![Ef
b ¼ Ef
bejβ • r′
ejωt
iy′ Hf
a ¼
ffiffiffi
ε
μ
r
Ef
bejβ • r′
ejωt
ix′ ð1:25Þ
Eb
b ¼ Eb
aeþjβ • r′
ejωt
iy′ Hb
a ¼
ffiffiffi
ε
μ
r
Eb
aeþjβ • r′
ejωt
ix′ ð1:26Þ
It is important to recognize that when there is a wave solution containing various
terms, any term that has the form shown in Eqs. (1.17) to (1.26) represents a plane wave
propagating in the direction of β.
1.1.3 Evanescent plane waves
Eqs. (1.22) to (1.26) described propagating plane waves that have real βxʹ, βyʹ, and βzʹ
values. The maximum real βxʹ and βyʹ values of propagating plane waves are limited to
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
βx′
2 þ βy′
2
q
ω
ffiffiffiffiffi
με
p
, i.e. 0 θxʹ, θyʹ, and θzʹ π/2. Nevertheless, Maxwell’s equation is
still satisfied even if
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
β2
x þ β2
y0
q
is larger than β. In that case Eq. (1.24) can only be
satisfied if βzʹ is imaginary. When βzʹ is imaginary, the zʹ variation is a real decaying or
growing exponential function, e
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
βx′
2þβy′
2β2
p
z′
. In any passive medium, the plane wave
cannot grow without energy input. Thus the solution must decay exponentially in the zʹ
direction. Any solution with imaginary βzʹ is called an evanescent wave. Such solutions
do not propagate in the z direction. They do not have a phase velocity. Evanescent
waves are excited usually in the vicinity of a boundary with an incident wave applied
across the boundary. It is only a near field, meaning that it is negligible at locations far
away from the boundary.1
It is interesting to note that when βxʹ = β, βyʹ = βzʹ = 0, it is no
longer a plane wave propagating in the zʹ direction. It is a plane wave propagating in
the +xʹ direction.
1.1.4 Intensity and power
In optics, only time-averaged power can be detected directly by means of detectors
or by recording media such as film. The time-averaged power per unit area is known
commonly as the intensity. In comparison with rf and microwaves, intensity analysis
plays a much more important role in optics. From text books on electromagnetic
theory, it is well known that the total time-averaged power in the direction of
propagation is [1]
Pav ¼
1
2
Re
ð
S
E H
• iz′ ds ¼
ð
S
I • iz′ ds; I ¼
1
2
Re½E H ð1:27Þ
1
It is important to note that although the mathematical solution of a plane wave exists for βx or βy values larger
than ω
ffiffiffiffiffi
με
p
, such a solution is important only if those plane waves are excited in specific applications such as
total internal reflection. Otherwise, the solutions have no practical significance.
1.1 Introduction to optical plane waves 9](https://image.slidesharecdn.com/4977104-250523144728-de6f10c6/85/Principles-Of-Optics-For-Engineers-Diffraction-And-Modal-Analysis-Chang-25-320.jpg)
![Eoejβ sin θx
ejβ cos θz
ejωt
iy þ Eoejβ sin ζx
ejβ cos ζz
ejωt
iy ð1:31Þ
According to Eq. (1.28), its time-averaged intensity in the z direction is
Iz ¼
ffiffiffi
ε
μ
r
Eo
j j2
1 þ cos β ðsin θ sin ζÞx þ ðcos θ cos ζÞz
ð Þ
½ ð1:32Þ
Therefore, for θ ≠ ζ, we would detect a sinusoidal intensity interference pattern of the
two waves in the x direction. As z changes, the interference pattern in x will change. If we
could record this intensity interference pattern, for example, by the transparency of a
film, we could reproduce the plane waves by illuminating this film with another input
plane wave. This is the very basic principle on which holography and phased array
detection are based [2,3,4].
However, if the two waves do not have the same ω or a definite phase relation between
them, then Iz ¼
ffiffiffiffiffiffiffi
ε=μ
p
Eo
j j2
. In other words, there is no interference pattern unless the
two waves are coherent.2
The total intensity of incoherent waves is just the sum of
the intensities of individual waves. It is also important to note that when the two coherent
plane waves have cross polarizations, the total intensity is also just the sum of the two
intensities without the interference effect.
(c) Representation by summation of plane waves
Let there be a linearly polarized TEM electric field propagating in the z direction with xy
variation g(x,y) at z = 0. It is well known that g can be represented by its Fourier
transform. Let G be the Fourier transform of g.
G fx; fy
¼ FtðgÞ ¼
ð
þ∞
∞
ð
þ∞
∞
gðx; yÞej2π fxxþfyy
ð Þdxdy ð1:33Þ
gðx; yÞ ¼ Ft
1
ðGÞ ¼
ð
þ∞
∞
ð
þ∞
∞
Gðfx; fyÞej2πðfxxþfyyÞ
dfxdfy ð1:34Þ
G is the magnitude of the Fourier component at (fx, fy). When g(x,y) contains only slow
variations in x and y, G will have significant values only at low spatial frequencies fx and
fy. In that case the integration in Eq. (1.34) could be approximated by just the integration
of G within a limited range of fx and fy.
Let
2πfx ¼ βx′ ¼ β cos θx′ ; 2πfy ¼ βy′ ¼ β cos θy′ ð1:35Þ
dfx ¼ β sin θx′dθx′ dfy ¼ β sin θy′dθy′ ð1:36Þ
2
If the two waves have a randomly time-varying relative phase relation, for example from two independent
lasers, the time-averaged detected intensity also will not have the interference pattern.
1.1 Introduction to optical plane waves 11](https://image.slidesharecdn.com/4977104-250523144728-de6f10c6/85/Principles-Of-Optics-For-Engineers-Diffraction-And-Modal-Analysis-Chang-27-320.jpg)
![Eq. (1.33) gives the magnitude of the Fourier component that has an xʹyʹ variation of
ejβ cos θx′x′
ejβ cos θy′y′
. If we also let β ¼ ð2πco=λoÞ
ffiffiffiffiffiffi
μεr
p
¼ 2πc=λ and consider only those
Fourier components with
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
βx′
2 þ βy′
2
q
β (i.e. 0 θxʹ and θyʹ π/2), then Eq. (1.33)
gives the magnitude of the Fourier component that has the xʹyʹ variation of a plane wave
propagating in a direction β, which has direction cosines θxʹ, θyʹ, and θzʹ. θzʹ is related to
θxʹ and θyʹ by Eq. (1.24). However, for propagating plane waves, θxʹ,θyʹ, and θzʹ must
be real. The maximum and minimum values of fx and fy for real values of θxʹ and θyʹ, are
–β/2π and +β/2π. Moreover,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
βx′
2 þ βy′
2
q
β.3
It means that, the Fourier components
that correspond to propagating waves with real values of θxʹ and θyʹ, are only Fourier
components that have low values of fxʹ and fyʹ.
In summary, when g contains only variations in x and y very much slower than λ,
G(fx,fy) has significant values only at those fx and fy less than the limit fmax. If fmax
β/2π, it means the electric field g(x,y) could be represented by superposition of just
plane waves propagating in different directions β. Under that condition, Eq. (1.34) can
be approximated by
gðx; yÞ ¼ Ft
1
ðGÞ ffi
ð
þfmax
fmax
ð
þfmax
fmax
Gðfx; fyÞej2πðfxxþfyyÞ
dfxdfy ð1:37Þ
There are three important concepts introduced here: (1) We have shown that plane
waves can be used to represent an arbitrary field with slow xʹ and yʹ variations. This is
equivalent to the modal expansion concept used in microwaves. Here, plane waves are
the modes of unbounded medium. (2) Fields with any xʹyʹ variation can be represented
by their Fourier components.4
This means that many modern Fourier analysis tech-
niques can be applied to optics. This is the basis of Fourier optics [2] and optical
image processing [4]. (3) Knowing the plane wave composition at z = 0, we have
determined the xy variation of each plane wave components at any distance z later.
Thus it allows us to predict the electric field that propagates to z via plane wave
analysis. Note that as component plane waves propagate the total optical radiation
spreads or contracts. This phenomenon is also known as the diffraction of optical
radiation. More details on diffraction will be presented in Chapter 3.
It is interesting to note that if G(fx,fy) contains frequency components with large fx and
fy such that fx
2
þ fy
2
β2
=4π2
, then the z variation of the plane waves for those
components will exponentially decay. This means that those components will contribute
only to the near field and they will not propagate far in the z direction. Only the frequency
components with fx
2
þ fy
2
β2
=4π2
will propagate, so the fields at some z distance
away will not be exactly the same as g(x,y).
3
Evanescent plane waves in the z direction could have βx′ or βy′ larger than β. However they are not
propagating waves.
4
Fields with rapid xy variation would yield Fourier components that are evanescent local waves in the z
direction.
12 Optical plane waves in unbounded medium](https://image.slidesharecdn.com/4977104-250523144728-de6f10c6/85/Principles-Of-Optics-For-Engineers-Diffraction-And-Modal-Analysis-Chang-28-320.jpg)
![then the refraction of a beam through a thin spherical lens placed at zʹ can be expressed in
a ray optical representation as
routðz
0
Þ
r′outðz
0
Þ
¼
1 0
1
f
1
riðz
0
Þ
r′iðz
0
Þ
ð1:90Þ
When the incident beam is parallel to the z axis, θi = 0. Parallel beams at different y
positions would all focus on the z axis at the position zʹ + f. Therefore z = f is commonly
known as the focal length of the lens. For parallel beams incident at small θi, θout will still be
related to θi by Eq. (1.88). Thus they will be focused to a point at z ¼ z
0
þ f and y ¼ θif .
The plane at z = zʹ + f is known as the focal plane of the lens. The preceding analysis has
only been carried out for optical beams incident in the y–z plane. However, in a
cylindrically symmetric configuration, the x and y axes can be rotated about the z axis.
Thus the results can be generalized to three dimensions for any beam incident in the
meridian plane.
The objective of the analysis presented here is only to demonstrate simple lens
properties by plane wave analysis. The analysis of practical lenses is much more
complex than the preceding discussion. It involves oblique rays, skewed rays, astigma-
tism, etc. and is beyond the scope of this book [5].
(b) Thin lens represented as a transparency with varying index
Similar to the discussion in Section 1.3.4 (c) for the prism, it is instructive to represent a
thin lens as a transparent planar medium with a varying phase change. Consider an
incident plane wave that has a beam size small compared to the size of the lens. It
propagates in the direction of z axis.
Ei ¼ Eoejβ1z
ejωt
Let us consider this small beam in the y–z plane near x = 0. At the transverse position
(x = 0, y), it passes through the lens beginning at z = z1 – r2 and ending at z = r1. Its phase
at the output will depend on y because the ray goes through a higher index region with
thickness, zʹʹ – zʹ, at y = yʹ= yʹʹ. The change in its phase, in comparison to a beam in free
space without the lens, is:
Δϕ ¼ β1ðn2 n1Þðz″ z0
Þ
¼ β1ðn2 n1Þ r1 1
y2
r1
2
8
:
9
=
;
1=2
z1 þ r2 1
y2
r2
2
8
:
9
=
;
1=2
0
B
@
1
C
A
ð1:91Þ
Here, zʹʹ zʹ and x and y r1 and r2 inside the lens. Binomial expansion can be used
again for the terms in the curly brackets. When the first-order approximation is used for a
thin lens, we obtain
Δϕ ¼ βðn2 n1Þ r1 þ r2 z1
y2
r1
y2
r2
: ð1:92Þ
1.3 Refraction of plane waves 27](https://image.slidesharecdn.com/4977104-250523144728-de6f10c6/85/Principles-Of-Optics-For-Engineers-Diffraction-And-Modal-Analysis-Chang-43-320.jpg)
![Exðz ¼ 0Þ ¼
ð
þ∞
∞
FAðf Þeþj2πðfof Þt
df ð1:109Þ
It is a sum of plane waves at different frequencies fo – f. Each component at fo – f
propagates with a different β to the position z. Therefore, at z
ExðzÞ ¼
ð
þ∞
∞
FAðf Þejβðfoof Þz
ej2πðfof Þt
df ð1:110Þ
Usually, fo – f fo. Therefore, β(f) can be represented by its Taylor’s series,
βðf Þ ¼ βðfoÞ þ
∂β
∂f
ðfoÞ ðf Þ þ
1
2
∂2
β
∂f 2
ðfoÞ ðf Þ2
þ ð1:111Þ
When the second- and higher-order terms can be neglected, we obtain
Ex ¼
ð
þ∞
∞
FAðf Þe j2π ∂β
∂ω
ð Þjfo
z
eþj2πft
df
2
4
3
5ejβðfoÞz
ejπfot
ð1:112Þ
where
vg ¼ ∂ω=∂β fo
ð1:113Þ
is known as the group velocity.
In a realistic situation, pulse distortion is important if the distance of propagation is
very long and the pulse duration of A(t) is short. If the group velocity is independent of f,
the quantity ej2π ∂β
∂ω
ð Þ fo z
j can be factored out of the integral. The pulse is then propagated to
z without distortion, i.e. A(t) is unchanged. The only change is a change of the phase of
Ex from ejβðfoÞz
which equals 2πð∂β=∂ωÞ fo
z
. Otherwise, there will be distortion, or
change of A(t). Clearly, when higher-order terms in Eq. (1.111) cannot be neglected,
there will be additional distortion.
Chapter summary
Basic plane wave analysis is presented. A plane wave is the simplest rigorous solution of
Maxwell’s equations. Yet it can be used to illustrate many basic concepts in optics.
Under appropriate circumstances, an optical ray could be represented locally approxi-
mately by a plane wave. Optical properties such as reflection, refraction, and focusing
can also be derived from plane wave analysis. The plane wave presented here shows how
the traditional analysis is related to Maxwell’s equations. However, much more complex
analyses are required for optical components design and image transfer [5]. Traditional
optics is better suited for these applications. On the other hand, plane wave analysis
shows optical properties that are not emphasized in traditional optics. These include the
32 Optical plane waves in unbounded medium](https://image.slidesharecdn.com/4977104-250523144728-de6f10c6/85/Principles-Of-Optics-For-Engineers-Diffraction-And-Modal-Analysis-Chang-48-320.jpg)
![is equal to ¼ of the wavelength. Identical results are obtained when the electric field is
polarized in the y–z plane. Note that for a given d the anti-reflection effect is wavelength
sensitive. As the wavelength deviates, the reflection will increase and the transmission
will decrease. Thus there is an effective wavelength range of the anti-reflection coating.
In reality, there may not be a coating material that has exactly the required nt. The
wavelength range within which anti-reflection, reflection, or beam splitting is required
for different applications may also need to be decreased or increased. Therefore, multi-
ple layer coatings are used in most commercial devices. However, the basic principle is
demonstrated by the above example. In a similar manner, coatings can be applied to
enhance reflection or to split the incident beam into desired ratios of reflected and
transmitted beams. Beam splitters can also be designed for beams incident at specific
incident angles.
Note that the analysis presented here is similar to impedance transformation analysis
of microwave transmission lines. The differential equation for E and H is identical to
that for V and I of microwave transmission lines [1]. In microwaves, anti-reflection is
called impedance matching.
Many of the analytical techniques developed for microwaves are also very useful for
optical analysis, especially when we need to analyze multi-layer transitions. It is
important for optics engineers to understand transmission line methods. However, a
detailed discussion of that is beyond the scope of this book.
2.2 Fabry–Perot resonance
2.2.1 Multiple reflections and Fabry–Perot resonance
Although plane waves propagating between two boundaries have already been analyzed
in the previous section by matching the total fields at the boundaries, an alternate way to
analyze it is to consider an incident plane wave multiply reflected and transmitted at the
two boundaries. Much more physical insight on resonance could be gained by present-
ing this alternate approach.
Let us first consider a plane wave incident on the first boundary at z = 0, without
considering the second boundary at z = d. This incident wave Ei would excite a reflected
backward wave Er1 in medium #1 and a transmitted forward wave Ef
1 in the transition
medium. Let the boundary at z = 0 have a reflection coefficient Γ1t and transmission
coefficient T1t for the incident wave. The x-y-z variations without showing the time
variation ejωt
are:
Ef
1 ¼ T1tEiejβtz
ix for d z 0; Er1 ¼ Γ1tEieþjβ1z
ix for z 0 ð2:20Þ
This boundary will have reflection coefficient Γt1 and transmission coefficient Tt1 for any
plane wave incident on it in the reverse direction from the transition medium.
As Ef
1 propagates to z = d, it excites a reflected wave Eb
1 in the transition medium and
a transmitted wave Eo1 in medium #2. Let the boundary at z = d have reflection
2.2 Fabry–Perot resonance 37](https://image.slidesharecdn.com/4977104-250523144728-de6f10c6/85/Principles-Of-Optics-For-Engineers-Diffraction-And-Modal-Analysis-Chang-53-320.jpg)
Principles Of Optics For Engineers Diffraction And Modal Analysis Chang
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- 7. Principles of Optics for Engineers Uniting historically different approaches by presenting optical analyses as solutions of Maxwell’s equations, this unique book enables students and practicing engineers to fully understand the similarities and differences between the various methods. The book begins with a thorough discussion of plane wave analysis, which provides a clear understanding of optics without considering boundary condition or device config- uration. It then goes on to cover diffraction analysis, including a rigorous analysis of TEM waves using Maxwell’s equations, and the use of Gaussian beams to analyze different applications. Modes of simple waveguides and fibers are also covered, as well as several approximation methods including the perturbation technique, the coupled mode analysis, and the super mode analysis. Analysis and characterization of guided wave devices, such as power dividers, modulators, and switches, are presented via these approximation methods. With theory linked to practical examples throughout, it provides a clear understanding of the interplay between plane wave, diffraction, and modal analysis, and how the different techniques can be applied to various areas such as imaging, spectral analysis, signal processing, and optoelectronic devices. William S. C. Chang is an Emeritus Professor of the Department of Electrical and Computer Engineering, University of California, San Diego (UCSD). After receiving his Ph.D. from Brown University in 1957, he pioneered maser and laser research at Stanford University, and he has been involved in guided-wave teaching and research at Washington University and UCSD since 1971. He has published over 200 technical papers and several books, including Fundamentals of Guided-Wave Optoelectronic Devices (Cambridge, 2009), Principles of Lasers and Optics (Cambridge, 2005) and RF Photonic Technology in Optical Fiber Links (Cambridge, 2002).
