OPTOMETRY –Part IV DIFFRACTION
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The document provides an extensive overview of diffraction, discussing key concepts such as Bragg's law, types of diffraction (Fraunhofer and Fresnel), and the use of diffraction gratings. It explains how light bends around obstacles, resulting in patterns of light and dark fringes, and highlights the different conditions and mathematical equations that govern these phenomena. Additionally, the differences between Fraunhofer and Fresnel diffraction are noted, with emphasis on their applications and the impact of aperture shapes on diffraction patterns.
OPTOMETRY –Part IV DIFFRACTION
- 1. OPTOMETRY – Part IV DIFFRACTION ER. FARUK BIN POYEN DEPT. OF AEIE, UIT, BU, BURDWAN, WB, INDIA FARUK.POYEN@GMAIL.COM
- 2. Contents: 1. Diffraction – Definition 2. Bragg’s Law 3. Diffraction due to Narrow Slit 4. Diffraction Grating 5. Fraunhofer Diffraction 6. Fraunhofer Geomety 7. Fresnel Diffraction 8. Fresnel Geometry 9. Dispersion 2
- 3. Diffraction of Light: Diffraction is the slight bending of light as it passes around the edge of an object. Diffracted light can produce fringes of light, dark or colored bands. Definition: The bending of light waves around the corners of an opening or obstacle and spreading of light waves into geometrical shadow is called diffraction. Types of Diffraction: Fraunhofer Diffraction Fresnel Diffraction 3
- 4. Bragg's Law: Bragg's Law: When x-rays are scattered from a crystal lattice, peaks of scattered intensity are observed which correspond to the following conditions: The angle of incidence = angle of scattering. The path length difference is equal to an integer number of wavelengths. It is important to understand crystal structure of all states of matter. First Order Diffraction: The first bright image to either side occurs when the difference in the path length of the light from neighboring slits of the grating is one wavelength, (the "first order" diffraction maximum). 4
- 5. Diffraction due to Narrow Slit: Figure shows the experimental arrangement for studying diffraction of light due to narrow slit. The slit AB of width a is illuminated by a parallel beam of monochromatic light of wavelength λ. The screen S is placed parallel to the slit for observing the effects of the diffraction of light. A small portion of the incident wave front passes through the narrow slit. Each point of this section of the wave front sends out secondary wavelets to the screen. These wavelets then interfere to produce the diffraction pattern. It becomes simple to deal with rays instead of wave fronts as shown in figure. In this figure, only seven rays have been drawn whereas actually there are a large number of them. 5
- 6. Diffraction due to Narrow Slit: Let us consider rays 1 and 5 which are in phase on in the wavefront AB. When these reach the wavefront AC, ray 5 would have a path difference ab say equal to λ/2. Thus, when these two rays reach point p on the screen; they will interfere destructively. Similarly, each pair 2 and 6, 3 and 7, 4 and 8 differ in path by λ/2 and will do the same. But the path difference 𝒂𝒃 = ( 𝒅 𝟐)𝒔𝒊𝒏𝝋 The equation for the first minimum is, then (𝒅 𝟐)𝒔𝒊𝒏𝝋 = 𝝀 𝟐 In general, the conditions for different orders of minima on either side of centre are given by: 𝑑𝑠𝑖𝑛𝜑 = 𝑚λ; 𝑚 = ±(1,2,3 … … ) The region between any two consecutive minima both above and below O will be bright. A narrow slit, therefore, produces a series of bright and dark regions with the first bright region at the center of the pattern. 6
- 7. Diffraction Grating: When there is a need to separate light of different wavelengths with high resolution, then a diffraction grating is most often the tool of choice. A large number of parallel, closely spaced slits constitutes a diffraction grating. The condition for maximum intensity is the same as that for the double slit or multiple slits, but with a large number of slits the intensity maximum is very sharp and narrow, providing the high resolution for spectroscopic applications. The peak intensities are also much higher for the grating than for the double slit. Thus a diffraction grating is the tool of choice for separating the colors in incident light. Resolving Power of Grating: 𝑅 = λ ∆λ 7
- 8. Diffraction Grating: The resolvance of such a grating depends upon how many slits are actually covered by the incident light source; i.e., if more slits are covered, a higher resolution in the projected spectrum is obtained. 8
- 9. Diffraction Grating – Double Slit Interface: The grating intensity expression gives a peak intensity which is proportional to the square of the number of slits illuminated. Increasing the number of slits not only makes the diffraction maximum sharper, but also much more intense. The intensity is given by the interference intensity expression 9
- 10. Difference - Fresnel Number: Fresnel Number: A parameter (dimensionless number) determining the regime of diffraction effects. Fresnel number F is defined as 𝑭 = 𝒂 𝟐 𝑳𝝀 a = characteristic size (e.g. radius) of the aperture; L = distance of the screen from the aperture; λ = incident wavelength; Fruanhofer Diffraction: 𝐹 ≪ 1 Fresnel Diffraction: 𝐹 ≤ 1 10
- 11. Rayleigh Criterion : Rayleigh Criterion: The generally accepted criterion for the minimum resolvable detail - the imaging process is said to be diffraction-limited when the first diffraction minimum of the image of one source point coincides with the maximum of another. 11
