Abstract

We derive expressions for the transfer functions of double- and multiple-cavity Fabry–Perot filters driven by laser sources with Lorentzian spectrum. These are of interest because of their applications in sensing and channel filtering in optical frequency-division multiplexing networks.

© 1996 Optical Society of America

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References

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  1. J. Stone, L. W. Stulz, “Pigtailed high finesse tunable Fabry–Perot interferometers with large, medium and small free spectral range,” Electron. Lett. 23, 781–783 (1987).
  2. A. A. M. Saleh, J. Stone, “Two-stage Fabry–Perot filters as demultiplexers in optical FDMA LAN’s,” IEEE J. Lightwave Technol. 7, 323–330 (1989).
  3. P. Humblet, W. M. Hamdy, “Crosstalk analysis and filter optimization of single and double cavity Fabry–Perot filters,” IEEE J. Select. Areas Commun. 8, 1095–1107 (1990).
  4. J. Capmany, J. Martí, H. Mangraham, “Impact of finite laser linewidth on the performance of OFDM networks employing single cavity Fabry–Perot demultiplexers,” IEEE J. Light-wave Technol. 13, 290–296 (1995).
  5. B. Crosignani, A. Yariv, P. Di Porto, “Time-dependent analysis of a fiber-optic passive loop resonator,” Opt. Lett. 11, 251–253 (1986).
  6. Y. Ohtsuka, “Analysis of a fiber-optic passive loop-resonator gyroscope: dependence on resonator parameters and light-source coherence,” IEEE J. Lightwave Technol. LT-3, 378–384 (1985).
  7. J. Capmany, “Time domain analysis of a direct coupled fiber ring resonator,” Opt. Commun. 92, 283–290 (1992).

1995 (1)

J. Capmany, J. Martí, H. Mangraham, “Impact of finite laser linewidth on the performance of OFDM networks employing single cavity Fabry–Perot demultiplexers,” IEEE J. Light-wave Technol. 13, 290–296 (1995).

1992 (1)

J. Capmany, “Time domain analysis of a direct coupled fiber ring resonator,” Opt. Commun. 92, 283–290 (1992).

1990 (1)

P. Humblet, W. M. Hamdy, “Crosstalk analysis and filter optimization of single and double cavity Fabry–Perot filters,” IEEE J. Select. Areas Commun. 8, 1095–1107 (1990).

1989 (1)

A. A. M. Saleh, J. Stone, “Two-stage Fabry–Perot filters as demultiplexers in optical FDMA LAN’s,” IEEE J. Lightwave Technol. 7, 323–330 (1989).

1987 (1)

J. Stone, L. W. Stulz, “Pigtailed high finesse tunable Fabry–Perot interferometers with large, medium and small free spectral range,” Electron. Lett. 23, 781–783 (1987).

1986 (1)

1985 (1)

Y. Ohtsuka, “Analysis of a fiber-optic passive loop-resonator gyroscope: dependence on resonator parameters and light-source coherence,” IEEE J. Lightwave Technol. LT-3, 378–384 (1985).

Capmany, J.

J. Capmany, J. Martí, H. Mangraham, “Impact of finite laser linewidth on the performance of OFDM networks employing single cavity Fabry–Perot demultiplexers,” IEEE J. Light-wave Technol. 13, 290–296 (1995).

J. Capmany, “Time domain analysis of a direct coupled fiber ring resonator,” Opt. Commun. 92, 283–290 (1992).

Crosignani, B.

Di Porto, P.

Hamdy, W. M.

P. Humblet, W. M. Hamdy, “Crosstalk analysis and filter optimization of single and double cavity Fabry–Perot filters,” IEEE J. Select. Areas Commun. 8, 1095–1107 (1990).

Humblet, P.

P. Humblet, W. M. Hamdy, “Crosstalk analysis and filter optimization of single and double cavity Fabry–Perot filters,” IEEE J. Select. Areas Commun. 8, 1095–1107 (1990).

