Abstract

Theoretical aspects of the frequency-translation ring circuit are considered through numerical simulations. We analyze the signal and noise propagation around an optical ring circuit that contains a frequency shifter, an erbium-doped fiber amplifier and a bandpass filter (BPF). The relations between the frequency-translation limit and some important parameters such as the BPF bandwidth and the polarization state are clarified. Numerical results for the frequency-translation limit are compared with reported experiments and a frequency translation of more than 100 GHz is predicted.

© 1994 Optical Society of America

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References

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  1. H. Toba, K. Oda, K. Nakanishi, N. Shibata, K. Nosu, N. Takato, M. Fukuda, “A 100-channel optical FDM transmission/distribution at 622 Mb/s over 50 km,” J. Lightwave Technol. 8, 1396–1401 (1990).
    [CrossRef]
  2. H. Toba, K. Oda, K. Nakanishi, K. Nosu, “Broad-band information distribution networks employing optical frequency division multiplexing technologies,” in Proceeding of the Global Telecommunications Conference (GLOBECOM) (Institute of Electrical and Electronics Engineers, San Diego, Calif., 1990), pp. 1512–1517.
    [CrossRef]
  3. B. Glance, “Frequency stabilization of FDM optical signals,” Electron. Lett. 23, 750–752 (1987).
    [CrossRef]
  4. H. Toba, K. Inoue, K. Nosu, G. Motosugi, “A multichannel laser diode frequency stabilizer for narrowly spaced optical frequency-division-multiplexing transmission,” J. Opt. Commun. 9, 3–7 (1988).
  5. K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, J. F. Weller, “6-34 GHz offset phase-locking of Nd:YAG 1319-nm nonplanar ring lasers,” Electron. Lett. 25, 1242–1243 (1989).
    [CrossRef]
  6. O. Ishida, H. Toba, “Lightwave synthesizer with lock-in-detected frequency references,” J. Lightwave Technol. 9, 1344–1352 (1991).
    [CrossRef]
  7. W. V. Sorin, D. K. Donald, S. A. Newton, M. Nazarathy, “Coherent FMCW reflectometry using a temperature-tuned Nd:YAG ring laser,” IEEE Photonics Technol. Lett. 2, 902–904 (1990).
    [CrossRef]
  8. T. G. Hodgkinson, P. Coppin, “Pulsed operation of an optical feedback frequency synthesizer,” Electron. Lett. 26, 1155–1157 (1990).
    [CrossRef]
  9. K. Shimizu, T. Horiguchi, Y. Koyamada, “Technique for translating light-wave frequency by using an optical ring circuit containing a frequency shifter,” Opt. Lett. 17, 1307–1309 (1992).
    [CrossRef] [PubMed]
  10. K. Shimizu, T. Horiguchi, Y. Koyamada, “Frequency translation of light waves by propagations around an optical ring circuit containing a frequency shifter: I. Experiment,” Appl. Opt. 32, 6718–6726 (1993).
    [CrossRef] [PubMed]
  11. K. Aida, J. Nakajima, K. Nakagawa, “Synthesis of optical pulsed sweep frequencies from a fiber loop with automatic gain controlled erbium-doped fiber amplifier and AOD,” in Proceedings of the European Conference on Optical Communication (Springer-Verlag, Berlin, 1992), Vol. 1, pp. 437–440.
  12. M. D. Levenson, S. S. Kano, Introduction to Nonlinear Laser Spectroscopy (Academic, San Diego, Calif., 1988).
  13. C. R. Giles, E. Desurvire, “Propagation of signal and noise in concatenated erbium-doped fiber optical amplifiers,” J. Lightwave Technol. 9, 147–154 (1991).
    [CrossRef]
  14. A. Yariv, Quantum Electronics, 3rd ed., (Wiley, New York, 1988), pp. 560–565.
  15. E. Desurvire, C. R. Giles, J. R. Simpson, “Gain saturation effects in high-speed multichannel erbium-doped fiber amplifiers at λ = 1.53 μm,” J. Lightwave Technol. LT-7, 2095–2105 (1989).
    [CrossRef]
  16. K. Inoue, H. Toba, K. Nosu, “Multichannel amplification utilizing an Er3+-doped fiber amplifier,” J. Lightwave Technol. 9, 368–374 (1991).
    [CrossRef]

