Prisms       1 2/1 1 221 1 sincossinsinsin iii n Using the Snell’s Law 11 sinsin ti n   21 it   This equation is used for determining the refractive index of transparent substances.   2i 1i 1t  m  11 ti   2t  22 it   21 ti   21 it   21 it     21 ti 21 ti   m  2/1  t   12 im   2/1   mi    2/sin 2/sin     m n Find the angle θi1 for which the deviation angle δ is minimal; i.e. δm (continue – 2).
  • 119. 119 OpticsSOLO Prisms The refractive index of transparent substances varies with the wavelength λ.        1 2/1 1 221 1 sincossinsinsin iii n   2i1i 1t    11 ti   2t  22 it  
  • 120. 120 OpticsSOLO http://physics.nad.ru/Physics/English/index.htm Prisms υ [THz]λ0 (nm)Color 384 – 482 482 – 503 503 – 520 520 – 610 610 – 659 659 - 769 780 - 622 622 - 597 597 - 577 577 - 492 492 - 455 455 - 390 Red Orange Yellow Green Blue Violet 1 nm = 10-9m, 1 THz = 1012 Hz        1 2/1 1 221 1 sincossinsinsin iii n In 1672 Newton wrote “A New Theory about Light and Colors” in which he said that the white light consisted of a mixture of various colors and the diffraction was color dependent. Isaac Newton 1542 - 1727
  • 121. 121 SOLO Dispersing Prisms Pellin-Broca Prism Abbe Prism Ernst Karl Abbe 1840-1905 At Pellin-Broca Prism an incident ray of wavelength λ passes the prism at a dispersing angle of 90°. Because the dispersing angle is a function of wavelength the ray at other wavelengths exit at different angles. By rotating the prism around an axis normal to the page different rays will exit at the 90°. At Abbe Prism the dispersing angle is 60°. Optics - Prisms
  • 122. 122 SOLO Dispersing Prisms (continue – 1) Amici Prism Optics - Prisms
  • 123. 123 SOLO Reflecting Prisms  2i 1i 1t  2t E B D G A F C BED  180  360 ABEBEDADE 1 90 i ABE   2 90 t ADE    3609090 12  it BED  12 180 it BED      21 180 ti BED The bottom of the prism is a reflecting mirror Since the ray BC is reflected to CD DCGBCF  Also CGDBFC  CDGFBC  FBCt   901  CDGi   902 21 it   202 sinsin ti nn  Snell’s Law Snell’s Law 110 sinsin ti nn   21 ti     12 i CDGFBC  ~ Optics - Prisms
  • 124. 124 SOLO Reflecting Prisms Porro Prism Porro-Abbe Prism Schmidt-Pechan Prism Penta Prism Optics - Prisms Roof Penta Prism
  • 125. 125 SOLO Reflecting Prisms Abbe-Koenig Prism Dove Prism Amici-roof Prism Optics - Prisms
  • 126. 126 SOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Nicol Prism The Nicol Prism is made up from two prisms of calcite cemented with Canada balsam. The ordinary ray can be made to totally reflect off the prism boundary, leving only the extraordinary ray.. Polarizing Prisms Optics - Prisms
  • 127. 127 SOLO Polarizing Prisms A Glan-Foucault prism deflects polarized light transmitting the s-polarized component. The optical axis of the prism material is perpendicular to the plane of the diagram. A Glan-Taylor prism reflects polarized light at an internal air-gap, transmitting only the p-polarized component. The optical axes are vertical in the plane of the diagram. A Glan-Thompson prism deflects the p-polarized ordinary ray whilst transmitting the s-polarized extraordinary ray. The two halves of the prism are joined with Optical cement, and the crystal axis are perpendicular to the plane of the diagram. Optics - Prisms Return to TOC
  • 128. 128 OpticsSOLO Lens Definitions Optical Axis: the common axis of symmetry of an optical system; a line that connects all centers of curvature of the optical surfaces. FFL First Focal Point Second Focal Point Principal Planes Second Principal Point First Principal Point Light Rays from Left EFL BFL Optical System Optical Axis Lateral Magnification: the ratio between the size of an image measured perpendicular to the optical axis and the size of the conjugate object. Longitudinal Magnification: the ratio between the lengthof an image measured along the optical axis and the length of the conjugate object. First (Front) Focal Point: the point on the optical axis on the left of the optical system (FFP) to which parallel rays on it’s right converge. Second (Back) Focal Point: the point on the optical axis on the right of the optical system (BFP) to which parallel rays on it’s left converge.
