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Implementation of a Paraxial Optical Propagation Method for
Large Photonic Devices
James E. ToneyPenn State Electro-Optics Center
October 8-10, 2009
Presented at the COMSOL Conference 2009 Boston
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Outline
• Computational Limitations of EM Propagation Modes
• Review of Beam Propagation Methods• Implementation of BPM-Like Mode in
Comsol• Representative Results
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EM Modes Require Mesh Size
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Beam Propagation Method
• Assume steady-state (time harmonic) oscillation
U(r,t)=U(r) e-iωt ∇2U+k2U = 0 (k=2πn/λ)
• Assume propagation is primarily along the z-axis
U(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 field• Field can then be “propagated” in the z-direction
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Applications of BPM
• Good for relatively large, waveguide-based devices– Couplers, splitters, interferometers, array waveguide
gratings• Not as good for high-index contrast systems• Cannot handle systems with arbitrary
propagation directions:– Photonic crystals (photonic band gaps)– Ring resonators– Tight bends
• Cannot do frequency mixing/nonlinear effects
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BPM Example : 3 dB Coupler
Equal splitting between arms Perturbation of refractive index
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PDE Implementation of BPM-Like Mode
∇2u+2ik0 ∂u/∂z+k02[(n/n0)2-1] u= 0[No assumption of ∂2u/∂z2~0 necessary]
Recall the basic paraxial wave equation:
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Geometry for 2D BPM-Like Mode
Active area
refractive index distribution n(x,z)=n0+∆n f(x,z)
defined to specify device structure
Absorbing layer to prevent reflections
n=n0 [1+ie(x-x0)/∆x ]
n=n0 [1+ie(y-y0)/∆y ]
e.g. step-index WG of width w:
f(x,z)=(x>-w/2)*(x
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Specification of Refractive Index Distribution
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BPM-Like Mode Allows a Much Coarser Mesh
3λ/2 λ λ/2Maximum Mesh Size
100 µm
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Examples of 2D Models
Y-splitter Mach-Zehnder Interferometer
∆φ=π
450 µm
500 µm
Graded-Index Waveguide
400 µm
∆φ=0
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Directional Coupler
0 0.5 1 1.5 2 2.5 30.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gap (microns)
Tran
sfer
Len
gth
(mm
)
Transfer length
Coupled mode theory (TE mode)
Scalar Paraxial mode
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Multi-Mode Interference Splitter
•According to theory, two-fold image occurs at L~400 µm for symmetrical excitation•Illustrates that the paraxial model (like BPM) accounts for reflections at moderate anglesMulti-
mode section
Output waveguides
410 µm
27 µm
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Beam Coupling via a Circular Microlens
Gaussian beam input
•High index contrasts makes this problem more challenging•Reduced mesh was used (~0.45 λ)•Field in low-intensity regions is somewhat grainy•Focusing distance (~200 µm from center) agrees fairly well with focal length from geometrical optics (175 µm)
n=1.4
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Scalar Paraxial Mode with Axial Symmetry
∂u2/∂z2+(1/r)∂(r ∂u/∂r)/∂r+2ik0∂u/∂z+k02[(n/n0)2-1]u= 0Scalar Wave Equation in 2D, Cylindrical Coordinates:
z
rSmall offset (
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Conclusions
• BPM-Like mode can be implemented easily in Comsol via a PDE mode
• Enables a much coarser mesh and therefore larger devices to be simulated
• Accounts for interference, evanescent wave coupling, refraction, glancing reflections, but not back reflections
• Can be integrated with thermal, RF and strain effects for complex devices
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Thank You!
James E. Toney, Ph.D.Research Engineer, Fiber Optics, Photonics and
Engineering DivisionPenn State University Electro-Optics Center222 Northpointe Blvd.Freeport, PA [email protected]
mailto:[email protected]�http://www.electro-optics.org/