Abstract

A novel approach to generating an optical frequency comb based on repeated frequency shifting is proposed and experimentally demonstrated. The frequency shifting is implemented via optical carrier suppression and single-sideband modulation using two Mach-Zehnder modulators in conjunction with a bidirectional asymmetric Mach-Zehnder interferometer with wavelength-shifted transmission spectra along the opposite directions. A theoretical analysis is performed, which is confirmed by a proof-of-concept experiment. A stable optical comb covering a spectral range of 0.18 THz is generated.

©2009 Optical Society of America

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References

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  1. P. J. Delfyett, F. Quinlan, S. Ozharar, and W. Lee, “Stabilized optical frequency combs from diode lasers–applications in optical communications, signal processing and instrumentation,” in Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OThN6.
  2. A. J. Seeds, C. C. Renaud, M. Pantouvaki, M. Robertson, I. Lealman, D. Rogers, R. Firth, P. J. Cannard, R. Moore, and R. Gwilliam, “Photonic synthesis of THz signals,” in Proceedings of the 36th European Microwave Conf. (2006), pp. 1107–1110.
  3. Z. Jiang, D. E. Leaird, C. Huang, H. Miao, M. Kourogi, K. Imai, and A. M. Weiner, “Spectral line-by-line pulse shaping on an optical frequency comb generator,” IEEE J. Quantum Electron. 43(12), 1163–1174 (2007).
    [Crossref]
  4. K. P. Ho and J. M. Kahn, “Optical frequency comb generator using phase modulation in amplified circulating loop,” IEEE Photon. Technol. Lett. 5(6), 721–725 (1993).
    [Crossref]
  5. S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
    [Crossref]
  6. P. Shen, N. J. Gomes, P. A. Davies, P. G. Huggard, and B. N. Ellison, “Analysis and demonstration of a fast tunable fiber-ring based optical frequency comb generator,” J. Lightwave Technol. 25(11), 3257–3264 (2007).
    [Crossref]
  7. M. Kourogi, K. Nakagawa, and M. Ohtsu, “Wide-span optical frequency comb generator for accurate optical frequency difference measurement,” IEEE J. Quantum Electron. 29(10), 2693–2701 (1993).
    [Crossref]
  8. T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Opt. Lett. 32(11), 1515–1517 (2007).
    [Crossref] [PubMed]
  9. S. Ozharar, F. Quinlan, I. Ozdur, S. Gee, and P. J. Delfyett, “Ultraflat optical comb generation by phase-only modulation of continuous-wave light,” IEEE Photon. Technol. Lett. 20(1), 36–38 (2008).
    [Crossref]
  10. K. Imai, M. Kourogi, and M. Ohtsu, “30-THz span optical frequency comb generation by self-phase modulation in an optical fiber,” IEEE J. Quantum Electron. 34(1), 54–60 (1998).
    [Crossref]
  11. R. P. Scott, N. K. Fontaine, J. P. Heritage, B. H. Kolner, and S. J. B. Yoo, “3.5-THz wide, 175 mode optical comb source,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper OWJ3.
  12. T. Kawanishi, T. Sakamoto, S. Shinada, and M. Izutsu, “Optical frequency comb generator using optical fiber loops with single sideband modulation,” IEICE Electron. Express 1(8), 217–221 (2004).
    [Crossref]
  13. T. Kawanishi and M. Izutsu, “Linear single-sideband modulation for high-SNR wavelength conversion,” IEEE Photon. Technol. Lett. 16(6), 1534–1536 (2004).
    [Crossref]
  14. X. Zhang and A. Mitchell, “A simple black box model for erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 12(1), 28–30 (2000).
    [Crossref]
  15. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
    [Crossref]
  16. S. Pan and J. P. Yao, “A frequency-doubling optoelectronic oscillator using a polarization modulator,” IEEE Photon. Technol. Lett. 21(13), 929–931 (2009).
    [Crossref]

2009 (1)

