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Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates

M. Skorobogatiy, Steven A. Jacobs, Steven G. Johnson, and Yoel Fink

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Abstract

Perturbation theory formulation of Maxwell’s equations gives a theoretically elegant and computationally efficient way of describing small imperfections and weak interactions in electro-magnetic systems. It is generally appreciated that due to the discontinuous field boundary conditions in the systems employing high dielectric contrast profiles standard perturbation formulations fail when applied to the problem of shifted material boundaries. In this paper we developed a novel coupled mode and perturbation theory formulations for treating generic non-uniform (varying along the direction of propagation) perturbations of a waveguide cross-section based on Hamiltonian formulation of Maxwell equations in curvilinear coordinates. We show that our formulation is accurate and rapidly converges to an exact result when used in a coupled mode theory framework even for the high index-contrast discontinuous dielectric profiles. Among others, our formulation allows for an efficient numerical evaluation of induced PMD due to a generic distortion of a waveguide profile, analysis of mode filters, mode converters and other optical elements such as strong Bragg gratings, tapers, bends etc., and arbitrary combinations of thereof. To our knowledge, this is the first time perturbation and coupled mode theories are developed to deal with arbitrary non-uniform profile variations in high index-contrast waveguides.

©2002 Optical Society of America

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Figures (3)
Fig. 1.
Fig. 1. a) Dielectric profile of a cylindrically symmetric fiber. Concentric dielectric interfaces are characterized by their radii ρi . b) Scaling variation - linear tapers. Fiber profile remains cylindrically symmetric, while the radii of the dielectric interfaces along the direction of propagation s become ρ i ( 1 + δ s L ) . c) Scaling variation - sinusoidal Bragg gratings. Fiber profile remains cylindrically symmetric, while the radii of the dielectric interfaces along the direction of propagation s become ρ i ( 1 + δ Sin ( 2 π s Λ ) ) . d) Non-concentric variation. Each dielectric interface stays cylindrically symmetric, while the center line is bent.

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Fig. 2.
Fig. 2. a) Transmitted power T 1 in the fundamental m = 1 mode, along with the transmitted powers T 2, T 3 in the second and third m = 1 parasitic modes as a function of the taper length L, calculated by our coupled mode theory. Results of anas ymptotically exact transfer matrix based CAMFR code are presented in circles. b) Convergence of the errors in the transmitted and reflected coefficients for a taper length of L = 10a as a function of the number of expansion modes. Solid lines correspond to the relative errors in the transmission coefficients while dotted lines correspond to the relative errors in the reflected coefficients. Calculated by our coupled mode theory, errors inthe transmission and reflection coefficients exhibit faster thana quadratic convergence.

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Fig. 3.
Fig. 3. a) Transmitted powers T 2, T 3, T 4 in the second third and forth m = 1 modes for the grating lengths [ Λ 2 , 3 Λ ] inthe Λ 2 increments are plotted in crosses, calculated by our coupled mode theory. In this geometry the incoming and outgoing waveguides are the same. Results of anas ymptotically exact transfer matrix based CAMFR code are presented in circles. When grating length is increased the power transfer to the first excited mode is monotonically increased as expected. b) Convergence of the errors in the transmitted and reflected coefficients for a grating of L = Λ 2 as a function of the number of expansion modes. Solid lines correspond to the relative errors inthe transmissionco efficients. Calculated by our coupled mode theory, errors in the transmission and reflection coefficients exhibit faster than a linear convergence.

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Equations (50)