1993 (1)

1992 (1)

1991 (3)

K. Inoue, H. Toba, K. Nosu, “Multichannel amplification utilizing an Er3+-doped fiber amplifier,” J. Lightwave Technol. 9, 368–374 (1991).
[CrossRef]

C. R. Giles, E. Desurvire, “Propagation of signal and noise in concatenated erbium-doped fiber optical amplifiers,” J. Lightwave Technol. 9, 147–154 (1991).
[CrossRef]

O. Ishida, H. Toba, “Lightwave synthesizer with lock-in-detected frequency references,” J. Lightwave Technol. 9, 1344–1352 (1991).
[CrossRef]

1990 (3)

W. V. Sorin, D. K. Donald, S. A. Newton, M. Nazarathy, “Coherent FMCW reflectometry using a temperature-tuned Nd:YAG ring laser,” IEEE Photonics Technol. Lett. 2, 902–904 (1990).
[CrossRef]

T. G. Hodgkinson, P. Coppin, “Pulsed operation of an optical feedback frequency synthesizer,” Electron. Lett. 26, 1155–1157 (1990).
[CrossRef]

H. Toba, K. Oda, K. Nakanishi, N. Shibata, K. Nosu, N. Takato, M. Fukuda, “A 100-channel optical FDM transmission/distribution at 622 Mb/s over 50 km,” J. Lightwave Technol. 8, 1396–1401 (1990).
[CrossRef]

1989 (2)

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, J. F. Weller, “6-34 GHz offset phase-locking of Nd:YAG 1319-nm nonplanar ring lasers,” Electron. Lett. 25, 1242–1243 (1989).
[CrossRef]

E. Desurvire, C. R. Giles, J. R. Simpson, “Gain saturation effects in high-speed multichannel erbium-doped fiber amplifiers at λ = 1.53 μm,” J. Lightwave Technol. LT-7, 2095–2105 (1989).
[CrossRef]

1988 (1)

H. Toba, K. Inoue, K. Nosu, G. Motosugi, “A multichannel laser diode frequency stabilizer for narrowly spaced optical frequency-division-multiplexing transmission,” J. Opt. Commun. 9, 3–7 (1988).

1987 (1)

B. Glance, “Frequency stabilization of FDM optical signals,” Electron. Lett. 23, 750–752 (1987).
[CrossRef]

Aida, K.

K. Aida, J. Nakajima, K. Nakagawa, “Synthesis of optical pulsed sweep frequencies from a fiber loop with automatic gain controlled erbium-doped fiber amplifier and AOD,” in Proceedings of the European Conference on Optical Communication (Springer-Verlag, Berlin, 1992), Vol. 1, pp. 437–440.

Coppin, P.

T. G. Hodgkinson, P. Coppin, “Pulsed operation of an optical feedback frequency synthesizer,” Electron. Lett. 26, 1155–1157 (1990).
[CrossRef]

Dagenais, M.

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, J. F. Weller, “6-34 GHz offset phase-locking of Nd:YAG 1319-nm nonplanar ring lasers,” Electron. Lett. 25, 1242–1243 (1989).
[CrossRef]

Desurvire, E.

C. R. Giles, E. Desurvire, “Propagation of signal and noise in concatenated erbium-doped fiber optical amplifiers,” J. Lightwave Technol. 9, 147–154 (1991).
[CrossRef]

E. Desurvire, C. R. Giles, J. R. Simpson, “Gain saturation effects in high-speed multichannel erbium-doped fiber amplifiers at λ = 1.53 μm,” J. Lightwave Technol. LT-7, 2095–2105 (1989).
[CrossRef]

Donald, D. K.

W. V. Sorin, D. K. Donald, S. A. Newton, M. Nazarathy, “Coherent FMCW reflectometry using a temperature-tuned Nd:YAG ring laser,” IEEE Photonics Technol. Lett. 2, 902–904 (1990).
[CrossRef]

Esman, R. D.

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, J. F. Weller, “6-34 GHz offset phase-locking of Nd:YAG 1319-nm nonplanar ring lasers,” Electron. Lett. 25, 1242–1243 (1989).
[CrossRef]

Fukuda, M.