S. Pan and J. P. Yao, “A frequency-doubling optoelectronic oscillator using a polarization modulator,” IEEE Photon. Technol. Lett. 21(13), 929–931 (2009).
[Crossref]

2008 (1)

S. Ozharar, F. Quinlan, I. Ozdur, S. Gee, and P. J. Delfyett, “Ultraflat optical comb generation by phase-only modulation of continuous-wave light,” IEEE Photon. Technol. Lett. 20(1), 36–38 (2008).
[Crossref]

2007 (3)

2004 (3)

D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

T. Kawanishi, T. Sakamoto, S. Shinada, and M. Izutsu, “Optical frequency comb generator using optical fiber loops with single sideband modulation,” IEICE Electron. Express 1(8), 217–221 (2004).
[Crossref]

T. Kawanishi and M. Izutsu, “Linear single-sideband modulation for high-SNR wavelength conversion,” IEEE Photon. Technol. Lett. 16(6), 1534–1536 (2004).
[Crossref]

2000 (1)

X. Zhang and A. Mitchell, “A simple black box model for erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 12(1), 28–30 (2000).
[Crossref]

1999 (1)

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[Crossref]

1998 (1)

K. Imai, M. Kourogi, and M. Ohtsu, “30-THz span optical frequency comb generation by self-phase modulation in an optical fiber,” IEEE J. Quantum Electron. 34(1), 54–60 (1998).
[Crossref]

1993 (2)

M. Kourogi, K. Nakagawa, and M. Ohtsu, “Wide-span optical frequency comb generator for accurate optical frequency difference measurement,” IEEE J. Quantum Electron. 29(10), 2693–2701 (1993).
[Crossref]

K. P. Ho and J. M. Kahn, “Optical frequency comb generator using phase modulation in amplified circulating loop,” IEEE Photon. Technol. Lett. 5(6), 721–725 (1993).
[Crossref]

Bennett, S.

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[Crossref]

Bull, D.

D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Burr, E.

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[Crossref]

Cai, B.

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[Crossref]

Davies, P. A.

Delfyett, P. J.

S. Ozharar, F. Quinlan, I. Ozdur, S. Gee, and P. J. Delfyett, “Ultraflat optical comb generation by phase-only modulation of continuous-wave light,” IEEE Photon. Technol. Lett. 20(1), 36–38 (2008).
[Crossref]

Ellison, B. N.

Fairburn, M.

D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Gee, S.

S. Ozharar, F. Quinlan, I. Ozdur, S. Gee, and P. J. Delfyett, “Ultraflat optical comb generation by phase-only modulation of continuous-wave light,” IEEE Photon. Technol. Lett. 20(1), 36–38 (2008).
[Crossref]

Ghanipour, P.

D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Gomes, N. J.

Gough, O.

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[Crossref]

Ho, K. P.

K. P. Ho and J. M. Kahn, “Optical frequency comb generator using phase modulation in amplified circulating loop,” IEEE Photon. Technol. Lett. 5(6), 721–725 (1993).
[Crossref]

Huang, C.

Z. Jiang, D. E. Leaird, C. Huang, H. Miao, M. Kourogi, K. Imai, and A. M. Weiner, “Spectral line-by-line pulse shaping on an optical frequency comb generator,” IEEE J. Quantum Electron. 43(12), 1163–1174 (2007).
[Crossref]

Huggard, P. G.

Imai, K.

Z. Jiang, D. E. Leaird, C. Huang, H. Miao, M. Kourogi, K. Imai, and A. M. Weiner, “Spectral line-by-line pulse shaping on an optical frequency comb generator,” IEEE J. Quantum Electron. 43(12), 1163–1174 (2007).
[Crossref]

K. Imai, M. Kourogi, and M. Ohtsu, “30-THz span optical frequency comb generation by self-phase modulation in an optical fiber,” IEEE J. Quantum Electron. 34(1), 54–60 (1998).
[Crossref]

Izutsu, M.

