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Pseudo-spectral analysis of radially-diagonalized Maxwell’s equations in cylindrical co-ordinates

K. Varis and A. R. Baghai-Wadji

Open Access Open Access

Abstract

We present a robust and accuracy enhanced method for analyzing the propagation behavior of EM waves in z-periodic structures in (r,ϕ,z)-cylindrical co-ordinates. A cylindrical disk, characterized by the radius a and the periodicity length Lz , defines the fundamental cell in our problem. The permittivity of the dielectric inside this cell is characterized by an arbitrary, single-valued function ε(r,ϕ,z) of all three spatial co-ordinates. We consider both open and closed boundary problems. Irrespective of the type of the boundary conditions on the surface r=a, our method requires the discretization of the fields in the interior of the disk only. Inside the disk volume, we expand the fields in terms of planewaves on discrete cylindrical surfaces ri =iΔ, with Δ being the discretization step length. The fields on the nested surfaces ri =iΔ in the interior of the simulation domain are interrelated by the application of a simple, yet, powerful finite difference scheme. In free space outside the disk, the fields are expanded in terms of the closed-form eigensolutions of the Maxwell’s equations in cylindrical co-ordinates. In order to uniquely determine the involved unknown coefficients, the solutions in the interior- and exterior domains are matched on the disk’s bounding surface. Our formulation utilizes a radially-diagonalized form of Maxwell’s equations, and merely involves four (out of the six) field components. It is demonstrated that our formulation is perfectly suited, but by no means limited, to cylindrical symmetric problems.

©2003 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. The geometry of test case 1. Image shows one unit cell, which is periodically replicated in the z-direction.

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Fig. 2.
Fig. 2. Dispersion diagram for the four guided modes in the geometry shown in Fig. 1. Curves with the marker “o” are computed with our method while curves with the marker “x” are obtained using the planewave method. Inset shows a zoom of the bands near the Brillouin zone edge.

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Fig. 3.
Fig. 3. The converge of the eigenmode propagation constant as a function of discrete basis size. The curve with circular tags is computed with our method, while the curve with cross tags is obtained with the eigenmode expansion method. For our method, the abscissa denotes the discretization step in the radial direction and also the number of planewave components in the z-direction, which are both equal here. For the eigenmode expansion method the horizontal axis denotes the number of eigenmodes used.

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Fig. 4.
Fig. 4. The real part of the z-directional eigenmode field pattern in a fiber with circular air holes.

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Equations (42)

