Abstract

We use a tight-binding formalism in the time domain to analyze the effect of resonant gain enhancement and spontaneous emission noise in amplifying coupled-resonator optical waveguides (CROWs). We find the net amplification of a wave propagating in a CROW does not always vary with the group velocity, and depends strongly on the termination and excitation of these structures. The signal-to-noise ratio and noise figure of CROW amplifiers are derived in the tight-binding formalism as well. The physical interpretations and practical consequences of the theoretical results are discussed.

© 2007 Optical Society of America

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    [CrossRef]
  6. S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Engineering photonic crystal impurity bands for waveguides, all-optical switches and optical delay lines," IEICE Trans. Electron. E85C, 181-189 (2002).
  7. B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
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    [CrossRef]
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  23. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press, 1997).
  24. P. Chak and J. Sipe, "Minimizing finite-size effects in artificial resonance tunneling structures," Opt. Lett. 13, 2568-2570 (2006).
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  25. W. C. Yueh, "Explicit inverses of several tridiagonal matrices," Appl. Math. E-Notes 6, 74-83 (2006).
  26. U. Peschel, A. L. Reynolds, B. Arredondo, F. Lederer, P. J. Roberts, T. F. Krauss, and P. J. I. de Maagt, "Transmission and reflection analysis of functional coupled cavity components," IEEE J. Quantum Electron. 38, 830-836 (2002).
    [CrossRef]
  27. T. Mukai and Y. Yamamoto, "Noise in an AlGaAs semiconductor-laser amplifier," IEEE J. Quantum Electron. 18, 564-575 (1982).
    [CrossRef]
  28. N. A. Olsson, "Heterodyne gan and noise measurement of a 1.5μm resonasnt semiconductor-laser amplifier," IEEE J. Quantum Electron. 22, 671-676 (1986).
    [CrossRef]
  29. R. J. Lang and A. Yariv, "Semiclassical theory of noise in multielement semiconductor lasers," IEEE J. Quantum Electron. 22, 436-449 (1986).
    [CrossRef]
  30. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989).
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    [CrossRef] [PubMed]
  32. G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold Company, 1986).
  33. S. Mookherjea, "Spectral characteristics of coupled resonators," J. Opt. Soc. Am. B 23, 1137-1145 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]

2007 (3)

2006 (7)

2004 (3)

2002 (3)

L. Florescu, K. Busch, and S. John, "Semiclassical theory of lasing in photonic crystals," J. Opt. Soc. Am. B 19, 2215-2223 (2002).
[CrossRef]

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Engineering photonic crystal impurity bands for waveguides, all-optical switches and optical delay lines," IEICE Trans. Electron. E85C, 181-189 (2002).

U. Peschel, A. L. Reynolds, B. Arredondo, F. Lederer, P. J. Roberts, T. F. Krauss, and P. J. I. de Maagt, "Transmission and reflection analysis of functional coupled cavity components," IEEE J. Quantum Electron. 38, 830-836 (2002).
[CrossRef]

2001 (2)

M. Bayindir, S. Tanriseven, and E. Ozbay, "Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures," Appl. Phys. A 72, 117-119 (2001).
[CrossRef]

S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdre, and U. Osterle, "Miniband transmission in a photonic crystal waveguide coupled-resonator optical waveguide," Opt. Lett. 26, 1019-1051 (2001).
[CrossRef]

1999 (1)

1998 (2)

S. Nojima, "Enhancement of optical gain in two-dimensional photonic crystals with active lattice points," Jpn. J. Appl. Phys., Part 2 37, L565-L567 (1998).
[CrossRef]

N. Stefanou and A. Modinos, "Impurity bands in photonic insulators," Phys. Rev. B 57, 12,127-12,133 (1998).
[CrossRef]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, "Microring resonator channel dropping filter," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

1994 (1)

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, "The photonic band-edge laser--a new approach to gain enhancement," Appl. Phys. Lett. 75, 1896-1899 (1994).
[CrossRef]

1988 (1)

S. S. Wang and H. G. Winful, "Dynamics of phase-locked semiconductor laser arrays," Appl. Phys. Lett. 52, 1774-1776 (1988).
[CrossRef]

1986 (3)

C. H. Henry, "Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers," J. Lightwave Technol. 4, 288-297 (1986).
[CrossRef]

N. A. Olsson, "Heterodyne gan and noise measurement of a 1.5μm resonasnt semiconductor-laser amplifier," IEEE J. Quantum Electron. 22, 671-676 (1986).
[CrossRef]

R. J. Lang and A. Yariv, "Semiclassical theory of noise in multielement semiconductor lasers," IEEE J. Quantum Electron. 22, 436-449 (1986).
[CrossRef]

1982 (1)

T. Mukai and Y. Yamamoto, "Noise in an AlGaAs semiconductor-laser amplifier," IEEE J. Quantum Electron. 18, 564-575 (1982).
[CrossRef]