- 9. Principles of Optics for Engineers Diffraction and Modal Analysis BY WILLIAM S. C. CHANG University of California, San Diego
- 10. University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107074903 © Cambridge University Press 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data Chang, William S. C. (William Shen-chie), 1931– Principles of optics for engineers: diffraction and modal analysis / by William S. C. Chang, University of California, San Diego. pages cm ISBN 978-1-107-07490-3 1. Optical engineering. 2. Diffraction. 3. Modal analysis. I. Title. TA1520.C47 2015 535–dc23 2014042202 ISBN 978-1-107-07490-3 Hardback Additional resources for this publication at www.cambridge.org/9781107074903 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 11. Contents Introduction 1 1 Optical plane waves in an unbounded medium 4 1.1 Introduction to optical plane waves 4 1.1.1 Plane waves and Maxwell’s equations 4 (a) The y-polarized plane wave 5 (b) The x-polarized plane wave 6 1.1.2 Plane waves in an arbitrary direction 7 1.1.3 Evanescent plane waves 9 1.1.4 Intensity and power 9 1.1.5 Superposition and plane wave modes 10 (a) Plane waves with circular polarization 10 (b) Interference of coherent plane waves 10 (c) Representation by summation of plane waves 11 1.1.6 Representation of plane waves as optical rays 13 1.2 Mirror reflection of plane waves 14 1.2.1 Plane waves polarized perpendicular to the plane of incidence 14 1.2.2 Plane waves polarized in the plane of incidence 15 1.2.3 Plane waves with arbitrary polarization 15 1.2.4 The intensity 15 1.2.5 Ray representation of reflection 15 1.2.6 Reflection from a spherical mirror 16 1.3 Refraction of plane waves 17 1.3.1 Plane waves polarized perpendicular to the plane of incidence 17 1.3.2 Plane waves polarized in the plane of incidence 19 1.3.3 Properties of refracted and transmitted waves 20 (a) Transmission and reflection at different incident angles 20 (b) Total internal reflection 21 (c) Refraction and reflection of arbitrary polarized waves 21 (d) Ray representation of refraction 21 1.3.4 Refraction and dispersion in prisms 22 (a) Plane wave analysis of prisms 22
- 12. (b) Ray analysis of prisms 24 (c) Thin prism represented as a transparent layer with a varying index 24 1.3.5 Refraction in a lens 25 (a) Ray analysis of a thin lens 25 (b) Thin lens represented as a transparency with varying index 27 1.4 Geometrical relations in image formation 28 1.5 Reflection and transmission at a grating 30 1.6 Pulse propagation of plane waves 31 Chapter summary 32 2 Superposition of plane waves and applications 34 2.1 Reflection and anti-reflection coatings 34 2.2 Fabry–Perot resonance 37 2.2.1 Multiple reflections and Fabry–Perot resonance 37 2.2.2 Properties of Fabry–Perot resonance 39 2.2.3 Applications of the Fabry–Perot resonance 41 (a) The Fabry–Perot scanning interferometer 41 (b) Measurement of refractive properties of materials 42 (c) Resonators for filtering and time delay of signals 43 2.3 Reconstruction of propagating waves 43 2.4 Planar waveguide modes viewed as internal reflected plane waves 46 2.4.1 Plane waves incident from the cladding 46 2.4.2 Plane waves incident from the substrate 48 (a) Incident plane waves with sin1 nc=ns ð Þ θs π=2 48 (b) Incident plane waves with 0 θs sin1 nc=ns ð Þ 48 2.4.3 Plane waves incident within the waveguide: the planar waveguide modes 48 2.4.4 The hollow dielectric waveguide mode 50 Chapter summary 51 3 Scalar wave equation and diffraction of optical radiation 53 3.1 The scalar wave equation 54 3.2 The solution of the scalar wave equation: Kirchhoff’s diffraction integral 55 3.2.1 Kirchhoff’s integral and the unit impulse response 57 3.2.2 Fresnel and Fraunhofer diffractions 57 3.2.3 Applications of diffraction integrals 58 (a) Far field diffraction pattern of an aperture 58 (b) Far field radiation intensity pattern of a lens 60 vi Contents
- 13. (c) Fraunhofer diffraction in the focal plane of a lens 62 (d) The lens viewed as a transformation element 65 3.2.4 Convolution theory and other mathematical techniques 65 (a) The convolution relation 66 (b) Double slit diffraction 66 (c) Diffraction by an opaque disk 67 (d) The Fresnel lens 67 (e) Spatial filtering 67 Chapter summary 71 4 Optical resonators and Gaussian beams 73 4.1 Integral equations for laser cavities 74 4.2 Modes in confocal cavities 75 4.2.1 The simplified integral equation for confocal cavities 75 4.2.2 Analytical solutions of the modes in confocal cavities 77 4.2.3 Properties of resonant modes in confocal cavities 78 (a) The transverse field pattern 78 (b) The resonance frequency 79 (c) The orthogonality of the modes 79 (d) A simplified analytical expression of the field 80 (e) The spot size 81 (f) The diffraction loss 81 (g) The line width of resonances 82 4.2.4 Radiation fields inside and outside the cavity 83 (a) The far field pattern of the TEM modes 84 (b) A general expression for the TEMlm Gaussian modes 84 (c) An example to illustrate confocal cavity modes 85 4.3 Modes of non-confocal cavities 86 4.3.1 Formation of a new cavity for known modes of confocal resonator 86 4.3.2 Finding the virtual equivalent confocal resonator for a given set of reflectors 88 4.3.3 A formal procedure to find the resonant modes in non-confocal cavities 89 4.3.4 An example of resonant modes in a non-confocal cavity 91 4.4 The propagation and transformation of Gaussian beams (the ABCD matrix) 91 4.4.1 A Gaussian mode as a solution of Maxwell’s equation 92 4.4.2 The physical meaning of the terms in the Gaussian beam expression 94 4.4.3 The analysis of Gaussian beam propagation by matrix transformation 95 4.4.4 Gaussian beam passing through a lens 97 Contents vii
- 14. 4.4.5 Gaussian beam passing through a spatial filter 98 4.4.6 Gaussian beam passing through a prism 100 4.4.7 Diffraction of a Gaussian beam by a grating 102 4.4.8 Focusing a Gaussian beam 103 4.4.9 An example of Gaussian mode matching 104 4.4.10 Modes in complex cavities 105 4.4.11 An example of the resonance mode in a ring cavity 106 Chapter summary 107 5 Optical waveguides and fibers 109 5.1 Introduction to optical waveguides and fibers 109 5.2 Electromagnetic analysis of modes in planar optical waveguides 112 5.2.1 The asymmetric planar waveguide 112 5.2.2 Equations for TE and TM modes 112 5.3 TE modes of planar waveguides 113 5.3.1 TE planar guided-wave modes 114 5.3.2 TE planar guided-wave modes in a symmetrical waveguide 115 5.3.3 The cut-off condition of TE planar guided-wave modes 117 5.3.4 An example of TE planar guided-wave modes 118 5.3.5 TE planar substrate modes 119 5.3.6 TE planar air modes 119 5.4 TM modes of planar waveguides 121 5.4.1 TM planar guided-wave modes 121 5.4.2 TM planar guided-wave modes in a symmetrical waveguide 122 5.4.3 The cut-off condition of TM planar guided-wave modes 123 5.4.4 An example of TM planar guided-wave modes 123 5.4.5 TM planar substrate modes 124 5.4.6 TM planar air modes 125 5.4.7 Two practical considerations for TM modes 126 5.5 Guided waves in planar waveguides 126 5.5.1 The orthogonality of modes 126 5.5.2 Guided waves propagating in the y–z plane 127 5.5.3 Convergent and divergent guided waves 127 5.5.4 Refraction of a planar guided wave 128 5.5.5 Focusing and collimation of planar guided waves 129 (a) The Luneberg lens 129 (b) The geodesic lens 129 (c) The Fresnel diffraction lens 130 5.5.6 Grating diffraction of planar guided waves 131 5.5.7 Excitation of planar guided-wave modes 134 5.5.8 Multi-layer planar waveguides 135 viii Contents
- 15. 5.6 Channel waveguides 135 5.6.1 The effective index analysis 136 5.6.2 An example of the effective index method 140 5.6.3 Channel waveguide modes of complex structures 141 5.7 Guided-wave modes in optical fibers 142 5.7.1 Guided-wave solutions of Maxwell’s equations 142 5.7.2 Properties of the modes in fibers 144 5.7.3 Properties of optical fibers in applications 145 5.7.4 The cladding modes 146 Chapter summary 146 6 Guided-wave interactions 148 6.1 Review of properties of the modes in a waveguide 149 6.2 Perturbation analysis 150 6.2.1 Derivation of perturbation analysis 150 6.2.2 A simple application of perturbation analysis: perturbation by a nearby dielectric 152 6.3 Coupled mode analysis 153 6.3.1 Modes of two uncoupled parallel waveguides 153 6.3.2 Modes of two coupled waveguides 154 6.3.3 An example of coupled mode analysis: the grating reflection filter 155 6.3.4 Another example of coupled mode analysis: the directional coupler 160 6.4 Super mode analysis 163 6.5 Super modes of two parallel waveguides 163 6.5.1 Super modes of two well-separated waveguides 164 6.5.2 Super modes of two coupled waveguides 164 6.5.3 Super modes of two coupled identical waveguides 166 (a) Super modes obtained from the effective index method 166 (b) Super modes obtained from coupled mode analysis 168 6.6 Directional coupling of two identical waveguides viewed as super modes 169 6.7 Super mode analysis of the adiabatic Y-branch and Mach-Zehnder interferometer 170 6.7.1 The adiabatic horn 170 6.7.2 Super mode analysis of a symmetric Y-branch 171 (a) A single-mode Y-branch 171 (b) A double-mode Y-branch 173 6.7.3 Super mode analysis of the Mach–Zehnder interferometer 173 Chapter summary 175 Contents ix
- 16. 7 Passive waveguide devices 176 7.1 Waveguide and fiber tapers 176 7.2 Power dividers 176 7.2.1 The Y-branch equal-power splitter 177 7.2.2 The directional coupler 177 7.2.3 The multi-mode interference coupler 178 7.2.4 The Star coupler 182 7.3 The phased array channel waveguide frequency demultiplexer 186 7.4 Wavelength filters and resonators 188 7.4.1 Grating filters 188 7.4.2 DBR resonators 189 7.4.3 The ring resonator wavelength filter 189 (a) Variable-gap directional coupling 190 (b) The resonance condition of the couple ring 191 (c) Power transfer 192 (d) The free spectral range and the Q-factor 192 7.4.4 The ring resonator delay line 194 Chapter summary 195 8 Active opto-electronic guided-wave components 196 8.1 The effect of electro-optical χ 197 8.1.1 Electro-optic effects in plane waves 197 8.1.2 Electro-optic effects in waveguides at low frequencies 198 (a) Effect of Δχʹ 198 (b) Effect of Δχʹʹ 199 8.2 The physical mechanisms to create Δχ 200 8.2.1 Δχʹ 200 (a) The LiNbO3 waveguide 202 (b) The polymer waveguide 203 (c) The III–V compound semiconductor waveguide 203 8.2.2 Δχʹʹ in semiconductors 205 (a) Stimulated absorption and the bandgap 205 (b) The quantum-confined Stark effect, QCSE 206 8.3 Active opto-electronic devices 211 8.3.1 The phase modulator 211 8.3.2 The Mach–Zhender modulator 212 8.3.3 The directional coupler modulator/switch 213 8.3.4 The electro-absorption modulator 214 8.4 The traveling wave modulator 215 Chapter summary 217 Appendix 219 Index 225 x Contents
- 17. Introduction Optics is a very old field of science. It has been taught traditionally as propagation, imaging, and diffraction of polychromatic natural light, then as interference, diffraction, and propagation of monochromatic light. Books like Principles of Optics by E. Wolf in 1952 gave a comprehensive and extensive in-depth discussion of properties of polychromatic and monochromatic light. Topics such as optical waveguide, fiber optics, optical signal processing, and holograms for laser light have been presented separately in more recent books. There appears to be no need for any new book in optics. However, there are several reasons to present optics differently, such as is done in this book. Many contemporary optics books are concerned with components and instruments such as lenses, microscopes, interferometers, gratings, etc. Reflection, refraction, and diffraction of optical radiation are emphasized in these books. Other books are concerned with the propagation of laser light in devices and systems such as optical fibers, optical waveguides, and lasers, where they are analyzed more like microwave devices and systems. The mathematical techniques used in the two approaches are very different. In one case, diffraction integrals and their analysis are important. In the other case, modal analysis is important. Students usually learn optical analysis in two separate ways and then reconcile, if they can, the similarities and differences between them. Practicing engineers are also not fully aware of the interplay of these two different approaches. These difficulties can be resolved if optical analyses are presented from the beginning as solutions of Maxwell’s equations and then applied to various applications using different techniques, such as diffraction or modal analysis. The major difficulty to present optics from the solutions of Maxwell’s equations is the complexity of the mathematics. Complex mathematical analyses often obscure the basic differences and similarities of the mathematical techniques and mask the understanding of basic concepts. Optical device configurations vary from simple mirrors to complex waveguide devices. How to solve Maxwell’s equations depends very much on the configuration of the components to be analyzed. The more complex the configuration, the more difficult the solution. Optics is presented in this book in the order of the complexity of the configuration in which the analysis is carried out. In this manner, the reasons for using different analytical techniques can be easily understood, and basic principles are not masked by any unnecessary mathematical complexity. Optics in unbounded media is first presented in this book in the form of plane wave analysis. A plane wave is the simplest solution of Maxwell’s equations. Propagation,
- 18. refraction, diffraction, and focusing of optical radiation, even optical resonators and planar waveguides, can be analyzed and understood by plane wave analysis. It leads directly to ray optics, which is the basis of traditional optics. It provides a clear demonstration and understanding of optics without considering boundary condition or device configuration. Even sophisticated concepts such as modal expansion can also be introduced using plane waves. Plane wave analysis is the focus of the first two chapters. Realistically, wave propagation in bulk optical components involves a finite boundary such as a lens that has a finite aperture. Plane wave analysis can no longer be used in this configuration. However, in these situations, the waves are still transverse electric and magnetic (TEM). Therefore, TEM waves are rigorously analyzed using Maxwell’s equations in Chapter 3. The diffraction analysis presented in Chapter 3 is identical to traditional optical analysis. Since applications of diffraction analysis are already covered extensively in existing optics books, only a few basic applications of diffraction theory are presented here. The distinct features of our presentation here are: (1) Both the TEM assumption of the Kirchoff’s integral analysis and the relation between diffraction theory and Maxwell’s equations are clearly presented. (2) Modern engineering concepts such as convolution, unit impulse response, and spatial filtering are introduced. Diffraction integrals are again used to analyze laser cavities in the first part of Chapter 4, for three reasons: (1) Laser modes are used in many applications. (2) The diffraction analysis leads directly to the concept of modes. It is instructive to recognize that they are inter-related. (3) An important consequence of laser cavity analysis is that laser modes are Gaussian. A Gaussian mode retains its functional form not only inside, but also outside of the cavity. The second part of Chapter 4 is focused on Gaussian beams and how different applications can be analyzed using Gaussian beams. Gaussian modes are also natural solutions of the Maxwell’s equations. It constitutes a complete set. Just like any other set of modes, such as plane waves, any radiation can be represented as summation of Gaussian modes. When the diffraction integral is used in Chapter 3 to analyze waves propagating through components with finite apertures, the diffraction loss needs to be calculated by the Kirchoff’s integral for each aperture. In comparison, the diffraction loss of a Gaussian beam propagating through an aperture can be calculated without any integration. Therefore, a Gaussian beam is used to represent TEM waves in many engineering applications. Although TEM modes exist in solid-state and gas laser cavities, waves propagating in waveguides and fibers are no longer transverse electric and magnetic. Microwave-like modal analysis needs to be used to analyze optical devices that have dimensions of the order of optical wavelength. Optical waveguides and fibers are dielectric devices. They are different from microwave devices. Microwave waveguides have closed metallic boundaries. The mathematical complexity of finding microwave waveguide modes is much simpler than that of optical waveguides. The distinct features in the analysis of dielectric waveguides are: (1) There are analytical solutions for very few basic device configurations because of the complex boundary conditions. Analyses of practical devices need to be carried out by 2 Introduction
- 19. approximation techniques. (2) There is a continuous set of radiation modes in addition to the discrete guided-wave modes. Any abrupt discontinuity will excite radiation modes. (3) The evanescent tail of the guided-wave modes not only reduces propagation loss, but also provides access to excite the modes by coupling through evanescent fields. (4) Multiple modes are often excited in devices. The performance of the device depends on what modes have been excited. Because of the complexity of modal analysis of optical waveguides and fibers, it is presented here in four parts. In the first part, modes of simple waveguides and fibers are discussed in Chapter 5. Analytical solutions for planar waveguides and step–index fiber are presented. Although these are not realistic devices, they are the only solutions that can be obtained from Maxwell’s equations. Modes of these simple basic devices are very useful for demonstrating various properties of the guided waves. Approximation methods are then presented to discuss modes of realistic devices. For example, the effective index method is used here to analyze channel waveguides. Guided-wave devices operate by mutual interactions among modes. These interactions need to be analyzed in the absence of exact solutions. Therefore, several approximation methods, the perturbation technique, the coupled mode analysis, and the super mode analysis, are presented in Chapter 6. The differences and similarities of the three methods are compared and explained. Examples in applications are used to demonstrate these techniques. In the third and fourth parts, modal analyses of passive and active guided-wave devices are presented. Passive guided-wave devices function mainly as power dividers, wavelength filters, resonators, and wavelength multiplexers. In each of these system functions, there are several different devices that could be used. Thus, devices that perform the same system function are discussed and analyzed together. Their performance is compared. Active devices utilize electro-optical effects of the electrical signals to operate. Discussion of active guided-wave devices is complex because there are different physical mechanisms involved. How these mechanisms work is reviewed.The electrical performance, as well as the optical performance of these devices are analyzed. In summary, when optics are presented as solutions of Maxwell’s equations, the inter-relation between plane wave, diffraction, and modal analysis becomes clear. For example, the use of modal analysis is not limited to waveguides and fibers. There can be modes and modal expansion in plane wave analysis, as well as in diffraction optics. As we learn optics step by step in the order of the mathematical complexity and device configuration, we learn optical analysis from various perspectives. Introduction 3