  • 129. 129 OpticsSOLO Definitions (continue – 1) Aperture Stop (AS): the physical diameter which limits the size of the cone of radiation which the optical system will accept from an axial point on the object. Field Stop (FS): the physical diameter which limits the angular field of view of an optical system. The Field Stop limit the size of the object that can be seen by the optical system in order to control the quality of the image. Entrance Pupil: the image of the Aperture Stop as seen from the object through the elements preceding the Aperture Stop. Exit Pupil: the image of the Aperture Stop as seen from an axial point on the image plane. A.S. F.S. I Aperture and Field Stops Entrance pupil Exit pupil A.S. I xpE npE Chief Ray Entrance and Exit pupils Entrance pupilExit pupil A.S. I xpE npE Chief Ray
  • 130. 130 OpticsSOLO Definitions (continue – 2) Aperture Stop (AS): the physical diameter which limits the size of the cone of radiation which the optical system will accept from an axial point on the object. Field Stop (FS): the physical diameter which limits the angular field of view of an optical system. The Field Stop limit the size of the object that can be seen by the optical system in order to control the quality of the image. Entrance Pupil: the image of the Aperture Stop as seen from the object through the elements preceding the Aperture Stop. Exit Pupil: the image of the Aperture Stop as seen from an axial point on the image plane. Entrance pupil Exit pupil A.S. I Chief Ray Marginal Ray Exp Enp
  • 131. 131 OpticsSOLO Definitions (continue – 3) Principal Planes: the two planes defined by the intersection of the parallel incident rays entering an optical system with the rays converging to the focal points after passing through the optical system. FFL First Focal Point Second Focal Point Principal Planes Second Principal Point First Principal Point Light Rays from Left EFL BFL Optical System Optical Axis Principal Points: the intersection of the principal planes with the optical axes. Nodal Points: two axial points of an optical system, so located that an oblique ray directed toward the first appears to emerge from the second, parallel to the original direction. For systems in air, the Nodal Points coincide with the Principal Points. Cardinal Points: the Focal Points, Principal Points and the Nodal Points.
  • 132. 132 OpticsSOLO Definitions (continue – 4) Relative Aperture (f# ): the ratio between the effective focal length (EFL) f to Entrance Pupil diameter D. Numerical Aperture (NA): sine of the half cone angle u of the image forming ray bundles multiplied by the final index n of the optical system. If the object is at infinity and assuming n = 1 (air): Dff /:#  unNA sin:  # 1 2 1 2 1 sin ff D uNA        EFL EFL D u Last Principal Plane of the Optical System (Spherical)
  • 133. 133 OpticsSOLO Perfect Imaging System • All rays originating at one object point reconverge to one image point after passing through the optical system. • All of the objects points lying on one plane normal to the optical axis are imaging onto one plane normal to the axis. • The image is geometrically similar to the object. Object Image SystemOptical Object Image SystemOptical Object Image SystemOptical
  • 134. 134 OpticsSOLO Lens Convention of Signs 1. All Figures are drawn with the light traveling from left to right. 2. All object distances are considered positive when they are measured to the left of the vertex and negative when they are measured to the right. 3. All image distances are considered positive when they are measured to the right of the vertex and negative when they are measured to the left. 4. Both focal length are positive for a converging system and negative for a diverging system. 5. Object and Image dimensions are positive when measured upward from the axis and negative when measured downward. 6. All convex surfaces are taken as having a positive radius, and all concave surfaces are taken as having a negative radius. Return to TOC
