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Causality and Kramers-Kronig relations for waveguides

Magnus W. Haakestad and Johannes Skaar

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Abstract

Starting from the condition that optical signals propagate causally, we derive Kramers-Kronig relations for waveguides. For hollow waveguides with perfectly conductive walls, the modes propagate causally and Kramers-Kronig relations between the real and imaginary part of the mode indices exist. For dielectric waveguides, there exists a Kramers-Kronig type relation between the real mode index of a guided mode and the imaginary mode indices associated with the evanescent modes. For weakly guiding waveguides, the Kramers-Kronig relations are particularly simple, as the modal dispersion is determined solely from the profile of the corresponding mode field.

©2005 Optical Society of America

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Figures (7)
Fig. 1.
Fig. 1. The real and imaginary part of the mode index for a hollow waveguide with perfectly conducting walls is shown in (a). The derivative of the real part of the mode index with respect to frequency is shown in (b). Also shown are results obtained using the Kramers-Kronig relations, where the real part of the mode index is determined from the imaginary part and vice versa. In all cases the results from the Kramers-Kronig relations are in excellent agreement with the exact results. We find that any discrepancy is only dependent on the numerical resolution in the calculations.

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Fig. 2.
Fig. 2. Real and imaginary part of the mode index for a dielectric-filled waveguide with perfectly conducting walls. Exact results and results obtained using the Kramers-Kronig relations for waveguides are shown.

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Fig. 3.
Fig. 3. The fundamental mode in a planar index-guiding waveguide for ωd/c = 40 (a), and the resulting integral Eq. (38) over evanescent modes (b).

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Fig. 4.
Fig. 4. Mode index for the fundamental TE mode in a dielectric waveguide (a). In (b) dn0r/dω is shown, based on a solution of the wave equation, together with results based on the Kramers-Kronig approach.

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Fig. 5.
Fig. 5. Mode index for the second order symmetric TE mode in a dielectric waveguide (a). In (b) dn2r/dω is shown, based on a solution of the wave equation, together with results based on the Kramers-Kronig approach.

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Fig. 6.
Fig. 6. The refractive index profile of a symmetric Bragg reflection waveguide is shown in (a). The fundamental guided mode for ωΛ/c = 35 is shown in (b).

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Fig. 7.
Fig. 7. The mode index n0r of the fundamental guided mode as a function of frequency is shown in (a) together with the band edges of the first bandgap. dnr 0/dω is shown in (b), based on a solution of the wave equation, together with results based on the Kramers-Kronig approach.

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Equations (44)