- 20. 1 Optical plane waves in an unbounded medium Engineers involved in design and the use of optical and opto-electronic systems are often required to analyze theoretically the propagation and the interaction of optical waves using different methods. Sometimes it is diffraction analysis; on other occasions, modal analysis. They are all solutions of Maxwell’s equations, yet they appear to be very different. All optical analyses should be presented as solutions of Maxwell’s equations so that the inter-relations between different analytical techniques are clear. In order to avoid unnecessary mathematical complexity, the simplest analysis should be presented first. In this book, optics will be presented first by plane wave analysis, followed by diffraction and modal analyses, in increasing order of complexity. Plane waves are the simplest form of optical waves that can be derived rigorously from Maxwell’s equations. Plane wave analysis can be used to derive ray analysis, which is the basis of traditional optics. It can be applied directly to analyze many optical phenomena such as refraction, reflection, dispersion, etc. It can also be used to demonstrate sophis- ticated concepts such as superposition, interference, resonance, guided waves, and Fourier optics. Plane wave analyses will be the focus of discussion in Chapters 1 and 2. However, plane wave analysis cannot be used to analyze diffraction, laser modes, optical signal processing, and propagation in small optical components such as fibers and waveguides, etc. These analyses will be the focus of discussion in subsequent chapters. 1.1 Introduction to optical plane waves Plane wave analysis is presented here in full detail, so that the mathematical derivations and details can be fully exhibited and the physical significances of these analyses are fully explained. 1.1.1 Plane waves and Maxwell’s equations All optical waves are solutions of the Maxwell’s equations (assuming there are no free carriers), ∇ E ¼ ∂B ∂t ; ∇ H ¼ ∂D ∂t ð1:1Þ
- 21. Here E is the electric field vector, H is the magnetic field vector, D is the displacement vector, and B is the magnetic induction vector. For isotropic media, B ¼ μH; D ¼ εE ð1:2Þ Let ix , iy , and iz , be unit vectors in the x, y, and z directions of an x-y-z rectangular coordinate system. Then E, H and the position vector r can be written as E ¼ Exix þ Eyiy þ Eziz H ¼ Hxix þ Hyiy þ Hziz ð1:3aÞ r ¼ xix þ yiy þ ziz ð1:3bÞ A special solution of Eqs. (1.1) and (1.2) is a plane wave that has no amplitude variation transverse to its direction of propagation. If we designate the z direction as the direction of propagation, this means that ∂ ∂x ¼ 0 and ∂ ∂y ¼ 0 ð1:4Þ Substituting ∂/∂x = 0 and δ/δy = 0 into the ∇ × E and ∇ × H equations leads to two distinct groups of equations: ∂Ey ∂z ¼ μ ∂Hx=∂t; ∂Hx ∂ z ¼ ε∂Ey=∂t; or ∂Ey 2 ∂z2 ¼ με ∂2 ∂t2 Ey ð1:5aÞ and ∂Hy ∂ z ¼ ε∂Ex=∂ t; ∂Ex ∂z ¼ μ∂Hy=∂ t; or ∂Hy 2 ∂z2 ¼ με ∂2 ∂t2 Hy ð1:5bÞ Clearly, these are two separate independent sets of equations. Ey and Hx are related only to each other, and Hy and Ex are related only to each other. Solutions of Eq. (1.5a) are plane waves with y polarization of the electric field (or x polarization in magnetic field). Solutions of Eq. (1.5b) are plane waves with x polarization in the electric field E (or y polarization in magnetic field H). (a) The y-polarized plane wave For a cw optical plane wave with a single angular frequency ω that has a time variation, ejωt , and for lossless media (i.e. the medium has a real value of ε), there is a well-known solution of Eq. (1.5a) in the complex notation. It is Ey ¼ Ef yejβz ejωt ; Hx ¼ Hf x ejβz ejωt ; Hf x ¼ ffiffiffi ε μ r Ef y; ð1:6aÞ where β ¼ ω ffiffiffiffiffi με p . The real time domain expression for the complex Ey shown in (1.6a) is Ef y cos βz ωt þ φ ð Þ where φ is the phase of Ef y at z = 0 and t = 0. The angular frequency ω is related to the optical frequency f by ω ¼ 2πf. This wave is known as a y-polarized forward propagating wave in the +z direction. The phase of 1.1 Introduction to optical plane waves 5
- 22. Ey, i.e. βz ωt ¼ β z vpt , is a constant when z ¼ vpt. Thus vp is known as the phase velocity of the plane wave. If the medium in which the plane wave propagates is free space, then ε ¼ εo and the free space phase velocity is co ¼ 1= ffiffiffiffiffiffiffi μεo p ≡3 108 m s1 . In free space, the optical wave length for a frequency f is λo, where f λo ¼ co. If the medium is a lossless dielectric material with a permittivity ε, then its index of refraction is n ¼ ffiffiffiffiffiffiffiffiffi ε=εo p , β ¼ nβo ¼ nω ffiffiffiffiffiffiffi μεo p . If ε is a function of wavelength, the medium is said to be dispersive. There is also a second solution for the same polarization of the electric field, Ey ¼ Eb y ejβz ejωt ; Hx ¼ Hb x ejβz ejωt ; Hb x ¼ ffiffiffi ε μ r Eb y ð1:6bÞ This solution is a backward propagating wave because the phase of Ey, i.e. βz þ ωt ¼ β z þ vpt , at any time t is a constant when z ¼ vpt and vp ¼ ω=β. If the permittivity has a loss component, ε ¼ εr jεσ, then β ¼ ω ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ εr jεσ ð Þ p ¼ βr jβσ ð1:7Þ The phase velocity of light is now vp ¼ c ¼ ω=βr. The amplitude of the plane wave decays as eβσz′ for forward waves and eþjβσz′ for backward waves. In comparison with the phase velocity of free space, the ratio of the phase velocities, co/c, is the effective refractive index of the plane wave, n ¼ coβr=ω ¼ co=c. The wavelength in the medium is λ ¼ λo=n. In addition to β, or phase velocity, the loss of optical waves in the medium is an important consideration in applications. (b) The x-polarized plane wave A similar solution exists for the x-polarized electric field and Hy. For the forward wave, Hy ¼ Hf y ejβz ejωt ; Ex ¼ Ef xejβz ejωt ; Ef x ¼ ffiffiffi μ ε r Hf y ð1:8aÞ For the backward wave, Hy ¼ Hb y eþjβz ejωt ; Ex ¼ Eb x ejβz ejωt ; Eb x ¼ ffiffiffi μ ε r Hb y ð1:8bÞ In summary, both equations (1.5a) and (1.5b) are second-order differential equations. Mathematically, each of them has two independent solutions, which are the forward and the backward propagating waves. However, Eqs. (1.5a) and (1.5b) are also two separate set of equations. The solution for Eq. (1.5a) describes a plane wave polarized in the y direction. The solution of Eq. (1.5b) describes a plane wave polarized in the x direction. Both waves have the same direction of propagation. β is usually designated as a propagation vector along the direction of propagation z that has magnitude β, β ¼ βiz; z ¼ ziz; βz ¼ β • z ð1:9Þ The forward wave has +β, the backward wave has –β. 6 Optical plane waves in unbounded medium
- 23. It is important to note that, along any direction of propagation, there are always plane waves with two orthogonal polarizations. In each polarization, there are always two solutions, the forward wave and the backward wave. The propagation constant β and phase velocity will depend on the medium and the frequency. 1.1.2 Plane waves in an arbitrary direction Frequently, plane waves in other directions of propagation need to be expressed math- ematically for analysis. As an example, let there be another xʹ-yʹ-zʹ rectangular coordi- nate which is related to the x-y-z coordinate by ix′ ¼ ix; iy′ ¼ cos θiy cos π 2 θ iz; iz′ ¼ cos π 2 θ iy þ cos θiz ð1:10Þ The x-y-z and the xʹ-yʹ-zʹ coordinates are illustrated in Figure 1.1. The xʹ-yʹ-zʹ coordinate is just the x-y-z coordinate rotated by angle θ about the x axis. The x and xʹ axes are the same. Let there be a plane wave propagating along the zʹ direction. The solutions for the yʹ and xʹ polarized plane waves have already been given in Eqs. (1.6) and (1.8). However, these solutions could also be expressed in the x, y, and z coordi- nates, where βz′ ¼ β • z′ ¼ β cos θz þ β cos π 2 θ y ð1:11Þ β ¼ βiz′ ¼ β cos θiz þ β cos π 2 θ iy ð1:12Þ ejβz′ ¼ ejβ • z′ ¼ ejβ • r ð1:13Þ For the yʹ polarized plane wave propagating in the +zʹ direction, y x z y’ x’ z’ θ θ Figure 1.1 Illustration of x-y-z and xʹ-yʹ-zʹ coordinates. 1.1 Introduction to optical plane waves 7
- 24. Ey′ ¼ Ef y0 iy′ ejβ • z′ ejωt ¼ Ef y0 ejβ • r ejωt ¼ Ef y0 cos θiy Ef y0 sin θiz ejβ • r ejωt ð1:14Þ Hx ¼ Hx′ ¼ ffiffiffi ε μ r Ef y0 ejβ • r ejωt ix ð1:15Þ For the yʹ polarized backward plane wave propagating in the –zʹ direction, Ey′ ¼ Eb y0 eþjβ • r ejωt iy′ ; Hb x0 ¼ ffiffiffi ε μ r Eb y0 eþjβ • r ejωt ix′ ð1:16Þ For the xʹ polarized plane wave propagating in the +zʹ direction, Ex′ ¼ Ex ¼ Ef x0 ejβ • r ejωt ix′ ð1:17Þ Hy′ ¼ ffiffiffi ε μ r Ef x0 ejβ • r ejωt iy′ ¼ ffiffiffi ε μ r Ef x0 cos θiy sin θiz ejβ • r ejωt ð1:18Þ For the xʹ polarized backward wave plane wave propagating in the –zʹ direction, Ex′ ¼ Ex ¼ Eb x0 eþjβ • r ejωt ix0 ð1:19Þ Hy′ ¼ ffiffiffi ε μ r Eb x0 eþjβ • r ejωt iy′ ð1:20Þ The preceding example can be generalized for any orientation of the xʹ, yʹ, and zʹ coordinates with respect to the x, y, and z coordinates. Any plane wave propagating in the zʹ direction can have two mutually perpendicular polarizations, ia and ib . iz′ , ia and ib are mutually perpendicular to each other, i.e. ia • ib ¼ ia • β ¼ ib • β ¼ 0. Let ia ¼ ix′ and ib ¼ iy′ ð1:21Þ Then the general solutions for the case of ia polarization are: Ef a ¼ Ef aejβ • r′ ejωt ix′ Hf a ¼ ffiffiffi ε μ r Ef aejβ • r′ ejωt iy′ ð1:22Þ Eb a ¼ Eb aeþjβ • r′ ejωt ix′ Hb a ¼ ffiffiffi ε μ r Eb aeþjβ • r′ ejωt iy′ ð1:23Þ β ¼ βx′ix′ þ βy′iy′ þ βz′iz′ β2 ¼ βx′ 2 þ βy′ 2 þ βz′ 2 ð1:24Þ Here, β makes angles θxʹ, θyʹ, and θzʹ with respect to the xʹ, yʹ, and zʹ axes, with βx′=β ¼ cos θx′; βy′=β ¼ cos θy′; and βz′=β ¼ cos θz′. The general solutions for the case of ib polarization are: 8 Optical plane waves in unbounded medium
- 25. Ef b ¼ Ef bejβ • r′ ejωt iy′ Hf a ¼ ffiffiffi ε μ r Ef bejβ • r′ ejωt ix′ ð1:25Þ Eb b ¼ Eb aeþjβ • r′ ejωt iy′ Hb a ¼ ffiffiffi ε μ r Eb aeþjβ • r′ ejωt ix′ ð1:26Þ It is important to recognize that when there is a wave solution containing various terms, any term that has the form shown in Eqs. (1.17) to (1.26) represents a plane wave propagating in the direction of β. 1.1.3 Evanescent plane waves Eqs. (1.22) to (1.26) described propagating plane waves that have real βxʹ, βyʹ, and βzʹ values. The maximum real βxʹ and βyʹ values of propagating plane waves are limited to ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi βx′ 2 þ βy′ 2 q ω ffiffiffiffiffi με p , i.e. 0 θxʹ, θyʹ, and θzʹ π/2. Nevertheless, Maxwell’s equation is still satisfied even if ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β2 x þ β2 y0 q is larger than β. In that case Eq. (1.24) can only be satisfied if βzʹ is imaginary. When βzʹ is imaginary, the zʹ variation is a real decaying or growing exponential function, e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi βx′ 2þβy′ 2β2 p z′ . In any passive medium, the plane wave cannot grow without energy input. Thus the solution must decay exponentially in the zʹ direction. Any solution with imaginary βzʹ is called an evanescent wave. Such solutions do not propagate in the z direction. They do not have a phase velocity. Evanescent waves are excited usually in the vicinity of a boundary with an incident wave applied across the boundary. It is only a near field, meaning that it is negligible at locations far away from the boundary.1 It is interesting to note that when βxʹ = β, βyʹ = βzʹ = 0, it is no longer a plane wave propagating in the zʹ direction. It is a plane wave propagating in the +xʹ direction. 1.1.4 Intensity and power In optics, only time-averaged power can be detected directly by means of detectors or by recording media such as film. The time-averaged power per unit area is known commonly as the intensity. In comparison with rf and microwaves, intensity analysis plays a much more important role in optics. From text books on electromagnetic theory, it is well known that the total time-averaged power in the direction of propagation is [1] Pav ¼ 1 2 Re ð S E H • iz′ ds ¼ ð S I • iz′ ds; I ¼ 1 2 Re½E H ð1:27Þ 1 It is important to note that although the mathematical solution of a plane wave exists for βx or βy values larger than ω ffiffiffiffiffi με p , such a solution is important only if those plane waves are excited in specific applications such as total internal reflection. Otherwise, the solutions have no practical significance. 1.1 Introduction to optical plane waves 9
- 26. The integration is carried out over the entire surface of the plane wave, S. The * designates the complex conjugate of the variable. Re designates the real part of the complex quantity. Therefore the time-averaged power per unit area in the direction of propagation zʹ in either polarization is Iz′ ¼ 1 2 Re EaHa ¼ 1 2 ffiffiffi ε μ r EaEa or Iz′ ¼ 1 2 Re EbHb ¼ 1 2 ffiffiffi ε μ r EbEb ð1:28Þ Note that although the total I is the sum of the Is in each polarization, the total I carries no information about polarization breakdown. Although the complex amplitude of the plane wave has a phase, its intensity I has no phase information. For plane waves in a lossless medium, i.e. εσ = 0, its intensity I is a constant. In media with loss, the decay of the time-averaged power is e2βσz′ for a forward wave and eþ2βσz′ for a backward wave. In microwaves, I is known as the Poynting vector. In x-y-z coordinates, the intensity along the z direction is ½ Re ExHy , the intensity along the y direction is ½ Re EzHx , and the intensity along the x direction is ½Re EyHz . 1.1.5 Superposition and plane wave modes Plane waves in different direction of propagation (or plane wave modes) can be super- imposed simultaneously. This is known as the superposition theory in linear media. Many interesting optical phenomena can be understood by superposition of plane waves. Three examples are presented here to illustrate the effects of superposition. They are important concepts in many applications. (a) Plane waves with circular polarization Let us consider superposition of two plane waves of equal magnitude, polarized in x and y, with a π/2 phase difference. E ¼ Eo ix þ jiy ð1:29Þ The real time domain form of this wave is E ¼ Eo cosðβz ωt þ φÞix þ sinðβz ωt þ φÞiy h i ð1:30Þ So that, at any time t, the polarization rotates at different z positions. This type of wave is known as a circular polarized optical wave because the polarization of E rotates as it propagates. When these two waves have unequal amplitudes they give rise to an elliptical polarized plane wave. (b) Interference of coherent plane waves Let us consider two plane waves of equal amplitude at the same ω and y polarization. They propagate at different directions of propagation β in the x–z plane. Their βs lie in the x–z plane and make angles, θ and ζ, with respect to the z axis. Mathematically, the waves are 10 Optical plane waves in unbounded medium
- 27. Eoejβ sin θx ejβ cos θz ejωt iy þ Eoejβ sin ζx ejβ cos ζz ejωt iy ð1:31Þ According to Eq. (1.28), its time-averaged intensity in the z direction is Iz ¼ ffiffiffi ε μ r Eo j j2 1 þ cos β ðsin θ sin ζÞx þ ðcos θ cos ζÞz ð Þ ½ ð1:32Þ Therefore, for θ ≠ ζ, we would detect a sinusoidal intensity interference pattern of the two waves in the x direction. As z changes, the interference pattern in x will change. If we could record this intensity interference pattern, for example, by the transparency of a film, we could reproduce the plane waves by illuminating this film with another input plane wave. This is the very basic principle on which holography and phased array detection are based [2,3,4]. However, if the two waves do not have the same ω or a definite phase relation between them, then Iz ¼ ffiffiffiffiffiffiffi ε=μ p Eo j j2 . In other words, there is no interference pattern unless the two waves are coherent.2 The total intensity of incoherent waves is just the sum of the intensities of individual waves. It is also important to note that when the two coherent plane waves have cross polarizations, the total intensity is also just the sum of the two intensities without the interference effect. (c) Representation by summation of plane waves Let there be a linearly polarized TEM electric field propagating in the z direction with xy variation g(x,y) at z = 0. It is well known that g can be represented by its Fourier transform. Let G be the Fourier transform of g. G fx; fy ¼ FtðgÞ ¼ ð þ∞ ∞ ð þ∞ ∞ gðx; yÞej2π fxxþfyy ð Þdxdy ð1:33Þ gðx; yÞ ¼ Ft 1 ðGÞ ¼ ð þ∞ ∞ ð þ∞ ∞ Gðfx; fyÞej2πðfxxþfyyÞ dfxdfy ð1:34Þ G is the magnitude of the Fourier component at (fx, fy). When g(x,y) contains only slow variations in x and y, G will have significant values only at low spatial frequencies fx and fy. In that case the integration in Eq. (1.34) could be approximated by just the integration of G within a limited range of fx and fy. Let 2πfx ¼ βx′ ¼ β cos θx′ ; 2πfy ¼ βy′ ¼ β cos θy′ ð1:35Þ dfx ¼ β sin θx′dθx′ dfy ¼ β sin θy′dθy′ ð1:36Þ 2 If the two waves have a randomly time-varying relative phase relation, for example from two independent lasers, the time-averaged detected intensity also will not have the interference pattern. 1.1 Introduction to optical plane waves 11
