July 2003 Chuck DiMarzio, Northeastern University ECEG105 Optics for Engineers Course Notes Part 7: Diffraction Prof. Charles A. DiMarzio Northeastern University Fall 2007 August 2007

July 2003 Chuck DiMarzio, Northeastern University 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 August 2007 ? /D D

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

July 2003 Chuck DiMarzio, Northeastern University Quantum Diffraction Examples 200 Random Paths Aperture 1 Aperture 2 Aperture 5 Aperture 10

July 2003 Chuck DiMarzio, Northeastern University Maxwell’s Eqs & Diffraction z x y 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 z x y 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

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

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

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

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

July 2003 Chuck DiMarzio, Northeastern University Kirchoff Integral Thm. Solution

July 2003 Chuck DiMarzio, Northeastern University Helmholtz-Kirchoff Integral P Surface A 0 Surface A P r’ r A0A0

July 2003 Chuck DiMarzio, Northeastern University H-K Integral Approximations

July 2003 Chuck DiMarzio, Northeastern University Some Approximations

July 2003 Chuck DiMarzio, Northeastern University Paraxial Approximation x1x1 x z

July 2003 Chuck DiMarzio, Northeastern University Integral Expressions (Hankel Transform)

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

July 2003 Chuck DiMarzio, Northeastern University 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 August 2007

July 2003 Chuck DiMarzio, Northeastern University Fraunhofer Diffraction (1) Very Important Parameter

July 2003 Chuck DiMarzio, Northeastern University Fraunhofer Diffraction (2)

July 2003 Chuck DiMarzio, Northeastern University Fraunhofer Lens (1)

July 2003 Chuck DiMarzio, Northeastern University Fraunhofer Lens (2) z z

July 2003 Chuck DiMarzio, Northeastern University Fraunhofer Diffraction Summary z z

July 2003 Chuck DiMarzio, Northeastern University Numerical Computation (1)

July 2003 Chuck DiMarzio, Northeastern University 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 DiMarzio, Northeastern University Circular Aperture, Uniform Field D h

July 2003 Chuck DiMarzio, Northeastern University Square Aperture, Uniform Field z D

July 2003 Chuck DiMarzio, Northeastern University No Aperture, Gaussian Field D

July 2003 Chuck DiMarzio, Northeastern University Fraunhoffer Examples

July 2003 Chuck DiMarzio, Northeastern University Imaging: Rayleigh Criterion R/d 0 is f# August 2007

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

July 2003 Chuck DiMarzio, Northeastern University Diffraction Grating ii dd Reflection Example d

July 2003 Chuck DiMarzio, Northeastern University Grating Equation sin(  d ) sin(  i ) degrees -sin(  i ) n= Reflected Orders Transmitted Orders

July 2003 Chuck DiMarzio, Northeastern University Grating Fourier Analysis GratingDiffraction Pattern Slit Convolve Sinc Multiply Repetition Pattern MultiplyConvolve Apodization Result

July 2003 Chuck DiMarzio, Northeastern University Grating for Laser Tuning f Gain f Cavity Modes ii August 2007

July 2003 Chuck DiMarzio, Northeastern University Monochrometer ii sin  n=1n=2n=3 Aliasing August 2007

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

July 2003 Chuck DiMarzio, Northeastern University 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 DiMarzio, Northeastern University Rectangular Aperture

July 2003 Chuck DiMarzio, Northeastern University Cornu Spiral C(u), Fresnel Cosine Integral S(u), Fresnel Sine Integral <u<5 u=0 u=1 u=2

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

July 2003 Chuck DiMarzio, Northeastern University Small Aperture =500 nm, 2a=100  m, z=5m. Fraunhofer Diffraction would have worked here. position, mm

July 2003 Chuck DiMarzio, Northeastern University Large Aperture =500 nm, 2a=1mm, z=5m. position, m

July 2003 Chuck DiMarzio, Northeastern University Circular Aperture Fresnel Cosine Integrand Output of Fresnel Zone Plate kr/2z

July 2003 Chuck DiMarzio, Northeastern University Phase in Pupil (1) Linear Phase Shift is tilt  D/2 Quadratic Phase Shift is focus

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

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