  • 135. 135 OpticsSOLO Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle Karl Friederich Gauss 1777-1855  s 's n 'n h l 'l    ' M T CA M’ R The optical path connecting points M, T, M’ is ''lnlnpathOptical  Applying cosine theorem in triangles MTC and M’TC we obtain:      2/122 cos2 RsRRsRl       2/122 cos'2'' RsRRsRl            2/1222/122 cos'2''cos2  RsRRsRnRsRRsRnpathOptical  Therefore According to Fermat’s Principle when the point T moves on the spherical surface we must have   0 d pathOpticald       0 ' sin''sin      l RsRn l RsRn d pathOpticald   from which we obtain           l sn l sn Rl n l n ' ''1 ' ' For small α and β we have ''& slsl  and we obtain R nn s n s n   ' ' ' Gaussian Formula for a Single Spherical Surface Return to TOC
  • 136. 136 OpticsSOLO Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law Apply Snell’s Law: 'sin'sin  nn  If the incident and refracted rays MT and TM’ are paraxial the angles and are small and we can write Snell’s Law:  ' From the Figure    ' '' nn        nnnnnn  ''' For paraxial rays α, β, γ are small angles, therefore '/// shrhsh     r h nn s h n s h n  ' ' ' or   r nn s n s n   ' ' ' Gaussian Formula for a Single Spherical Surface Karl Friederich Gauss 1777-1855 Willebrord van Roijen Snell 1580-1626  s 's n 'n h l 'l    ' M T CA M’ r
  • 137. 137 OpticsSOLO Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law for s → ∞ the incoming rays are parallel to optical axis and they will refract passing trough a common point called the focus F’.   r nn s n s n   ' ' ' s '' fs  n 'n h 'l    ' T C A F’ R  fs  's n 'n h l   F T CA R '   r nn f nn    ' ' ' r nn n f   ' ' ' for s’ → ∞ the refracting rays are parallel to optical axis and therefore the incoming rays passes trough a common point called the focus F.   r nnn f n     '' r nn n f   ' '' n n f f  Return to TOC
  • 138. 138 OpticsSOLO Derivation of Lens Makers’ Formula We have a lens made of two spherical surfaces of radiuses r1 and r2 and a refractive index n’, separating two media having refraction indices n a and n”. Ray MT1 is refracted by the first spherical surface (if no second surface exists) to T1M’.   111 ' ' ' r nn s n s n   11111 ''& sMAsTA  Ray T1T2 is refracted by the second spherical surface to T2M”. 2222 ""&'' sMAsMA    222 '" " " ' ' r nn s n s n   Assuming negligible lens thickness we have , and since M’ is a virtual object for the second surface (negative sign) we have 21 '' ss  21 '' ss    221 '" " " ' ' r nn s n s n   M’ M '1f1f 1s Axis T1 T2 A1 A2 C1 1rC2 F’1F’’2 M’’ F’2F1 ''2f'2f '1s '2s ''2s 2r n 'n ''n
  • 139. 139 OpticsSOLO Derivation of Lens Makers’ Formula (continue – 1) M” M f s Axis A1 A2 C1 1rC2 F” F ''f ''s 2r n 'n ''n   111 ' ' ' r nn s n s n   Add those equations   221 '" " " ' ' r nn s n s n       2121 '"' " " r nn r nn s n s n     M’ M '1f1f 1s Axis T1 T2 A1 A2 C1 1rC2 F’1F’’2 M’’ F’2F1 ''2f'2f '1s '2s ''2s 2r n 'n ''n The focal lengths are defined by tacking s1 → ∞ to obtain f” and s”2 → ∞ to obtain f     f n r nn r nn f n      212 '"' " " Let define s1 as s and s”2 as s” to obtain     21 '"' " " r nn r nn s n s n         f n r nn r nn f n      21 '"' " "
  • 140. 140 OpticsSOLO Derivation of Lens Makers’ Formula (continue – 2) M” M f s Axis A1 A2 C1 1rC2 F” F ''f ''s 2r n 'n n If the media on both sides of the lens is the same n = n”.              21 11 1 ' " 11 rrn n ss              21 11 1 '1 " 1 rrn n ff Therefore " 11 " 11 ffss  Lens Makers’ Formula
  • 141. 141 OpticsSOLO First Order, Paraxial or Gaussian Optics In 1841 Gauss gave an exposition in “Dioptrische Untersuchungen” for thin lenses, for the rays arriving at shallow angles with respect to Optical axis (paraxial). Karl Friederich Gauss 1777-1855 Derivation of Lens Formula 'y s 's M’A F’ M T F 'ffx 'x Q Q’ 'y y S Axis yFrom the similarity of the triangles and using the convention:   '' ' '~' f y s yy TAFTSQ    Lens Formula in Gaussian form     f y s yy FASQTS '' ~       0'  y Sum of the equations:       ' ' ' '' f y f y s yy s yy       since f = f’ fss 1 ' 11  Return to TOC
  • 142. 142 OpticsSOLO First Order, Paraxial or Gaussian Optics (continue – 1) Gauss explanation can be extended to the first order approximation to any optical system. Karl Friederich Gauss 1777-1855 'y s 's M’P1 F’ M T F 'ffx 'x Q Q’ 'y y Axis y P2 First Focal Point First Principal Point Second Focal PointSecond Principal Point Optical System Object Image Lens Formula in Gaussian form 'y s 's M’A F’ M T F 'ffx 'x Q Q’ 'y y S Axis y fss 1 ' 11  s – object distance (from the first principal point to the object). s’ – image distance (from the second principal point to the image). f – EFL (distance between a focal point to the closest principal plane).