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x = ρCos ( θ ) y = ρSin ( θ ) z = s .
x = ρCos ( θ ) ( 1 + f ( s ) ) y = ρSin ( θ ) ( 1 + f ( s ) ) z = s .
x = ρCos ( θ ) Cos ( s R ) + R ( Cos ( s R ) − 1 ) y = ρSin ( θ ) z = ρCos ( θ ) Sin ( s R ) + R Sin ( s R ) ,
− i ∂ ∂ z B ̂ ∣ ψ 〉 = A ̂ ∣ ψ 〉 ,
B ̂ = ( 0 − z ̂ × z ̂ × 0 ) ,
A ̂ = ( ω c ∊ − c ω ∇ t × [ z ̂ ( 1 μ z ̂ ∙ ( ∇ t × ) ) ] 0 0 ω c μ − c ω ∇ t × [ z ̂ ( 1 ∊ z ̂ ∙ ( ∇ t × ) ) ] ) ,
〈 ψ β * 0 ∣ B ̂ ∣ ψ β ′ 0 〉 = β ′ ∣ β ′ ∣ δ β , β′ ,
β B ̂ ∣ ψ β 0 〉 = A ̂ 0 ∣ ψ β 0 〉 .
− i ∂ ∂ z B ̂ ∣ ψ 〉 = ( A ̂ 0 + Δ A ̂ ( z ) ) ∣ ψ 〉 ,
∣ ψ 〉 = ∑ i C i ( z ) exp i β i z ∣ ψ β i 0 〉 ,
− i B ∂ C → ∂ z = Δ A C → ,
C j ( z ) = 〈 ψ β j 0 ∣ Δ A ̂ ∣ ψ β n 0 〉 〈 ψ β j 0 ∣ B ̂ ∣ ψ β j 0 〉 〈 ψ β n 0 ∣ B ̂ ∣ ψ β n 0 〉 exp i ( β n − β j ) z − 1 β n − β j ,
a → i = ∂ x 1 ∂ q i ∂ x 2 ∂ q i ∂ x 3 ∂ q i .
a → i = 1 g a → j × a → k , k j ≠ i ,
g i j = ∂ x k ∂ q i ∂ x k ∂ q j ,
a → i ∙ a → j = δ i , j , a → i ∙ a → j = g i j , a → i ∙ a → j = g i j ,
x = ρCos ( θ ) ( 1 + δ s L ) y = ρSin ( θ ) ( 1 + δ s L ) z = s .
i → ρ = ( Cos ( θ ) , Sin ( θ ) , 0 ) ; i → ρ = Cos ( θ ) , Sin ( θ ) , − δ ρ L ) i → θ = ( − Sin ( θ ) , Cos ( θ ) , 0 ) ; i → θ = ( − Sin ( θ ) , Cos ( θ ) , 0 ) i → s = ( δ ρ L Cos ( θ ) , δ ρ L Sin ( θ ) , 1 ) 1 + ( δ ρ L ) 2 ; i → s = 0,0,1 .
x ρ θ s , y ρ θ s , z ρ θ s ,
∊ q 1 q 2 q 3 ∂ E i g i i ∂ c t = 1 g e ijk ∂ H k g k k ∂ q j − μ q 1 q 2 q 3 ∂ H i g i i ∂ c t = 1 g e ijk ∂ E k g k k ∂ q j ,
( E ρ θ s t H ρ θ s t ) = ( E ρ θ s H ρ θ s ) exp − iωt .
i ∂ E θ g θ θ ∂ s = ∂ i E s g s s ∂ θ + ω c g ( g ρ ρ H ρ + g ρ θ g θ θ H θ + g ρ s g s s H s ) − i ∂ E ρ g ρ ρ ∂ s = − ∂ i E s g s s ∂ ρ + ω c g ( g θ ρ g ρ ρ H ρ + g θ θ H θ + g θ s g s s H s ) i ∂ H θ g θ θ ∂ s = − ∂ i H s g s s ∂ θ + ω c ∊ g ( g ρ ρ E ρ + g ρ θ g θ θ E θ + g ρ s g s s E s ) i ∂ H ρ g ρ ρ ∂ s = ∂ i H s g s s ∂ ρ + ω c ∊ g ( g θ ρ g ρ ρ E ρ + g θ θ E θ + g θ s g s s E s )
i E s g s s = − c ω ∊ g g s s ( ∂ H θ g θ θ ∂ ρ − ∂ H ρ g ρ ρ ∂ θ ) − i g s s ( g s ρ g ρ ρ E ρ + g s θ g θ θ E θ ) i H s g s s = c ω g g s s ( ∂ E θ g θ θ ∂ ρ − ∂ E ρ g ρ ρ ∂ θ ) − i g s s ( g s ρ g ρ ρ H ρ + g s θ g θ θ H θ ) .