H. Toba, K. Oda, K. Nakanishi, N. Shibata, K. Nosu, N. Takato, M. Fukuda, “A 100-channel optical FDM transmission/distribution at 622 Mb/s over 50 km,” J. Lightwave Technol. 8, 1396–1401 (1990).
[CrossRef]

Giles, C. R.

C. R. Giles, E. Desurvire, “Propagation of signal and noise in concatenated erbium-doped fiber optical amplifiers,” J. Lightwave Technol. 9, 147–154 (1991).
[CrossRef]

E. Desurvire, C. R. Giles, J. R. Simpson, “Gain saturation effects in high-speed multichannel erbium-doped fiber amplifiers at λ = 1.53 μm,” J. Lightwave Technol. LT-7, 2095–2105 (1989).
[CrossRef]

Glance, B.

B. Glance, “Frequency stabilization of FDM optical signals,” Electron. Lett. 23, 750–752 (1987).
[CrossRef]

Goldberg, L.

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, J. F. Weller, “6-34 GHz offset phase-locking of Nd:YAG 1319-nm nonplanar ring lasers,” Electron. Lett. 25, 1242–1243 (1989).
[CrossRef]

Hodgkinson, T. G.

T. G. Hodgkinson, P. Coppin, “Pulsed operation of an optical feedback frequency synthesizer,” Electron. Lett. 26, 1155–1157 (1990).
[CrossRef]

Horiguchi, T.

Inoue, K.

K. Inoue, H. Toba, K. Nosu, “Multichannel amplification utilizing an Er3+-doped fiber amplifier,” J. Lightwave Technol. 9, 368–374 (1991).
[CrossRef]

H. Toba, K. Inoue, K. Nosu, G. Motosugi, “A multichannel laser diode frequency stabilizer for narrowly spaced optical frequency-division-multiplexing transmission,” J. Opt. Commun. 9, 3–7 (1988).

Ishida, O.

O. Ishida, H. Toba, “Lightwave synthesizer with lock-in-detected frequency references,” J. Lightwave Technol. 9, 1344–1352 (1991).
[CrossRef]

Kano, S. S.

M. D. Levenson, S. S. Kano, Introduction to Nonlinear Laser Spectroscopy (Academic, San Diego, Calif., 1988).

Koyamada, Y.

Levenson, M. D.

M. D. Levenson, S. S. Kano, Introduction to Nonlinear Laser Spectroscopy (Academic, San Diego, Calif., 1988).

Motosugi, G.

H. Toba, K. Inoue, K. Nosu, G. Motosugi, “A multichannel laser diode frequency stabilizer for narrowly spaced optical frequency-division-multiplexing transmission,” J. Opt. Commun. 9, 3–7 (1988).

Nakagawa, K.

K. Aida, J. Nakajima, K. Nakagawa, “Synthesis of optical pulsed sweep frequencies from a fiber loop with automatic gain controlled erbium-doped fiber amplifier and AOD,” in Proceedings of the European Conference on Optical Communication (Springer-Verlag, Berlin, 1992), Vol. 1, pp. 437–440.

Nakajima, J.

K. Aida, J. Nakajima, K. Nakagawa, “Synthesis of optical pulsed sweep frequencies from a fiber loop with automatic gain controlled erbium-doped fiber amplifier and AOD,” in Proceedings of the European Conference on Optical Communication (Springer-Verlag, Berlin, 1992), Vol. 1, pp. 437–440.

Nakanishi, K.

H. Toba, K. Oda, K. Nakanishi, N. Shibata, K. Nosu, N. Takato, M. Fukuda, “A 100-channel optical FDM transmission/distribution at 622 Mb/s over 50 km,” J. Lightwave Technol. 8, 1396–1401 (1990).
[CrossRef]

H. Toba, K. Oda, K. Nakanishi, K. Nosu, “Broad-band information distribution networks employing optical frequency division multiplexing technologies,” in Proceeding of the Global Telecommunications Conference (GLOBECOM) (Institute of Electrical and Electronics Engineers, San Diego, Calif., 1990), pp. 1512–1517.
[CrossRef]

Nazarathy, M.