Appl. Math. E-Notes (1)

W. C. Yueh, "Explicit inverses of several tridiagonal matrices," Appl. Math. E-Notes 6, 74-83 (2006).

Appl. Phys. A (1)

M. Bayindir, S. Tanriseven, and E. Ozbay, "Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures," Appl. Phys. A 72, 117-119 (2001).
[CrossRef]

Appl. Phys. Lett. (2)

S. S. Wang and H. G. Winful, "Dynamics of phase-locked semiconductor laser arrays," Appl. Phys. Lett. 52, 1774-1776 (1988).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, "The photonic band-edge laser--a new approach to gain enhancement," Appl. Phys. Lett. 75, 1896-1899 (1994).
[CrossRef]

IEEE J. Quantum Electron. (4)

U. Peschel, A. L. Reynolds, B. Arredondo, F. Lederer, P. J. Roberts, T. F. Krauss, and P. J. I. de Maagt, "Transmission and reflection analysis of functional coupled cavity components," IEEE J. Quantum Electron. 38, 830-836 (2002).
[CrossRef]

T. Mukai and Y. Yamamoto, "Noise in an AlGaAs semiconductor-laser amplifier," IEEE J. Quantum Electron. 18, 564-575 (1982).
[CrossRef]

N. A. Olsson, "Heterodyne gan and noise measurement of a 1.5μm resonasnt semiconductor-laser amplifier," IEEE J. Quantum Electron. 22, 671-676 (1986).
[CrossRef]

R. J. Lang and A. Yariv, "Semiclassical theory of noise in multielement semiconductor lasers," IEEE J. Quantum Electron. 22, 436-449 (1986).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

S. Mookherjea, "Using gain to tune the dispersion relation of coupled-resonator optical waveguides," IEEE Photon. Technol. Lett. 18, 715-717 (2006).
[CrossRef]

IEICE Trans. Electron. (1)

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Engineering photonic crystal impurity bands for waveguides, all-optical switches and optical delay lines," IEICE Trans. Electron. E85C, 181-189 (2002).

J. Lightwave Technol. (3)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, "Microring resonator channel dropping filter," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

C. H. Henry, "Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers," J. Lightwave Technol. 4, 288-297 (1986).
[CrossRef]

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, "Polymer microring coupled-resonator optical waveguides," J. Lightwave Technol. 24, 1843-1849 (2006).
[CrossRef]

J. Opt. Soc. Am. B (4)

Jpn. J. Appl. Phys., Part 2 (1)

S. Nojima, "Enhancement of optical gain in two-dimensional photonic crystals with active lattice points," Jpn. J. Appl. Phys., Part 2 37, L565-L567 (1998).
[CrossRef]

Nat. Photonics (1)

F. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Phys. Rev. B (1)

N. Stefanou and A. Modinos, "Impurity bands in photonic insulators," Phys. Rev. B 57, 12,127-12,133 (1998).
[CrossRef]

Other (6)

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).
[CrossRef]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press, 1997).

J. K. S. Poon, P. Chak, J. M. Choi, and A. Yariv, "Slowing light with Fabry-Perot resonator arrays," J. Opt. Soc. Am. B, submitted for publication.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989).

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold Company, 1986).

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Figures (6)

Fig. 1
Fig. 1

Various configurations of coupled resonators: (a) infinitely long CROWs, (b) finite CROWs in isolation, (c) finite CROWs with out-coupling at the ends, and (d) finite CROWs with an input optical field with out-coupling at the ends.

Fig. 2
Fig. 2

Im [ ω n ] versus Re [ ω n ] Ω for the CROW with out-coupling at two ends. τ e = 10 4 , τ i = 5 × 10 4 , κ = 0.1 , and N = 20 .

Fig. 3
Fig. 3

The transmittance, S t S i n 2 , of CROWs for various values of τ e . The other parameters are τ i = 5 × 10 4 , κ = 0.1 , and N = 10 . Only the portion of S t S i n 2 2 is shown for comparison.

Fig. 4
Fig. 4

The exact solution of the transmittance from Eq. (21) and the approximation given by Eq. (22) as a function of the number of resonators ( N ) at the band-center frequency with optical loss. The other parameters are τ i = 5 × 10 3 , κ = 0.01 , τ e = 1 κ = 100 .

Fig. 5
Fig. 5

The normalized SNR factor, G, as a function of wavelength at various gain levels. For the calculations, τ l = 10 4 , κ = 0.01 , τ e = 100 , N = 10 .

Fig. 6
Fig. 6

The NF as a function of the number of resonators ( N ) in an active CROW at the band-center frequency where the losses are exactly compensated. The parameters for the calculations are in the text.