T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Opt. Lett. 32(11), 1515–1517 (2007).
[Crossref] [PubMed]

T. Kawanishi, T. Sakamoto, S. Shinada, and M. Izutsu, “Optical frequency comb generator using optical fiber loops with single sideband modulation,” IEICE Electron. Express 1(8), 217–221 (2004).
[Crossref]

T. Kawanishi and M. Izutsu, “Linear single-sideband modulation for high-SNR wavelength conversion,” IEEE Photon. Technol. Lett. 16(6), 1534–1536 (2004).
[Crossref]

Jaeger, N. A. F.

D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Jiang, Z.

Z. Jiang, D. E. Leaird, C. Huang, H. Miao, M. Kourogi, K. Imai, and A. M. Weiner, “Spectral line-by-line pulse shaping on an optical frequency comb generator,” IEEE J. Quantum Electron. 43(12), 1163–1174 (2007).
[Crossref]

Kahn, J. M.

K. P. Ho and J. M. Kahn, “Optical frequency comb generator using phase modulation in amplified circulating loop,” IEEE Photon. Technol. Lett. 5(6), 721–725 (1993).
[Crossref]

Kato, H.

D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Kawanishi, T.

T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Opt. Lett. 32(11), 1515–1517 (2007).
[Crossref] [PubMed]

T. Kawanishi and M. Izutsu, “Linear single-sideband modulation for high-SNR wavelength conversion,” IEEE Photon. Technol. Lett. 16(6), 1534–1536 (2004).
[Crossref]

T. Kawanishi, T. Sakamoto, S. Shinada, and M. Izutsu, “Optical frequency comb generator using optical fiber loops with single sideband modulation,” IEICE Electron. Express 1(8), 217–221 (2004).
[Crossref]

Kourogi, M.

Z. Jiang, D. E. Leaird, C. Huang, H. Miao, M. Kourogi, K. Imai, and A. M. Weiner, “Spectral line-by-line pulse shaping on an optical frequency comb generator,” IEEE J. Quantum Electron. 43(12), 1163–1174 (2007).
[Crossref]

K. Imai, M. Kourogi, and M. Ohtsu, “30-THz span optical frequency comb generation by self-phase modulation in an optical fiber,” IEEE J. Quantum Electron. 34(1), 54–60 (1998).
[Crossref]

M. Kourogi, K. Nakagawa, and M. Ohtsu, “Wide-span optical frequency comb generator for accurate optical frequency difference measurement,” IEEE J. Quantum Electron. 29(10), 2693–2701 (1993).
[Crossref]

Leaird, D. E.

Z. Jiang, D. E. Leaird, C. Huang, H. Miao, M. Kourogi, K. Imai, and A. M. Weiner, “Spectral line-by-line pulse shaping on an optical frequency comb generator,” IEEE J. Quantum Electron. 43(12), 1163–1174 (2007).
[Crossref]

Miao, H.

Z. Jiang, D. E. Leaird, C. Huang, H. Miao, M. Kourogi, K. Imai, and A. M. Weiner, “Spectral line-by-line pulse shaping on an optical frequency comb generator,” IEEE J. Quantum Electron. 43(12), 1163–1174 (2007).
[Crossref]

Mitchell, A.

X. Zhang and A. Mitchell, “A simple black box model for erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 12(1), 28–30 (2000).
[Crossref]

Nakagawa, K.

M. Kourogi, K. Nakagawa, and M. Ohtsu, “Wide-span optical frequency comb generator for accurate optical frequency difference measurement,” IEEE J. Quantum Electron. 29(10), 2693–2701 (1993).
[Crossref]

Ohtsu, M.

K. Imai, M. Kourogi, and M. Ohtsu, “30-THz span optical frequency comb generation by self-phase modulation in an optical fiber,” IEEE J. Quantum Electron. 34(1), 54–60 (1998).
[Crossref]

M. Kourogi, K. Nakagawa, and M. Ohtsu, “Wide-span optical frequency comb generator for accurate optical frequency difference measurement,” IEEE J. Quantum Electron. 29(10), 2693–2701 (1993).
[Crossref]

Ozdur, I.