W. V. Sorin, D. K. Donald, S. A. Newton, M. Nazarathy, “Coherent FMCW reflectometry using a temperature-tuned Nd:YAG ring laser,” IEEE Photonics Technol. Lett. 2, 902–904 (1990).
[CrossRef]

Newton, S. A.

W. V. Sorin, D. K. Donald, S. A. Newton, M. Nazarathy, “Coherent FMCW reflectometry using a temperature-tuned Nd:YAG ring laser,” IEEE Photonics Technol. Lett. 2, 902–904 (1990).
[CrossRef]

Nosu, K.

K. Inoue, H. Toba, K. Nosu, “Multichannel amplification utilizing an Er3+-doped fiber amplifier,” J. Lightwave Technol. 9, 368–374 (1991).
[CrossRef]

H. Toba, K. Oda, K. Nakanishi, N. Shibata, K. Nosu, N. Takato, M. Fukuda, “A 100-channel optical FDM transmission/distribution at 622 Mb/s over 50 km,” J. Lightwave Technol. 8, 1396–1401 (1990).
[CrossRef]

H. Toba, K. Inoue, K. Nosu, G. Motosugi, “A multichannel laser diode frequency stabilizer for narrowly spaced optical frequency-division-multiplexing transmission,” J. Opt. Commun. 9, 3–7 (1988).

H. Toba, K. Oda, K. Nakanishi, K. Nosu, “Broad-band information distribution networks employing optical frequency division multiplexing technologies,” in Proceeding of the Global Telecommunications Conference (GLOBECOM) (Institute of Electrical and Electronics Engineers, San Diego, Calif., 1990), pp. 1512–1517.
[CrossRef]

Oda, K.

H. Toba, K. Oda, K. Nakanishi, N. Shibata, K. Nosu, N. Takato, M. Fukuda, “A 100-channel optical FDM transmission/distribution at 622 Mb/s over 50 km,” J. Lightwave Technol. 8, 1396–1401 (1990).
[CrossRef]

H. Toba, K. Oda, K. Nakanishi, K. Nosu, “Broad-band information distribution networks employing optical frequency division multiplexing technologies,” in Proceeding of the Global Telecommunications Conference (GLOBECOM) (Institute of Electrical and Electronics Engineers, San Diego, Calif., 1990), pp. 1512–1517.
[CrossRef]

Shibata, N.

H. Toba, K. Oda, K. Nakanishi, N. Shibata, K. Nosu, N. Takato, M. Fukuda, “A 100-channel optical FDM transmission/distribution at 622 Mb/s over 50 km,” J. Lightwave Technol. 8, 1396–1401 (1990).
[CrossRef]

Shimizu, K.

Simpson, J. R.

E. Desurvire, C. R. Giles, J. R. Simpson, “Gain saturation effects in high-speed multichannel erbium-doped fiber amplifiers at λ = 1.53 μm,” J. Lightwave Technol. LT-7, 2095–2105 (1989).
[CrossRef]

Sorin, W. V.

W. V. Sorin, D. K. Donald, S. A. Newton, M. Nazarathy, “Coherent FMCW reflectometry using a temperature-tuned Nd:YAG ring laser,” IEEE Photonics Technol. Lett. 2, 902–904 (1990).
[CrossRef]

Takato, N.

H. Toba, K. Oda, K. Nakanishi, N. Shibata, K. Nosu, N. Takato, M. Fukuda, “A 100-channel optical FDM transmission/distribution at 622 Mb/s over 50 km,” J. Lightwave Technol. 8, 1396–1401 (1990).
[CrossRef]

Toba, H.

O. Ishida, H. Toba, “Lightwave synthesizer with lock-in-detected frequency references,” J. Lightwave Technol. 9, 1344–1352 (1991).
[CrossRef]

K. Inoue, H. Toba, K. Nosu, “Multichannel amplification utilizing an Er3+-doped fiber amplifier,” J. Lightwave Technol. 9, 368–374 (1991).
[CrossRef]

H. Toba, K. Oda, K. Nakanishi, N. Shibata, K. Nosu, N. Takato, M. Fukuda, “A 100-channel optical FDM transmission/distribution at 622 Mb/s over 50 km,” J. Lightwave Technol. 8, 1396–1401 (1990).
[CrossRef]

H. Toba, K. Inoue, K. Nosu, G. Motosugi, “A multichannel laser diode frequency stabilizer for narrowly spaced optical frequency-division-multiplexing transmission,” J. Opt. Commun. 9, 3–7 (1988).

