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Theoretical and computational concepts for periodic optical waveguides

G. Lecamp, J. P. Hugonin, and P. Lalanne

Open Access Open Access

Abstract

We present a general, rigorous, modal formalism for modeling light propagation and light emission in three-dimensional (3D) periodic waveguides and in aggregates of them. In essence, the formalism is a generalization of well-known modal concepts for translation-invariant waveguides to situations involving stacks of periodic waveguides. By surrounding the actual stack by perfectly-matched layers (PMLs) in the transverse directions, reciprocity considerations lead to the derivation of Bloch-mode orthogonality relations in the sense of E×H products, to the normalization of these modes, and to the proof of the symmetrical property of the scattering matrix linking the Bloch modes. The general formalism, which rigorously takes into account radiation losses resulting from the excitation of radiation Bloch modes, is implemented with a Fourier numerical approach. Basic examples of light scattering like reflection, transmission and emission in periodic-waveguides are accurately resolved.

© 2007 Optical Society of America

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Figures (7)
Fig. 1.
Fig. 1. Actual and computational geometries considered in this work. (a) Sketch of a geometry composed of aggregate of different periodic-waveguide sections. The geometry is assumed to be surrounded by infinite uniform claddings in the x- and y-transverse directions. (b) Associated computational system obtained by bounding the actual waveguide with PMLs (in blue) in the transverse directions. (c) A periodic-waveguide section of (a). (d) Associated periodic waveguide bound with PMLs. In (b) and (d), the PMLs have a finite thickness that is not represented for the sake of clarity.

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Fig. 2.
Fig. 2. The two solutions used for deriving the S-matrix reciprocity relation. At a given frequency ω, the QNBMs of the left periodic-waveguide section (z<L1), labelled by “L” for “left”, are denoted by Φ (p,L) and those on the right side (z>L2), labelled by “R” for “right”, by Φ (p,R), p being a relative integer. The geometries are arbitrary and may contain sources for the left and right ends of the structure and for L1<z<L2. The whole system is assumed to be surrounded by PMLs (not shown) everywhere in the transverse directions.

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Fig. 3.
Fig. 3. Some implications of Eq. 21. (a) The complex modal transmission-coefficients do not depend on the propagation sense. (b) Same property for the cross modal reflection-coefficients. (c) The excitation amplitude of a QNBM by a dipole source J δ(r-r0) located at point r=r0 is equal to the scalar product between the source J and the field E(r0) scattered at the dipole location by exciting the same geometry with the reciprocal QNBM.

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Fig. 4.
Fig. 4. Excitation of QNBMs by a Dirac dipole source J δ(r-r 0) located at point r=r 0. The D(p,R) and D(p,L) coefficients represent the modal amplitude coefficients of the excited forward- and backward-QNBMs, respectively. The periodic waveguide is not necessarily symmetric for the study, as shown by the échelette profile.

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Fig. 5.
Fig. 5. QNBM calculation of a PhC waveguide. (a) Schematic view of the PhC waveguide formed by removing a line defect in the ΓK direction of a 2D PhC structure composed of a triangular lattice of air holes (lattice constant a=0.24 µm) etched into a silicon slab (n=3.55). The slab thickness is 0.6a and the air holes radii 0.29a. The inset shows the dispersion relation of the fundamental guided QNBM Φ(-1). (b) Display of the 300-first normalized propagation constants of the QNBMs for a frequency a/λ=0.255, point A in the inset. Blue dots and red squares are obtained for (fPML)-1=(1+i) and (fPML)-1=5(1+i), respectively.

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Fig. 6.
Fig. 6. Scattering at the interface between two periodic sections. (a) Schematic top view of the 3D scattering problem. The PhC parameters are the same as in the caption of Fig. 5. (b) Convergence of the a-FMM for the modal reflectivity R of the fundamental guided QNBM Φ(-1). The calculation is performed for a/λ=0.255, point A in the inset of Fig. 5(a).

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Fig. 7.
Fig. 7. Dipole emission into PhC waveguides closed at one extremity by a PhC mirror. (a) Schematic top view of the 3D problem. The PhC parameters are the same as in the caption of Fig. 5. The dipole is parallel to the x-axis and is located in the central plane of the membrane at z0=0. (b) Convergence of the a-FMM for the β-factor defined as the power emitted into Φ(1) normalized to the total power emitted. The calculation is performed for a/λ=0.255, point A in the inset of Fig. 5(a).

