ECEG 105 Optics for Engineers Course Notes Part

ECEG 105 Optics for Engineers Course Notes Part 7: Diffraction Prof. Charles A. Di. Marzio Northeastern University Fall 2007 July 2003 August 2007 Chuck Di. Marzio, Northeastern University 1

Diffraction Overview • General Equations • Fraunhofer – Fourier Optics – Special Cases – Image Resolution – Diffraction Gratings – Acousto-Optical Modulators • Fresnel – Cornu Spiral – Circular Apertures • Summary It's All About /D July 2003 August 2007 ? /D Chuck Di. Marzio, Northeastern University D 2

Difraction: Quantum Approach • Uncertainty • Angle of Flight • Photon Momentum • Uncertainty in p July 2003 • For a Better Result – Use Exact PDF – Gaussian is best • Satisfies the equality • Minimum-uncertainty wavepacket Chuck Di. Marzio, Northeastern University 3

Quantum Diffraction Examples 200 Random Paths July 2003 Aperture 1 Aperture 5 Aperture 2 Aperture 10 Chuck Di. Marzio, Northeastern University 4

Maxwell’s Eqs & Diffraction y y z z x x z-component of curl is zero y-component of curl is zero x-component is not E in y direction, B in -x direction Propagation in z direction July 2003 z-component of curl is not zero if E changes in x direction Now, B has a z component, so Propagation is along both z and x Chuck Di. Marzio, Northeastern University 5

Summary of Diffraction Math Maxwell’s Equations Scalar Fields “Simple Systems” Yee Numerical Methods Spheres Mie Scattering General Problems July 2003 Green’s Theorem Helmholtz Equation r>>λ Kirchoff Integral Theorem All Scalar Wave Problems Fresnel. Kirchoff Integral Formula Obliquity=2, Fresnel Diffraction Paraxial Approximation Fraunhofer Conditions Fourier Transforms Shadows and Zone Plates x, y Separable Problems Polar Symmetry Fields Far From Aperture Chuck Di. Marzio, Northeastern University Hankel Transforms Circular Apertures 6

Kirchoff Integral Theorem (1) • General Wave Probs. – Solve Maxwell's Eqs. – Use Boundary Conditions – Hard or Impossible • Kirchoff Integral Approach – Algorithmic – Correct (Almost) July 2003 – – • Based on Maxwell's Equations • Scalar Fields Complete • Amplitude and Phase Amenable to Approximation Comp. Efficient? Intuitive • Similar to Huygens Chuck Di. Marzio, Northeastern University 7

Kirchoff Integral Theorem (2) • The Idea – Green's Theorem – Consider Point of Interest – Helmholtz Equation – Correlate Wavefronts • Ties to Maxwell's • “Best Wavefront” Equations (Scalar Field) – Converging Uniform – Various Approximations Spherical Wave – Numerical Techniques • Actual Wavefront • Results • The Mathematics – Fresnel Diffraction – Start with Converging Spherical – Fraunhofer Diffraction Wave July 2003 Chuck Di. Marzio, Northeastern University 8

Kirchoff Integral Setup Surface A 0 P The Goal: A Green’s Function Approach. July 2003 Chuck Di. Marzio, Northeastern University 9

Kirchoff Integral Thm. Solution July 2003 Chuck Di. Marzio, Northeastern University 10

Helmholtz-Kirchoff Integral Surface A 0 P Surface A A 0 r’ r P July 2003 Chuck Di. Marzio, Northeastern University 11

H-K Integral Approximations July 2003 Chuck Di. Marzio, Northeastern University 12

Some Approximations July 2003 Chuck Di. Marzio, Northeastern University 13

Paraxial Approximation x 1 x z July 2003 Chuck Di. Marzio, Northeastern University 14

Integral Expressions (Hankel Transform) July 2003 Chuck Di. Marzio, Northeastern University 15

Fraunhofer and Fresnel • Fraunhofer works – in far field or – at focus. • Fresnel works – everywhere else. – For example, it predicts effects at edges of shadows. July 2003 August 2007 Chuck Di. Marzio, Northeastern University z z 16

Fraunhofer Diffraction • Equations • A Hint of Fourier Optics • Numerical Computations • Special Cases (Gaussian, Uniform) • Imaging • Brief Comment on SM and MM Fibers • Gratings • Brief Comment on Acousto-Optics July 2003 August 2007 Chuck Di. Marzio, Northeastern University 17

Fraunhofer Diffraction (1) Very Important Parameter July 2003 Chuck Di. Marzio, Northeastern University 18

Fraunhofer Diffraction (2) July 2003 Chuck Di. Marzio, Northeastern University 19

Fraunhofer Lens (1) July 2003 Chuck Di. Marzio, Northeastern University 20

Fraunhofer Lens (2) z z July 2003 Chuck Di. Marzio, Northeastern University 21

Fraunhofer Diffraction Summary z z July 2003 Chuck Di. Marzio, Northeastern University 22

Numerical Computation (1) July 2003 Chuck Di. Marzio, Northeastern University 23

Numerical Computation (2) • Quadratic Phase of Integrand – Near Focus (z=f): Not a problem – Otherwise • Many cycles in integrating over aperture • Contributions tend to cancel, so • roundoff error becomes significant • but geometric optics is pretty good here, – except at edges. – We will approach this problem later. July 2003 Chuck Di. Marzio, Northeastern University 24