- 28. Eq. (1.33) gives the magnitude of the Fourier component that has an xʹyʹ variation of ejβ cos θx′x′ ejβ cos θy′y′ . If we also let β ¼ ð2πco=λoÞ ffiffiffiffiffiffi μεr p ¼ 2πc=λ and consider only those Fourier components with ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi βx′ 2 þ βy′ 2 q β (i.e. 0 θxʹ and θyʹ π/2), then Eq. (1.33) gives the magnitude of the Fourier component that has the xʹyʹ variation of a plane wave propagating in a direction β, which has direction cosines θxʹ, θyʹ, and θzʹ. θzʹ is related to θxʹ and θyʹ by Eq. (1.24). However, for propagating plane waves, θxʹ,θyʹ, and θzʹ must be real. The maximum and minimum values of fx and fy for real values of θxʹ and θyʹ, are –β/2π and +β/2π. Moreover, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi βx′ 2 þ βy′ 2 q β.3 It means that, the Fourier components that correspond to propagating waves with real values of θxʹ and θyʹ, are only Fourier components that have low values of fxʹ and fyʹ. In summary, when g contains only variations in x and y very much slower than λ, G(fx,fy) has significant values only at those fx and fy less than the limit fmax. If fmax β/2π, it means the electric field g(x,y) could be represented by superposition of just plane waves propagating in different directions β. Under that condition, Eq. (1.34) can be approximated by gðx; yÞ ¼ Ft 1 ðGÞ ffi ð þfmax fmax ð þfmax fmax Gðfx; fyÞej2πðfxxþfyyÞ dfxdfy ð1:37Þ There are three important concepts introduced here: (1) We have shown that plane waves can be used to represent an arbitrary field with slow xʹ and yʹ variations. This is equivalent to the modal expansion concept used in microwaves. Here, plane waves are the modes of unbounded medium. (2) Fields with any xʹyʹ variation can be represented by their Fourier components.4 This means that many modern Fourier analysis tech- niques can be applied to optics. This is the basis of Fourier optics [2] and optical image processing [4]. (3) Knowing the plane wave composition at z = 0, we have determined the xy variation of each plane wave components at any distance z later. Thus it allows us to predict the electric field that propagates to z via plane wave analysis. Note that as component plane waves propagate the total optical radiation spreads or contracts. This phenomenon is also known as the diffraction of optical radiation. More details on diffraction will be presented in Chapter 3. It is interesting to note that if G(fx,fy) contains frequency components with large fx and fy such that fx 2 þ fy 2 β2 =4π2 , then the z variation of the plane waves for those components will exponentially decay. This means that those components will contribute only to the near field and they will not propagate far in the z direction. Only the frequency components with fx 2 þ fy 2 β2 =4π2 will propagate, so the fields at some z distance away will not be exactly the same as g(x,y). 3 Evanescent plane waves in the z direction could have βx′ or βy′ larger than β. However they are not propagating waves. 4 Fields with rapid xy variation would yield Fourier components that are evanescent local waves in the z direction. 12 Optical plane waves in unbounded medium
- 29. 1.1.6 Representation of plane wave as optical rays When an optical wave has a finite beam size, the diffraction effect will spread the beam as it propagates. Diffraction analysis allows us to analyze the fields at various positions that are not at the center of the beam. Diffraction will be discussed in Chapters 3 and 4. However, in many situations, we are only interested in analyzing the optical beam near the center; then diffraction is not important. In those situations, a local optical beam with finite beam size can be approximated by a plane wave, as long as the beam size is much larger than the optical wavelength and the variation of the beam within a distance of wavelengths is very small.5 The plane waves could then be considered simply as optical rays. Furthermore, in the analysis of natural light, which has many wavelengths or frequency components, with no specific phase relation among the different compo- nents, the phase interference effects of optical light are not important. Only the location, intensity, and direction of the propagated beam are detectable and important. Let the beam propagates in a y–z plane at an angle θ with respect to the z axis. The y–z plane is located at a constant x position. Let the beam originate at z = 0. In the traditional ray analysis used in the literature, its position at variance z distance from the z axis, i.e its y position at z, is given by r(z) and its direction is given by rʹ(z) which is drðyÞ=dz. Note that, for a ray making an angle θ with respect to the z axis, r′ ¼ sin θ. Then the ray can be represented at a given z by its ray matrix rðzÞ r′ðzÞ ð1:38Þ Note that for a ray making an angle θ with respect to the z axis, r′ ¼ sin θ. When the beam reaches a new y position later at zʹ, where d ¼ z′ z, its r(zʹ) and rʹ(zʹ) at zʹ are related to r(z) and rʹ(z) by r′ðz′Þ ¼ r′ðzÞ; rðz′Þ ¼ rðzÞ þ r′ðzÞd ð1:39Þ In other words, the relation can be expressed by a ray matrix, rðz′Þ r′ðz′Þ ¼ 1 d 0 1 rðzÞ r′ðzÞ ð1:40Þ A ray in an arbitary direction in the xyz coordinate could be considered as a ray in the yʹ–z plane of a new xʹyʹzʹ coordinate. Similar to Section 1.1.2 the xʹyʹzʹ coordinate is a rotation of the xyz coordinate. The expressions for r and rʹ in xʹyʹzʹ have been given in Eqs. (1.39) and (1.40). They could be expressed in terms of xyz through coordinate transformation as we have done in Section 1.1.2. The ray representation is only an approximation. It ignores the size of the optical beam and the size of the medium in which the beam propagates. It ignores diffraction effects. It does not give the intensity of the beam unless it is specified separately. When the 5 For example, the free space wavelength of visible light ranges from 0.4 to 0.7 μm. A uniform visible light beam a fraction of a millimeter wide can be approximated by a plane wave near the center of the beam. The approximation is good within short distances of propagation, such as a few centimeters or more. 1.1 Introduction to optical plane waves 13
- 30. polarization of the ray is important in some applications, it must be specified in addition to the ray matrix for r and rʹ. 1.2 Mirror reflection of plane waves Reflection properties of optical light can be analyzed very simply by plane waves. 1.2.1 Plane waves polarized perpendicular to the plane of incidence Let there be a plane wave, polarized in the x direction and propagating in the β direction in the y–z plane that makes an angle θi with respect to the z axis, β ¼ β sin θiiy þ β cos θiiz; Ei⊥ ¼ Eoejβ sin θiy ejβ cos θiz ejωt ix ð1:41Þ In this case, the electric field is perpendicular to the plane of incidence, which is the y–z plane. Thus the electric and magnetic fields are designated by E⊥ and H⊥. The incident wave generates a reflected electric field plane wave, βr ¼ β sin θriy þ β cos θriz; Er⊥ ¼ Erejβ sin θry eþjβ cos θrðzz′Þ ejωt ix ð1:42Þ When this wave is incident on a ideal planar mirror (or an ideal conductor with infinite conductivity) at z = zʹ, extending from x = –∞ to +∞ and from y = –∞ to +∞, the boundary condition at z = zʹ is that the total electric field tangential to the boundary, i.e. Ei⊥ þ Er⊥, be zero at z = zʹ. The boundary condition at z = zʹ demands that θr ¼ π θi and Er ¼ Eoejβ cos θiz′ ð1:43Þ or Er⊥ ¼ Eoejβ sin θiy eþjβ cos θiðzz′Þ ejβ cos θiz′ ejωt ix ð1:44Þ From Eqs. (1.18) and (1.20) of Section 1.1, the magnetic field for the incident wave is Hi⊥ ¼ ffiffiffi ε μ r Eoðcos θiiy sin θiizÞejβ sin θiy ejβ cos θiz ejωt ð1:45Þ The magnetic field for the reflected wave is Hr⊥ ¼ ffiffiffi ε μ r Eoðcos θiiy þ sin θiizÞejβ sin θiy eþjβ cos θiðzz′Þ ejβ cos θiz′ ejωt ð1:46Þ The relation given in Eq. (1.43) is commonly known as the law of reflection. The reflection changes the direction of propagation from β to βr which is a mirror reflection of β. The polarizations of the incident and reflected electric field are the same, but the orientations of the incident and reflected magnetic field are different. The magnetic field will induce surface current in the conductor at z = zʹ. For ideal mirrors, the ratio Er j j= Eo j j, called the reflectivity R of the mirror, is one. For actual mirrors with reflectivity R 1, Er ¼ REoejβ cos θiz′ . 14 Optical plane waves in unbounded medium
- 31. 1.2.2 Plane waves polarized in the plane of incidence The second independent solution for plane wave propagating in the same β direction has H directed along the x direction and E polarized in the y–z plane, which is the plane of incidence. The incident wave is designated as E== and H== . Ei== ¼ Eo½cos θiiy þ sin θiizejβ sin θiy ejβ cos θiz ejωt ð1:47Þ Hi== ¼ Eo ffiffiffi ε μ r ejβ sin θiy ejβ cos θiz ejωt ix ð1:48Þ The reflected plane wave is Er== ¼ ΓEo½þcos θiiy þ sin θiizejβ sin θiy eþjβ cos θiðzz′Þ ejβ cos θiz′ ejωt ð1:49Þ Hr== ¼ ΓEo ffiffiffi ε μ r ejβ sin ϑiy eþjβ cos ϑiðzz′Þ ejβ cos ϑiz′ ejωt ix ð1:50Þ Note that only the y component of the total electric field is zero at z = zʹ for ideal mirrors with Γ = 1. 1.2.3 Plane waves with arbitrary polarization For plane waves with an electric field polarized in any other direction, it can always be decomposed into the summation of two mutually perpendicular polarized electric field plane wave components, one polarized perpendicular to the plane of incidence and one polarized in the plane of incidence. There is a change in the reflected electric and magnetic field from that of the incident field at the reflection boundary. According to Sections 1.2.1 and 1.2.2, the polarization of the reflected beam will depend on the decomposition. Results obtained in Eqs. (1.41) to (1.50) could be applied to any plane waves in any direction of propagation in any polarization by a change of the x-y-z coordinates to new xʹ-y-zʹ coordinates. In the new coordinates the direction of the incident beam is in the yʹ-zʹ plane. 1.2.4 The intensity According to Eq. (1.28) of Section 1.1, the intensities of the incident and reflected waves along their directions of propagation are Ii ¼ 1 2 ffiffiffi ε μ r Eo j j2 ; Ir ¼ 1 2 ffiffiffi ε μ r Er j j2 ð1:51Þ 1.2.5 Ray representation of reflection The reflection at zʹ of a light beam could again be described by the ray matrix representa- tion discussed in Section 1.1.5. In that case, the r and the rʹ of the incident and the reflected beams at zʹ in the plane of incidence are related by 1.2 Mirror reflection of plane waves 15
- 32. rrðz′Þ r′rðz′Þ ¼ 1 0 0 1 riðz′Þ r′iðz′Þ ð1:52Þ The phase of the reflected beam and the polarizations of the beams are not included in the ray representation. 1.2.6 Reflection from a spherical mirror Let there be a mirror that is a section of a sphere with radius R. Here, in this section, R is not the reflectivity of the mirror as it is commonly used in the literature. Figure 1.2 shows the cross-sectional view of a spherical mirror in the y–z plane and an incident beam. The spherical mirror is centered at the origin of the x-y-z coordinate. It is much larger than the size of the incident beam. Consider a small incident beam, uniform within a lateral area that is much larger than the optical wavelength. It is sufficiently wide so that it can be approximated by a plane wave near the center of the beam. Let the incident beam be represented approximately by a plane wave polarized in the x direction at an angle θi with respect to the z axis in the y–z plane. β ¼ β sin θiiy þ β cos θiiz; Ei⊥ ¼ Eoeþjβ sin θiy ejβ cos θiz ejωt ix ð1:53Þ Hi⊥ ¼ ffiffiffi ε μ r Eoðcos θiiy sin θiizÞeþjβ sin θiy ejβ cos θiz ejωt ð1:54Þ In Figure 1.2, θi is shown as a negative angle. The slope of the incident beam is tanθi. When the beam is incident on the mirror at zʹ, the mirror at that location can be approximated by a planar mirror tangential to the sphere. This flat tangential mirror makes an angle φ with respect to the y axis. According to Sections 1.2.1 and 1.2.5, the reflected beam will make an angle π þ 2φ þ θi with respect to the z axis. The slope of ϕ r y z ϕ Spherical mirror Incident beam Reflected beam –θi –π + 2ϕ + θi Figure 1.2 The cross-sectional view in the y–z plane for an optical beam reflected by a spherical mirror. The local incident beam at the incident angle –θ is reflected by the curved mirror. The plane tangential to the spherical mirror at the incident location makes an angle –φ with respect to the vertical axis. The reflected beam makes an angle π þ 2φ þ θi with respect to the +z axis. 16 Optical plane waves in unbounded medium
- 33. the reflected beam is tanðπ þ 2φ þ ϑiÞ ¼ tan 2φ þ tan θi 1 þ tan 2φ tan θi ffi 2rðz′Þ R þ tan θi. The same conclusion is obtained when the electric field is polarized in the y–z plane.6 Therefore, the reflection from a spherical mirror in the ray representation is rrðz′Þ r′rðz′Þ ¼ 1 0 2 R 1 riðz′Þ r′iðz′Þ ð1:55Þ For incident beams parallel to the z axis, the reflected beams will be focused at z = R/2. Therefore, this location is called the focus of the spherical mirror.7 1.3 Refraction of plane waves Refraction properties of optical radiation could also be derived directly by plane wave analysis. The law of refraction is concerned with the reflection and the change of direction of propagation of optical light incident obliquely onto a planar boundary of two materials that have different dielectric permittivities, ε1 and ε2, or indices of refraction, n1 and n2. For the sake of simplicity, the media are assumed to be lossless in Sections 1.3.1 to 1.3.4.8 Refraction is used in designing optical components ranging from eye glasses and cameras, to telescopes. Refraction and reflection of plane waves will be discussed first, followed by ray optical analysis and analysis of components such as prisms, lenses, and gratings. 1.3.1 Plane waves polarized perpendicular to the plane of incidence Let there be an incident plane wave polarized in the x direction, perpendicular to the plane of incidence, and propagating in a direction in the y–z plane with an angle θi with respect to the z axis in media 1. β ¼ β1sin θiiy þ β1cos θiiz; Ei⊥ ¼ Eoejβ1sin θix ejβ1cos θiz ejωt ix ð1:56Þ Hi⊥ ¼ ffiffiffiffi ε1 μ r Eoðcos θiiy sin θiizÞejβ1sin θiy ejβ1cos θiz ejωt ð1:57Þ Let there be a plane boundary at zʹ, extending from x = –∞ to +∞ and from y = –∞ to +∞, with medium #1 at z zʹ and medium #2 at z zʹ. The boundary separates medium #1 from medium #2. In addition to the transmitted wave in medium #2, there is a reflected wave in medium #1. The boundary condition at z = zʹ is that the electric field E and the magnetic field H tangential to the boundary, i.e. Ex, Ey, Hx, and Hy, must be continuous across the boundary at zʹ. Since the incident wave is polarized in the x 6 Since the mirror is curved, the locally reflected beam is no longer strictly a plane wave. The use of plane wave for local analysis is an approximation. 7 The analysis presented here does not include rays at angles of incidence oblique to meridian planes. It is presented here only to demonstrate the very basic concept. 8 In media with losses, β1 and β2 will be complex. Waves will be attenuated as they propagate. The matching of attenuated waves at the boundary becomes much more complex than the simple relationship presented here. 1.3 Refraction of plane waves 17
- 34. direction, the reflected and transmitted wave must also be polarized in the x direction. The reflected wave in medium #1 is9 βr ¼ β1sin θriy þ β1cos θriz; Er⊥ ¼ Γ⊥12Eoejβ1sin θry ejβ1cos θrz ejωt ix ð1:58Þ Hr⊥ ¼ ffiffiffiffi ε1 μ r Γ⊥12Eoðcos θriy þ sin θrizÞejβ1sin θry ejβ1cos θrz ejωt ð1:59Þ The transmitted plane wave in media #2 is βt ¼ β2sin θtiy þ β2cos θtiz; Et⊥ ¼ T⊥12Eoejβ2sin θty ejβ2cos θtz ejωt ix ð1:60Þ Ht⊥ ¼ ffiffiffiffi ε2 μ s T⊥12Eo cos θtiy sin θtiz ejβ2sin θty ejβ2cos θtz ejωt ð1:61Þ Figure 1.3 illustrates the incident wave, the reflected wave, and the transmitted wave in media #1 and #2, plus the boundary at z = zʹ. The continuity conditions of tangential E and H at z = zʹ at all time t demand that θr ¼ π θi; β2 sin θt ¼ β1 sin θi or n2 sin θt ¼ n1 sin θi ð1:62Þ y z x Reflected wave θr θi θt Incident wave Transmitted wave z = z’ Medium #1 Medium #2 Figure 1.3 Reflection and transmission at a planar dielectric interface. The incident beam makes an angle θ with respect to the +z axis. The transmitted beam refracted from the vertical interface makes an angle θt. The reflected beam makes an angle θr. 9 Note the notations. Γ⊥12 and T⊥12 stand for reflection and transmission coefficients of the electric field perpendicular to the plane of incidence from medium #1 to medium #2. The coefficients may be different when the polarization is changed or the direction of propagation is reversed. 18 Optical plane waves in unbounded medium
- 35. T⊥12 ¼ 1 þ Γ⊥12; T⊥12 ¼ ð1 Γ⊥12Þ n1 cos θi n2 cos θt ð1:63Þ or Γ⊥12 ¼ n1 cos θi n2 cos θt n1 cos θi þ n2 cos θt ¼ sinðθi θtÞ sinðθi þ θtÞ ; T⊥12 ¼ 2n1cos θi n1cos θi þ n2 cos θt ð1:64Þ In this case, the intensities of the incident, reflected, and transmitted waves in the z direction are: I⊥i ¼ ffiffiffiffi ε1 μ r Eo j j2 cos θi; I⊥r ¼ ffiffiffiffi ε1 μ r Γ⊥12Eo j j2 cos θr; I⊥t ¼ ffiffiffiffi ε2 μ r T⊥12Eo j j2 cos θt ð1:65Þ The intensities are conserved in the z direction, i.e. I⊥i ¼ I⊥t þ I⊥r 1.3.2 Plane waves polarized in the plane of incidence The second independent solution of the plane wave propagating in the same β direction has the electric field polarized in the plane of incidence. Its H is directed along the x direction, and its E is polarized in the y–z plane. Ei== ¼ Eo½cos θiiy þ sin θiizejβ1sin θi1y ejβ1cos θiz ejωt ð1:66Þ Hi== ¼ Eo ffiffiffiffi ε1 μ r ejβ1sin θiy ejβ1cos θiz ejωt ix ð1:67Þ The reflected wave in medium #1 and the transmitted wave in medium #2 are: Er== ¼ Γ==12Eo½þ cos θiiy þ sin θiizejβ1sin θiy eþjβ1cos θiðzz′Þ ejβ1cos θiz′ ejωt ð1:68Þ Hr== ¼ þ Γ==12Eo ffiffiffiffi ε1 μ r ejβ1sin θi1y eþjβ1cos θiðzz′Þ ejβ1cos θiz′ ejωt ix ð1:69Þ Et== ¼ T==12Eo½ cos θtiy þ sin θtizejβ2sin θty ejβ2cos θtz ejωt ð1:70Þ Ht== ¼ T==12Eo ffiffiffiffi ε2 μ r ejβ2sin θty ejβ2cos θtz ejωt ix ð1:71Þ The boundary conditions at z = zʹ requires: β2 sin θt ¼ β1sin θi or n2 sin θt ¼ n1sin θi ð1:72Þ ðΓ==12 1Þcos θi ¼ T==12 cos θt; ð1 þ Γ==12Þn1 ¼ T==12n2 ð1:73Þ 1.3 Refraction of plane waves 19
- 36. In other words, T==12 ¼ 2n1 cos θi n2 cos θi þ n1 cos θt ; Γ==12 ¼ n2 cos θi n1 cos θt n2 cos θi þ n1 cos θt ¼ tanðθi θtÞ tanðθi þ θtÞ ð1:74Þ The intensities in the z direction for the incident, transmitted, and reflected plane waves are: I==i ¼ ffiffiffiffi ε1 μ r Eo j j2 cos θi; I==r ¼ ffiffiffiffi ε1 μ r Γ==12Eo 2 cos θi; I==t ¼ ffiffiffiffi ε2 μ r T==12Eo 2 cos θt ð1:75Þ Again, the intensities in the z direction are conserved, i.e. I==i ¼ I==r þ I==t. It is important to note that all optical reflection, refraction, and diffraction effects are calculated based on meeting the boundary conditions by waves that satisfy Maxwell’s equations. The energy in the waves is conserved. Note that the transmission and reflection coefficients of optical waves are dependent on the polarization of the electric field, while the intensity of the wave is not affected. 1.3.3 Properties of refracted and transmitted waves (a) Transmission and reflection at different incident angles It is interesting to note that, at normal incidence, θi ¼ θt ¼ 0 and θr ¼ π. Tand Γ are the same for polarizations either perpendicular to the plane of incidence or in the plane of incidence. The direction of propagation of the transmitted wave is the same as the incident wave, while the reflected wave has a reverse direction of propagation. There is no change of polarization of the reflected and transmitted waves from the incident wave. T12 ¼ 2n1 n1 þ n2 ; Γ12 ¼ n2 n1 n2 þ n1 ð1:76Þ Ii ¼ It þ Ir ð1:77Þ It is important to realize the relative importance of this result in practical applications. At an interface of free space with n1 = 1 and glass with n2 = 1.5, T12 = 0.8 and Γ12 = 0.2, which is small. Therefore in many applications of glass components, such as imaging through a lens, the reflection may not be analyzed. The situation is very different when medium #2 has a large index of refraction such as a III–V semiconductor. If n2 = 3.5, then T12 = 0.56 and Γ12 = 0.44 at normal incidence. At other angles of incidence, Γ and T will vary dependent on the angle of incidence and the polarization. The magnitude of reflection increases at large θi. It is interesting to note that when θi þ θt ¼ π=2, Γ==12 ¼ 0 in Eq. (1.74). The θi that satisfies this condition is traditionally known as the Brewster angle. At this angle the incident and the reflected plane waves are polarized perpendicular to each other in the plane of incidence. The Brewster angle has many practical applications because at this angle the reflection is zero without any anti-reflection coating. 20 Optical plane waves in unbounded medium
