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.

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References

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  1. A. Yariv, Optical Electronics in Modern Communications, (Oxford University Press, 1997).
  2. N.W. Ashcroft and N.D. Mermin, Solid State Physics, (Harcourt, 1976).
  3. A.Yariv, Y.Xu, R.K. Lee, and A.Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711 (1999).
    [CrossRef]
  4. Y. Xu, R.K. Lee, and A. Yariv, "Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide," J. Opt. Soc. Am. B 17, 387 (2000).
    [CrossRef]
  5. S. Mookherjea and A. Yariv, "Optical pulse propagation and holographic storage in a coupled-resonator optical waveguide," Submitted to Phys. Rev. E (June 2001).
  6. M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140 (2000).
    [CrossRef] [PubMed]
  7. N.G.R. Broderick and C.M. de Sterke, "Theory of grating superstructures," Phys. Rev. E 55, 3634 (1997).
    [CrossRef]
  8. C.M. de Sterke, "Superstructure gratings in the tight-binding approximation," Phys. Rev. E 57, 3502 (1998).
    [CrossRef]
  9. J.D. Jackson, Classical Electrodynamics, third edition, (John Wiley & Sons, 1999).
  10. T.M. Apostol, Mathematical Analysis, (Addison-Wesley, 1964).
  11. A.V. Oppenheim, A.S. Willksy, and I.T. Young, Signals and Systems, (Prentice-Hall, 1995).
  12. J.M. Wozencraft and I.M. Jacobs, Principles of Communication Engineering, (John Wiley & Sons, 1965).

Other (12)

A. Yariv, Optical Electronics in Modern Communications, (Oxford University Press, 1997).

N.W. Ashcroft and N.D. Mermin, Solid State Physics, (Harcourt, 1976).

A.Yariv, Y.Xu, R.K. Lee, and A.Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711 (1999).
[CrossRef]

Y. Xu, R.K. Lee, and A. Yariv, "Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide," J. Opt. Soc. Am. B 17, 387 (2000).
[CrossRef]

S. Mookherjea and A. Yariv, "Optical pulse propagation and holographic storage in a coupled-resonator optical waveguide," Submitted to Phys. Rev. E (June 2001).

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140 (2000).
[CrossRef] [PubMed]

N.G.R. Broderick and C.M. de Sterke, "Theory of grating superstructures," Phys. Rev. E 55, 3634 (1997).
[CrossRef]

C.M. de Sterke, "Superstructure gratings in the tight-binding approximation," Phys. Rev. E 57, 3502 (1998).
[CrossRef]

J.D. Jackson, Classical Electrodynamics, third edition, (John Wiley & Sons, 1999).

T.M. Apostol, Mathematical Analysis, (Addison-Wesley, 1964).

A.V. Oppenheim, A.S. Willksy, and I.T. Young, Signals and Systems, (Prentice-Hall, 1995).

J.M. Wozencraft and I.M. Jacobs, Principles of Communication Engineering, (John Wiley & Sons, 1965).

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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.

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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