( E 0 ρ θ s H 0 ρ θ s ) β m = ( E 0 ( ρ ) H 0 ( ρ ) ) β m exp iβs + iθm .
− β E θ β 0 m = − g 0 θ θ m E s β 0 m + ω c g 0 g 0 ρ ρ g 0 θ θ H ρ β 0 m β E ρ β 0 m = − g 0 ρ ρ ∂ i E s β 0 m ∂ ρ + ω c g 0 g 0 ρ ρ g 0 θ θ H θ β 0 m β H θ β 0 m = g 0 θ θ m H s β 0 m + ω c ∊ g 0 g 0 ρ ρ g 0 θ θ E ρ β 0 m − β H ρ β 0 m = g 0 ρ ρ ∂ i H s β 0 m ∂ ρ + ω c ∊ g 0 g 0 ρ ρ g 0 θ θ E θ β 0 m
i E s β 0 m = − c ω ∊ g 0 ( ∂ H θ β 0 m g 0 θ θ ∂ ρ − i m H ρ β 0 m g 0 ρ ρ ) i H s β 0 m = c ω g 0 ( ∂ E θ β 0 m g 0 θ θ ∂ ρ − i m E ρ β 0 m g 0 ρ ρ ) ,
〈 ψ β * , m 0 ∣ B ̂ ∣ ψ β ′ , m ′ 0 〉 = β ′ ∣ β ′ ∣ δ β , β ′ =
∫ ( ( H θ β * 0 m ) * E ρ β ′ 0 m ′ − ( H θ β * 0 m ) * E θ β ′ 0 m ′ + H θ β ′ 0 m ′ ( E ρ β * 0 m ) * − H ρ β ′ 0 m ′ ( E θ β * 0 m ) * ) J 0 ( ρ ) dρdθ ,
∣ Ψ β , m 〉 = ( g ρ ρ g 0 ρ ρ E ρ 0 ( ρ x y z ) i → ρ + g θ θ g 0 θ θ E θ 0 ( ρ x y z ) i → θ g ρ ρ g 0 ρ ρ H ρ 0 ( ρ x y z ) i → ρ + g θ θ g 0 θ θ H θ 0 ( ρ x y z ) i → θ ) β m exp imθ x y z .
( E ρ E θ H ρ H θ ) = ∑ β ′ , m ′ C m ′ β ′ ( s ) ( g ρ ρ g 0 ρ ρ E ρ 0 ( ρ ) g θ θ g 0 θ θ E θ 0 ( ρ ) g ρ ρ g 0 ρ ρ H ρ 0 ( ρ ) g θ θ g 0 θ θ H θ 0 ( ρ ) ) β ′ m ′ exp im ′ θ .
− i B ∂ C → ( s ) ∂ s = M C → ( s ) ,
M β * , m ; β ′ , m ′ = 〈 ψ β * m ∣ M ̂ ∣ ψ β ′ , m ′ 〉 = ω c ∫ exp i ( m ′ − m ) θ ×
( E ρ 0 ( ρ ) E θ 0 ( ρ ) E s 0 ( ρ ) H ρ 0 ( ρ ) H θ 0 ( ρ ) H s 0 ( ρ ) ) β * m † ( ∊ d ρ ρ ∊ d ρ θ ∊ d ρ s 0 0 0 ∊ d θ ρ ∊ d θ θ ∊ d θ s 0 0 0 ∊ d s ρ ∊ d s θ ∊ d s s 0 0 0 0 0 0 d ρ ρ d ρ θ d ρ s 0 0 0 d θ ρ d θ θ d θ s 0 0 0 d s ρ d s θ d s s ) ( E ρ 0 ( ρ ) E θ 0 ( ρ ) E s 0 ( ρ ) H ρ 0 ( ρ ) H θ 0 ( ρ ) H s 0 ( ρ ) ) β ′ m′ J 0 ( ρ ) dρdθ ,
d ρ ρ = g g 0 θ θ g 0 ρ ρ ( g ρ ρ − ( g ρ s ) 2 g s s ) d ρ θ = d θ ρ = g ( g ρ θ − g ρ s g θ s g s s ) d ρ s = d s ρ = g 0 g 0 θ θ g 0 s s g ρ s g s s d θ θ = g g 0 ρ ρ g 0 θ θ ( g θ θ − ( g θ s ) 2 g s s ) d θ s = d s θ = g 0 g 0 ρ ρ g 0 s s g θ s g s s d s s = − g 0 g 0 s s g s s g 0 ρ ρ g 0 θ θ g .
i 〈 H ρ β * 0 m ∣ g 0 θ θ ∂ E θ g θ θ ∂ s 〉 = 〈 H ρ β * 0 m ∣ g 0 θ θ ∂ i E s g s s ∂ θ 〉 +
ω c ( 〈 H ρ β * 0 m ∣ g g ρ ρ g 0 θ θ H ρ 〉 + 〈 H ρ β * 0 m ∣ g 0 θ θ g g θ θ g ρ θ H θ ) 〉 + 〈 H ρ β * 0 m ∣ g 0 θ θ g g s s g ρ s H s ) 〉 )