- 37. (b) Total internal reflection When n1 n2, at the angle of incidence θi, such that n1 sin θi ¼ n2, θt ¼ π=2; Γ ¼ 1 and It ¼ 0. This means that, for a plane wave with any polarization there is no energy transmitted in the z direction. For θi sin1 n2=n1 and Ei polarized perpendicular to the plane of incidence, the boundary condition in Eq. (1.62) demands that cos θt ¼ j ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn2 1 sin2 θ1=n2 2Þ 1 q . Therefore, we have an evanescent wave in medium 2 in which the propagation constant in the z direction of the transmitted wave shown in Eq. (1.60) is imaginary. The reflection coefficient is Γ12 ¼ Γ12 j jejϕ12 . From Eq. (1.64), we obtain Γ⊥12 j j ¼ 1 and φ⊥12 ¼ tan1 2n1 cos θi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2 sin2 θi n2 2 p n1 2 cos2 θi ðn1 2 sin2 θi n2 2Þ From Eq. (1.74), a similar conclusion can be reached for plane waves polarized in the plane of incidence. Again, Γ==12 ¼ 1. However, the phase angle is different from ϕ⊥12. φ==12 ¼ tan1 2n2 cos θi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 2 sin2 θi n1 2 p n2 2 cos2 θi ðn2 2 sin2 θi n1 2Þ In summary, the incident plane wave with any polarization is said to be totally internally reflected at the boundary for θi sin1 n2=n1, but the phase angle is dependent on polarization. Total internally reflected waves have only an evanescent tail in the lower index medium. Total internal reflection is utilized extensively in optical fibers and waveguides to minimize the loss due to the surroundings, by using a cladding layer that has a lower index of refraction so that losses in the surrounding media at distances further away from the interface than the length of the evanescent tail do not cause much propagation loss to the totally internal reflected optical wave. (c) Refraction and reflection of arbitrary polarized waves For plane waves with arbitrary polarization, results derived in Eqs. (1.56) to (1.75) are applicable when the electric field is first decomposed into two components, one polar- ized perpendicular to the plane of incidence and the second polarized in the plane of incidence. Although these two components have the same direction of propagation of reflected and transmitted waves (see Eqs. (1.62) and (1.72)), their polarization, transmis- sion coefficient T, and reflection coefficient Γ are different. (d) Ray representation of refraction It was shown in Sections 1.1.6 and 1.2.5 that natural light with finite beam width and location can be represented by its ray matrix. There is also a matrix representation of the refracted (i.e. transmitted) beam as follows 1.3 Refraction of plane waves 21
- 38. rðz′Þ r′ðz′Þ ¼ 1 0 0 n1 n2 rinðz′Þ rin′ðz′Þ ¼ 1 0 0 sin θ2 sin θ1 rinðz′Þ rin′ðz′Þ ð1:78Þ Note that the ray matrix representation is independent of polarization. It does not tell us the magnitude, the size, or the polarization of the refracted beam. The reflected beam is not included in Eq. (1.78). The ray matrix representation of the reflected beam for a mirror is given in Eq. (1.52) in Section 1.2.5. 1.3.4 Refraction and dispersion in prisms Prisms are optical components used to redirect the direction of propagation of an optical beam. Note that the permittivity ε and the index of refraction n2 of materials are usually wavelength (i.e. ω or λ) dependent. This means that the transmitted plane wave at the boundary will have different directions of propagation at different wavelengths. This is known commonly as the dispersion. In a prism spectrometer, the incoming optical beam may have many wavelength components. The collimated incident beam passes through a prism. At the exit of the prism, different wavelength components propagate in different directions due to the dispersion effect. The exit beams in different directions are focused by a lens to different positions. An exit slit located on the focal plane of the lens selects the radiation in a specific wavelength range to be detected. The spectral width of the detected radiation is determined by the width of the slit. As the prism rotates, the detected radiation displays the spectral component of the incident radiation as a function of the prism angle. (a) Plane wave analysis of prisms A prism is usually a dielectric cylinder with a triangular cross-section made from material with a refractive index n2. This index n2 is larger than the index of the surrounding medium, which has index n1. Usually the surrounding medium is free space with n1 = 1. The triangular cross section of a prism in the y–z plane is shown in Figure 1.4. The prism is uniform in the x direction. It has a vertex angle, A +B, and a base angle, ðπ=2 AÞ, for the front surface, and base angle, ðπ=2 BÞ, for the back surface. The dimensions of the surfaces of the triangle are larger than the width of the optical beam, which is much larger than the optical wavelength itself. Let there be an optical incident beam propagating in a direction θi from the z axis in the y–z plane. For uniform beams that have a beam width much larger than the optical wavelength, the beams can be represented by plane waves near the center of the beam. The incident beam, the refracted beam in the prism, and the transmitted beam of the prism are also illustrated in Figure 1.4. The analysis of the wave propagation in prisms is simply a detailed analysis of the directions of the refracted beams at each dielectric interface, as follows. In order to analyze the beam propagation, let us designate xʹ-yʹ-zʹ coordinates and xʹʹ- yʹʹ-zʹʹ coordinates, as shown in Figure 1.4. The yʹ axis is parallel to the front prism surface 22 Optical plane waves in unbounded medium
- 39. and the zʹ axis is perpendicular to the front surface. The yʹʹ axis is parallel to the back prism surface and the zʹʹ axis is perpendicular to the back surface. The xʹ-yʹ-zʹ and xʹʹ-yʹʹ- zʹʹ coordinates are related to the x-y-z coordinates by: iy′ ¼ cos Aiy þ sin Aiz; iz′ ¼ sin Aiy þ cos Aiz ð1:79Þ iy″ ¼ cos Biy sin Biz′ iz″ ¼ þsin Biy þ cos Biz ð1:80Þ In Figure 1.4, the incident beam in material #1 is directed in the θ 0 i direction in the yʹ–zʹ plane, where θ 0 i ¼ θi þ A. The refracted beam from the front surface is directed in the θʹt direction in the yʹ–zʹ plane. According to Eqs. (1.62) and (1.72), θ 0 t ¼ sin1 ðn1 sin θ 0 i =n2Þ. The angle θt that this beam makes with respect to the z axis in the x–y plane is θt ¼ A θ 0 t . In the xʹʹ-yʹʹ-zʹʹ coordinates, the refracted beam at the angle θt in the x-y-z coordinates makes an angle θ″ i with respect to the zʹʹ axis, where θ″ i ¼ θt þ B ¼ A θ 0 t þ B. Its exit beam in medium #1 makes an angle θ″ t with respect to the zʹʹ axis, where θ″ t ¼ sin1 ðn2 sin θ″ =n1Þ. In the x-y-z coordinates, this exit beam makes an angle θout with respect to the z axis, θout ¼ B θ″ t . This analysis of beam direction is independent of polarization. There are also reflected beams at each surface. Reflections need to be considered whenever the difference of refractive indices at the interface of the prism is large. y z x, x’, x’’ z’ y’’ z’’ θi y’ θ’t θ’’t θ’’i θout A B A B Front surface Back surface Cross-section of a prism Incident beam Exit output beam Refracted beam θ’i Figure 1.4 Incident, refracted, and transmitted wave in a prism. The prism has a vertex angle A + B. The incident beam angle is θi. It is refracted by the front prism surface. The transmitted beam from the front surface makes an angle θʹt with respect to the vertical axis of the front prism surface. It is refracted again by the back prism surface. The output beam angle is θout. 1.3 Refraction of plane waves 23
- 40. Reflection and transmission at each surface can be calculated according to Eqs. (1.63), (1.64) and (1.74) from θ 0 i , θ 0 t , θ″ i , and θ″ t . However, in most applications, only the analysis of the transmitted beam is important, the reduction of the amplitude of the transmitted wave due to reflections is not important. (b) Ray analysis of prisms In the ray matrix representation, the incident ray enters the front prism surface at the zi location with the y position ri(zi). It has a riʹ(zi) in the x-y-z coordinates, r 0 i ðziÞ ¼ sin θi. The refracted beam at zi has the same y position, rtðziÞ ¼ riðziÞ, and a slope, rt′ðziÞ ¼ sin θt. When the refracted beam is incident on the back surface at the zout position, its y location is routðzoutÞ ¼ riðziÞ þ ðzout ziÞtan θt. The slope of the output beam is r 0 outðyoutÞ ¼ sin θout. In matrix notation the relationship is: rout r 0 out ¼ 1 0 0 sin θout sin θt 1 ðzout ziÞ=cos θt 0 1 1 0 0 sin θt sin θi ri r 0 i : ð1:81Þ In the case of thin prisms, A+B is small, and zout zi ≈ 0. rout r 0 out ¼ 1 0 0 sin θout sin θi ri r 0 i ð1:82Þ In other words, a thin prism does not change the position of the beam. It only changes its direction. Furthermore, for a small incident angles θi, θout ¼ θi n2 n1 n1 ðA þ BÞ: ð1:83Þ Note that the reflected beams are not included in the ray representation above. The magnitude and polarization of the beams are also not included in the ray representation. These quantities may not be important in applications that use natural light in compo- nents that have low value of n2. For applications, such as image formation, the ray representation is a simple method for analyzing the direction, position, and propagation distance of the beam that are most important. (c) Thin prism represented as a transparent layer with a varying index It is interesting to view this result from another viewpoint. In a thin prism we could also consider the prism as a dielectric layer that has an n2 layer with thickness τ embedded in a medium with index n1. The thickness τ varies at different position y. From Figure 1.4, we obtain τ ¼ ðyvert yÞðtan A þ tan BÞ ffi ðyvert yÞðA þ BÞ. Here, yvert is the vertex of the prism. Let there be a plane wave propagating in the z direction in a medium that has index n1. The beam is centered at yi. After transmitting through this composite dielectric layer, the electric field for this beam is 24 Optical plane waves in unbounded medium
- 41. Eout ¼ Eoejn1βoz ejðn2n1Þβoτ ejωt ¼ Eoejn1βoz ejðn2n1Þβoyvert ejn1βosin ϕouty ejωt ; ð1:84Þ where n1 sin φout ¼ ðn2 n1ÞðA þ BÞ: Any plane wave that has an expðjn1βo sin φoutyÞ variation in y is a plane wave propagating at an angle φout in the y–z plane. This φout agrees with the θout given in Eq. (1.83) above. In other words, we have just introduced an important new concept. Transmission through a thin prism could also be represented by transmission of a plane wave through a medium with a phase transmission coefficient that is a linear function of y, t ¼ toejðn2n1ÞβoðyvertyÞðtan Aþtan BÞ ð1:85Þ Eout ¼ tEin. Note that the results in Eqs. (1.84) and (1.85) are independent of polarization. In other words, when a plane wave is transmitted through a refractive medium with variable refractive index given in Eq. (1.85), it produces an output beam in a different direction of propagation. The conclusion is also valid for a small incidence angle θi. Conversely, any transmission medium with a phase transmission coefficient that has a linear y variation will tilt the incident beam to a new direction of propagation like a prism. 1.3.5 Refraction in a lens A lens is probably the most commonly used optical component. It is used principally for imaging and instrumentation. Ray analysis is the principle tool used for lens design. The design of a compound lens is very complex. A detailed discussion on ray analysis of lens design is beyond the scope of this book. However, an analysis of a simple spherical lens for meridian rays will be beneficial to illustrate the basic principle of a lens.10 It will be presented first by ray analysis, then as a transparent medium with a quadratic varying phase in transmission. (a) Ray analysis of a thin lens Let us consider a simple spherical lens whose geometrical configuration is shown in Figure 1.5. The right surface of the lens is described by x″2 þ y″2 þ z″2 ¼ r2 1 ð1:86Þ The left surface of the lens is described by x 0 2 þ y 0 2 þ ðz 0 z1Þ2 ¼ r2 2 ð1:87Þ 10 Like prism analysis, reflection exists at any dielectric interface. There are reductions of the amplitude of the transmitted wave as it propagates through the lens. Reflections in lenses are analyzed when it is necessary. Thus only ray analysis of the transmitted beam will be presented here. 1.3 Refraction of plane waves 25
- 42. The origin of the spherical surfaces are at z = 0 and z = z1. The refractive index of the lens is n2. It is placed in a medium that has refractive index n1. In free space, n1 = 1. The thin lens and the dotted spheres shown in Figure 1.5 are the cross-sectional view of the lens and the spheres in the y–z plane. In order to demonstrate the properties of a lens with simple ray analysis, let us consider an optical beam incident on the lens in the y–z plane at an angle –θi with respect to the z axis. This beam is incident on the lens at the zʹ and yʹ positions. The refracted beam is transmitted through the lens and excites an output beam. In the thin lens approximation, the yʹ positions of the beam at the front and the back surfaces of the lens are the same. At the yʹ position of the front surface, the curved spherical lens surface can be approximated locally by a plane tangential to the sphere centered at z1. This plane makes an angle Awith respect to the Yaxis. Similarly, at the back surface of the lens, the curved surface can be approximated by a plane tangential to the sphere centered at z = 0. This plane makes an angle B with respect to the y axis. Thus the change of direction of the beam going through the lens at this location is approximately the same as a beam going through a prism with the vertex angle, A+B. From Section 1.3.4, we obtain θout ¼ θi n2 n1 n1 ðA þ BÞ ¼ θi n2 n1 n1 1 r1 þ 1 r2 y ð1:88Þ If we designate 1 f ¼ n2 n1 n1 1 r1 þ 1 r2 ð1:89Þ The thin lens r2 r1 . . z = z1 z = 0 x’’2 +y’’2 +z’’2 = r1 2 x’2 +y’2 +(z’− z1)2 = r2 2 z y x Figure 1.5 The geometrical configuration of a spherical lens. Two spherical surfaces centered about z = z1 and z = 0 are shown. The front (left) and back (right) surfaces are made from the interception section of these two spherical surfaces. 26 Optical plane waves in unbounded medium
- 43. then the refraction of a beam through a thin spherical lens placed at zʹ can be expressed in a ray optical representation as routðz 0 Þ r′outðz 0 Þ ¼ 1 0 1 f 1 riðz 0 Þ r′iðz 0 Þ ð1:90Þ When the incident beam is parallel to the z axis, θi = 0. Parallel beams at different y positions would all focus on the z axis at the position zʹ + f. Therefore z = f is commonly known as the focal length of the lens. For parallel beams incident at small θi, θout will still be related to θi by Eq. (1.88). Thus they will be focused to a point at z ¼ z 0 þ f and y ¼ θif . The plane at z = zʹ + f is known as the focal plane of the lens. The preceding analysis has only been carried out for optical beams incident in the y–z plane. However, in a cylindrically symmetric configuration, the x and y axes can be rotated about the z axis. Thus the results can be generalized to three dimensions for any beam incident in the meridian plane. The objective of the analysis presented here is only to demonstrate simple lens properties by plane wave analysis. The analysis of practical lenses is much more complex than the preceding discussion. It involves oblique rays, skewed rays, astigma- tism, etc. and is beyond the scope of this book [5]. (b) Thin lens represented as a transparency with varying index Similar to the discussion in Section 1.3.4 (c) for the prism, it is instructive to represent a thin lens as a transparent planar medium with a varying phase change. Consider an incident plane wave that has a beam size small compared to the size of the lens. It propagates in the direction of z axis. Ei ¼ Eoejβ1z ejωt Let us consider this small beam in the y–z plane near x = 0. At the transverse position (x = 0, y), it passes through the lens beginning at z = z1 – r2 and ending at z = r1. Its phase at the output will depend on y because the ray goes through a higher index region with thickness, zʹʹ – zʹ, at y = yʹ= yʹʹ. The change in its phase, in comparison to a beam in free space without the lens, is: Δϕ ¼ β1ðn2 n1Þðz″ z0 Þ ¼ β1ðn2 n1Þ r1 1 y2 r1 2 8 : 9 = ; 1=2 z1 þ r2 1 y2 r2 2 8 : 9 = ; 1=2 0 B @ 1 C A ð1:91Þ Here, zʹʹ zʹ and x and y r1 and r2 inside the lens. Binomial expansion can be used again for the terms in the curly brackets. When the first-order approximation is used for a thin lens, we obtain Δϕ ¼ βðn2 n1Þ r1 þ r2 z1 y2 r1 y2 r2 : ð1:92Þ 1.3 Refraction of plane waves 27
- 44. The focal length of a thin spherical lens is given in Eq. (1.89) as 1/f = (n2 – n1)(1/r1 + 1/r2). Thus, for any wave passing through a thin lens near x = 0, we can now multiply the incident wave on the lens by a phase function, tl ¼ ejβ1ðn2n1Þðr1þr2z1Þ ej β1 2f ðy2 Þ ð1:93Þ to obtain the wave that has passed through the lens. The preceding result can be extended to incident rays in any meridian plane by rotating the x and y axes with respect to the z axis. Therefore the general result in the x-y-z coordinates for a plane wave incident on the lens in the z direction is t ¼ ej β1 2f ðx2 þy2 Þ ð1:94Þ Eout ¼ tEoejβ1r1 ejωt ð1:95Þ The output electric field at the back side of the lens is Eout ¼ Eoejβ1ðr1þf Þ ejβ1f ej β1 2f ðx2 þy2 Þ ejωt h i ð1:96Þ The quantity in the brackets represents a spherical wave, ejβ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2þx2þy2 p ejωt , in the form of the first term of a binomial expansion at z = r1. The spherical wave is focused on to the location z = r1 + f. This is a very simple result that can be applied to any incident wave passing perpendicularly through a lens. It should be emphasized that this is a thin lens approx- imation. Only an ideal lens can be represented by Eq. (1.94). A practical lens will have other higher-order phase shifts, which are considered as distortions from an ideal lens. Note that the output after leaving the lens is no longer a plane wave. Fourier analysis discussed in Section 1.1.5 (c) must be used to find its plane wave components. The importance of this representation is to recognize that whenever a medium has a quadratic phase variation in transmission, it functions as a lens. 1.4 Geometrical relations in image formation Image formation is one of the most important applications in optics. It has been presented extensively in traditional optical literatures. It is also a very specialized topic. The geometrical relation between an object and its image is presented here only to demonstrate the basic relation between ray analysis and image formation. Consider a point optical source placed at x = 0 and z = –p at the position y = hob, a thin lens with focal length f is placed at z = 0, centered at x = 0 and y = 0 and perpendicular to the z axis. Figure 1.6 illustrates the configuration. From the discussion in Section 1.1.5 (c), we can consider that the point source yields a summation of plane wave component beams in different directions. Let us consider two incident component rays. (a) A component ray that propagates parallel to the z axis. According to discussion in the previous section, this 28 Optical plane waves in unbounded medium
