Abstract

The simple method for modeling of circuits of weakly coupled lossy resonant cavities, previously developed in quantum mechanics, is generalized to enable calculation of the transmission and reflection amplitudes and group delay of light. Our result is the generalized Breit-Wigner formula, which has a clear physical meaning and is convenient for fast modeling and optimization of complex resonant cavity circuits and, in particular, superstructure gratings in a way similar to modeling and optimization of electric circuits. As examples, we find the conditions when a finite linear chain of cavities and a linear chain with adjacent cavities act as bandpass and double bandpass filters, and the condition for a Y-shaped structure to act as a bandpass 50/50 light splitter. The group delay dependencies of the considered structures are also investigated.

© 2003 Optical Society of America

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References

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Appl. Phys. Lett.

M. Sumetskii, �??Narrow current dip for the double quantum dot resonant tunneling structure with three leads: Sensitive nanometer Y-branch switch,�?? Appl. Phys. Lett. 63, 3185 (1993).
[CrossRef]

IEEE J. Quant. Electron.

E. Ozbay, M. Bayindir, I. Bulu, and E. Cubukcu, �??Investigation of localized coupled-cavity modes in twodimensional photonic bandgap structures,�?? IEEE J. Quant. Electron. 38, 837 (2002).
[CrossRef]

K. Hosomi and T. Katsuyama, �??A dispersion compenstor using coupled defects in a photonic crystal,�?? IEEE J. Quant. Electron. 38, 825 (2002).
[CrossRef]

IEEE J. Quantum Electron

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, �??Optical delay lines based on optical filters,�?? IEEE J. Quant. Electron. 27, 525 (2001).
[CrossRef]

IEEE J. Sel. Topics in Quant. Electron.

S. Mookherjea and A. Yariv, �??Coupled resonator optical waveguides,�?? IEEE J. Sel. Topics in Quant. Electron. 3, 448 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

R. Slavic and S. LaRochelle, �??Large-band periodic filters for DWDM using multiple-superimposed fiber Bragg gratings,�?? IEEE Photon. Technol. Lett. 14, 1704 (2002).
[CrossRef]

J. Appl. Phys.

S. Lan, S. Nishikawa, and H. Ishikawa, �??Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses,�?? J. Appl. Phys. 90, 4321 (2001).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

J. Phys: Condens. Matter

M. Sumetskii, �??Modeling of complicated nanometer resonant tunneling devices with quantum dots,�?? J. Phys: Condens. Matter 3, 2651 (1991).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

M. Sumetskii, �??Forming of wave packets by one-dimensional tunneling structures having a time-dependent potential,�?? Phys. Rev. B 46, 4702 (1992).
[CrossRef]

M. Sumetskii, �??Resistance resonances for resonant-tunneling structures of quantum dots," Phys. Rev. B 48, 4586 (1993).
[CrossRef]

N. Stefanou and A. Modinos, �??Impurity bands in photonic insulators,�?? Phys. Rev. B 57, 12127 (1998).
[CrossRef]

Phys. Rev. E

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

Phys. Rev. Lett.

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

Y. Meir and N. S. Wingreen, �??Landauer Formula for the current through an interacting electron region,�?? Phys. Rev. Lett. 68, 2512 (1992).
[CrossRef] [PubMed]

Other

S. Datta, Electronic transport in mesoscopic systems, (Cambridge University Press, 1995).

Yu. N. Demkov and V. N. Ostrovskii, Zero-range potentials and their applications in atomic physics, (Plenum Press, 1988).
[CrossRef]

R. Kashyap, Fiber Bragg Gratings, (Academic Press, 1999).

N. W. Ashcroft and N. D. Mermin, Solid state physics, (Saunders College, Philadelphia, 1976).

. L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R. M. Westervelt, and N. S. Wingreen, �??Electron Transport in Quantum Dots,�?? Proceedings of the NATO Advanced Study Institute on Mesoscopic Electron Transport, edited by L. L. Sohn, L. P. Kouwenhoven, and G. Schön, 1997, pp.105-214.

L. D. Landau and E. M. Lifshitz, Quantum mechanics, (Pergamon Press, 1958, pp. 440-449).

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Figures (7)

Fig. 1.
Fig. 1.

Resonant cavity structure.

Fig. 2.
Fig. 2.

Single cavity structures: a – a cavity between two ports, b and c – transmission and group delay spectrum of this structure for γ11121/2.

Fig. 3.
Fig. 3.

A structure of three resonant cavities, a – not optimized, b – optimized, λ0=1500 nm.

Fig. 4.
Fig. 4.

Transmission and group delay spectrum for the apodized 50-cavity structure, λ0=1500 nm; a – transmission spectrum for the lossless cavities, b – transmission spectrum for the cavities having internal loss γ int = 0.01δ0.

Fig. 5.
Fig. 5.

Transmission and group delay spectrum for the double bandpass structure, λ0=1500 nm.

Fig. 6.
Fig. 6.

a – Y-splitter assembled of resonant cavities; b – transmission and group delay spectrum for the coupling parameters shown in Fig. 6a, λ0=1500 nm.

Fig. 7.
Fig. 7.

a – all-reflecting single port device; b – model of a device consisting of cavities having hexagonal internal symmetry; c – generalized model of a port.

Equations (14)

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u ( r ) = n = 1 N C n u n ( r ) + u m ( in ) ( r )
H ( u ) = λ 2 u ,
H m ( port ) ( u m ( in ) ) = λ 2 u m ( in ) , H n ( cav ) ( u n ) = ( λ n ( 0 ) + i 2 γ n ) 2 u n ,
ΛC = χ ( m ) , C = C 1 C 2 C N , χ ( m ) = χ 1 m χ 2 m χ Nm .
Λ ( λ ) = λ λ 1 + i 2 γ 1 δ 12 δ 1 N δ 21 λ λ 2 + i 2 γ 2 δ 2 N δ N 1 δ N 2 λ λ N + i 2 γ N
γ jm = χ jm 2 .
A lm ( λ ) = χ il χ jm * t ij , R ll ( λ ) = 1 i χ ll 2 t ii , l m
τ lm = λ 2 2 πc Im [ d ln [ A lm ( λ ) ] ] , τ ll = λ 2 2 πc Im [ d ln [ R ll ( λ ) ] ] .
A 12 ( λ ) = χ 11 χ 12 * λ λ 1 + i 2 γ 1 , R 11 ( λ ) = 1 i χ 11 2 λ λ 1 + i 2 γ 1 ,
τ 12 ( λ ) = λ 2 πc γ 1 4 ( λ λ 1 ) 2 + γ 1 2 , τ 11 ( λ ) = τ 12 ( λ ) .
λ n = λ n ( 0 ) + Δ λ n
Δ λ n = 1 2 ( λ n ( 0 ) ) 3 d r u n ( r ) ( H H n ) u n ( r )
δ ij = 1 2 ( λ i ( 0 ) ) 3 d r u i ( r ) ( H H j ) u j ( r )
χ im = 1 2 ( λ m ( 0 ) ) 3 d r u i ( r ) ( H H m ( port ) ) u m ( in ) ( r ) .

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