4 Wave Optics Spherical Wave Image Formation and

4. Wave Optics

Spherical Wave, Image Formation, and Huygens’ Principle Wavefront: a surface over which the phase of a wave is constant Huygens’ Principle

Linear Polarization

Circular/Elliptical Polarization

Unpolarized Light and Polarizer

Liquid Crystal Display (LCD)

3 D Imaging by Polarizers

Reflection and Transmittance of Polarized Lights Fresnel equations: Note: p-polarization: E-field plane of incidence s-polarization: E-field plane of incidence

Goos-Haenchen Shift

Optical Transfer Matrix to Analyze Three-layer Film

Optical Transfer Matrix to Analyze Three-layer Film (Cont’)

Antireflection Film Antireflection Coatings on Solar Cells

High-reflectance Film

High-reflectance Film (Cont’)

Interference Young’s Experiment Interference — superposition of two light wave result in bright and dark fringes Conditions for Interference: • same polarization • same frequency • constant phase relationship (coherence)

Conditions for Interference If 1 = 2 = Bright fringes: = 0, 2 , 4 , …(in phase) Dark fringes: = , 3 , 5 , …(out of phase)

Interferences of Coherent/Incoherent Waves • Coherence: All component electromagnetic waves are in phase or in the same phase difference. • Interference of coherent waves: Waves of different frequencies interfere to form a pulse if they are coherent. • Interference of incoherent waves: Spectrally incoherent light interferes to form continuous light with a randomly varying phase and amplitude.

Fabry-Perot Interferometer

Fabry-Perot Interferometer (Cont’)

Fabry-Perot Interferometer (Cont’) Ga. As’s natural cleavage plane is (1, 1, 0)-plane. Si’s and Ge’s natural cleavage plane are (1, 1, 1)-plane.

Mach-Zehnder Interferometer

Holography/Hologram Recording process Reconstruction process

Approximate 3 D Hologram Videos

Michelson Interferometer

Sagnac Effect and Ring Interferometer N: Fringe number

Fresnel (Near-field) Diffraction

Fraunhofer (Far-field) Diffraction

Fraunhofer Diffraction Pattern of a Rectangular Aperture

Fraunhofer Diffraction Pattern of a Circular Aperture

Resolving Power of Imaging Systems Rayleigh criterion

Resolution Limit • Rayleigh criterion two object point can be resolved by the lens of an optical system Minimum resolvable angular: D: diameter of open aperture : wavelength of light source Note: if < min, images cannot be resolved Minimum resolvable separation: For objective lens, where =h/d 1 numerical aperture NA=sin 1

Resolution of Human Eye Resolving power of human eye 0. 3 mrad Resolution limit of human eye 0. 075 mm

Fourier Transform by a Convex Lens

Optical Fourier Transform Input Plane a(x, y) Fourier Plane FTL f f A(u, v)

Optical Signal Processing

Examples of Optical Signal Processing

Examples of Optical Signal Processing (Cont’)

Fourier Optics and Its applications Optical Computing

Comparison between Phase Contrast Microscopy by Optical Signal Processing (Left) and Conventional Image Processing of Sharpening Edges (Right)

Appendix 4 -1 Coherence

Coherence Function Mutually coherent: point sources u 1(t 1, x 1, y 1, z 1) and u 2(t 1, x 2, y 2, z 2 ) maintain a fixed phase relation Mutual coherence function: Normalized mutual coherence function: (complex degree of coherence or degree of correlation) where 11( ) and 22( ) are the self-coherence functions of u 1(t) and u 2(t)

Demonstration of Coherence extended source interference pattern Visibility of fringe: If I 1 = I 2= I (best condition), = 12( ) i. e. , visibility of the fringe is a measure of the degree of coherence

Spatial Coherence extended source Intensity distribution of the resultant fringe of two points on the extended source: extended source

Measurement of Spatial Coherence

Temporal Coherence Visibility of the fringe is a measure of the degree of temporal coherence 11( ) at same point Coherence length of the light source

Measurement of Temporal Coherence

Appendix 4 -2 Fourier Transform

Fourier Transform Pairs

Basic Theorems of Fourier Transforms

Basic Theorems of Fourier Transforms (Cont’)

Application of Fourier Transform. Distinguishing Similar Signals

Appendix 4 -3 Phase Transform Function of a Lens

Usage of a Thin Lens Phase Transformation Phase transform function: T(x, y)=exp[j (x, y)] and Phase variation: (x, y)=knt(x, y) where t(x, y): thickness function of lens To find thickness function t(x, y)

Phase Transform Function of a Lens where Thickness function of lens:

Phase Transform Function of a Lens Phase transform function: Note: f > 0, convergent effect f < 0, divergent effect