Chapter 8 Diffraction (1) Fraunhofer diffraction A coherent line source Diffraction by single slit, double slit and many slits Rectangular aperature Circular aperature Diffraction grating (2) Fresnel diffraction Circular obstacles Diffraction by a slit, narrow obstacle (3) Kirchhoff’s scalar diffraction theory

Fraunhofer Diffraction

Huygens-Fresnel Principle Every unobstructed point of a wavefront, at a given instant, serves as a source of spherical secondary wavelets (with the same frequency as that of the primary wave). The amplitude of the optical field at any point beyond is the superposition of all these wavelets (considering their amplitudes and relative phases).

Fraunhofer & Fresnel Diffraction (Far-field diffraction) Fraunhofer diffraction Aperature size Source (observation) distance Wavelength  Phase (Near-field diffraction) Fresnel diffraction

phase (SA) – phase (SB) < /4 Aspect of Phase for Fraunhofer Diffraction A B S a R R phase (SA) – phase (SB) < /4  R > a2 / 

A Coherent Line Source

Intensity Profile of a Coherent Line Source (1) When D >> ,  is large  I (  0)  0 A long coherent line source (D >> ) can be treated as a single-point emitter radiating (a circular wave) predominantly in the forward,  = 0, direction. (2) When D << ,  is small  I()  I(0) A point source emitting sphereical waves.

Fraunhofer Diffraction by a Single Slit z

Fraunhofer Diffraction Pattern of a Single Slit

Fraunhofer Diffraction b Fraunhofer Diffraction by Double Slits a

Fraunhofer Diffraction Pattern of Double Slits (Young’s experiment) (max) kasin = m

Fraunhofer Diffraction by Many Slits

Fraunhofer Diffraction by Many Slits (Principal maxima)

Experimental and Theoritical Results Comparison between Experimental and Theoritical Results a=4b, N=4

Fraunhofer Diffraction by Rectangular Aperture Assimption: coherent secondary point sources within the aperture. Source point (0, y, z), Observation point (X, Y, Z)

Fraunhofer Pattern of a Square Aperture

Matlab Demonstration

Fraunhofer Diffraction by Circular Aperture

Bessel Function of the First Kind

Diffracted Irradiance of a Circular Aperture

Airy Disk & Airy Rings 84%

Resolution of Imaging Systems Just resolved when the center of one Airy disk falls on the first minimum of the Airy pattern of the other. Angular limit of resolution Limit of resolution

An Interesting Experiment on Image Resolution

Revision of Fraunhofer Diffraction Fraunhofer Difraction functions as “Fourier Transform” from geomety (space) domain to wave-vector (wave-number) domain.  Uncertainty principle

The Diffraction Grating Reflection phase grating Transmission phase grating (Grating equation)

Grating Spectroscopy (I) (Angular width of a line) (Angular dispersion)

Grating Spectroscopy (II) (limit of resolution) (F-P spectroscopy) (free spectral range)

Two- and Three-Dimensional Gratings Why don’t use visible lights for diffraction of solid crystals? X-ray difraction pattern for SiO2

Fresnel Diffraction

Fresnel Zones Obliquity factor Inclination factor Half-period zones

Propagation of a Spherical Wavefront

Optical Disturbance from Fresnel Zones (I) If m is odd, If m is even,

Optical Disturbance from Fresnel Zones (II) The last contributing zone occurred at  = 90o i.e. K() = 0 for /2  | |   and Km(/2) = 0 (Primary wave, SP) (Secondary wavelets)

The Vibration Curve (I) The resultant phasor changes in phase by  while the aperture size increases by one zone. K() decreases rapidly only for the first few zones.

The Vibration Curve (II)

Circular Apertures Case I: m is even, E  0 Case II: m is odd, E  |E1|

Drffraction Patterns for Circular Apertures of Increasing Size The intensity at the center The intensity at the off-axis position

Method for Evaluating the Intensity at Off-Axis Positions

Radius of Aperture & Number of Zones >> Number of zones  Examples:  = 1 m, r0 = 1 m,  = 500 nm and R = 1 cm  400 zones If  and r0 are very large, such that only a small fraction of the first zone appears in the aperture, Fraunhofer diffraction occurs.

Circular Obstacles As P moves close to the disk,  increases, Kl+10, so E0

Fresnel Zone Plate

Rectangular Apertures (I)

Rectangular Apertures (II) Example: u1 = u2   and v1 = v2    E2 = I0

Diffraction Patterns for Increasing Square Apertures

The Cornu Spiral A larger size of aperture corresponds to a larger arc length w. For checking the E-field at off-axis positions, move two end points with constant w.

Fresnel Diffraction by a Slit

Fresnel Diffraction Pattern of a Slit

The Semi-Infinite Opaque Screen

Fresnel Diffraction Pattern of a Semi-Infinite Opaque Screen

Fresnel Diffraction by a Narrow Obstacle

Babinet’s Principle When the transparent regions on one diffraction screen exactly correspond to the opaque regions of the other and vice versa, these two screen are complementary.

Diffraction Patterns of Complementary Screens

Kirchhoff’s Scalar Diffraction Theory (I) Fresnel-Kirchhoff diffraction formula

Kirchhoff’s Scalar Diffraction Theory (II)

Homework: 11, 25, 28, 52