S. Ozharar, F. Quinlan, I. Ozdur, S. Gee, and P. J. Delfyett, “Ultraflat optical comb generation by phase-only modulation of continuous-wave light,” IEEE Photon. Technol. Lett. 20(1), 36–38 (2008).
[Crossref]

Ozharar, S.

S. Ozharar, F. Quinlan, I. Ozdur, S. Gee, and P. J. Delfyett, “Ultraflat optical comb generation by phase-only modulation of continuous-wave light,” IEEE Photon. Technol. Lett. 20(1), 36–38 (2008).
[Crossref]

Pan, S.

S. Pan and J. P. Yao, “A frequency-doubling optoelectronic oscillator using a polarization modulator,” IEEE Photon. Technol. Lett. 21(13), 929–931 (2009).
[Crossref]

Quinlan, F.

S. Ozharar, F. Quinlan, I. Ozdur, S. Gee, and P. J. Delfyett, “Ultraflat optical comb generation by phase-only modulation of continuous-wave light,” IEEE Photon. Technol. Lett. 20(1), 36–38 (2008).
[Crossref]

Reid, A.

D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Sakamoto, T.

T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Opt. Lett. 32(11), 1515–1517 (2007).
[Crossref] [PubMed]

T. Kawanishi, T. Sakamoto, S. Shinada, and M. Izutsu, “Optical frequency comb generator using optical fiber loops with single sideband modulation,” IEICE Electron. Express 1(8), 217–221 (2004).
[Crossref]

Seeds, A. J.

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[Crossref]

Shen, P.

Shinada, S.

T. Kawanishi, T. Sakamoto, S. Shinada, and M. Izutsu, “Optical frequency comb generator using optical fiber loops with single sideband modulation,” IEICE Electron. Express 1(8), 217–221 (2004).
[Crossref]

Weiner, A. M.

Z. Jiang, D. E. Leaird, C. Huang, H. Miao, M. Kourogi, K. Imai, and A. M. Weiner, “Spectral line-by-line pulse shaping on an optical frequency comb generator,” IEEE J. Quantum Electron. 43(12), 1163–1174 (2007).
[Crossref]

Yao, J. P.

S. Pan and J. P. Yao, “A frequency-doubling optoelectronic oscillator using a polarization modulator,” IEEE Photon. Technol. Lett. 21(13), 929–931 (2009).
[Crossref]

Zhang, X.

X. Zhang and A. Mitchell, “A simple black box model for erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 12(1), 28–30 (2000).
[Crossref]

IEEE J. Quantum Electron. (3)

Z. Jiang, D. E. Leaird, C. Huang, H. Miao, M. Kourogi, K. Imai, and A. M. Weiner, “Spectral line-by-line pulse shaping on an optical frequency comb generator,” IEEE J. Quantum Electron. 43(12), 1163–1174 (2007).
[Crossref]

M. Kourogi, K. Nakagawa, and M. Ohtsu, “Wide-span optical frequency comb generator for accurate optical frequency difference measurement,” IEEE J. Quantum Electron. 29(10), 2693–2701 (1993).
[Crossref]

K. Imai, M. Kourogi, and M. Ohtsu, “30-THz span optical frequency comb generation by self-phase modulation in an optical fiber,” IEEE J. Quantum Electron. 34(1), 54–60 (1998).
[Crossref]

IEEE Photon. Technol. Lett. (6)

S. Ozharar, F. Quinlan, I. Ozdur, S. Gee, and P. J. Delfyett, “Ultraflat optical comb generation by phase-only modulation of continuous-wave light,” IEEE Photon. Technol. Lett. 20(1), 36–38 (2008).
[Crossref]

T. Kawanishi and M. Izutsu, “Linear single-sideband modulation for high-SNR wavelength conversion,” IEEE Photon. Technol. Lett. 16(6), 1534–1536 (2004).
[Crossref]

X. Zhang and A. Mitchell, “A simple black box model for erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 12(1), 28–30 (2000).
[Crossref]

