1 Optical Diffraction Theory and Its Applications on Photonic Device Design

2 Diffraction Source Free Space Plane Wave Waveguide or resonator Wave guidance for power or signal transmission Wave manipulation for other purposes Diffraction as a natural evolving process towards the plane wave Direct applications: broadcasting, satellite comm., radar, etc.

3 Models for Diffraction and Scattering DiffractionScattering >>D D D Characteristics of the diffracted/scattered EM fields and the simulation models differ depending on the relative magnitude of the wavelength and feature size of the object. <<D Quasi-Static Geometric Optics ≈D Full Wave Single PoleMulti Poles Ray

4 Generalized Maxwell’s equations in homogeneous medium Generalized wave equations in homogeneous medium Generalized Wave Equations

5 Kirchhoff’s Diffraction Formula Scalar Green’s Function n E,HE,H Region 1 V S J,MJ,M Note: The integral is for closed surface only!

6 Diffraction Formulas for Open Surfaces E,HE,H S dl ds' C r r' O Line integrals enclosing the surface must be added to amend Kirchhoff’s formula:

7 P 1 (x 1,y 1,z 1 ) P 2 (x 2,y 2,z 2 ) Q(ξ,η,0) O x y z R1R1 r'r'R2R2 r2r2 r1r1 Diffraction by Aperture Spherical wave at Q from P1Spherical wave at P2 from Q

8 Diffracted Field: Ignore terms of 1/R³ if R>> Diffracted Field R1R1 R2R2 o Q P1P1 P2P2 r1r1 r2r2 r'r' 22 11

9 Spherical wave at the aperture Q from P1 Spherical wave at the point P2 from Q Inclination factor corrected by Kirchhoff Physical Interpretation of Kirchhoff’s Integral Original Huygens Principle The diffracted field is a superposition of spherical waves emanating from the wavefront of another spherical wave originating from the point source. R1R1 R2R2 o Q P1P1 P2P2 r1r1 r2r2 r'r' 22 11 For backward wave: For forward wave:

10 Free-space field at P 2 from P 1 Diffraction Coefficient F: Focus length

11 Different Approximations for Kirchhoff’s Integral R1R1 R2R2 o Q P1P1 P2P2 r1r1 r2r2 r'r' Wavefront curvature correction Plane wavefront correction Spherical wave centered at P i i=1,2 Fresnel Diffraction Fraunhofer Diffraction Geometrical Optics

12 Expression of Diffraction Coefficient

13 Fresnel Diffraction R1R1 R2R2 o Q P1P1 P2P2 r1r1 r2r2 r'r' x2x2 z2z2 z1z1 x1x1 n ˆ 22 11

14 Fraunhofer Diffraction Neglect the higher order terms in η and ξ

15 Fraunhofer Diffraction P1 on axis,  2 small ( < 30°) Uniform illumination over aperture 22 Equivalent to the stationary phase approximation

16 Fresnel and Fraunhofer Diffraction In Fraunhofer diffraction, both incident and diffracted waves may be considered to be plane (i.e. both S and P are a large distance away) If either S or P are close enough that wavefront curvature is not negligible, then we have Fresnel diffraction P S

17 Fraunhofer Diffraction For non-uniform illumination at the aperture, and by restoring the free space field factor, we find: where The coordinates in the new system are all normalized by the wavelength, i.e.,

18 Plane Wave Propagation in Free Space In homogeneous medium, the plane wave solution to Maxwell’s equations can be expressed as: By assuming the plane wave is propagating along z, we have: For an arbitrary wave propagating along z, upon its passing through an aperture at z=0, we can expand the wave in terms of the plane waves: Upon its further propagating from z=0 to z=z 0 Once the distribution of a propagating wave is known at a specific location z=0, we will be able to compute its distribution at any location z.

19 Plane Wave Expression At far field, i.e., z 0 is a large number, we can introduce the stationary-phase approximation: We obtain the same Fraunhofer diffraction formula, i.e., the far field angular distribution is the Fourier transform of the field distribution at the aperture. to obtain

20 Applications Aperture: phaser or diaphragm θ Low spatial frequency region region High spatial frequency region region Far field (Fraunhofer diffraction) condition: (all normalized by the wavelength) To ensure this condition without going too far: insert a Fresnel lens to bring the Fourier transform closer to f – focal length of the lens

21 Functional Photonic Component Design with Slab-waveguide: Building Block Substrate Normal slab-waveguide: low index region Slab-waveguide with thicker cladding: high index region Fabrication: standard lithography + etching Vertically confined by the slab-waveguide Propagation 2D diffraction Aperture: phaser

22 Wavelength Multiplexer/Demultiplexer Input ridge waveguide Output ridge waveguides W1, W2 W1 W2 Slab-waveguide low index region Slab-waveguide high index region Input lens for beam expanding Output lens for beam refocusing Blazed diffractional grating

23 Optical Switch Output ridge waveguides Input ridge waveguide Focused beam with no bias Slab-waveguide low index region Slab-waveguide high index region Input lens for beam expanding Output lens for beam refocusing Biased wedge for beam steering Focused beam under bias

24 Beam Splitter Output ridge waveguides Input ridge waveguide Slab-waveguide low index region Slab-waveguide high index region Input lens for beam expanding Output lens for beam refocusing 1 st phase plate DC filter plateFourier transform lens

Channel Beam Splitter: Design y A L0 L1 L2 L5 W z x L3 G L4 F S00 S10 S20 S21 S01 S11 p Q1 Q2 L4 L5 x z

Channel Beam Splitter: Simulation Result

27 Polarization Beam Combiner/Splitter

28 Form Birefringence δ t Ex Ey kz

29 Polarization Beam Combiner/Splitter X-pol loss: dB Crosstalk dB Y-pol loss: 0.66 dB Crosstalk dB

30 Time Lens

31 Input/Output Simulation Result