Abstract

Resonator micro optic gyro (RMOG) is a promising candidate for applications requiring small, light and robust gyros. In optical passive ring resonator gyros, clockwise and counter clockwise lightwaves are modulated at different frequencies to reduce the backscattering induced noise. The effectiveness of this technique, however, is determined by the carrier suppression level. Accurate modulation index and high environmental temperature stability is required for achieving high total carrier suppression for the traditional single phase modulation technique (SPMT). In this paper, we propose an RMOG based on the double phase modulation technique (DPMT). Compared with the traditional SPMT, two additional phase modulations are added to provide additional carrier suppression. It is found that the control accuracy of the modulation index and temperature stability is relaxed more than 30 times. It is easily performed for reducing the backscattering error below the shot noise limited sensitivity. The modulation parameters in the DPMT are analyzed and optimized. Based on the optimum parameters of the DPMT, a bias stability of 1.85 × 10−4 rad/s is successfully demonstrated in the polarization maintaining silica waveguide resonator with the length of 7.9 cm. This is the best result reported to date, to the best of our knowledge, for a waveguide type passive ring resonator gyro.

© 2011 OSA

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2010

2009

2008

H. Ma, S. Wang, and Z. Jin, “Silica waveguide ring resonators with multi-turn structure,” Opt. Commun. 281, 2509–2512 (2008).

2007

2006

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[CrossRef]

M. Oehme, J. Werner, E. Kasper, M. Jutzi, and M. Berroth, “High bandwidth Ge p-i-n photodetector integrated on Si,” Appl. Phys. Lett. 89(7), 071117 (2006).
[CrossRef]

X. L. Zhang, H. I. Ma, Z. H. Jin, and C. Ding, “Open-loop operation experiments in a resonator fiber-optic gyro using the phase modulation spectroscopy technique,” Appl. Opt. 45(31), 7961–7965 (2006).
[CrossRef] [PubMed]

2005

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[CrossRef] [PubMed]

C. Ciminelli, F. Peluso, and M. N. Armenise, “A new integrated optical angular velocity sensor,” Proc. SPIE 5728, 93–100 (2005).
[CrossRef]

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

2004

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

2003

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

2001

2000

K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator microoptic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000).
[CrossRef]

C. Monovoukas, A. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).
[CrossRef]

1999

K. Taguchi, K. Fukushima, A. Ishitani, and M. Ikeda, “Experimental investigation of a semiconductor ring laser as an optical gyroscope,” IEEE Trans. Instrum. Meas. 48(6), 1314–1318 (1999).
[CrossRef]

K. Suzuki, K. Takiguchi, and K. Hotate, “Reduction of backscattering-induced noise by ternary phase shift keying in resonator micro-optic gyro integrated on silica planar lightwave circuit,” Electron. Lett. 35(13), 1076–1077 (1999).
[CrossRef]

1998

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Progress on the GaAlAs ring laser gyroscope,” Alta Freq., Rivista Di Electron. 10, 45 (1998).

M. Armenise and P. J. R. Laybourn, “Design and Simulation of a Ring Laser for Miniaturised Gyroscopes,” Proc. SPIE 3464, 81–90 (1998).
[CrossRef]

1994

R. Adar, M. R. Serbin, and V. Mizrahi, “Less than 1 dB per meter propagation loss of silica waveguides measured using a ring resonator,” J. Lightwave Technol. 12(8), 1369–1372 (1994).
[CrossRef]

1991

T. J. Kaiser, D. Cardarelli, and J. Walsh, “Experimental developments in the RFOG,” Proc. SPIE 1367, 121–126 (1991).
[CrossRef]

1990

K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photon. Technol. Lett. 2(1), 75–77 (1990).
[CrossRef]

1989

K. Iwatsuki, M. Saruwatari, M. Kawachi, and H. Yamazaki, “Waveguide-type optical passive ring-resonator gyro using time division detection scheme,” Electron. Lett. 25(11), 688–689 (1989).
[CrossRef]

1988

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5μm wavelength,” Proc. SPIE 985, 202–210 (1988).

1983

J. Haavisto, “Thin film waveguides for inertial sensors,” Proc. Soc. Photo Opt. Instrum. Eng. 412, 221–228 (1983).

1976

1963

W. M. Macek and D. T. M. Davis., “Rotation rate sensing with traveling wave ring laser,” Appl. Phys. Lett. 2(3), 67 (1963).
[CrossRef]

1913

G. Sagnac, “L’ether lumineux demontre par l’effet du vent relatif d’ether dans un interferometer en rotation uniforme,” C. R. Acad. Sci. 95, 708 (1913).

Adar, R.

R. Adar, M. R. Serbin, and V. Mizrahi, “Less than 1 dB per meter propagation loss of silica waveguides measured using a ring resonator,” J. Lightwave Technol. 12(8), 1369–1372 (1994).
[CrossRef]

Armenise, M.

Armenise, M. N.

Benavidez, M.

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

Berroth, M.