- 45. ray will be redirected after the lens. It passes through the focus of the lens at z = f. (b) A component ray that propagates toward the center of the lens at y = 0. This ray is not redirected in its direction of propagation because it passes locally through two parallel dielectric interfaces with negligible separation at y = 0. The two rays meet at the image point. If ray analysis is carried out for rays in other directions of propagation, they will also meet at the same image point. In other words, the optical light from the object point source is refocused by the lens to the image point. The relation between the positions of the object and the image is determined geometrically. From Figure 1.6, it is clear that hob f ¼ him q f for ray ðaÞ; hob p ¼ him q for ray ðbÞ ð1:97Þ or, 1 p þ 1 q ¼ 1 f ; him hob ¼ q p ð1:98Þ Any extended object at z = –p can be represented by the summation of point objects at different h. Therefore the magnification ratio of the image to the extended object is q/p. When p = ∞, q = f, and him = 0. Thus the object is focused by the lens to z = f. Conversely, when p = f, q = ∞. A point source is collimated by the lens to a parallel beam. Eqs. (1.97) and (1.98) represent the geometrical relations between an object and its image. –p +q +f z y x Object Image Thin lens hob –him Figure 1.6 Illustration of the geometrical relations in imaging. The object hob long is placed at –p. The lens with focal length f is placed at 0. The image him long appears at q. 1.4 Geometrical relations in image formation 29
- 46. 1.5 Reflection and transmission at a grating An important property of optical radiation used in many applications is the diffraction of an optical wave by a grating. Analysis of grating diffraction in traditional optical analysis is often complex. However, it can be easily understood by plane wave analysis. A grating is either a transmission or reflection medium with a periodic variation of amplitude or phase, or a mirror with a periodic variation of reflectivity. If an optical wave is incident on a grating then its transmitted or reflected wave will have different Fourier plane wave components that correspond to different directions of propagation. These Fourier components are known as different orders of grating diffraction. Consider first a thin medium that has a periodic sinusoidal amplitude transmission coefficient t of the electric field such that t ¼ toð1 þ Δt cos 2πfgyÞ ¼ to þ to Δt 2 ej2πfgy þ to Δt 2 ej2πfgy ð1:99Þ The medium is placed at z = 0, parallel to the x–y plane. to is its averaged transmission, and Δt is the magnitude of the periodic variation in the y direction. Δt ≤ 1, so t is always positive. The minimum transmission is to – Δt; the maximum transmission is to + Δt. The periodic variation has a grating period Tg per unit length in the y direction where Tg ¼ 1=fg Let there be an incident plane wave polarized in the x direction and propagating at an angle θ with respect to the z axis. Eix ¼ Exix ¼ Eoejβ cos θz ejβ sin θy ejωt ix ; Hiy ¼ Hyiy ¼ ffiffiffi ε μ r Eoejβ cos θz ejβ sin θy ejωt iy ð1:100Þ The output plane wave at z 0 after the grating is Eox ¼ Eoxix;Eox ¼ Eoto ejβsinθy þ Δt 2 ejβðsinθ2πfg=βÞy þ Δt 2 ejβðsinθþ2πfg=βÞy ejβcosθz ejωt ð1:101Þ Hoy ¼ Hoyiy; Hoy ¼ ffiffiffi ε μ r Eox ð1:102Þ The output wave has three components, a plane wave propagating in the incident direction, a plane wave propagating at an angle θ+1 , called the +1 order diffracted wave where θþ1 ¼ sin1 ðsin θ þ 2πfg=βÞ, and a plane wave propagating at an angle θ−1 , called the –1 order diffracted wave, where θ1 ¼ sin1 ðsin θ 2πfg=βÞ. Note that θ−1 and θ+1 of any propagating diffracted wave must be less than π=2, otherwise that order of the diffracted wave is an evanescent wave. When the diffracted wave for a specific order is evanescent, we say that the grating is cut off for that order. A similar result is obtained for a y polarized incident wave. If t depends on polarization, the magnitude of the diffracted wave will be polarization dependent. However, the diffrac- tion angles will not be polarization dependent. 30 Optical plane waves in unbounded medium
- 47. The sinusoidal grating transmission function in Eq. (1.99) is used here because it is simple to analyze. When the periodic transmission function t has a non-sinusoidal periodic variation, it can be expressed as a Fourier series with periodicity Tg. For example, for a grating with an on–off periodic variation of Δt, Δt can be expressed as a repetition of individual on–off sections. ΔtðyÞ ¼ Δto X m rect mTg y δ=2 ¼ X n Δtncosð2πnfgyÞ ð1:103Þ rect(τ) is defined as rectðτÞ ¼ 1 for τ j j ≤ 1 and rectðτÞ ¼ 0 for τ j j 1 ð1:104Þ Here, δ is the width of individual on-section, δ Tg. Tg δ is the width of individual off-section. ΔtðyÞ can also be expressed by its Fourier series. Each Fourier component has a magnitude Δtk. For the kth Fourier component, Δtk ¼ 2 ð Tg 0 ΔtðyÞcosð2πkfgyÞdy ð1:105Þ Each ±n Fourier component has an angle of diffraction, θn ¼ sin1 ðsin θ 2nπfg=βÞ They are the ±nth order of diffracted waves. Only those with θn π=2 are propagat- ing waves. Plane wave analysis provides a simple way to understand grating diffraction. Note that the direction of the nth-order diffracted wave is dependent on β, which is propor- tional to the optical frequency ω, or wavelength. This is known as the dispersion of the grating diffraction. Different orders of diffraction will have different angles for the diffracted beam. Some orders of diffraction may be cut off. 1.6 Pulse propagation of plane waves When the amplitude of the plane wave is time dependent, the wave is a pulse. Let there be a plane wave pulse in the z direction, polarized in the x direction, Ex ¼ Exix ¼ AðtÞejβðωoÞz ejωot ix ð1:106Þ Here, in order to emphasize the dispersion effect, we have written β as β(ω). At z = 0, Ex ¼ AðtÞej2πfot ð1:107Þ A(t) can be represented by its Fourier transform pairs, FAðf Þ ¼ ð þ∞ ∞ AðtÞej2πft dt; AðtÞ ¼ ð þ∞ ∞ FAðf Þeþj2πft df ð1:108Þ Here FA is the component of A at frequency f. Therefore, at z = 0, 1.6 Pulse propagation of plane waves 31
- 48. Exðz ¼ 0Þ ¼ ð þ∞ ∞ FAðf Þeþj2πðfof Þt df ð1:109Þ It is a sum of plane waves at different frequencies fo – f. Each component at fo – f propagates with a different β to the position z. Therefore, at z ExðzÞ ¼ ð þ∞ ∞ FAðf Þejβðfoof Þz ej2πðfof Þt df ð1:110Þ Usually, fo – f fo. Therefore, β(f) can be represented by its Taylor’s series, βðf Þ ¼ βðfoÞ þ ∂β ∂f ðfoÞ ðf Þ þ 1 2 ∂2 β ∂f 2 ðfoÞ ðf Þ2 þ ð1:111Þ When the second- and higher-order terms can be neglected, we obtain Ex ¼ ð þ∞ ∞ FAðf Þe j2π ∂β ∂ω ð Þjfo z eþj2πft df 2 4 3 5ejβðfoÞz ejπfot ð1:112Þ where vg ¼ ∂ω=∂β fo ð1:113Þ is known as the group velocity. In a realistic situation, pulse distortion is important if the distance of propagation is very long and the pulse duration of A(t) is short. If the group velocity is independent of f, the quantity ej2π ∂β ∂ω ð Þ fo z j can be factored out of the integral. The pulse is then propagated to z without distortion, i.e. A(t) is unchanged. The only change is a change of the phase of Ex from ejβðfoÞz which equals 2πð∂β=∂ωÞ fo z . Otherwise, there will be distortion, or change of A(t). Clearly, when higher-order terms in Eq. (1.111) cannot be neglected, there will be additional distortion. Chapter summary Basic plane wave analysis is presented. A plane wave is the simplest rigorous solution of Maxwell’s equations. Yet it can be used to illustrate many basic concepts in optics. Under appropriate circumstances, an optical ray could be represented locally approxi- mately by a plane wave. Optical properties such as reflection, refraction, and focusing can also be derived from plane wave analysis. The plane wave presented here shows how the traditional analysis is related to Maxwell’s equations. However, much more complex analyses are required for optical components design and image transfer [5]. Traditional optics is better suited for these applications. On the other hand, plane wave analysis shows optical properties that are not emphasized in traditional optics. These include the 32 Optical plane waves in unbounded medium
- 49. dependence of refraction on polarization, the differentiation between the amplitude (including phase) and intensity of the wave, the importance of change in polarization, the phase interference effects, etc. Sophisticated engineering analytical techniques can be illustrated by plane wave analysis. Concepts such as evanescent waves are introduced. Thin refractive compo- nents are representable by a transparent medium with phase variation. Grating diffrac- tion is presented as another example of how phase variation can be used to understand simply a complex phenomenon. Note that, an arbitrary optical field can be represented by summation of plane waves in the form of Fourier transformation, which is the basis of optical signal processing. Representation of an arbitrary radiation pattern by superposition of plane waves is also probably the simplest form of modal analysis in which the modes are just the plane waves. Plane wave analysis is also an important vehicle to learn the basic mathematics of wave solutions. For example, there are always two independent solutions, the forward and the backward waves, and two mutually perpendicular polarizations for each direc- tion of propagation. Optical interactions in all components are analyzed by matching the boundary conditions at the interfaces. References 1. David M. Pozar, Microwave Engineering, John Wiley Sons, 2005. 2. Joseph W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1968. 3. P. Hariharan, Optical Holography Principles, Techniques, and Applications, Cambridge University Press, 1996. 4. W. Thomas Cathey, Optical Information Processing and Holography, John Wiley Sons, 1974. 5. M. Born and E. Wolf, Principles of Optics, Pergamon Press, 1959. References 33
- 50. 2 Superposition of plane waves and applications The basics of many applications such as anti-reflection and reflection coatings, beam splitters, interferometers, resonators, holography, and planar waveguides, etc. can be analyzed by superposition and multiple reflections of plane waves. These analyses demonstrate the usefulness of simple plane wave analysis in another dimension. This is the focus of Chapter 2. However, there are many shortcomings of plane wave analysis. It does not provide a full characterization of many applications because it ignores lateral variation of beams that occur in real components. For example, the consequence of the finite size of the beam is not included in plane wave analysis. Other analytical tools such as Fourier transform and convolution theory also cannot be presented by plane wave analysis. Laser cavity modes and Gaussian beams are not plane waves. These analyses are presented in Chapters 3 and 4. 2.1 Reflection and anti-reflection coatings Reflection and transmission of plane waves at a dielectric interface can often be increased or reduced by coating the surface with transparent dielectric layers that have appropriate refractive indices. It is an anti-reflection coating when it is designed for maximum transmission and a reflection coating when it is designed for maximum reflection. It is a beam splitter when a specific ratio of reflected and transmitted intensities is required for some applications. Consider an x-polarized plane wave propagating in the +z direction in a medium with refractive index n1. If this wave is incident perpendicularly onto another unbounded medium that has a refractive index n2 at z d, the reflection Γ12 and transmission T12 of this plane wave at the boundary is given by Eq. (1.64) in Section 1.3.1 as: Γ12 ¼ 1 n2 n1 1 þ n2 n1 ; T12 ¼ 2 1 þ n2 n1 ð2:1Þ The magnitude of the transmitted and reflected waves in Eq. (2.1) can be changed by adding layers of transparent materials with appropriate refractive indices in front of medium #2. Let us consider a single transition layer of a transparent dielectric material that has a refractive index nt and thickness d. It is placed from z = 0 to z = d in front of the medium
- 51. with n2, as shown in Figure 2.1. Let there be an x-polarized wave for z 0, incident on the interface along the z direction. Its x-y-z variation is: Ei ¼ Eiejβ1z ix ð2:2Þ Hi ¼ ffiffiffiffi ε1 μ r Eiejβ1z iy ð2:3Þ For simplicity, the time variation of ejωt is not shown here explicitly. There will also be a reflected wave in z 0, Er ¼ REieþjβ1z ix ð2:4Þ Hr ¼ ffiffiffiffi ε1 μ r REieþjβ1z iy ð2:5Þ There are two plane waves in the transition layer in 0 z d, a forward wave and a backward wave. Et ¼ ðEf ejβtz þ Eb eþjβtz Þix ð2:6Þ Ht ¼ ffiffiffiffi εt μ r ðE f ejβtz Eb eþjβtz Þiy ð2:7Þ There is a transmitted wave in the unbounded medium with refractive index n2 at z d. Medium # 1, n = n1 Medium # 2, n = n2 Coating, n = nt Incident wave Transmitted wave Reflected wave Forward wave Backward wave y z x d Figure 2.1 Anti-reflection and reflection coatings. A coating with index nt is placed between z = 0 and z = d. The incident and reflected waves in medium #1, the forward and backward waves in the coating, and the transmitted wave in medium #2 are shown. 2.1 Reflection and anti-reflection coating 35
- 52. Eo ¼ Eoejβ2ðzdÞ ix ð2:8Þ Ho ¼ ffiffiffiffi ε2 μ r Eoejβ2ðzdÞ iy ð2:9Þ In order to meet the boundary conditions at z = 0 and at z = d, it is required that Ei þ REi ¼ Ef þ Eb ; 1 Z1 ðEi REiÞ ¼ 1 Zt ðEf Eb Þ ð2:10Þ Ef ejβtd þ Eb eþjβtd ¼ Eo; 1 Zt ðEf ejβtd Eb eþjβtd Þ ¼ 1 Z2 Eo ð2:11Þ Zt ¼ ffiffiffiffi μ εt r ; Z1 ¼ ffiffiffiffi μ ε1 r ; Z2 ¼ ffiffiffiffi μ ε2 r ; Zt Z1 ¼ n1 nt ; Z2 Zt ¼ nt n2 ð2:12Þ The solution of Ef and Eb in Eq. (2.11) is: Eb Ef ej2βtd ¼ 1 n2 nt 1 þ n2 nt ; or Eb Ef ¼ ej2βtd Γt2 ð2:13Þ Γt2 ¼ 1 n2 nt 1 þ n2 nt ¼ Γ2t; Tt2 ¼ 2 1 þ n2 nt ; Γt2 þ Tt2 ¼ 1 ð2:14Þ From Eq. (2.10), R ¼ Γ1t þ Γt2ej2βtd 1 þ Γ1tΓt2ej2βtd ð2:15Þ Γ1t ¼ 1 nt n1 1 þ nt n1 ¼ Γt1; T1t ¼ 2 1 þ nt n1 ; Γ1t þ T1t ¼ 1 ð2:16Þ Eo Ei ¼ T1tTt2 1 þ Γ1tΓt2ej2βtd ejβtd ð2:17Þ If we choose nt and d such that nt ¼ ffiffiffiffiffiffiffiffiffi n1n2 p and ej2βtd ¼ 1 ð2:18Þ Then nt n1 ¼ n2 nt ¼ ffiffiffiffiffi n2 n1 r ; R ¼ 0; Eo ¼ jEi ð2:19Þ In this manner, we have obtained an anti-reflection coating that has no reflection in medium #1 and 100% transmission into medium #2 at a specific wavelength, at which d 36 Superposition of plane waves and applications
- 53. is equal to ¼ of the wavelength. Identical results are obtained when the electric field is polarized in the y–z plane. Note that for a given d the anti-reflection effect is wavelength sensitive. As the wavelength deviates, the reflection will increase and the transmission will decrease. Thus there is an effective wavelength range of the anti-reflection coating. In reality, there may not be a coating material that has exactly the required nt. The wavelength range within which anti-reflection, reflection, or beam splitting is required for different applications may also need to be decreased or increased. Therefore, multi- ple layer coatings are used in most commercial devices. However, the basic principle is demonstrated by the above example. In a similar manner, coatings can be applied to enhance reflection or to split the incident beam into desired ratios of reflected and transmitted beams. Beam splitters can also be designed for beams incident at specific incident angles. Note that the analysis presented here is similar to impedance transformation analysis of microwave transmission lines. The differential equation for E and H is identical to that for V and I of microwave transmission lines [1]. In microwaves, anti-reflection is called impedance matching. Many of the analytical techniques developed for microwaves are also very useful for optical analysis, especially when we need to analyze multi-layer transitions. It is important for optics engineers to understand transmission line methods. However, a detailed discussion of that is beyond the scope of this book. 2.2 Fabry–Perot resonance 2.2.1 Multiple reflections and Fabry–Perot resonance Although plane waves propagating between two boundaries have already been analyzed in the previous section by matching the total fields at the boundaries, an alternate way to analyze it is to consider an incident plane wave multiply reflected and transmitted at the two boundaries. Much more physical insight on resonance could be gained by present- ing this alternate approach. Let us first consider a plane wave incident on the first boundary at z = 0, without considering the second boundary at z = d. This incident wave Ei would excite a reflected backward wave Er1 in medium #1 and a transmitted forward wave Ef 1 in the transition medium. Let the boundary at z = 0 have a reflection coefficient Γ1t and transmission coefficient T1t for the incident wave. The x-y-z variations without showing the time variation ejωt are: Ef 1 ¼ T1tEiejβtz ix for d z 0; Er1 ¼ Γ1tEieþjβ1z ix for z 0 ð2:20Þ This boundary will have reflection coefficient Γt1 and transmission coefficient Tt1 for any plane wave incident on it in the reverse direction from the transition medium. As Ef 1 propagates to z = d, it excites a reflected wave Eb 1 in the transition medium and a transmitted wave Eo1 in medium #2. Let the boundary at z = d have reflection 2.2 Fabry–Perot resonance 37
- 54. coefficient Γt2 and transmission coefficient Tt2 for any forward wave propagating in the transition medium. Then we obtain: Eo1 ¼ Tt2ðT1tEiejβtd Þejβ2ðzdÞ ix for z d ð2:21Þ Eb 1 ¼ Γt2ðT1tEiejβtd ÞeþjβtðzdÞ ix for 0 z d ð2:22Þ The reflected wave Eb 1 propagates back to z = 0 and excites another transmitted backward wave Er2 in medium #1 and a reflected forward wave Ef 2 in the transition medium. Ef 2 ¼ Γt1Γt2ðT1tEiejβtd Þejβtd ejβtz ix for 0 z d ð2:23Þ Eb 2 ¼ Γt1Γt2 2 ðT1tEiejβtd Þe2jβtd e þ jβtðzdÞ ix for 0 z d ð2:24Þ Er2 ¼ Tt1Γt2ðT1tEiejβtd Þejβtd ejβ1z ix ð2:25Þ As Ef 2 reaches z = d, it excites a forward Eo2 in the transition medium and an Eb 2 in medium #2. Eo2 ¼ Tt2ðΓt1Γt2T1tEiej3βtd Þejβ2ðzdÞ ix for z d ð2:26Þ As Eb 2 reaches z = 0, it excites a forward Ef 3 in the transition region and an Er2 in medium #1. Er3 ¼ Tt1Γt1Γt2 2 ðT1tEiejβtd Þe3jβtd e þ jβ1z ix for 0 z ð2:27Þ Consequentially, the forward and the backward waves in the transition medium continue to generate backward reflected waves at z 0 and transmitted output waves at z d. The amplitudes of the total forward and backward propagating waves are related to the incident wave by: Eo Ei ¼ T1tTt2ejβtd ½1 þ Γt1Γt2ej2βtd þ Γt1 2 Γt2 2 ej4βtd þ . . . ¼ T1tTt2ejβtd 1 Γt1Γt2ej2βtd ð2:28Þ for z d, Er Ei ¼ Γ1t þ Γt2T1tTt1ej2βtd 1 1 Γt1Γt2ej2βtd ¼ Γ1t þ Γt2ej2βtd 1 Γt1Γt2ej2βtd ð2:29Þ for z 0, and Ef Ei ¼ T1t 1 1 Γt1Γt2ej2βtd ; Eb Ei ¼ T1tΓt2 ej2βtd 1 Γt1Γt2ej2βtd ð2:30Þ for 0 z d. When the Γs and Ts in Eqs. (2.14) and (2.16) in the previous section are used in Eqs. (2.28) and (2.29) the solutions of Eo and Er become identical to the results in Eqs. (2.15) and (2.17) of the previous section. However, the above results are more general. They 38 Superposition of plane waves and applications