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ℒ ( ∂ ∂ ϕ , ∂ ∂ z , μ , ε , ω ) [ h ϕ h z e ϕ e z ] = ∂ ∂ r [ h ϕ h z e ϕ e z ]
ℒ ψ → = ∂ ∂ r ψ →
ℒ ( 2 ) ψ → = ∂ ∂ r ℒ ψ →
ℒ ( 2 ) ψ → = ∂ 2 ∂ r 2 ψ → .
ℒ ( n ) ψ → = ∂ n ∂ r n ψ →
{ ∂ n ψ → ( r , ϕ , z ) ∂ r n } | r = r 0 = ℒ ( n ) ( ∂ ∂ ϕ , ∂ ∂ z ) ψ → ( r 0 , ϕ , z ) .
ψ → ( r 0 + h , ϕ , z ) = ∑ n = 0 ∞ h n n ! { ℒ ( n ) ( ∂ ∂ ϕ , ∂ ∂ z ) ψ → ( r 0 , ϕ , z ) } .
∇ × E = j ω μ H
∇ × H = − j ω ε H
∇ × f = 1 r ∣ u r r u ϕ u z ∂ ∂ r ∂ ∂ ϕ ∂ ∂ z f r r f ϕ f z ∣ ,
r f ϕ = f ˜ ϕ
1 r [ ∂ ∂ ϕ e z − ∂ ∂ z e ˜ ϕ − r ∂ ∂ r e z + r ∂ ∂ z e r ∂ ∂ r e ˜ ϕ − ∂ ∂ ϕ e r ] = j ω μ [ h r 1 r h ˜ ϕ h z ] .
e r = − 1 jωεr ∂ ∂ ϕ h z + 1 jωεr ∂ ∂ z h ˜ ϕ
− ∂ ∂ z 1 jωεr ∂ ∂ ϕ h z + ∂ ∂ z 1 jωεr ∂ ∂ z h ˜ ϕ − jωμ r h ˜ ϕ = ∂ ∂ r e z
jωμr h z − ∂ ∂ ϕ 1 jωεr ∂ ∂ ϕ h z + ∂ ∂ ϕ 1 jωεr ∂ ∂ z h ˜ ϕ = ∂ ∂ r e ˜ ϕ
[ 0 0 A 11 A 12 0 0 A 21 A 22 B 11 B 12 0 0 B 21 B 22 0 0 ] [ e z e ˜ ϕ h z h ˜ ϕ ] = ∂ ∂ r [ e z e ˜ ϕ h z h ˜ ϕ ]
A 11 = − ∂ ∂ z 1 jωεr ∂ ∂ ϕ
A 12 = ∂ ∂ z 1 jωεr ∂ ∂ z − jωμ r
A 21 = − ∂ ∂ ϕ 1 jωεr ∂ ∂ ϕ + jωμr
A 22 = ∂ ∂ ϕ 1 jωεr ∂ ∂ z
B 11 = ∂ ∂ ϕ 1 jωεr ∂ ∂ z
B 12 = − 1 jωμr ∂ 2 ∂ z ∂ ϕ + jωε r
B 21 = 1 jωμr ∂ 2 ∂ ϕ 2 − jωεr
B 22 = − 1 jωμr ∂ 2 ∂ ϕ ∂ z
[ e r h r ] = [ 0 1 jωμr ∂ ∂ ϕ 0 1 jωμr ∂ ∂ z − 1 jωεr ∂ ∂ ϕ 0 1 jωεr ∂ ∂ z 0 ] T [ e z e ˜ ϕ h z h ˜ ϕ ] .
f ( r , ϕ , z ) = ∑ m , n f m , n ( r ) e j k m z e j n ϕ .
Δ [ A 11 i A 12 i A 21 i A 22 i ] [ h ˜ ϕ i h z i ] + [ e ˜ ϕ i − 1 2 e z i − 1 2 ] = [ e ˜ ϕ i + 1 2 e z i + 1 2 ]
Δ [ B 11 i + 1 2 B 12 i + 1 2 B 21 i + 1 2 B 22 i + 1 2 ] [ e ˜ ϕ i + 1 2 e z i + 1 2 ] + [ h ˜ ϕ i h z i ] = [ h ˜ ϕ i + 1 h z i + 1 ]
h ˜ ϕ 0 ≡ 0 .
∮ l E · d l = jωμ ∫ S H · d S
E · d l = e ˜ ϕ d ϕ .
H · d S = h z d S .
jωμ ∫ S H · d S = jωμ π Δ 2 4 h z 0 .
∫ 0 2 π ∑ m , n e ˜ ϕ , m , n 1 2 e j k m z e jnϕ dϕ = jωμ π Δ 2 4 ∑ m , n h z , m , n 1 2 e j k m z e jnϕ .
∑ m , n e ˜ ϕ , m , n 1 2 2 π δ [ n ] e j k m z .
∑ m , n e ˜ ϕ , m , n 1 2 δ [ n ] δ [ n ̂ ] 4 π 2 e j k m z = jωμ π Δ 2 4 ∑ m , n h 0 , m , n 1 2 2 π δ [ n − n ̂ ] e j k m z .
h z , m ̂ , n ̂ 0 − δ [ n ̂ ] 8 jωμ Δ 2 e ϕ , m ̂ , 0 1 2 = 0 .
Φ m , n 1 = [ e ˜ ϕ e z h ˜ ϕ h z ] = [ j n H n ( λ m r ) λ 2 j k m H n ( λ m r ) rωε k m ∂ ∂ r H n ( λ m r ) 0 ] ,
Φ m , n 2 = [ e ˜ ϕ e z h ˜ ϕ h z ] = [ j rωμ k m ∂ ∂ r H n ( λ m r ) 0 n H n ( λ m r ) − λ 2 k m H n ( λ m r ) ] .
Ψ ( r , ϕ , z ) = ∑ m , n [ a m , n Φ m , n 1 ( r ) + b m , n Φ m , n 2 ( r ) ] e j k m z e jnϕ .
M ( ω , K z ) f = 0 .
M ( ω , K z ) f = ρ ( ω , K z )
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