Mangraham, H.

J. Capmany, J. Martí, H. Mangraham, “Impact of finite laser linewidth on the performance of OFDM networks employing single cavity Fabry–Perot demultiplexers,” IEEE J. Light-wave Technol. 13, 290–296 (1995).

Martí, J.

J. Capmany, J. Martí, H. Mangraham, “Impact of finite laser linewidth on the performance of OFDM networks employing single cavity Fabry–Perot demultiplexers,” IEEE J. Light-wave Technol. 13, 290–296 (1995).

Ohtsuka, Y.

Y. Ohtsuka, “Analysis of a fiber-optic passive loop-resonator gyroscope: dependence on resonator parameters and light-source coherence,” IEEE J. Lightwave Technol. LT-3, 378–384 (1985).

Saleh, A. A. M.

A. A. M. Saleh, J. Stone, “Two-stage Fabry–Perot filters as demultiplexers in optical FDMA LAN’s,” IEEE J. Lightwave Technol. 7, 323–330 (1989).

Stone, J.

A. A. M. Saleh, J. Stone, “Two-stage Fabry–Perot filters as demultiplexers in optical FDMA LAN’s,” IEEE J. Lightwave Technol. 7, 323–330 (1989).

J. Stone, L. W. Stulz, “Pigtailed high finesse tunable Fabry–Perot interferometers with large, medium and small free spectral range,” Electron. Lett. 23, 781–783 (1987).

Stulz, L. W.

J. Stone, L. W. Stulz, “Pigtailed high finesse tunable Fabry–Perot interferometers with large, medium and small free spectral range,” Electron. Lett. 23, 781–783 (1987).

Yariv, A.

Electron. Lett. (1)

J. Stone, L. W. Stulz, “Pigtailed high finesse tunable Fabry–Perot interferometers with large, medium and small free spectral range,” Electron. Lett. 23, 781–783 (1987).

IEEE J. Light-wave Technol. (1)

J. Capmany, J. Martí, H. Mangraham, “Impact of finite laser linewidth on the performance of OFDM networks employing single cavity Fabry–Perot demultiplexers,” IEEE J. Light-wave Technol. 13, 290–296 (1995).

IEEE J. Lightwave Technol. (2)

A. A. M. Saleh, J. Stone, “Two-stage Fabry–Perot filters as demultiplexers in optical FDMA LAN’s,” IEEE J. Lightwave Technol. 7, 323–330 (1989).

Y. Ohtsuka, “Analysis of a fiber-optic passive loop-resonator gyroscope: dependence on resonator parameters and light-source coherence,” IEEE J. Lightwave Technol. LT-3, 378–384 (1985).

IEEE J. Select. Areas Commun. (1)

P. Humblet, W. M. Hamdy, “Crosstalk analysis and filter optimization of single and double cavity Fabry–Perot filters,” IEEE J. Select. Areas Commun. 8, 1095–1107 (1990).

Opt. Commun. (1)

J. Capmany, “Time domain analysis of a direct coupled fiber ring resonator,” Opt. Commun. 92, 283–290 (1992).

Opt. Lett. (1)

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Figures (7)

Fig. 1
Fig. 1

Layout of the double-cavity Fabry–Perot filter.

Fig. 2
Fig. 2

Transfer function of a double-cavity Fabry–Perot filter with equal cavity lengths and R1 = R2 = 0.97 with the source coherence Δ1 = Δ2 = Δ as a parameter: Δ = 1, solid curve; Δ = 0.9245, broken curve; Δ = 0.8546, dotted curve; Δ = 0.73, broken dotted curve.

Fig. 3
Fig. 3

Transfer function of a double-cavity Fabry–Perot filter with different but similar cavity lengths (Vernier configuration), where k1 = 6, k2 = 7, and R1 = R2 = 0.97 with Δ1 as a parameter: Δ = 1, solid curve; Δ = 0.9245, broken curve; Δ = 0.8546, dotted curve; Δ = 0.73, broken dotted curve.