H. Toba, K. Oda, K. Nakanishi, K. Nosu, “Broad-band information distribution networks employing optical frequency division multiplexing technologies,” in Proceeding of the Global Telecommunications Conference (GLOBECOM) (Institute of Electrical and Electronics Engineers, San Diego, Calif., 1990), pp. 1512–1517.
[CrossRef]

Weller, J. F.

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, J. F. Weller, “6-34 GHz offset phase-locking of Nd:YAG 1319-nm nonplanar ring lasers,” Electron. Lett. 25, 1242–1243 (1989).
[CrossRef]

Williams, K. J.

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, J. F. Weller, “6-34 GHz offset phase-locking of Nd:YAG 1319-nm nonplanar ring lasers,” Electron. Lett. 25, 1242–1243 (1989).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed., (Wiley, New York, 1988), pp. 560–565.

Appl. Opt. (1)

Electron. Lett. (3)

B. Glance, “Frequency stabilization of FDM optical signals,” Electron. Lett. 23, 750–752 (1987).
[CrossRef]

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, J. F. Weller, “6-34 GHz offset phase-locking of Nd:YAG 1319-nm nonplanar ring lasers,” Electron. Lett. 25, 1242–1243 (1989).
[CrossRef]

T. G. Hodgkinson, P. Coppin, “Pulsed operation of an optical feedback frequency synthesizer,” Electron. Lett. 26, 1155–1157 (1990).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

W. V. Sorin, D. K. Donald, S. A. Newton, M. Nazarathy, “Coherent FMCW reflectometry using a temperature-tuned Nd:YAG ring laser,” IEEE Photonics Technol. Lett. 2, 902–904 (1990).
[CrossRef]

J. Lightwave Technol. (5)

E. Desurvire, C. R. Giles, J. R. Simpson, “Gain saturation effects in high-speed multichannel erbium-doped fiber amplifiers at λ = 1.53 μm,” J. Lightwave Technol. LT-7, 2095–2105 (1989).
[CrossRef]

K. Inoue, H. Toba, K. Nosu, “Multichannel amplification utilizing an Er3+-doped fiber amplifier,” J. Lightwave Technol. 9, 368–374 (1991).
[CrossRef]

C. R. Giles, E. Desurvire, “Propagation of signal and noise in concatenated erbium-doped fiber optical amplifiers,” J. Lightwave Technol. 9, 147–154 (1991).
[CrossRef]

O. Ishida, H. Toba, “Lightwave synthesizer with lock-in-detected frequency references,” J. Lightwave Technol. 9, 1344–1352 (1991).
[CrossRef]

H. Toba, K. Oda, K. Nakanishi, N. Shibata, K. Nosu, N. Takato, M. Fukuda, “A 100-channel optical FDM transmission/distribution at 622 Mb/s over 50 km,” J. Lightwave Technol. 8, 1396–1401 (1990).
[CrossRef]

J. Opt. Commun. (1)

H. Toba, K. Inoue, K. Nosu, G. Motosugi, “A multichannel laser diode frequency stabilizer for narrowly spaced optical frequency-division-multiplexing transmission,” J. Opt. Commun. 9, 3–7 (1988).

Opt. Lett. (1)

Other (4)

H. Toba, K. Oda, K. Nakanishi, K. Nosu, “Broad-band information distribution networks employing optical frequency division multiplexing technologies,” in Proceeding of the Global Telecommunications Conference (GLOBECOM) (Institute of Electrical and Electronics Engineers, San Diego, Calif., 1990), pp. 1512–1517.
[CrossRef]

A. Yariv, Quantum Electronics, 3rd ed., (Wiley, New York, 1988), pp. 560–565.

K. Aida, J. Nakajima, K. Nakagawa, “Synthesis of optical pulsed sweep frequencies from a fiber loop with automatic gain controlled erbium-doped fiber amplifier and AOD,” in Proceedings of the European Conference on Optical Communication (Springer-Verlag, Berlin, 1992), Vol. 1, pp. 437–440.