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Equations (49)

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∇ × E = j ω μ ( r ) H and ∇ × H = − j ω ε ( r ) E + J δ ( r − r 0 ) ,
∇ × E 1 = j ω 1 μ H 1 and ∇ × H 1 = − j ω 1 ε E 1 + J 1 δ ( r − r 1 ) ,
∇ × E 2 = j ω 2 μ H 2 and ∇ × H 2 = − j ω 2 ε E 2 + J 2 δ ( r − r 2 ) .
∬ S ( E 2 × H 1 ) • dS = ∭ V j ( ω 1 E 2 T ε E 1 + ω 2 H 1 T μ H 2 ) d V − E 2 ( r 1 ) • J 1 .
∬ S ( E 2 × H 1 − E 1 × H 2 ) • d S = ∭ V j [ ω 1 ( E 2 T ε E 1 − H 2 T μ H 2 ) − ω 2 ( E 1 T ε E 2 − H 1 T μ H 2 ) ] d V
− [ E 2 ( r 1 ) • J 1 − E 1 ( r 2 ) • J 2 ] .
∬ z = z 2 ( E 2 × H 1 − E 1 × H 2 ) • z dS − ∬ z = z 1 ( E 2 × H 1 − E 1 × H 2 ) • z dS =
j ( ω 1 − ω 2 ) ∭ V ( E 1 T ε E 2 − H 1 T μ H 2 ) dV − [ J 1 • E 2 ( r 1 ) − J 2 • E 1 ( r 2 ) ] .
F z ( Φ 1 , Φ 2 ) = ∬ S ( E 2 × H 1 − E 1 × H 2 ) • z dS ,
E V ( Φ 1 , Φ 2 ) = ∭ V ( E 1 T ε E 2 − H 1 T μ H 2 ) dV .
F z 2 ( Φ 1 , Φ 2 ) − F z 1 ( Φ 1 , Φ 2 ) = j ( ω 1 − ω 2 ) E V + J 2 • E 1 ( r 2 ) − J 1 • E 2 ( r 1 ) .
∣ E ( m ) ( r + a z ) , H ( m ) ( r + a z ) > = ∣ E ( m ) ( r ) , H ( m ) ( r ) > ,
F z ( Φ ( p , ω ’ ) , Φ ( − q , ω ) ) ( 1 − exp { j [ k p ( ω ’ ) − k q ( ω ) ] a } ) = j ( ω − ω ’ ) E Cell ( z ) ( Φ ( p , ω ’ ) , Φ ( − q , ω ) ) .
F z ( Φ ( p , ω ) , Φ ( − q , ω ) ) = ∬ z ( E ( − q , ω ) × H ( p , ω ) − E ( p , ω ) × H ( − q , ω ) ) • z dS = F ( p , ω ) δ p , q ,
F z ( Φ ( p , ω ’ ) , Φ ( − p , ω ) ) { 1 − exp [ j { k p ( ω ’ ) − k p ( ω ) } a ] } = j ( ω − ω ’ ) E Cell ( z ) ( Φ ( p , ω ’ ) , Φ ( − p , ω ) ) .
F ( p , ω ) = E ( p , ω ) v g ( p ) ⁄ a ,
E ( p , ω ) = 2 ∭ Cell E ( − p ) T ε E ( p ) dV = − 2 ∭ Cell H ( p ) T μ H ( − p ) dV .
∣ E ( 1 , r ) , H ( 1 , r ) > = ∣ E ( − 1 , r ) * , − H ( − 1 , r ) * > ,
2 ∬ z Re ( E ( 1 , r ) × H ( 1 , r ) * ) • z dS = F ( 1 , ω ) , and
∭ Cell [ E ( 1 , r ) * ε E ( 1 , r ) + H ( 1 , r ) * μ H ( 1 , r ) ] dV = E ( 1 , ω ) ,
F L 2 ( Φ 1 , Φ 2 ) − F L 1 ( Φ 1 , Φ 2 ) = E 1 ( r 2 ) • J 2 − E 2 ( r 1 ) • J 1 .
Φ 1 = ∑ p > 0 I 1 ( p , L ) Φ ( p , L ) + D 1 ( p , L ) Φ ( − p , L ) ,