  • 143. 143 OpticsSOLO Derivation of Lens Formula (continue) 'y s 's M’A F’ M T F 'ffx 'x Q Q’ 'y y S Axis y From the similarity of the triangles and using the convention:   f y x y FASQMF ' ~   Lens Formula in Newton’s form   f y x y QMFTAF    ' ' '''~'   0'  y Multiplication of the equations:     2 ' ' ' f yy xx yy     or 2 ' fxx  Isaac Newton 1643-1727 First Order, Paraxial or Gaussian Optics (continue – 2) Published by Newton in “Opticks” 1710
  • 144. 144 OpticsSOLO Derivation of Lens Formula (continue) 'h s 's M’A F’ M T F 'ffx 'x Q Q’ 'h h S Axis h First Order, Paraxial or Gaussian Optics (continue – 3) Lateral or Transverse Magnification f x x f s s h h mT '''  (-) sign(+) signQuantity virtual objectreal objects virtual imagereal images’ diverging lensconverging lensf inverted objecterect objecth inverted imageerect imageh’ inverted imageerect imagemT
  • 145. 145 OpticsSOLO Concave Spherical Convex Spherical Paraboloidal Conic Ellipsoidal General Aspherical Plane Converging : General use Diverging : General use Accurately focuses a parallel beam or produces a parallel beam from a point source Refocuses a diverging bundle at another point (P) displaced from the point of origin (O) Change the direction of beam Used mostly in combination systems of two or more components BASIC MIRRORS FORMS
  • 146. 146 OpticsSOLO Convex Plano Convex Meniscus Concave Plano Concave Meniscus Doublet Multi- Element Aspheric Converging: General Use, Magnification Converging: Used often in opposed doubles to reduce spherical aberration Converging: reduced spherical aberration Diverging: General Use, Demagnification Diverging: Used in multi-element combinations Diverging: reduced spherical aberration Corrected for chromatic aberration High order of aberration correction used in complex systems Corrected for spherical aberration used in condenser systems BASIC LENS FORMS Return to TOC
  • 147. 147 OpticsSOLO Ray Tracing F C O I Object Virtual Image Convex Mirror R/2 R/2 R F F’C O I Object Real Image Converging Lens FC O I Object Real Image Concave Mirror F F’C O I Object Virtual Image Diverging Lens Ray Tracing is a graphically implementation of paralax ray analysis. The construction doesn’t take into consideration the nonideal behavior, or aberration of real lens. The image of an off-axis point can be located by the intersection of any two of the following three rays: 1. A ray parallel to the axis that is reflected through F’. 2. A ray through F that is reflected parallel to the axis. 3. A ray through the center C of the lens that remains undeviated and undisplaced (for thin lens).
  • 148. 148 OpticsSOLO Infinity Principal focus SUMMARY OF SIMPLE IMAGING LENSES f f2f2 f 0 's 'ss fs 2 fsf 2' fs 2 fs 2' fsf 2 fs 2' 's 's s s fs  's s s 's fs  fs ' s 's fsf 2 fs ' Real, inverted small Telescope Real, inverted smaller Camera Real, inverted same size Photocopier Real, inverted larger Projector No image Searchlight Virtual, erect larger Microscope Virtual, erect smaller Various Figure Object Location Image Location Image Properties Example L.J. Pinson, “Electro-Optics”, John Wiley & Sons, 1985, pg.54 Return to TOC
  • 149. 149 OpticsSOLO Matrix Formulation The Matrix Formulation of the Ray Tracing method for the paraxial assumption was proposed at the beginning of nineteen-thirties by T.Smith. Assuming a paraxial ray entering at some input plane of an optical system at the distance r1 from the symmetry axis and with a slope r1’ and exiting at some output plane at the distance r2 from the symmetry axis and with a slope r2’, than the following linear (matrix) relation applies: Principal PlanesInput plane Output plane Ray path 1h 2h 1r 2r '1r '2r Symmetry axis                         ''' 1 1 1 1 2 2 r r M r r DC BA r r        DC BA Mwhere ray transfer matrix When the media to the left of the input plane and to the right of the output plane have the same refractive index, we have: 1det  CBDAM
  • 150. 150 OpticsSOLO Matrix Formulation (continue -1) Uniform Optical Medium In an Uniform Optical Medium of length d no change in ray angles occurs: Ray path d 1r 2r '1r '2r Symmetry axis 1 2 '' ' 12 112 rr rdrr          10 1 d M Medium Optical Uniform Planar Interface Between Two Different Media Ray path 1r 2r '1r '2r Symmetry axis 1 2 1n 2n 12 rr  '' 1 2 1 2 12 r n n r rr   Apply Snell’s Law: 2211 sinsin  nn  paraxial assumption:   tan'sin r From Snell’s Law: '' 1 2 1 2 r n n r         21 /0 01 nn M Interface Planar 1det 2 1  n n M Interface Planar 1det  Medium Optical UniformM The focal length of this system is infinite and it has not specific principal planes.