M. Oehme, J. Werner, E. Kasper, M. Jutzi, and M. Berroth, “High bandwidth Ge p-i-n photodetector integrated on Si,” Appl. Phys. Lett. 89(7), 071117 (2006).
[CrossRef]

Borel, P. I.

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Caldwell, R. B.

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

Campanella, C. E.

Cao, H.-

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

Cardarelli, D.

T. J. Kaiser, D. Cardarelli, and J. Walsh, “Experimental developments in the RFOG,” Proc. SPIE 1367, 121–126 (1991).
[CrossRef]

Chen, Q. Y.

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5μm wavelength,” Proc. SPIE 985, 202–210 (1988).

Ciminelli, C.

Cohen, O.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Davis, D. T. M.

W. M. Macek and D. T. M. Davis., “Rotation rate sensing with traveling wave ring laser,” Appl. Phys. Lett. 2(3), 67 (1963).
[CrossRef]

De Leonardis, F.

Dell'Olio, F.

Demma, N. A.

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5μm wavelength,” Proc. SPIE 985, 202–210 (1988).

Deng, H.

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

Ding, C.

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[CrossRef]

X. L. Zhang, H. I. Ma, Z. H. Jin, and C. Ding, “Open-loop operation experiments in a resonator fiber-optic gyro using the phase modulation spectroscopy technique,” Appl. Opt. 45(31), 7961–7965 (2006).
[CrossRef] [PubMed]

Donati, S.

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Progress on the GaAlAs ring laser gyroscope,” Alta Freq., Rivista Di Electron. 10, 45 (1998).

Eliseev, P. G.

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

Fage-Pedersen, J.

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Fang, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[CrossRef] [PubMed]

Fukushima, K.

K. Taguchi, K. Fukushima, A. Ishitani, and M. Ikeda, “Experimental investigation of a semiconductor ring laser as an optical gyroscope,” IEEE Trans. Instrum. Meas. 48(6), 1314–1318 (1999).
[CrossRef]

Giuliani, G.

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Progress on the GaAlAs ring laser gyroscope,” Alta Freq., Rivista Di Electron. 10, 45 (1998).

Haavisto, J.

J. Haavisto, “Thin film waveguides for inertial sensors,” Proc. Soc. Photo Opt. Instrum. Eng. 412, 221–228 (1983).

Hak, D.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[CrossRef] [PubMed]

Hirose, A.

K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photon. Technol. Lett. 2(1), 75–77 (1990).
[CrossRef]

Hotate, K.

K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator microoptic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000).
[CrossRef]

K. Suzuki, K. Takiguchi, and K. Hotate, “Reduction of backscattering-induced noise by ternary phase shift keying in resonator micro-optic gyro integrated on silica planar lightwave circuit,” Electron. Lett. 35(13), 1076–1077 (1999).
[CrossRef]

K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photon. Technol. Lett. 2(1), 75–77 (1990).
[CrossRef]

Hsiao, H. K.

Hubner, J

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Hübner, J.

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Ikeda, M.

K. Taguchi, K. Fukushima, A. Ishitani, and M. Ikeda, “Experimental investigation of a semiconductor ring laser as an optical gyroscope,” IEEE Trans. Instrum. Meas. 48(6), 1314–1318 (1999).
[CrossRef]

Ishitani, A.

K. Taguchi, K. Fukushima, A. Ishitani, and M. Ikeda, “Experimental investigation of a semiconductor ring laser as an optical gyroscope,” IEEE Trans. Instrum. Meas. 48(6), 1314–1318 (1999).
[CrossRef]

Iwatsuki, K.

K. Iwatsuki, M. Saruwatari, M. Kawachi, and H. Yamazaki, “Waveguide-type optical passive ring-resonator gyro using time division detection scheme,” Electron. Lett. 25(11), 688–689 (1989).
[CrossRef]

Jin, Z.

H. Ma, S. Wang, and Z. Jin, “Silica waveguide ring resonators with multi-turn structure,” Opt. Commun. 281, 2509–2512 (2008).

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[CrossRef]

Jin, Z. H.

Jones, R.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Jutzi, M.

M. Oehme, J. Werner, E. Kasper, M. Jutzi, and M. Berroth, “High bandwidth Ge p-i-n photodetector integrated on Si,” Appl. Phys. Lett. 89(7), 071117 (2006).
[CrossRef]

Kaiser, T. J.

T. J. Kaiser, D. Cardarelli, and J. Walsh, “Experimental developments in the RFOG,” Proc. SPIE 1367, 121–126 (1991).
[CrossRef]

Kasper, E.

M. Oehme, J. Werner, E. Kasper, M. Jutzi, and M. Berroth, “High bandwidth Ge p-i-n photodetector integrated on Si,” Appl. Phys. Lett. 89(7), 071117 (2006).
[CrossRef]

Kawachi, M.