Equations (62)

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P ( r , t ) = ϵ 0 χ ( r ) E + ϵ 0 p ( r , t ) ,
× × E ( r , t ) + 1 c 2 [ ϵ ( r ) + i σ ( r ) ] E ̈ ( r , t ) = 1 c 2 p ̈ ( r , t ) .
E ( r , t ) = exp ( i ω t ) n = 1 N a n ( t ) E Ω ( r n Λ z ̂ ) ,
× × E Ω ( r ) = Ω 2 c 2 ϵ Ω ( r ) E Ω ( r ) ,
d 3 rE Ω * ( r Λ z ̂ ) f ( r ) E Ω ( r ) d 3 rE Ω * ( r ) f ( r ) E Ω ( r ) ,
2 i ω a ̇ m ( 1 + Δ α + i σ m ) = a m [ ( ω 2 Ω 2 ) + ω 2 ( Δ α + i σ m ) ] + a m + 1 [ ω 2 ( d + i Δ σ ) Ω 2 b ] + a m 1 [ ω 2 ( d * + i Δ σ * ) Ω 2 b * ] p ̈ m exp ( i ω t ) ,
Δ α = d 3 rE Ω * ( r ) [ ϵ ( r ) ϵ Ω ( r ) ] E Ω ( r ) ,
b = d 3 rE Ω * ( r ) ϵ Ω ( r Λ z ̂ ) E Ω ( r Λ z ̂ ) ,
d = d 3 rE Ω * ( r ) ϵ ( r ) E Ω ( r Λ z ̂ ) ,
σ m = d 3 rE Ω * ( r ) σ ( r ) E Ω ( r ) ,
Δ σ m = d 3 rE Ω * ( r ) σ ( r ) E Ω ( r Λ z ̂ ) ,
p m = d 3 rE Ω * ( r m Λ z ̂ ) p ( r ) .
i a ̇ m = a m [ ( ω Ω ) + i ω σ m 2 ] + κ a m + 1 + κ * a m 1 p ̈ m 2 ω exp ( i ω t ) ,
κ = ω 2 d 3 rE Ω * ( r ) [ ϵ ( r + m Λ z ̂ ) ϵ Ω ( r Λ z ̂ ) ] E Ω ( r Λ z ̂ ) .
a ̇ m = a m [ i ( ω Ω ) + 1 τ i ] i κ ( a m + 1 + a m 1 ) i ω s m ( t ) ,
( ω Ω ) = 2 κ cos ( K R Λ ) cosh ( K I Λ ) ,
1 τ i + 2 κ sin ( K R Λ ) sinh ( K I Λ ) = 0 .
sinh ( K I Λ ) = Λ 2 τ i v ¯ g .
i ω a = [ i Ω + 1 τ i i κ 0 0 0 0 i κ i Ω + 1 τ i i κ 0 0 0 . . . . . . . . . . . . . . . . . . . i κ i Ω + 1 τ i ] a .
ω n = ( Ω i τ i ) 2 κ cos ( n π N + 1 ) , n = 1 N ,
a m = sin ( m n π N + 1 ) , m = 1 N .
i ω a = [ i Ω + 1 τ i 1 τ e i κ 0 0 0 0 i κ i Ω + 1 τ i i κ 0 0 0 . . . . . . . . . . . . . . . . . . . i κ i Ω + 1 τ i 1 τ e ] a ,
ω n = Ω + 2 κ cos ( n π N + 1 ) + i [ 1 τ i + 2 τ e sin 2 ( n π N + 1 ) m = 1 N sin 2 ( m ( n π N + 1 ) ) ] .
i ω a = [ i Ω + 1 τ i 1 τ e i κ 0 0 0 0 i κ i Ω + 1 τ i i κ 0 0 0 . . . . . . . . . . . . . . . . . . . i κ i Ω + 1 τ i 1 τ e ] a i μ [ S i n 0 . . 0 ] M a i μ s i n .
a = i μ ( i ω I M ) 1 s i n i μ T s i n .
2 τ e a 1 i μ S i n = 0 .
S t S i n 2 = 2 τ e T N , 1 2 .
T N , 1 = κ sin ( ϕ ) i τ e 2 sin ( ( N 1 ) ϕ ) + 2 κ τ e sin ( N ϕ ) i κ 2 sin ( ( N + 1 ) ϕ ) ,
cos ( ϕ ) = ( ω Ω ) 2 κ i 2 κ τ i ,
S t S i n b c 4 e N 2 κ τ i 1 ( κ τ e ) + 2 + κ τ e τ i < 0 .
f ( t ) = f ̃ ( ω ̃ ) exp ( i ω ̃ t ) d ω ̃ ,
f ̃ ( ω ̃ ) = 1 2 π f ( t ) exp ( i ω ̃ t ) d t .