  • 151. 151 OpticsSOLO Matrix Formulation (continue -2) A Parallel-Sided Slab of refractive index n bounded on both sides with media of refractive index n1 = 1 Ray path d 21 rr  43 rr  '1r '4r Symmetry axis '2r '3r nn 211 n 11 n We have three regions: • on the right of the slab (exit of ray):                   '/0 01 ' 3 3 124 4 r r nnr r • in the slab:                   '10 1 ' 2 2 3 3 r rd r r • on the left of the slab (entrance of ray):                   '/0 01 ' 1 1 212 2 r r nnr r Therefore:                               '/0 01 10 1 /0 01 ' 1 1 21124 4 r r nn d nnr r                                21 21 122112 /0 /1 /0 01 /0 01 10 1 /0 01 nn nnd nnnn d nn M mediaentranceslabmediaexit Slab Sided Parallel         10 /1 21 nnd M Slab Sided Parallel 1det  Slab Sided ParallelM
  • 152. 152 OpticsSOLO Matrix Formulation (continue -3) Spherical Interface Between Two Different Media Ray path 21 rr  '1r '2r Symmetry axis 1n 2n i r 1 12 rr  Apply Snell’s Law: rnin sinsin 21  paraxial assumption: rrii  sin&sin From Snell’s Law: rnin 21                           2 1 2 1 2 1 12 21 0101 n n n D n n Rn nnM Interface Spherical 1det 2 1  n n M Interface Spherical 12 11 ' '     rr ri From the Figure:    122111 ''   rnrn 111 / Rr   12 121 2 11 2 ' ' Rn rnn n rn r     1 12 11 112 2 12 ' ' n rn Rn rnn r rr       1 12 1 : R nn D  where: Power of the surface If R1 is given in meters D1 gives diopters
  • 153. 153 OpticsSOLO Matrix Formulation (continue -4) Thick Lens 21 rr  43 rr  '1r i 2 1 '2r '3r r 2R 1R f '4r1C 2F IO 1F 2C Principal planes 2n 1n s 's d We have three regions: • on the right of the slab (exit of ray):                        ' 01 ' 3 3 1 2 1 2 4 4 r r n n n D r r • in the slab:                   '10 1 ' 2 2 3 3 r rd r r • on the left of the slab (entrance of ray):                        ' 01 ' 1 1 2 1 2 1 2 2 r r n n n D r r Therefore:                                                                       ' 101 ' 01 10 1 01 ' 1 1 2 1 2 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 4 4 r r n n n D n n d n D d n n n D r r n n n D d n n n D r r                          2 2 21 21 1 21 2 1 2 1 1 1 n D d nn DD d n DD n n d n D d M Lens Thick   2 21 2 R nn D     1 12 1 : R nn D                            2 1 21 21 1 21 2 1 2 2 1 1 1 n D d nn DD d n DD n n d n D d M Lens Thick 1det  Lens Thick M or 21 DD 
  • 154. 154 OpticsSOLO Matrix Formulation (continue -5) Thick Lens (continue -1) 21 rr  43 rr  '1r i 2 1 '2r '3r r 2R 1R f '4r1C 2F IO 1F 2C Principal planes 2n 1n 2R 1R 2f 1C 2F I O 1F 2C Principal planes 2n 1n 1h 2h s s 's 's d Ray 2 Ray 1 1f Let use the second Figure where Ray 2 is parallel to Symmetry Axis of the Optical System that is refracted trough the Second Focal Point.                                              '1 1 ' 1 1 2 2 21 21 1 21 2 1 2 1 4 4 r r n D d nn DD d n DD n n d n D d r r We found: 2141 /'&0' frrr Ray 2: By substituting Ray2 parameters we obtain: 1 2 1 21 21 1 21 4 1 ' r f r nn DD d n DD r          1 21 21 1 21 2           nn DD d n DD f frrr /'&0' 414 Ray 1: We found:                                             '1 1 ' 4 4 2 1 21 21 1 21 2 1 2 2 1 1 r r n D d nn DD d n DD n n d n D d r r 4 1 4 21 21 1 21 1 1 ' r f r nn DD d n DD r          2 1 21 21 1 21 1 f nn DD d n DD f          