M. D. Levenson, S. S. Kano, Introduction to Nonlinear Laser Spectroscopy (Academic, San Diego, Calif., 1988).

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Figures (10)

Fig. 1
Fig. 1

Basic configuration of the frequency-translation ring circuit.

Fig. 2
Fig. 2

Normalized optical transmittance of the 1-mm BPF used in the experiments. The dashed curve is a Lorentzian fitting curve.

Fig. 3
Fig. 3

Numerical results of the ASE noise output power with respect to the circulation number for the BPF center frequency deviation: curve a is for 0 GHz, curve b is for 8 GHz, curve c is for 16 GHz, and curve d is for 18 GHz.

Fig. 4
Fig. 4

Numerical results of (a) the saturated amplification gain G i , (b) the optical transmittance for the signal pulse L(fs i ), and (c) the net gain of the loop for the signal pulse (dashed curve) and the averaged net gain of the loop for the ASE noise (solid curve).

Fig. 5
Fig. 5

Profiles of the ASE noise spectrum for the circulation numbers of (a) 80, (b) 240, and (c) 400. The dashed line indicates the initial signal frequency deviation. The solid line shows the signal frequency deviation for each circulation number.

Fig. 6
Fig. 6

Numerical results of the ASE noise output power with respect to the circulation number for different input signal power and noise factors: curve a is for Ps1 = 0.1 mW and nsp − 1.5, curve b is for Ps1 = 1 mW and nsp = 4, and curve c is for Ps1 = 1 mW and nsp = 1.5.

Fig. 7
Fig. 7

Numerical results of the ASE noise output power with respect to the circulation number for the best and worst signal-polarization directions: curve a is for the best polarization state of the signal pulse with a BPF center frequency deviation value of 16 GHz, curve b is for the worst polarization state of the signal pulse with a BPF center frequency deviation value of 16 GHz, and curve c is for the worst polarization state of the signal pulse with a BPF center frequency deviation value of 0 GHz.

Fig. 8
Fig. 8

Experimental and numerical results of the ASE noise output power with respect to circulation number in the optimum frequency translation. Curve a is the experimental result for the 1536-nm wavelength, curve b is the experimental result for the 1552-nm wavelength, and curve c is the numerical result based on the homogeneous amplification gain assumption.

Fig. 9
Fig. 9

Numerical results of the ASE noise output power with respect to the circulation number for a 5-nm BPF and a + 80-MHz unit frequency shift. The BPF center frequency deviations is optimized at +40 GHz.

Fig. 10
Fig. 10

Numerical results of the optimum frequency translation limit with respect to the BPF bandwidth for different unit frequency shifts of +80 MHz (curve a), +160 MHz (curve b), +240 MHz (curve c), +320 MHz (curve d), and +400 MHz (curve e).

Tables (1)

Tables Icon

Table 1 Conditions Forming the Basis for the Numerical Calculations

Equations (12)

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

P s out i + P n out i = P s out i + 1 + P n out i + 1 = P const ,
P s in i + 1 = P s out i = L s i G i P s in i ,
P n in i + 1 = P n out i = L n i G i P n in i + L n i S i .
G i P const L s i P s in i + L n i P n in i ,
L ( f ) = L ( f c ) 1 1 + ( f f c B / 2 ) 2 ,
f s i = f s 1 + ( i 1 ) Δ f .
P s i + 1 ( f s i + Δ f ) = L ( f s i ) G i P s i ( f s i ) ,
p n i + 1 ( f + Δ f ) = L ( f ) G i p n i ( f ) + L ( f ) S i B .
P n i + 1 = f c B / 2 f c + B / 2 p n i + 1 ( f ) d f .
S i = 2 n sp ( G i 1 ) h ν B ,
G i = P const L ( f s i ) P s i ( f s i ) + f c B / 2 f c + B / 2 L ( f ) [ p n i ( f ) + 2 n sp ( G i 1 G i ) h ν ] d f .
G i = 1 L ( f s i ) , P s i P n i .

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