i ω ̃ a ̃ m = a ̃ m [ i Δ + 1 τ i ] i κ ( a ̃ m + 1 + a ̃ m 1 ) i ω s ̃ m ,
1 τ i = 1 τ g 1 τ l ,
i ω ̃ a ̃ s p , m = 1 τ l a ̃ s p , m i Ω s ̃ m ,
a ̃ s p , m 2 = Ω 2 ω ̃ 2 + 1 τ l 2 s ̃ m 2 .
U s p , m ( t ) = a s p , m ( t ) 2 d 3 r ϵ 0 ϵ Ω ( r ) E Ω ( r ) 2 = a s p , m ( t ) 2 V ,
U s p , m = lim T 1 T T 2 T 2 d t U s p , m ( t )
= lim T 2 π V T d ω ̃ a ̃ s p , m ( ω ̃ ) 2 ,
U s p , m = P s p , m τ l 2 = R s p , m Ω τ l 2 .
lim T s ̃ m ( 0 ) 2 T = R s p , m 4 π 2 V Ω ,
[ i ( ω ̃ + Δ ) + 1 τ i 1 τ e i κ 0 0 0 0 i κ i ( ω ̃ + Δ ) + 1 τ i i κ 0 0 0 . . . . . . . . . . . . . . . . . . . i κ i ( ω ̃ + Δ ) + 1 τ i 1 τ e ] a ̃ i ω [ s ̃ 1 s ̃ 2 . . s ̃ N ] i μ [ S ̃ i n 0 . . 0 ] = 0 ,
a ̃ = i ω P 1 s ̃ i μ P 1 s ̃ i n ,
a N ( ω ̃ ) 2 = μ 2 P N , 1 1 S ̃ i n 2 + ω 2 j = 1 N P N , j 1 s ̃ j 2 [ ω μ ( P N , 1 1 ) * j = 1 N P N , j 1 s ̃ j S ̃ i n * + c.c. ] .
i n ( ω ̃ ) = η ω μ ( P N , 1 1 ) * j = 1 N P N , j 1 s ̃ j S ̃ i n * + c.c. ,
i n 2 = η 2 lim T 1 T d ω ̃ 2 ω 2 μ 2 P N , 1 1 S ̃ i n 2 j , k = 1 N P N , j 1 ( P N , k 1 ) * s ̃ j s ̃ k * + ω 2 μ 2 2 Re [ ( P N , 1 1 ) * 2 S ̃ i n * 2 j , k = 1 N P N , j 1 P N , k 1 s ̃ j s ̃ k ] .
i n 2 = η 2 lim T 1 T d ω ̃ 2 ω 2 μ 2 P N , 1 1 s ̃ i n 2 j = 1 N P N , j 1 s ̃ j 2 .
i n 2 η 2 lim T 1 T [ 2 ω 2 μ 2 P N , 1 1 S ̃ i n 2 j = 1 N P N , j 1 s ̃ j 2 Δ ω ̃ ] ω ̃ = 0 = ω 2 μ 2 T N , 1 S ̃ i n ( 0 ) 2 j = 1 N T N , j 2 R s p , j 2 π 2 V Ω Δ ω ̃ ,
i i n 2 = η 2 μ 4 lim T 2 π T d ω ̃ P N , 1 1 S ̃ i n 4 η 2 μ 4 lim T 2 π T T N , 1 S ̃ i n ( 0 ) 4 Δ ω ̃ ,
SNR = 4 π 3 V Ω μ 2 T N , 1 2 ω 2 R s p j = 1 N T N , j 2 lim T S ̃ i n ( 0 ) 2 T .
NF = SNR i n SNR o u t ,
SNR i n = S ̃ s i g ( 0 ) 2 2 S ̃ δ ( 0 ) 2 .
SNR o u t = 4 π 3 V Ω μ 2 T N , 1 2 ω 2 R s p j = 1 N T N , j 2 lim T S ̃ s i g ( 0 ) 2 + S ̃ δ ( 0 ) 2 T .
NF = ω 2 R s p j = 1 N T N , j 2 4 π 3 V Ω μ 2 T N , 1 2 lim T T 2 S ̃ δ ( 0 ) 2 .
S ̂ δ ( t ) = ω V A ̂ ( t ) ,
1 2 S ̂ δ S ̂ δ + S ̂ δ S ̂ δ = ω 2 V ,
S δ 2 = lim T 2 π T d ω ̃ S ̃ δ ( ω ̃ ) 2 lim T 2 π T S ̃ δ ( 0 ) 2 Δ ω ̃ .
lim T S ̃ δ ( 0 ) 2 T = ω 4 π V Δ ω ̃ .
NF = κ R s p j = 1 N T N , j 2 π 2 μ 2 T N , 1 2 .
I ( ω ) = c n p h ω n V ,
n p h = ρ ( ω ) d ω = ρ ( K ) d K ,
n p h = N Λ 2 π δ ω v g ( ω ) .

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