− i 〈 H θ β * 0 m ∣ g 0 ρ ρ ∂ E ρ g ρ ρ ∂ s 〉 = − 〈 H θ β * 0 m ∣ g 0 ρ ρ ∂ i E s g s s ∂ ρ 〉 +
ω c ( 〈 H θ β * 0 m ∣ g 0 ρ ρ g g ρ ρ g θ ρ H ρ 〉 + 〈 H θ β * 0 m ∣ g g 0 ρ ρ g θ θ H θ ) 〉 + 〈 H θ β * 0 m ∣ g 0 ρ ρ g g s s g θ s H s 〉 )
− i 〈 E ρ β * 0 m ∣ g 0 θ θ ∂ H θ g θ θ ∂ s 〉 = − 〈 E ρ β * 0 m ∣ g 0 θ θ ∂ i H s g s s ∂ θ 〉 +
ω c ∊ ( 〈 E ρ β * 0 m ∣ g g ρ ρ g 0 θ θ E ρ 〉 + 〈 E ρ β * 0 m ∣ g 0 θ θ g g θ θ g ρ θ E θ 〉 + 〈 E ρ β * 0 m ∣ g 0 θ θ g g s s g ρ s E s 〉 )
i 〈 E θ β * 0 m ∣ g 0 ρ ρ ∂ H ρ g ρ ρ ∂ s 〉 = 〈 E θ β * 0 m ∣ g 0 ρ ρ ∂ i H s g s s ∂ ρ 〉 +
ω c ∊ ( 〈 E θ β * 0 m ∣ g 0 ρ ρ g g ρ ρ g θ ρ E ρ 〉 + 〈 E θ β * 0 m ∣ g g 0 ρ ρ g θ θ E θ 〉 + 〈 E θ β * 0 m ∣ g 0 ρ ρ g g s s g θs E ρ 〉 ) .
− i B ∂ C m β ( s ) ∂ s = ∑ β ′ m ′ C m ′ β ′ ( s ) 〈 ψ β * , m 0 ∣ M ̂ ∣ ψ β ′ , m ′ 0 〉 = ∑ β ′ m ′ C m ′ β ′ [ 〈 H ρ β * 0 m ∣ g 0 θ θ ∂ i E sβ ′ m ′ g s s ∂ θ 〉
− 〈 H θ β * 0 m ∣ g 0 ρ ρ ∂ i E s β ′ m ′ g s s ∂ ρ 〉 − 〈 E ρ β * 0 m ∣ g 0 θ θ ∂ i H sβ ′ m ′ g s s ∂ θ 〉 + 〈 E θ β * 0 m ∣ g 0 ρ ρ ∂ i H sβ ′ m ′ g s s ∂ ρ 〉
+ ω c ( 〈 H ρ β * 0 m ∣ g g ρ ρ g 0 θ θ H ρ β ′ 0 m ′ 〉 + 〈 H ρ β * 0 m ∣ g 0 θ θ g g θ θ g ρ θ H θ β ′ 0 m ′ 〉 + 〈 H ρ β * 0 m ∣ g 0 θ θ g g s s g ρ s H s β ′ 0 m ′ 〉 )
+ ω c ( 〈 H θ β * 0 m ∣ g 0 ρ ρ g g ρ ρ g θ ρ H ρ β ′ 0 m ′ 〉 + 〈 H θ β * 0 m ∣ g g 0 ρ ρ g θ θ H θ β ′ 0 m ′ 〉 + 〈 H θ β * 0 m ∣ g 0 ρ ρ g g s s g θ s H s β ′ 0 m ′ 〉 )
+ ω c ∊ ( 〈 E ρ β * 0 m ∣ g g ρ ρ g 0 θ θ E ρ β ′ 0 m ′ 〉 + 〈 E ρ β * 0 m ∣ g 0 θ θ g g θ θ g ρ θ E θ β ′ 0 m ′ 〉 + 〈 E ρ β * 0 m ∣ g 0 θ θ g g s s g ρ s H s β ′ 0 m ′ 〉 )
+ ω c ∊ ( 〈 E θ β * 0 m ∣ g 0 ρ ρ g g ρ ρ g θ ρ E ρ β ′ 0 m ′ 〉 + 〈 E θ β * 0 m ∣ g g 0 ρ ρ g θ θ H θ β ′ 0 m ′ 〉 + 〈 H θ β * 0 m ∣ g 0 ρ ρ g g s s g θ s H s β ′ 0 m ′ 〉 ) ] ,
i E s β ′ m ′ g s s = − c ω ∊ g g s s ( ∂ H θ β ′ 0 m ′ g 0 θ θ ∂ ρ − ∂ H ρ β ′ 0 m ′ g 0 ρ ρ ∂ θ ) − i g s s ( g s ρ g 0 ρ ρ E ρ β ′ 0 m ′ + g s θ g 0 θ θ E θ β ′ 0 m ′ )
i H s β ′ m ′ g s s = c ω g g s s ( ∂ E θ β ′ 0 m ′ g 0 θ θ ∂ ρ − ∂ E ρ β ′ 0 m ′ g 0 ρ ρ ∂ θ ) − i g s s ( g s ρ g 0 ρ ρ H ρ β ′ 0 m ′ + g s θ g 0 θ θ E θ β ′ 0 m ′ ) .
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