S. Pan and J. P. Yao, “A frequency-doubling optoelectronic oscillator using a polarization modulator,” IEEE Photon. Technol. Lett. 21(13), 929–931 (2009).
[Crossref]

K. P. Ho and J. M. Kahn, “Optical frequency comb generator using phase modulation in amplified circulating loop,” IEEE Photon. Technol. Lett. 5(6), 721–725 (1993).
[Crossref]

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[Crossref]

IEICE Electron. Express (1)

T. Kawanishi, T. Sakamoto, S. Shinada, and M. Izutsu, “Optical frequency comb generator using optical fiber loops with single sideband modulation,” IEICE Electron. Express 1(8), 217–221 (2004).
[Crossref]

J. Lightwave Technol. (1)

Opt. Lett. (1)

Proc. SPIE (1)

D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Other (3)

R. P. Scott, N. K. Fontaine, J. P. Heritage, B. H. Kolner, and S. J. B. Yoo, “3.5-THz wide, 175 mode optical comb source,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper OWJ3.

P. J. Delfyett, F. Quinlan, S. Ozharar, and W. Lee, “Stabilized optical frequency combs from diode lasers–applications in optical communications, signal processing and instrumentation,” in Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OThN6.

A. J. Seeds, C. C. Renaud, M. Pantouvaki, M. Robertson, I. Lealman, D. Rogers, R. Firth, P. J. Cannard, R. Moore, and R. Gwilliam, “Photonic synthesis of THz signals,” in Proceedings of the 36th European Microwave Conf. (2006), pp. 1107–1110.

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic diagram of the proposed optical comb generator based on repeated frequency shifting; (b) wavelength-shifted transmission spectra of the AMZI along the opposite directions.

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Fig. 2
Fig. 2

Theoretically calculated envelope of the optical comb, in which the power distribution P(n) versus n for three different values of κGη is shown.

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Fig. 3
Fig. 3

(a) The configuration of the AMZI; (b) The measured transmission spectra of the AMZI along the two opposite directions.

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Fig. 4
Fig. 4

The measured optical spectra of the generated optical frequency comb. From (a) to (f), the gain of the EDFA is gradually increased.

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Equations (16)

Equations on this page are rendered with MathJax. Learn more.

f o + f e 1 − f e 2 = f o − f e 1 − f shift
f e 1 + f e 2 = f FSR
f o + f e 1 − f e 2 = f o − f e 1 + f FSR − f shift
f o − f e 1 + f e 2 = f o + f e 1 − f FSR − f shift
or   f o − f e 1 + f e 2 = f o + f e 1 + f FSR − f shift
P ' ( n + 1 ) = η P ( n )
P in = ∑ n = 0 ∞ P ' ( n + 1 ) = ∑ n = 0 ∞ η P ( n )
G = G ( P in ) = G max 1 + ( P in / P sat ) α
κ G P ' ( n + 1 ) = κ G η P ( n ) = P ( n + 1 )
P ( n ) = ( κ G η ) n P ( 0 )
P in = ∑ n = 1 ∞ η ( κ G η ) n − 1 P ( 0 ) = η P ( 0 ) 1 − κ G η
W = n f FSR = 3 f FSR 10 log ( 1 / κ G η )
P in ′ = ∑ n = 1 N η ( κ G η ) n − 1 P ( 0 ) = { η P ( 0 ) [ 1 − ( κ G η ) N − 1 ] / ( 1 − κ G η ) κ G η < 1 N η P ( 0 ) κ G η = 1
P FBG ( n ) = ( κ η ) n ∏ i = 1 n G ( i ) P ( 0 ) = [ κ G ( 1 ) η ] n ∏ i = 1 n G ( i ) G ( 1 ) P ( 0 )
or   10 log 10 P FBG ( n ) P ( 0 ) = 10 log 10 [ κ G ( 1 ) η ] n + ∑ i = 1 n 10 log 10 G ( i ) G ( 1 )
G ( n ) = P FBG ( n ) κ η P FBG ( n − 1 ) ( n > 1 )

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