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- 56. and to pray with us for our own poor lives. Mr. Royle, she said, Cornish will stay. And with an expression on her face of infinite sweetness and pathos, she drew him to one of the cushioned lockers and seated herself by his side. I saw that her charming wonderful grace, her cordial tender voice, and her condescension, which a man of his condition would feel, had deeply moved him. The steward seated himself on the other side of her, and I began to read from the open book before me, beginning the chapter which she had chosen for us during my absence on deck. This chapter was the eleventh of St. John, wherein is related the story of that sickness which was not unto death, but for the glory of God, that the Son of God might be glorified thereby. I read only to the thirty-sixth verse, for what followed that did not closely apply to our position; but there were passages preceding it which stirred me to the centre of my heart, knowing how they went home to the mourner, more especially those pregnant lines—Martha saith unto Him, I know that he shall rise again in the resurrection at the last day. Jesus said unto her, I am the resurrection and the life: he that believeth in Me, though he were dead, yet shall he live, which made me feel that the words I had formerly addressed to her were not wholly idle. I then turned to St. Matthew, and read from the eighth chapter those few verses wherein it is told that Christ entered a ship with His disciples, and that there arose a great storm. Only men in a tempest at sea, their lives in jeopardy, and worn out with anxiety and the fear of death, know how great is the comfort to be got out of this brief story of our Lord's power over the elements, and His love of those whom He died to save; and, taking this as a kind of text, I knelt down, the others imitating me, and prayed that He who rebuked the sea and the wind before His doubting disciples, would be with us who believed in Him in our present danger.
- 57. Many things I said (feeling that He whom I addressed was our Father, and that He alone could save us) which have gone from my mind, and tears stood in my eyes as I prayed; but I was not ashamed to let the others see them, even if they had not been as greatly affected as I, which was not the case. Nor would I conclude my prayer without entreating God to comfort the heart of the mourner, and to receive in heaven the soul of him for whom she was weeping. I then shook Cornish and the steward heartily by the hand, and I am sure, by the expression in Cornish's face, that he was glad he had stayed, and that his kneeling in prayer had done him good. Now, said I, you had best get your dinner, and relieve the boatswain; and you, steward, obtain what food you can, and bring it to us here, and then you and the bo'sun can dine together. The two men left the cabin, and I went and seated myself beside Miss Robertson, and said all that I could to comfort her. She was very grateful to me for my prayers for herself and her father, and already, as though she had drawn support from our little service, spoke with some degree of calmness of his death. It would have made her happy, she said, could she have kissed him before he died, and have been awake to attend to any last want. I told her that I believed he had died in his sleep, without a struggle; for, so recent as his death was, less placidity would have appeared in his face had he died awake or conscious. I added that secretly I had never believed he would live to reach Valparaiso, had the ship continued her voyage. He was too old a man to suffer and survive the physical and mental trials he had passed through; and sad though his death was under the circumstances which surrounded it, yet she must think that it had only been hastened a little; for he was already an old man, and his end might have been near, even had all prospered and he had reached England in his own ship.
- 58. By degrees I drew her mind away from the subject by leading her thoughts to our own critical position. At another time I should have softened my account of our danger: but I thought it best to speak plainly, as the sense of the insecurity of our lives would in some measure distract her thoughts from her father's death. She asked me if the storm was not abating. It is not increasing in violence, I answered, which is a good sign. But there is one danger to be feared which must very shortly take me on deck. The wind may suddenly lull and blow again hard from another quarter. This would be the worst thing that could happen to us, for we should then have what is called a cross sea, and the ship is so deeply loaded that we might have great difficulty in keeping her afloat. May I go on deck with you? You would not be able to stand. Feel this! I exclaimed, as the ship's stern rose to a sickening height and then came down, down, down, with the water roaring about her as high as our ears. Let me go with you! she pleaded. Very well, I replied, meaning to keep her under the companion, half-way up the ladder. I took a big top-coat belonging to the captain and buttoned her up in it, and also tied his fur cap over her head, so that she would be well protected from the wind, whilst the coat would keep her dress close against her. I then slipped on my oilskins, and taking a strong grip of her hand to steady her, led her up the companion ladder. Do not come any farther, said I. Wherever you go I will go, she answered, grasping my arm. Admiring her courage and stirred by her words, which were as dear to me as a kiss from her lips would have been, I led her right
- 59. on to the deck over to windward, and made her sit on a small coil of rope just under the rail. The sea was no heavier than it had been since the early morning, and yet my short absence below had transformed it into a sublime and stupendous novelty. You will remember that not only was the Grosvenor a small ship, but that she lay deep, with a free board lower by a foot and a half than she ought to have shown. The height from the poop rail to the water was not above twelve feet; and it is therefore no exaggeration to say that the sea, running from fifteen to twenty feet high, stood like walls on either side of her. To appreciate the effect of such a sea upon a ship like the Grosvenor, you must have crossed the Atlantic in a hurricane, not in an immense and powerful ocean steamer, but in a yacht. But even this experience would not enable you to realise our danger; for the yacht would not be overloaded with cargo, she would probably be strong, supple, and light; whereas the Grosvenor was choked to the height of the hold with seven hundred and fifty tons of dead weight, and was a Nova Scotia soft wood ship, which means that she might start a butt at any moment and go to pieces in one of her frightful swoops downwards. Having lodged Miss Robertson in a secure and sheltered place, I crawled along the poop on to the main-deck and sounded the well again. I found a trifle over six inches of water in her, which satisfied me that she was still perfectly tight, and that the extra leakage was owing to the drainings from the decks. I regained the poop and communicated the good news to the boatswain, who nodded; but I noticed that there was more anxiety in his face than I liked to see, and that he watched the ship very closely each time she pitched with extra heaviness.
- 60. Miss Robertson was looking up at the masts with alarmed eyes; but I pointed to them and smiled, and shook my head to let her know that their wrecked appearance need not frighten her. I then took the telescope, and, making it fast over my back, clambered into the mizzen-top, she watching my ascent with her hands tightly clasped. The ensign still roared some half-dozen feet below the gaff-end; it was a brave bit of bunting to hold on as it did. I planted myself firmly against the rigging, and carefully swept the weather horizon, and finding nothing there, pointed the glass to leeward; but all that part of the sea was likewise a waste of foaming waves, with never a sign of a ship in all the raging seas. I was greatly disappointed, for though no ship could have helped us in such a sea, yet the sight of one hove to near us—and no ship afloat, sailer or steamer, but must have hove to in that gale—would have comforted us greatly, as a promise of help at hand, and rescue to come when the wind should have gone down.
- 61. CHAPTER V. All that day the wind continued to blow with frightful force, and the sky to wear its menacing aspect. On looking, however, at the barometer at four o'clock in the afternoon, I observed a distinct rise in the mercury; but I did not dare to feel elated by this promise of an improvement; for, as I have before said, the only thing the mercury foretells is a change of weather, but what kind of change you shall never be sure of until it comes. What I most dreaded was the veering of the gale to an opposite quarter, whereby, a new sea being set running right athwart, or in the eye of the already raging sea, our decks would be helplessly swept and the ship grow unmanageable. A little after eight the wind sensibly decreased, and, to my great delight, the sky cleared in the direction whence the gale was blowing, so that there was a prospect of the sea subsiding before the wind shifted, that is, if it shifted at all. When Cornish, who had been below resting after a long spell, came on deck and saw the stars shining, and that the gale was moderating, he stared upwards like one spell-bound, and then, running up to me, seized my hand and wrung it in silence. I heartily returned this mute congratulation, and we both went over and shook hands with the boatswain; and those who can appreciate the dangers of the frightful storm that had been roaring
- 62. about us all day, and feel with us in the sentiments of despair and helplessness which the peril we stood in awoke in us, will understand the significance of our passionate silence as we held each other's hand and looked upon the bright stars, which shone like the blessing of God upon our forlorn state. I was eager to show Mary Robertson those glorious harbingers, and ran below to bring her on deck. I found her again in the cabin in which her father lay, bending over his body in prayer. I waited until she turned her head, and then exclaimed that the wind was falling, and that all the sky in the north- west was bright with stars, and begged her to follow me and see them. She came immediately, and, after looking around her, cried out in a rapturous voice— Oh, Mr. Royle! God has heard our prayers! and, in the wildness of her emotions, burst into a flood of tears. I held her hand as I answered— It was your grief that moved me to pray to Him, and I consider you our guardian angel on board this ship, and that God who loves you will spare our lives for your sake. No, no; do not say so; I am not worthier than you—not worthier than the brave boatswain and Cornish, whose repentance would do honour to the noblest heart. Oh, if my poor father had but been spared to me! She turned her pale face and soft and swimming eyes up to the stars and gazed at them intently, as though she witnessed a vision there. But though the wind had abated, it still blew a gale, and the sea boiled and tumbled about us and over our decks in a manner that would have been terrifying had we not seen it in a greater state of fury.
- 63. I sent the steward forward to see if he could get the galley fire to burn, so as to boil us some water for coffee, for though the ship was in a warm latitude, yet the wind, owing to its strength, was at times piercingly cold, and we all longed for a hot drink—a cup of hot coffee or cocoa being infinitely more invigorating, grateful, and warming than any kind of spirits drunk cold. All that the steward did, however, was to get wet through; and this he managed so effectually that he came crawling aft, looking precisely as if he had been fished out of the water with grappling- hooks. I lighted a bull's-eye lamp and went to the pumps and sounded the well. On hauling up the rod I found to my consternation that there were nine inches of water in the ship. I was so much startled by this discovery that I stood for some moments motionless; then, bethinking me that one of the plugged auger holes might be leaking, I slipped forward without saying a word to the others, and, getting a large mallet from the tool-chest, I entered the forecastle, so as to get into the fore peak. I had not been in the forecastle since the men had left the ship, and I cannot describe the effect produced upon me by this dark deserted abode, with its row of idly swinging hammocks glimmering in the light shed by the bull's-eye lamp; the black chests of the seamen which they had left behind them; here and there a suit of dark oilskins suspended by a nail and looking like a hanged man; the hollow space resonant with the booming thunder of the seas and the mighty wash of water swirling over the top-gallant deck. The whole scene took a peculiarly ghastly significance from the knowledge that of all the men who had occupied those hammocks and bunks, one only survived; for four of them we ourselves had killed, and I could not suppose that the long-boat had lived ten minutes after the gale had broken upon her.
- 64. I made my way over the cable-ranges, stooping my head to clear the hammocks, and striking my shins against the sea-chests, and swung myself into the hold. Here I found myself against the water casks, close against the cargo, and just beyond was the bulk-head behind which the boatswain had hidden while Stevens bored the holes. Carefully throwing the light over the walls, I presently perceived the plugs or ends of the broom-stick protruding; and going close to them I found they were perfectly tight, that no sign of moisture was visible around them. It may seem strange that this discovery vexed and alarmed me. And yet this was the case. It would have made me perfectly easy in my mind to have seen the water gushing in through one of these holes, because not only would a few blows of the mallet have set it to rights, but it would have acquainted me with the cause of the small increase of water in the hold. Now that cause must be sought elsewhere. Was it possible that the apprehensions I had felt each time the ship had taken one of her tremendous headers were to be realised? —that she had strained a butt or started a bolt in some ungetatable place? Here where I stood, deep in the ship, below the water-line, it was frightful to hear her straining, it was frightful to feel her motion. The whole place resounded with groans and cries, as if the hold had been filled with wounded men. What bolts, though forged by a Cyclops, could resist that horrible grinding?—could hold together the immense weight which the sea threw up as a child a ball, leaving parts of it poised in air, out of water, unsustained save by the structure that contained it, then letting the whole hull fall with a hollow, horrible crash into a chasm
- 65. between the waves, beating it first here, then there, with blows the force of which was to be calculated in hundreds of tons? I scrambled up through the fore scuttle, and perceiving Cornish smoking a pipe under the break of the poop, I desired he would go and relieve the boatswain at the wheel for a short while and send him to me, as I had something particular to say to him. I waited until the boatswain came, as here was the best place I could choose to conduct a conversation. Beyond all question the wind was falling, and though the ship still rolled terribly, she was not taking in nearly so much water over her sides. I re-trimmed the lamp in my hand, and in a few minutes the boatswain joined me. I said to him at once— I have just made nine inches of water in the hold. When was that? he inquired. Ten minutes ago. When you sounded the well before what did you find? Between five and six inches. I'll tell you what it is, sir, said he. You'll hexcuse me sayin' of it, but it's no easy job to get at the true depth of water in a ship's bottom when she's tumblin' about like this here. I think I got correct soundings. Suppose, he continued, you drop the rod when she's on her beam ends. Where's the water? Why, the water lies all on one side, and the rod 'll come up pretty near dry. I waited until the ship was level. Ah, you did, because you knows your work. But it's astonishin' what few persons there are as really does know how to sound the
- 66. pumps. You'll hexcuse me, sir, but I should like to drop the rod myself. Certainly, I replied, and I hope you'll make it less than I. In order to render my description clear to readers not acquainted with such details, I may state that in most large ships there is a pipe that leads from the upper deck, alongside the pumps, down to the bottom or within a few inches of the bottom of the vessel. The water in the hold necessarily rises to the height of its own level in this pipe; and in order to gauge the depth of water, a dry rod of iron, usually graduated in feet and inches, is attached to the end of a line and dropped down the tube, and when drawn up the depth of water is ascertained by the height of the water on the rod. It is not too much to say that no method for determining this essential point in a ship's safety could well be more susceptible of inaccuracy than this. The immersed rod, on being withdrawn from the tube, wets the sides of the tube; hence, though the rod be dry when it is dropped a second time, it is wetted in its passage down the tube; and as the accuracy of its indication is dependent on its exhibiting the mark of the level of water, it is manifest that if it becomes wetted before reaching the water, the result it shows on being withdrawn must be erroneous. Secondly, as the boatswain remarked to me, if the well be sounded at any moment when the vessel is inclined at any angle on one side or the other, the water must necessarily roll to the side to which the vessel inclines, by which the height of the water in the well is depressed, so that the rod will not report the true depth. Hence, to use the sounding-rod properly, one must not only possess good sense, but exercise very great judgment. I held the lamp close to the sounding-pipe, and the boatswain carefully dried the rod on his coat preparatory to dropping it.