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E ( z , t ) = ∫ Ω 1 ∪ ( − Ω 1 ) ∑ j A j ( ω ) ψ j ( ω ) exp ⁡ [ i β j ( ω ) z ] exp ⁡ ( − iωt ) d ω
+ ∫ Ω 2 ∪ ( − Ω 2 ) B ( ω ) χ ( z , ω ) exp ⁡ [ iωz c ] exp ⁡ ( − iωt ) d ω .
∫ − ∞ ∞ ψ i * ( ω ) ψ j ( ω ) d y ≡ 〈 ψ i ( ω ) | ψ j ( ω ) 〉 = δ ij ,
∑ j ∣ 〈 ψ | ψ j ( ω ) 〉 ∣ 2 = 1
E ( 0 , t ) = ψ src ∫ − ∞ ∞ A in ( ω ) exp ⁡ ( − iωt ) d ω .
E ( L , t ) = ψ det ∫ − ∞ ∞ A out ( ω ) exp ⁡ ( − iωt ) d ω .
A out ( ω ) = G ( ω ) A in ( ω ) ,
G ( ω ) = { ∑ j 〈 ψ det ∣ ψ j ( ω ) 〉 〈 ψ j ( ω ) ∣ ψ src 〉 exp ⁡ [ i β j ( ω ) L ] if ω ∈ Ω 1 , 〈 ψ det ∣ χ ( L , ω ) 〉 exp ⁡ [ iωL c ] if ω ∈ Ω 2 .
a out ( t ) = 1 2 π ∫ − ∞ ∞ g ( t − t ′ ) a in ( t ′ ) d t ′ ,
G ( ω ) = 1 2 π ∫ L c ∞ g ( t ) exp ⁡ ( iωt ) d t ,
G ˜ ( ω ) ≡ G ( ω ) exp ⁡ ( − iωL c ) = 1 2 π ∫ 0 ∞ g ( t + L c ) exp ⁡ ( iωt ) d t .
ψ src = ψ det = ψ 0 ( ω 0 ) ,
G ˜ ( ω ) = { ∑ j ∣ 〈 ψ 0 ( ω 0 ) ∣ ψ j ( ω ) 〉 ∣ 2 exp ⁡ { i [ n j ( ω ) − 1 ] ωL c } if ω ∈ Ω 1 〈 ψ 0 ( ω 0 ) ∣ χ ( L , ω ) 〉 if ω ∈ Ω 2 ,
n j ( ω ) = c β j ( ω ) ω .
F ( ω ) ≡ c iL [ G ˜ ( ω ) − 1 ] = { ∑ j ∣ 〈 ψ 0 ( ω 0 ) ∣ ψ j ( ω ) 〉 ∣ 2 [ n j ( ω ) − 1 ] ω if ω ∈ Ω 1 c iL [ 〈 ψ 0 ( ω 0 ) ∣ χ ( L , ω ) 〉 − 1 ] if ω ∈ Ω 2 ,
Re F ( ω ) = 2 ω π P ∫ 0 ∞ Im F ( ω ′ ) ω ′ 2 − ω 2 d ω ′
Im F ( ω ) = − 2 π P ∫ 0 ∞ ω ′ Re F ( ω ′ ) ω ′ 2 − ω 2 d ω ′ ,
c j ( ω 0 , ω ) ≡ ∣ 〈 ψ 0 ( ω 0 ) | ψ j ( ω ) 〉 ∣ 2 .
2 ω π P ∫ 0 ∞ Im F ( ω ′ ) ω ′ 2 − ω 2 d ω ′ = 2 ω π P ∫ Ω 1 ∑ ′ j c j ( ω 0 , ω ′ ) n j i ( ω ′ ) ω ′ ω ′ 2 − ω 2 d ω ′
− 2 ωc πL ∫ Ω 2 Re 〈 ψ 0 ( ω 0 ) ∣ χ ( L , ω ′ ) 〉 − 1 ω ′ 2 − ω 2 d ω ′ ,
δ ( ω 0 , ω ) = − 2 c πL ∫ Ω 2 Re 〈 ψ 0 ( ω 0 ) ∣ χ ( L , ω ′ ) 〉 − 1 ω ′ 2 − ω 2 d ω ′ ,
∑ j c j ( ω 0 , ω ) [ n j r ( ω ) − 1 ] = 2 π P ∫ Ω 1 ∑ ′ j c j ( ω 0 , ω ′ ) n j i ( ω ′ ) ω ′ ω ′ 2 − ω 2 d ω ′ + δ ( ω 0 , ω ) ,
d n 0 r ( ω ) d ω ∣ ω = ω 0 = 4 ω 0 π P ∫ Ω 1 ∑ ′ j c j ( ω 0 , ω ′ ) n j i ( ω ′ ) ω ′ ( ω ′ 2 − ω 0 2 ) 2 d ω ′ + ∂ δ ( ω 0 , ω ) ∂ ω ∣ ω = ω 0 .
∣ ∂ δ ∂ ω ∣ = 4 cω πL ∣ ∫ Ω 2 Re 〈 ψ 0 ( ω 0 ) ∣ χ ( L , ω ′ ) 〉 − 1 ( ω ′ 2 − ω 2 ) 2 d ω ′ ∣ ≤ 8 cω πL ∫ Ω 2 d ω ′ ( ω ′ 2 − ω 2 ) 2 ,
d n 0 r ( ω ) d ω = 4 ω π P ∫ Ω 1 ∑ ′ j c j ( ω , ω ′ ) n j i ( ω ′ ) ω ′ ( ω ′ 2 − ω 2 ) 2 d ω ′ .
n 0 r ( ω ) − 1 = 2 π P ∫ 0 ∞ ∑ j c j ( ω , ω ′ ) n j i ( ω ′ ) ω ′ ω ′ 2 − ω 2 d ω ′ .
G ( ω ) = exp ⁡ [ i β j ( ω ) L ] .
β j ( ω ) = 1 c ω 2 − ω j , c 2 ,
n j r ( ω ) = { 0 if ω j , c > ω ≥ 0 1 − ( ω j , c ω ) 2 if ω ≥ ω j , c
n j i ( ω ) = { ( ω j , c ω ) 2 − 1 if ω j , c > ω ≥ 0 0 if ω ≥ ω j , c .
F ( ω ) = [ n j ( ω ) − 1 ] ω .
[ n j r ( ω ) − 1 ] ω = 2 ω π P ∫ 0 ∞ n j i ( ω ′ ) ω ′ ω ′ 2 − ω 2 d ω ′
n j i ( ω ) ω = − 2 π P ∫ 0 ∞ ω ′ [ n j r ( ω ′ ) − 1 ] ω ′ ω ′ 2 − ω 2 d ω ′ .
d n j r ( ω ) d ω = 4 ω π ∫ 0 ω j , c n j i ( ω ′ ) ω ′ ( ω ′ 2 − ω 2 ) 2 d ω ′ .
ε r ( ω ) = 1 + ω p 2 ω res 2 − ω 2 − iγω ,
ψ j ( ω ′ ) = 1 π cos ⁡ ( k t y ) ,
k t = ( ω ′ c ) 2 + ∣ β j ∣ 2 ,
c ( k t ) ≡ c j ( ω , ω ′ ) = 1 π ∣ ∫ − ∞ ∞ cos ⁡ ( k t y ) ψ 0 ( ω ) d y ∣ 2
= 1 π ∣ ∫ − ∞ ∞ ψ 0 ( ω ) exp ⁡ ( i k t y ) d y ∣ 2 ,
∑ ′ j c j ( ω , ω ′ ) n j i ( ω ′ ) ω ′ = c ∫ ω ′ c ∞ c ( k t ) k t 2 − ( ω ′ c ) 2 d k t ,
d n 0 r ( ω ) d ω = 4 ωc π P ∫ 0 ∞ ∫ ω ′ c ∞ c ( k t ) k t 2 − ( ω ′ c ) 2 d k t ( ω ′ 2 − ω 2 ) 2 d ω ′ .
E K ( y , z ) = E K ( y ) exp ⁡ ( iKy ) exp ⁡ ( iβz ) ,
K = m π Λ + i K i , m = 1,2 …
E c ( y , z ) = cos ⁡ ( k 1 y ) exp ⁡ ( iβz ) ,
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