- 12. Difference: Fresnel & Fraunhofer Diffraction: In Fraunhofer diffraction: 𝐹 ≪ 1 Source and the screen are far away from each other. Incident wave fronts on the diffracting obstacle are plane. Diffraction obstacle give rise to wave fronts which are also plane. Plane diffracting wave fronts are converged by means of a convex lens to produce diffraction pattern. In Fresnel diffraction: 𝐹 ≤ 1 Source and screen are not far away from each other. Incident wave fronts are spherical. Wave fronts leaving the obstacles are also spherical. Convex lens is not needed to converge the spherical wave fronts. 12
- 13. Fraunhofer Diffraction: Fraunhofer diffraction is the special case where the incoming light is assumed to be parallel and the image plane is assumed to be at a very large distance compared to the diffracting object. Fraunhofer diffraction deals with the limiting cases where the light approaching the diffracting object is parallel and monochromatic, and where the image plane is at a distance large compared to the size of the diffracting object. 13
- 14. Fraunhofer Diffraction: In Fraunhofer diffraction, the diffraction pattern is independent of the distance to the screen, depending only on the angles to the screen from the aperture. It is used when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens. 14
- 15. Fraunhofer Diffraction – Single Slit: This is an attempt to more clearly visualize the nature of single slit diffraction. The phenomenon of diffraction involves the spreading out of waves past openings which are on the order of the wavelength of the wave. The spreading of the waves into the area of the geometrical shadow can be modeled by considering small elements of the wavefront in the slit and treating them like point sources. One of the characteristics of single slit diffraction is that a narrower slit will give a wider spectrum. 15
- 16. Fraunhofer Diffraction – Single Slit: If light from symmetric elements near each edge of the slit travels to the centerline of the slit, as indicated by rays 1 and 2, their light arrives in phase and experiences constructive interference. The first minimum in intensity for the light through a single slit can be visualized in terms of rays 3 and 4. The condition for minimum light intensity is that light from these two elements arrive 180° out of phase. 16
- 17. Fraunhofer Double Slit Diffraction: The pattern formed by the interference and diffraction of coherent light is distinctly different for a single and double slit. The single slit intensity envelope is shown by the dashed line and that of the double slit for a particular wavelength and slit width is shown by the solid line. The number of bright maxima within the central maximum of the single-slit pattern is influenced by the width of the slit and the separation of the double slits. 17
- 18. Fraunhofer Circular Aperture Diffraction: When light from a point source passes through a small circular aperture, it does not produce a bright dot as an image, but rather a diffuse circular disc known as Airy's disc surrounded by much fainter concentric circular rings. This example of diffraction is of great importance because the eye and many optical instruments have circular apertures. If this smearing of the image of the point source is larger than that produced by the aberrations of the system, the imaging process is said to be diffraction-limited, and that is the best that can be done with that size aperture. This limitation on the resolution of images is quantified in terms of the Rayleigh criterion so that the limiting resolution of a system can be calculated. 18
- 19. Fresnel Diffraction: Fresnel diffraction refers to the general case where those restrictions are relaxed. This makes it much more complex mathematically. Some cases can be treated in a reasonable empirical and graphical manner to explain some observed phenomena. Also called Near-Field Diffraction. It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed from relatively close to the object. 19
- 20. Fresnel Geometry: For the Fresnel case, all length parameters are allowed to take comparable values, so all must be included as variables in the problem. The usual geometry assumes a monochromatic slit source and the problem is set up in terms of a parameter v. This parameter is used with the Cornu spiral or a table of elliptical integrals. To calculate the intensity at point P, the geometry is set up in terms of the parameter v which is used with the Cornu spiral. 20
- 21. Fresnel Diffraction – v Parameter: The v-parameter in Fresnel diffraction analysis can be thought of as the arclength along the amplitude vector diagram called the Cornu spiral. In the Fresnel diffraction case where the curvature of the wavefront is included, the relative phase is not constant and the amplitude elements bend into the spiral curve. 21
- 22. Fresnel Diffraction - Single Slit: The more accurate Fresnel treatment of the single slit gives a pattern which is similar in appearance to that of the Fraunhofer single slit except that the minima are not exactly zero. 22
- 23. Fresnel Diffraction – Opaque Barrier: The diffraction pattern produced by monochromatic light and an opaque edge includes light which penetrates into the geometric shadow and an alternating pattern of bright and dark fringes outside the shadow. 23
- 24. Fresnel Diffraction – Opaque Barrier: An opaque circular disk gives a concentric ring diffraction pattern similar to the circular aperture, but in addition it has a bright spot in the center referred to as either Poisson's spot or Fresnel's spot. 24
- 25. References: 1. http://www.andor.com/learning-academy/radiometry-and-photometry-an-overview-of- the-science-of-measuring-light 2. https://www.emedicalprep.com/study-material/physics/wave-optics/interference-light- waves-youngs-experiment/ 3. https://physicsabout.com/diffraction-of-light/ 4. https://physicsabout.com/polarization-of-light/ 5. http://www.physicsclassroom.com/class/light/Lesson-1/Polarization 6. http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html 25