- 67. He then let it fall some distance down the tube, keeping it, however, well above the bottom, until the ship, midway in a roll, stood for a moment on a level keel. He instantly dropped the rod, and hauling it up quickly, remarked that we had got the true soundings this time. He held the rod to the light, and I found it a fraction over nine inches. That's what it is, anyways, said he, putting down the rod. An increase of three inches since the afternoon. Well, there's nothen to alarm us in that, is there, Mr. Royle? he exclaimed. Perhaps its one o' my plugs as wants hammerin'. No, they're as tight as a new kettle, I answered. I have just come from examining them. Well, all we've got to do is to pump the ship out; and, if we can, make the pumps suck all right. That 'll show us if anything's wrong. This was just the proposition I was about to make; so I went into the cuddy and sang out for the steward, but he was so long answering that I lost my temper and ran into the pantry, where I found him shamming to be asleep. I started him on to his legs and had him on the main-deck in less time than he could have asked what the matter was. Look here! I cried, if you don't turn to and help us all to save our lives, I'll just send you adrift in that quarter-boat with the planks out of her bottom! What do you mean by pretending to be asleep when I sing out to you? And after abusing him for some time to let him know that I would have no skulking, and that if his life were worth having he must save it himself, for we were not going to do his work and our own as well, I bid him lay hold of one of the pump-handles, and we all three of us set to work to pump the ship.
- 68. If this were not the heaviest job we had yet performed, it was the most tiring; but we plied our arms steadily and perseveringly, taking every now and then a spell of rest, and shifting our posts so as to vary our postures; and after pumping I scarcely know how long, the pumps sucked, whereat the boatswain and I cheered heartily. Now, sir, said the boatswain, as we entered the cuddy to refresh ourselves with a drain of brandy and water after our heavy exertions, we know that the ship's dry, leastways, starting from the ship's bottom; if the well's sounded agin at half-past ten—its now half-past nine—that 'll be time enough to find out if anything's gone wrong. How about the watches? We're all adrift again. Here's Cornish at the wheel, and its your watch on deck. As I said this, Miss Robertson came out of the cabin where her father lay—do what I might I could not induce her to keep away from the old man's body—and approaching us slowly asked why we had been pumping. Why, ma'm, replied the boatswain, it's always usual to pump the water out o' wessels. On dry ships it's done sometimes in the mornin' watch, and t'others they pumps in the first dog watch. All accordin'. Some wessels as they calls colliers require pumpin' all day long; and the Heagle, which was the fust wessel as I went to sea in, warn't the only Geordie as required pumpin' not only all day long but all night long as well. Every wessel has her own custom, but it's a werry dry ship indeed as don't want pumpin' wunce a day. I was afraid, she said, when I heard the clanking of the pumps that water was coming into the ship. She looked at me earnestly, as though she believed that this was the case and that I would not frighten her by telling her so. I had learnt to interpret the language of her eyes by this time, and answered her doubts as though she had expressed them. I should tell you at once if there was any danger threatened in that way, I said. There was more water in the ship than I cared to
- 69. find in her, and so the three of us have been pumping her out. About them watches, Mr. Royle? exclaimed the boatswain. Well, begin afresh, if you like, I replied. I'll take the wheel for two hours, and then you can relieve me. Why will you not let me take my turn at the wheel? said Miss Robertson. The boatswain laughed. I have proved to you that I know how to steer. Well, that's right enough, said the boatswain. All three of you can lie down, then. I smiled and shook my head. Said the boatswain: If your arms wur as strong as your sperrit Miss, there'd be no reason why you shouldn't go turn and turn about with us. But I can hold the wheel. It 'ud fling you overboard. Listen to its kickin'. You might as well try to prewent one o' Barclay Perkins' dray hosses from bustin' into a gallop by catchin' hold o' it's tail. It 'ud be a poor look-out for us to lose you, I can tell yer. What, continued the boatswain, energetically, we want to know is that you're sleepin', and forgettin' all this here excitement in pleasing dreams. To see a lady like you knocked about by a gale o' wind is just one o' them things I have no fancy for. Mr. Royle, if I had a young and beautiful darter, and a Dook or a Barryonet worth a thousand a year, if that ain't sayin' too much, wos to propose marriage to her, an' ax her to come and be married to him in some fur-off place, wich 'ud oblige her to cross the water, blowed if I'd consent. No flesh an' blood o' mine as I had any kind o' feeling for should set foot on board ship without fust having a row with me. Make no mistake. I'm talkin' o' females, Miss. I say the sea ain't a fit place for women and gells. It does middlin' well for the likes of me and Mr. Royle here, as aren't afraid o' carryin' full-rigged
- 70. ships and other agreeable dewices in gunpowder and Hindian ink on our harms, and is seasoned, as the sayin' is, to the wexations o' the mariner's life. But when it comes to young ladies crossin' the ocean, an' I don't care wot they goes as—as passengers or skippers' wives, or stewardishes, or female hemigrants—then I say it ain't proper, and if I'd ha' been a lawyer I'd ha' made it agin the law, and contrived such a Act of Paleyment as 'ud make the gent as took his wife, darter, haunt, cousin, grandmother, female nephey, or any relations in petticoats to sea along with him, wish hisself hanged afore he paid her passage money. I was so much impressed by this vehement piece of rhetoric, delivered with many convulsions of the face, and a great deal of hand-sawing, that I could not forbear mixing him some more brandy and water, which he drank at a draught, having first wished Miss Robertson and myself long life and plenty of happiness. His declamation had quite silenced her, though I saw by her eyes that she would renew her entreaties the moment she had me alone. Then you'll go on deck, sir, and relieve Cornish, and I'll turn in? observed the boatswain. Yes. Right, said he, and was going. I added: We must sound the well again at half-past ten. Aye! aye! I shan't be able to leave the wheel, and I would rather you should sound than Cornish. I'll send the steward to rouse you. Very well, said he. And after waiting to hear if I had anything more to say, he entered his cabin, and in all probability was sound asleep two minutes after. Miss Robertson stood near the table, with her hands folded and her eyes bent down.
- 71. I was about to ask her to withdraw to her cabin and get some sleep. Mr. Royle, you are dreadfully tired and worn out, and yet you are going on deck to remain at the wheel for two hours. That is nothing. Why will you not let me take your place? Because—— Let the steward keep near that ladder there, so that I can call to him if I want you. Do you think I could rest with the knowledge you were alone on deck? You refuse because you believe I am not to be trusted, she said gently, looking down again. If your life were not dependent on the ship's safety, I should not think of her safety, but of yours. I refuse for your own sake, not for mine—no, I will not say that. For both our sakes I refuse. I have one dear hope—well, I will call it a great ambition, which I need not be ashamed to own: it is, that I may be the means of placing you on shore in England. This hope has given me half the courage with which I have fought on through danger after danger since I first brought you from the wreck. If anything should happen to you now, I feel that all the courage and strength of heart which have sustained me would go. Is that saying too much? I do not wish to exaggerate, I exclaimed, feeling the blood in my cheeks, and lamenting, without being able to control, the impulse that had forced this speech from me, and scarcely knowing whether to applaud or detest myself for my candour. She looked up at me with her frank, beautiful eyes, but on a sudden averted them from my face to the door of the cabin where her dead father lay. A look of indescribable anguish came over her, and she drew a deep, long, sobbing breath.
- 72. Without another word, I took her hand and led her to the cabin, and I knew the reason why she did not turn and speak to me was that I might not see she was weeping. But it was a time for action, and I dared not let the deep love that had come to me for her divert my thoughts from my present extremity. I summoned the steward, who tumbled out of his cabin smartly enough, and ordered him to bring his mattress and lay it alongside the companion ladder so as to be within hail. This done, I gained the poop and sent Cornish below.
- 73. CHAPTER VI. As I stood at the wheel I considered how I should act when the storm had passed. And I was justified in so speculating, because now the sky was clear right away round, and the stars large and bright, though a strong gale was still blowing and keeping the sea very heavy. Indeed, the clearness of the sky made me think that the wind would go to the eastward, but as yet there was no sign of it veering from the old quarter. We had been heading west ever since we hove to, and travelling broadside on dead south south-east. Now, if wind and sea dropped, our business would be to make sail if possible, and, with the wind holding north north-west, make an eight hours' board north-easterly, and then round and stand for Bermuda. This, of course, would depend upon the weather. It was, however, more than possible that we should be picked up very soon by some passing ship. It was not as though we were down away in the South Pacific, or knocking about in the poisonous Gulf of Guinea, or up in the North Atlantic at 60°. We were on a great ocean highway, crossed and re-crossed by English, American, Dutch, and French ships, to and from all parts of the world; and bad indeed would our fortune be, and baleful the star under which we
- 74. sailed, if we were not overhauled in a short time and assistance rendered us. A great though unexpressed ambition of mine was to save the ship and navigate her myself, not necessarily to England, but to some port whence I could communicate with her owners and ask for instructions. As I have elsewhere admitted, I was entirely dependent on my profession, my father having been a retired army surgeon, who had died extremely poor, leaving me at the age of twelve an orphan, with no other friend in the world than the vicar of the parish we dwelt in, who generously sent me to school for two years at his own expense, and then, after sounding my inclinations, apprenticed me to the sea. Under such circumstances, therefore, it would be highly advantageous to my interests to save the ship, since my doing so would prefer some definite claims upon the attention of the owners, or perhaps excite the notice of another firm more generous in their dealings with their servants, and of a higher commercial standing. Whilst I stood dreaming in this manner at the wheel, allowing my thoughts to run on until I pictured myself the commander of a fine ship, and ending in allowing my mind to become engrossed with thoughts of Mary Robertson, whom I believed I should never see again after we had bidden each other farewell on shore, and who would soon forget the young second mate, whom destiny had thrown her with for a little time of trouble and suffering and death, I beheld a figure advance along the poop, and on its approach I perceived the boatswain. I've been sounding the well, Mr. Royle, said he. I roused up on a sudden and went and did it, as I woke up anxious; and there's bad news, sir, twelve inches o' water. Twelve inches! I cried. It's true enough. I found the bull's-eye on the cuddy table, and the rod don't tell no lies when it's properly used.
- 75. The pumps suck at four inches, don't they? Yes, sir. Then that's a rise of eight inches since half-past nine o'clock. What time is it now? Twenty minutes arter ten. We must man the pumps at once. Call Cornish. You'll find the steward on a mattress against the companion ladder. He paused a moment to look round him at the weather, and then went away. I could not doubt now that the ship was leaky, and after what we had endured, and my fond expectation of saving the vessel—and the miserable death, after all our hopes, that might be in store for us—I felt that it was very very hard on us, and I yielded to a fit of despair. What struck most home to me was that my passionate dream to save Mary Robertson might be defeated. The miseries which had been accumulated on her wrung my heart to think of. First her shipwreck, and then the peril of the mutiny, and then the dreadful storm that had held us face to face with death throughout the fearful day, and then the death of her father, and now this new horror of the ship whereon we stood filling with water beneath our feet. Yet hope—and God be praised for this mercy to all men—springs eternal, and after a few minutes my despair was mastered by reflection. If the ship made no more water than eight inches in three-quarters of an hour, it would be possible to keep her afloat for some days by regular spells at the pump, and there were four hands to work them if Miss Robertson steered whilst we pumped. In that time it would be a thousand to one if our signal of distress was not seen and answered. Presently I heard the men pumping on the main-deck, and the boatswain's voice singing to encourage the others. What courage that man had! I, who tell this story, am ashamed to think of the prominence I give to my own small actions when all the heroism
- 76. belongs to him. I know not what great writer it was who, visiting the field of the battle of Waterloo, asked how it was that the officers who fell in that fight had graves and monuments erected to them, when the soldiers—the privates by whom all the hard work was done, who showed all the courage and won the battle—lay nameless in hidden pits? And so when we send ships to discover the North Pole we have little to say about poor Jack, who loses his life by scurvy, or his toes and nose by frost-bites, who labours manfully, and who makes all the success of the expedition so far as it goes. Our shouts are for Jack's officer; we title him, we lionize him—his was all the work, all the suffering, all the anxiety, we think. I, who have been to sea, say that Jack deserves as much praise as his skipper, and perhaps a little more; and if honour is to be bestowed, let Jack have his share; and if a monument is to be raised, let poor Jack's name be written on the stone as well as the other's; for be sure that Jack could have done without the other, but also be sure that the other couldn't have done without Jack. Chained to my post, which I dared not vacate for a moment, for the ship pitched heavily, and required close watching as she came to and fell off upon the swinging seas, I grew miserably anxious to learn how the pumping progressed, and felt that, after the boatswain, my own hands would do four times the work of the other two. It was our peculiar misfortune that of the four men on board the ship three only should be capable; and that as one of the three men was constantly required at the wheel, there were but two available men to do the work. Had the steward been a sailor our difficulties would have been considerably diminished, and I bitterly deplored my want of judgment in allowing Fish and the Dutchman to be destroyed; for though I would not have trusted Johnson and Stevens, yet the other two might have been brought over to work for us, and I had no doubt that the spectacle of the perishing wretches in the long-boat, as she was whirled past us, would have produced as salutary an effect upon them as it had upon Cornish; and with two extra hands of this kind we could not only have kept
- 77. the pumps going, but have made shift to sail the ship at the same time. The hollow thrashing sounds of the pump either found Miss Robertson awake or aroused her, for soon after the pumping had commenced she came on deck, swathed in the big warm overcoat and fur cap. Such a costume for a girl must make you laugh in the description; and yet, believe me, she lost in nothing by it. The coat dwarfed her figure somewhat, but the fur cap looked luxurious against her fair hair, and nothing could detract from the exquisite femininity of her face, manner, and carriage. I speak of the impression she had made on me in the daytime; the starlight only revealed her white face now to me. Is the water still coming into the ship? she asked. The bo'sun has reported to me that eight inches deep have come into her since half-past nine. Is that much? More than we want. I don't like to trouble you with my questions, Mr. Royle; but I am very, very anxious. Of course you are; and do not suppose that you can trouble. Ask me what you will. I promise to tell you the truth. If you find that you cannot pump the water out as fast as it comes in, what will you do? Leave the ship. How? she exclaimed, looking around her. By that quarter-boat there. But it would fill with water and sink in such waves as these. These waves are not going to last, and it is quite likely that by this time to-morrow the sea will be calm.
- 78. Will the ship keep afloat until to-morrow? If the water does not come in more rapidly than it does at present, the ship will keep afloat so long as we can manage to pump her out every hour. And so, said I, laughing to encourage her, we are not going to die all at once, you see. She drew quite close to me and said— I shall never fear death while you remain on board, Mr. Royle. You have saved me from death once, and, though I may be wicked in daring to prophesy, yet I feel certain—certain, she repeated, with singular emphasis, that you will save my life again. I shall try very hard, be sure of that, I answered. I believe—no, it is not so much a belief as a strong conviction, with which my mind seems to have nothing to do, that, whatever dangers may be before us, you and I will not perish. She paused, and I saw that she was looking at me earnestly. You will not think me superstitious if I tell you that the reason of my conviction is a dream? My poor father came and stood beside me: he was so real! I stretched out my arms to him, and he took my hand and said, 'Darling, do not fear! He who has saved your life once will save it again. God will have mercy upon you and him for the prayers you offered to Him.' He stooped and kissed me and faded away, and I started up and heard the men pumping. I went to look at him, for I thought ... I thought he had really come to my side.... Oh, Mr. Royle, his spirit is with us! Though my mind was of too prosaic a turn to catch at any significance in a dream, yet there was a strange, deep, solemn tenderness in her voice and manner as she related this vision, that impressed me. It made my heart leap to hear her own sweet lips pronounce her faith in me, and my natural hopes and longings for life gathered a new light and enthusiasm from her own belief in our future salvation.
- 79. Shipwrecked persons have been saved by a dream before now, I replied, gravely. Many years ago a vessel called the Mary went ashore on some rocks to the southward of one of the Channel Islands. A few of the crew managed to gain the rocks, where they existed ten or twelve days without water or any kind of food save limpets, which only increased their thirst without relieving their hunger. A vessel bound out of Guernsey passed the rocks at a distance too far away to observe the signals of distress made by the perishing men. But the son of the captain had twice dreamed that there were persons dying on those rocks, and so importuned his father to stand close to them that the man with great reluctance consented. In this way, and by a dream, those sailors were saved. Though I do not, as a rule, believe in dreams, I believe this story to be true, and I believe in your dream. She remained silent, but the ship presently giving a sudden lurch, she put her hand on my arm to steady herself, and kept it there. Had I dared I should have bent my head and kissed the little hand. She could not know how much she made me love her by such actions as this. The boatswain has told me, she said, after a short silence, that you want to save the ship. I asked him why? Are you angry with me for being curious? Not in the least. What did he answer? He said that you thought the owners would recompense you for your fidelity, and promote you in their service. Now how could he know this? I have never spoken such thoughts to him. It would not be difficult to guess such a wish. Well, I don't know that I have any right to expect promotion or recompense of any kind from owners who send their ship to sea so badly provisioned that the men mutiny.
- 80. But if the water gains upon the ship you will not be able to save her? No, she must sink. What will you do then? Put you on shore or on board another ship, I replied, laughing at my own evasion, for I knew what she meant. Oh, of course, if we do not reach the shore we shall none of us be able to do anything, she said, dropping her head, for she stood close enough to the binnacle light to enable me to see her movements and almost catch the expression on her face. I mean what will you do when we get ashore? I must try to get another ship. To command? Oh dear no! as second mate, if they'll have me. If command of a ship were given you would you accept it? If I could, but I can't. She asked quickly, Why not? Because I have not passed an examination as master. She was silent again, and I caught myself listening eagerly to the sound of the pumping going on on the main-deck and wondering at my own levity in the face of our danger. But I could not help forgetting a very great deal when she was at my side. All at once it flashed upon me that her father owned several ships, and that her questions were preliminary to her offering me the command of one of them. I give you my honour that all recollection of who and what she was, of her station on shore, of her wealth as the old man's heiress, had as absolutely gone out of my mind as if the knowledge had never been imparted. What she was to me—what love and the wonderful association of danger and death had endeared her to me
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