Φ 2 = ∑ p > 0 I 2 ( p , L ) Φ ( p , L ) + D 2 ( p , L ) Φ ( − p , L ) ,
F L 1 ( Φ 1 , Φ 2 ) = 4 ∑ p > 0 ( I 1 ( p , L ) D 2 ( p , L ) − I 2 ( p , L ) D 1 ( p , L ) ) .
F L 2 ( Φ 1 , Φ 2 ) = 4 ∑ p > 0 ( I 2 ( p , R ) D 1 ( p , R ) − I 1 ( p , R ) D 2 ( p , R ) ) .
4 ∑ p > 0 ( I 1 ( p , L ) D 2 ( p , L ) + I 1 ( p , R ) D 2 ( p , R ) ) − E 2 ( r 1 ) • J 1 = 4 ∑ p > 0 ( I 2 ( p , L ) D 1 ( p , L ) + I 2 ( p , R ) D 1 ( p , R ) ) − E 1 ( r 2 ) • J 2 .
( I 1 ) T S I 2 − E 2 ( r 1 ) • J 1 = ( I 2 ) T S I 1 − E 1 ( r 2 ) • J .
for z > z 0 , Φ = ∑ p > 0 D ( p , R ) Φ ( p ) ,
and for z < z 0 , Φ = ∑ p > 0 D ( p , L ) Φ ( −p ) ,
D ( m , L ) = − E ( m ) ( r 0 ) • J exp ( j k m z 0 ) ⁄ 4 , and
D ( m , R ) = − E ( −m ) ( r 0 ) • J exp ( −j k m z 0 ) ⁄ 4 ,
P 1 = P − 1 = ∣ I ∣ 2 ∣ E ( − 1 ) ( r 0 ) • u ∣ 2 ⁄ 16 .
H ( r ) = ∑ p , q ( U xpq x + U ypq y + U xpq z ) exp ( jp G x x + jq G y y ) ,
E ( r ) = ∑ p , q ( S xpq x + S y pq y + S zpq z ) exp ( j p G x x + j q G y y ) ,
1 k 0 d [ Ψ ] d z = Ω ( z ) [ Ψ ] ,
Ψ ( p ) = ∑ n = 1 N b n ( p ) exp ( − λ n ( p ) z ) W n ( p ) + f n ( p ) exp ( λ n ( p ) z ) W − n ( p ) ,
[ b ( i ) f ( t ) ] = [ S 11 S 12 S 21 S 22 ] [ b ( t ) f ( i ) ] ,
[ I − S 12 0 S 22 ] [ b ( i ) f ( i ) ] = ρ [ S 11 0 − S 21 I ] [ b ( i ) f ( i ) ] .
[ b QNM f QNBM ] = S T [ b QNBM f QNM ] = [ S 11 S 12 S 21 S 22 ] [ b QNBM f QNM ] .
S 22 = ( F + ) − 1 , S 12 = B + S 22 , S 21 = − S 22 F − and S 11 = B − − S 12 F − ,
[ b M f W − ] = S T [ b W − f M ] .
[ b W + f W + ] = [ − D ( L ) D ( R ) ] + [ b W − f W − ] ,
Φ ( r + a z , ω ) = Φ ( r , ω ) exp ( jk a ) .
F z 2 ( Φ , Φ m ) − F z 1 ( Φ , Φ m ) = ∣ E ( m ) ( r 0 ) ∣ 2 exp ( j k m z 0 ) .
F z 1 ( Φ , Φ m ) = ∣ E ( m ) ( r 0 ) ∣ 2 exp ( j k m z 0 ) { exp [ j ( k m + k ) a ] − 1 } − 1
X ˆ = X ( x ) , Y ˆ = Y ( y ) , Z ˆ = Z ( z ) ,
μ ˆ = L μ L ⁄ Det ( L ) , ε ˆ = L ε L ⁄ Det ( L ) , J ˆ = L J ,
∭ V ( E T ε E − H T μ H ) dV = ∭ V ˆ ( E ˆ T ε ˆ E ˆ − H ˆ T μ ˆ H ˆ ) d V ˆ .
∬ S ( E × H ) • z dS = ∬ S ˆ ( E ˆ × H ˆ ) • z d S ˆ .
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