Fig. 4
Fig. 4

Resonance to maximum secondary sidelobe ratio in decibels for the Vernier configuration of Fig. 3 as a function of the normalized coherence parameter of the first cavity Δ1. The various curves correspond to different mirror reflectivities or cavity finesses [R1 = R2 = R = exp(−π/F)].

Fig. 5
Fig. 5

Spectral sensitivity S for a Vernier filter with K1 = 6, K2 = 7 as a function of the phase shift with respect to that of a given resonance, with the normalized source coherence of the first cavity Δ1 as a parameter. (a), (b), and (c) correspond to different values of the mirror reflectivities as shown in the insets.

Fig. 6
Fig. 6

Maximum sensitivity of a Vernier filter with K1 = 6, K2 = 7 as a function of the normalized source coherence parameter of the first cavity Δ1 with mirror reflectivity R as a parameter.

Fig. 7
Fig. 7

Shift in the optimum operation point φ for maximum spectral sensitivity in a Vernier filter with K1 = 6, K2 = 7 as a function of the source coherence parameter of the first cavity Δ1 with mirror reflectivity R as a parameter.

Equations (11)

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h 1 ( t ) = k = 0 h 1 k δ ( t - k τ 1 ) ,             h 2 ( t ) = k = 0 h 2 k δ ( t - k τ 2 ) ,
h 1 k = ( 1 - R 1 - A 1 ) R 1 k exp ( j 2 π k δ 1 ) ,             δ 1 = 2 f n c L 1 / c , h 2 k = ( 1 - R 2 - A 2 ) R 2 k exp ( j 2 π k δ 2 ) ,             δ 2 = 2 f n c L 2 / c ,
E o ( t ) = k = 0 r = 0 h 1 k h 2 r E i n ( t - k τ 1 - r τ 2 ) ,
T = k = 0 r = 0 l = 0 s = 0 h 1 k h 1 l * h 2 r * h 2 s γ [ ( l - k ) τ 1 + ( s - r ) τ 2 ] .
γ [ ( l - k ) τ 1 + ( s - r ) τ 2 ] = γ [ ( l - k ) τ 1 ] γ [ ( s - r ) τ 2 ] = Δ 1 - l - k Δ 2 - s - r ,
T ( δ 1 , δ 2 , Δ 1 , Δ 2 ) = T 1 ( δ 1 , Δ 1 ) T 2 ( δ 2 , Δ 2 ) ,
T i ( δ i , Δ i ) = a ( Δ i ) 1 + b ( Δ i ) sin 2 ( π δ i ) , a ( Δ i ) = ( 1 - Δ i 2 R i 2 ) ( 1 - R i ) 2 ( 1 - R i 2 ) ( 1 - Δ i R i ) 2 , b ( Δ i ) = 4 Δ i R i ( 1 - Δ i R i ) 2 .
T ( δ 1 , δ 2 , δ N , Δ 1 , Δ 2 , Δ N ) = k = 1 N T k ( δ k , Δ k ) .
S = T ( φ ) φ .
S ( Δ 1 , Δ 2 , φ ) = - 2 φ T 2 ( φ ) a ( Δ 1 ) a ( Δ 2 ) [ b ( Δ 1 ) K 1 2 + b ( Δ 2 ) K 2 2 + 2 b ( Δ 1 ) b ( Δ 2 ) K 1 2 K 2 2 φ 2 ] .
1 + 2 φ S ( Δ 1 , Δ 2 , φ ) T ( Δ 1 , Δ 2 , φ ) + 4 φ 2 b ( Δ 1 ) K 1 2 b ( Δ 2 ) K 2 2 + b ( Δ 2 ) K 2 2 b ( Δ 1 ) K 1 2 + 2 φ 2 = 0.

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