  • 155. 155 OpticsSOLO Matrix Formulation (continue -6) Thin Lens 21 rr  43 rr  '1r i 2 1 '2r '3r r 2R 1R f '4r1C 2F IO 1F 2C Principal planes 2n 1n s 's d For thick lens we found                          2 2 21 21 1 21 2 1 2 1 1 1 n D d nn DD d n DD n n d n D d M Lens Thick          21 21 1 211 nn DD d n DD f For thin lens we can assume d = 0 and obtain             1 1 01 f M Lens Thin 1 211 n DD f     2 21 2 R nn D     1 12 1 : R nn D                  211 2 1 21 11 1 1 RRn n n DD f 21 rr  43 rr  2R 1R f '4r1C 2F IO 1F 2C Principal planes 2n 1n s 's '1r
  • 156. 156 OpticsSOLO Matrix Formulation (continue -7) Thin Lens (continue – 1) For a biconvex lens we have R2 negative                211 2 11 1 1 RRn n f For a biconcave lens we have R1 negative                211 2 11 1 1 RRn n f             1 1 01 f M Lens Thin
  • 157. 157 OpticsSOLO Matrix Formulation (continue -8) A Length of Uniform Medium Plus a Thin Lens                               f d f d d f MMM Medium Uniform Lens Thin Lens Thin Medium Uniform 1 1 1 10 1 1 1 01 21 rr  43 rr  2R 1R f '4r1C 2F IO 1F 2C Principal planes 2n 1n s 's '1r d Combination of Two Thin Lenses 2n 1d 1f 2d 2f 2n                                      21 21 2 2 2 1 1 1 21 2 21 1 21 21 2 2 1 1 1 1 22 2 1 11 1 1 1 1 1 1 1 1122 ff dd f d f d f d ff d ff f dd dd f d f d f d f d f d MMMMM dMedium Uniform fLens Thin dMedium Uniform fLens Thin Lenses Thin Two The Focal Length of the Combination of Two Thin Lenses is: 21 2 21 111 ff d fff 
  • 158. 158 OpticsSOLO Matrix Formulation (continue -9) Mirrors r Spherical Mirror i i ii  i i iy RSpherical Mirror Center of Curvature r   Ryiii /tan  Consider a Spherical Mirror of radius R. From the geometry: For small angles: Ryiii / also: iri  2   2/rii   Ryiri /2 Define by n the index of reflexion of the medium: Rynnn yy iir ir /2                      i i r r n y Rnn y  1/2 01 Therefore:                1/ 01 1/2 01 fnRn M Mirror Spherical
  • 159. 159 OpticsSOLO Matrix Formulation (continue -10) Cavity of two Mirrors d 12M 21M 2MirrorM 1MirrorM Spherical Mirror M1 Radius R1 Spherical Mirror M2 Radius R2 O                             10 1 1/2 01 10 1 1/2 01 12 1221 12 d Rn d Rn MMMMM MMirror Spherical MMirror SphericalCavity Figure shows two spherical mirrors facing each other forming an optical cavity. Light leaves point O, traverse the gap in the positive direction, is reflected by Mirror M1, retraces the gap in the negative direction, and is reflected by Mirror M2. The System Matrix is:                            21221 2 21 1 2 1 1122 /21/21/2/4/2/2 /22/21 /21/2 1 /21/2 1 RdnRdnRdnRRdnRnRn RdndRdn RdnRn d RdnRn d            21 22 2121 2 21 1 2 1 /4/4/21/4/2/2 /22/21 RRdnRdnRdnRRdnRnRn RdndRdn MCavity
  • 160. 160 Go to OPTICS Part II OpticsSOLO
  • 161. January 5, 2015 161 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA