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Optical pulse propagation in the tight-binding approximation

Shayan Mookherjea and Amnon Yariv

Open Access Open Access

Abstract

We formulate the equations describing pulse propagation in a one-dimensional optical structure described by the tight binding approximation, commonly used in solid-state physics to describe electrons levels in a periodic potential. The analysis is carried out in a way that highlights the correspondence with the analysis of pulse propagation in a conventional waveguide. Explicit expressions for the pulse in the waveguide are derived and discussed in the context of the sampling theorems of finite-energy space and time signals.

©2001 Optical Society of America

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Figures (1)
Fig. 1.
Fig. 1. (752 kB) Pulse propagation in a CROW structure described by the tight binding approximation. The envelope of the eigenmode of the structure is shown in red, and the Gaussian pulse envelope in blue, propagating from left to right, indexed by an arbitrary time coordinate at the upper-right corner.

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Equations (26)

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ϕ k ( z ) = ∑ n exp ( − i n k R ) ∑ l b l ψ l ( z − n R )
k m = m ( 2 π L )
ε ( z , t = 0 ) = ∫ d k 2 π c k ϕ k ( z )
ϕ k ( z ) = [ ∣ Δ k ∣ ∑ m = − ∞ ∞ δ ( k − m Δ k ) ] ∑ n exp ( − inkR ) ∑ l b l ψ l ( z − n R )
ε ( z , t ) = ∫ d k 2 π e i ω ( k ) t c k ϕ k ( z ) .
ω ( k 0 + K ) = ω ( k 0 ) + d ω d ω ∣ k = k 0 K + … ≈ ω 0 + v g K
ε ( z , t ) = e i ω 0 t ∫ d K 2 π e i v g t K c k 0 + K ϕ k 0 + K ( z ) .
ε ( z = 0 , t ) = e i ω 0 t E ( z = 0 , t ) ,
c k 0 + K = 1 ϕ k 0 + K ( 0 ) ∫ d ( ∣ v g ∣ t ′ ) E ( z = 0 , t ′ ) e − i v g t ′ K .
ε ( z , t ) = e i ω 0 t ∫ d ( ∣ v g ∣ t ′ ) E ( z = 0 , t ′ ) ∫ d K 2 π ϕ k 0 + K ( z ) ϕ k 0 + K ( 0 ) e i v g ( t − t ′ ) K .
ε ( z , t ) = e i ω 0 t ∫ d ( ∣ v g ∣ t ′ ) E ( z = 0 , t ′ ) ∫ d K 2 π e − i ( k 0 + K ) z e i v g ( t − t ′ ) K
= e i ( ω 0 t − k 0 z ) E ( z = 0 , t − z v g ) .
ϕ k 0 + K ( 0 ) = ∑ n e − i ( k 0 + K ) n R ∑ l b l ψ ( − n R )
= 1 + ∑ l b l ψ l ( R ) 2 cos ⁡ [ ( k 0 + K ) R ] + …
[ ϕ k 0 + K ( 0 ) ] − 1 ≈ 1 − ∑ l b l ψ l ( R ) 2 cos ⁡ [ ( k 0 + K ) R ] ,
ε ( z , t ) = e i ω 0 t ∑ n e − i k 0 n R ∑ l b l ψ l ( z − n R ) ∫ d ( ∣ v g ∣ t ′ ) E ( z = 0 , t ′ )
× ∫ d K 2 π [ ∣ Δ K ∣ ∑ m δ ( K − m Δ K ) ] e i T K
∣ Δ K ∣ ∑ m = − ∞ ∞ δ ( K − m Δ K ) ⥦ Ƒ Ʈ ∑ m = − ∞ ∞ δ ( T − m Δ T )
ε ( z , t ) = e i ω 0 t ∑ n e − i k 0 n R ∑ l b l ψ l ( z − n R ) ∑ m E ( z = 0 , t − n R + m L v g ) .
Δ ε ( z , t ) = − ∑ l ′ b l ′ ψ l ′ ( R ) e i ω 0 t { ∑ n e − i k 0 ( n − 1 ) R ∑ l b l ψ l ( z − n R ) ×
∑ m E ( z = 0 , t − ( n − 1 ) R + m L v g ) + ∑ n e − i k 0 ( n + 1 ) R ×
∑ l b l ψ l ( z − n R ) ∑ m E ( z = 0 , t − ( n + 1 ) R + m L v g ) } .
2 π ( 2 π / L ) v g = 2 T max which implies T max = 1 2 L v g .
2 π R = 2 K max which implies K max = 1 2 ( 2 π R ) .
1 2 v g T min = R which implies T min = 2 R v g .
D = ( 2 π R v g ) ( 1 2 L v g ) + 1 = π N + 1 ,
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