Circular Aperture, Uniform Field D h July 2003 Chuck Di. Marzio, Northeastern University 25

Square Aperture, Uniform Field D z July 2003 Chuck Di. Marzio, Northeastern University 26

No Aperture, Gaussian Field D July 2003 Chuck Di. Marzio, Northeastern University 27

Fraunhoffer Examples July 2003 Chuck Di. Marzio, Northeastern University 28

Imaging: Rayleigh Criterion R/d 0 is f# July 2003 August 2007 Chuck Di. Marzio, Northeastern University 29

Single-Mode Optical Fiber Beam too Large (lost power at edges) Beam too Small (lost power through cladding) July 2003 Chuck Di. Marzio, Northeastern University 30

Diffraction Grating Reflection Example d i d July 2003 Chuck Di. Marzio, Northeastern University 31

Grating Equation sin( d) 1 0. 5 5 4 3 sin( i) 2 1 0 -0. 5 -1 -100 Reflected Orders July 2003 n=0 -1 -sin( i) 0 Transmitted Orders Chuck Di. Marzio, Northeastern University 100 200 -2 -3 degrees 32

Grating Fourier Analysis Slit Sinc Convolve Multiply Diffraction Pattern Repetition Pattern Result Multiply Convolve Apodization Result July 2003 Chuck Di. Marzio, Northeastern University 33

Grating for Laser Tuning Gain f Cavity Modes f i July 2003 August 2007 Chuck Di. Marzio, Northeastern University 34

Monochrometer i Aliasing n=1 n=2 n=3 sin July 2003 August 2007 Chuck Di. Marzio, Northeastern University 35

Acousto-Optical Modulator • Acoustic Wave: – Sinusoidal Grating • Sound Source Wavefronts as Moving Mirrors – Signal Enhancement – Doppler Shift • Acoustic Frequency Multiplied by Order Absorber More Rigorous Analysis is Possible but Somewhat Complicated July 2003 August 2007 Chuck Di. Marzio, Northeastern University 36

Fresnel Diffraction • Fraunhofer Diffraction Assumed: – Obliquity = 2 – Paraxial Approximation – At focus or at far field • Relax the Last Assumption – More Complicated Integrals – Describe Fringes at edges of shadows July 2003 Chuck Di. Marzio, Northeastern University 37

Rectangular Aperture July 2003 Chuck Di. Marzio, Northeastern University 38

Cornu Spiral S(u), Fresnel Sine Integral -5<u<5 u=2 u=0 C(u), Fresnel Cosine Integral July 2003 Chuck Di. Marzio, Northeastern University 39 u=1

Using the Cornu Spiral S(u), Fresnel Sine Integral a=1 C(u), Fresnel Cosine Integral July 2003 Chuck Di. Marzio, Northeastern University 40

Small Aperture =500 nm, 2 a=100 m, z=5 m. Fraunhofer Diffraction would have worked here. 0. 8 1. 4 0. 6 1. 2 0. 4 1 0. 2 0. 8 0 0. 6 -0. 2 0. 4 -0. 4 0. 2 -0. 6 -0. 8 -0. 6 July 2003 -0. 4 -0. 2 0. 4 0. 6 0. 8 0 -6 -4 Chuck Di. Marzio, Northeastern University -2 0 2 position, mm 4 6 8 -3 41

Large Aperture =500 nm, 2 a=1 mm, z=5 m. 0. 8 3 0. 6 2. 5 0. 4 2 0 1. 5 -0. 2 1 -0. 4 0. 5 -0. 6 -0. 8 -0. 6 -0. 4 July 2003 -0. 2 0. 4 0. 6 0. 8 0 -0. 02 -0. 015 -0. 01 -0. 005 Chuck Di. Marzio, Northeastern University 0 0. 005 0. 01 position, m 0. 015 0. 02 42 0. 025

Circular Aperture Fresnel Cosine Integrand kr/2 z Output of Fresnel Zone Plate July 2003 Chuck Di. Marzio, Northeastern University kr/2 z 43

Phase in Pupil (1) Quadratic Phase Shift is focus Linear Phase Shift is tilt D/2 July 2003 Chuck Di. Marzio, Northeastern University 44

Phase in Pupil (2) Quartic Phase is Spherical Aberration Fresnel Lens has wrapped quadratic phase July 2003 Atmoshperic Turbulence can be modeled as random phase in the pupil plane Chuck Di. Marzio, Northeastern University 45

Summary of Diffraction Math Maxwell’s Equations Scalar Fields “Simple Systems” Yee Numerical Methods Spheres Mie Scattering General Problems July 2003 Green’s Theorem Helmholtz Equation r>>λ Kirchoff Integral Theorem All Scalar Wave Problems Fresnel. Kirchoff Integral Formula Obliquity=2, Fresnel Diffraction Paraxial Approximation Fraunhofer Conditions Fourier Transforms Shadows and Zone Plates Separable Problems Polar Symmetry Fields Far From Aperture Chuck Di. Marzio, Northeastern University Hankel Transforms Circular Apertures 46