Implementation Of A Paraxial Optical Propagation Method .

Presented at the COMSOL Conference 2009 BostonImplementation of a ParaxialOptical Propagation Method forLarge Photonic DevicesJames E. ToneyPenn State Electro-Optics CenterOctober 8-10, 2009

“We Honor Those Who Serve”Outline Computational Limitations of EMPropagation Modes Review of Beam Propagation Methods Implementation of BPM-Like Mode inComsol Representative Results2

“We Honor Those Who Serve”EM Modes RequireMesh Size λ2λ/3λ/3λ/6Maximum Mesh Size100µm3

“We Honor Those Who Serve”Beam PropagationMethod Assume steady-state (time harmonic) oscillationU(r,t) U(r) e-iωt 2U k2U 0 (k 2πn/λ) Assume propagation is primarily along the z-axisU(r) u(r) eik0z 2u 2ik0 u/ z (k2-k02) u 0 Assume that the field varies slowly along the z-axis ( 2u/ z2 0) u/ z i/2k [ xy2u (k2-k02) u] in0/(2k0n) { xy2u k02[(n/n0)2-1] u} Choose a form for the input fieldField can then be “propagated” in the z-direction4

“We Honor Those Who Serve”Applications of BPM Good for relatively large, waveguide-baseddevices– Couplers, splitters, interferometers, array waveguidegratings Not as good for high-index contrast systems Cannot handle systems with arbitrarypropagation directions:– Photonic crystals (photonic band gaps)– Ring resonators– Tight bends Cannot do frequency mixing/nonlinear effects5

“We Honor Those Who Serve”BPM Example : 3 dBCouplerEqual splitting between armsPerturbation of refractive index6

“We Honor Those Who Serve”PDE Implementation ofBPM-Like ModeRecall the basic paraxial wave equation: 2u 2ik0 u/ z k02[(n/n0)2-1] u 0[No assumption of 2u/ z2 0 necessary]7

“We Honor Those Who Serve”Geometry for 2D BPMLike Moden n0 [1 ie(y-y0)/ y ]Active arearefractive index distributionn(x,z) n0 n f(x,z)n n0 [1 ie(x-x0)/ x ]defined to specify devicestructuree.g. step-index WG of width w:f(x,z) (x -w/2)*(x w/2)Absorbing layer to prevent reflections8

“We Honor Those Who Serve”Specification of RefractiveIndex Distribution9

“We Honor Those Who Serve”BPM-Like Mode Allowsa Much Coarser Mesh3λ/2λλ/2MaximumMesh Size100µm10

“We Honor Those Who Serve”Examples of 2D ModelsGraded-IndexWaveguideY-splitterMach-Zehnder Interferometer φ 0400µm450µm φ π500µm11

“We Honor Those Who Serve”Directional Coupler10.9TransferlengthTransfer Length (mm)0.8Coupled mode theory(TE ap (microns)22.5312

“We Honor Those Who Serve”Multi-ModeInterference SplitterOutputwaveguidesMultimodesection According to theory, two-foldimage occurs at L 400 µm forsymmetrical excitation Illustrates that the paraxialmodel (like BPM) accounts forreflections at moderate angles410 µm27 µm13

“We Honor Those Who Serve”Beam Coupling via aCircular Microlensn 1.4 High index contrasts makesthis problem more challenging Reduced mesh was used ( 0.45λ) Field in low-intensity regions issomewhat grainy Focusing distance ( 200 µmfrom center) agrees fairly wellwith focal length fromgeometrical optics (175 µm)Gaussian beam input14

“We Honor Those Who Serve”Scalar Paraxial Modewith Axial SymmetryScalar Wave Equation in 2D, Cylindrical Coordinates: u2/ z2 (1/r) (r u/ r)/ r 2ik0 u/ z k02[(n/n0)2-1]u 0zPropagationin singlemode fiberSmall offset ( 1 µm)to help stabilityr15

“We Honor Those Who Serve”Conclusions BPM-Like mode can be implementedeasily in Comsol via a PDE mode Enables a much coarser mesh andtherefore larger devices to be simulated Accounts for interference, evanescentwave coupling, refraction, glancingreflections, but not back reflections Can be integrated with thermal, RF andstrain effects for complex devices16

“We Honor Those Who Serve”Thank You!James E. Toney, Ph.D.Research Engineer, Fiber Optics, Photonics andEngineering DivisionPenn State University Electro-Optics Center222 Northpointe Blvd.Freeport, PA 16229jtoney@eoc.psu.eduwww.electro-optics.org17

Penn State Electro-Optics Center. . Good for relatively large, waveguide-based devices – Couplers, splitters, interferometers, array waveguide . Engineering Division. Penn State University Electro-Optics Center. 222