K. Iwatsuki, M. Saruwatari, M. Kawachi, and H. Yamazaki, “Waveguide-type optical passive ring-resonator gyro using time division detection scheme,” Electron. Lett. 25(11), 688–689 (1989).
[CrossRef]

Kristensen, M

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Kristensen, M.

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Laybourn, P. J. R.

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Progress on the GaAlAs ring laser gyroscope,” Alta Freq., Rivista Di Electron. 10, 45 (1998).

M. Armenise and P. J. R. Laybourn, “Design and Simulation of a Ring Laser for Miniaturised Gyroscopes,” Proc. SPIE 3464, 81–90 (1998).
[CrossRef]

Liao, L.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Ling, H.

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

Liu, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Liu, C.-y.

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

Ma, H.

H. Ma, S. Wang, and Z. Jin, “Silica waveguide ring resonators with multi-turn structure,” Opt. Commun. 281, 2509–2512 (2008).

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[CrossRef]

Ma, H. I.

Macek, W. M.

W. M. Macek and D. T. M. Davis., “Rotation rate sensing with traveling wave ring laser,” Appl. Phys. Lett. 2(3), 67 (1963).
[CrossRef]

Maseeh, F.

C. Monovoukas, A. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).
[CrossRef]

Miesel, K. A.

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5μm wavelength,” Proc. SPIE 985, 202–210 (1988).

Mizrahi, V.

R. Adar, M. R. Serbin, and V. Mizrahi, “Less than 1 dB per meter propagation loss of silica waveguides measured using a ring resonator,” J. Lightwave Technol. 12(8), 1369–1372 (1994).
[CrossRef]

Monovoukas, C.

C. Monovoukas, A. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).
[CrossRef]

Nicolaescu, R.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Oehme, M.

M. Oehme, J. Werner, E. Kasper, M. Jutzi, and M. Berroth, “High bandwidth Ge p-i-n photodetector integrated on Si,” Appl. Phys. Lett. 89(7), 071117 (2006).
[CrossRef]

Osinski, M.

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

Paniccia, M.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Passaro, V. M. N.

Peake, G. M.

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

Pedersen, J. F

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Peluso, F.

C. Ciminelli, F. Peluso, and M. N. Armenise, “A new integrated optical angular velocity sensor,” Proc. SPIE 5728, 93–100 (2005).
[CrossRef]

Poulsen, M. R.

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Povlsen, J. H.

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Rong, H.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[CrossRef] [PubMed]

Rottwitt, K

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Rottwitt, K.

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Rouse, G. F.

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5μm wavelength,” Proc. SPIE 985, 202–210 (1988).

Rubin, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Sagnac, G.

G. Sagnac, “L’ether lumineux demontre par l’effet du vent relatif d’ether dans un interferometer en rotation uniforme,” C. R. Acad. Sci. 95, 708 (1913).

Samara-Rubio, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Sanders, G. A.

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5μm wavelength,” Proc. SPIE 985, 202–210 (1988).

Saruwatari, M.

K. Iwatsuki, M. Saruwatari, M. Kawachi, and H. Yamazaki, “Waveguide-type optical passive ring-resonator gyro using time division detection scheme,” Electron. Lett. 25(11), 688–689 (1989).
[CrossRef]

Serbin, M. R.

R. Adar, M. R. Serbin, and V. Mizrahi, “Less than 1 dB per meter propagation loss of silica waveguides measured using a ring resonator,” J. Lightwave Technol. 12(8), 1369–1372 (1994).
[CrossRef]

Shorthill, R. W.

Smagley, V. A.

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

Smolyakov, G. A.

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

Sorel, M.

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Progress on the GaAlAs ring laser gyroscope,” Alta Freq., Rivista Di Electron. 10, 45 (1998).

Strandjord, L. K.

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5μm wavelength,” Proc. SPIE 985, 202–210 (1988).

Suzuki, K.

K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator microoptic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000).
[CrossRef]

K. Suzuki, K. Takiguchi, and K. Hotate, “Reduction of backscattering-induced noise by ternary phase shift keying in resonator micro-optic gyro integrated on silica planar lightwave circuit,” Electron. Lett. 35(13), 1076–1077 (1999).
[CrossRef]

Svalgaard, M

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Svalgaard, M.

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Svendsen, W.

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

Swiecki, A.

C. Monovoukas, A. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).
[CrossRef]

Taguchi, K.

K. Taguchi, K. Fukushima, A. Ishitani, and M. Ikeda, “Experimental investigation of a semiconductor ring laser as an optical gyroscope,” IEEE Trans. Instrum. Meas. 48(6), 1314–1318 (1999).
[CrossRef]

Takiguchi, K.

K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator microoptic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000).
[CrossRef]

K. Suzuki, K. Takiguchi, and K. Hotate, “Reduction of backscattering-induced noise by ternary phase shift keying in resonator micro-optic gyro integrated on silica planar lightwave circuit,” Electron. Lett. 35(13), 1076–1077 (1999).
[CrossRef]

K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photon. Technol. Lett. 2(1), 75–77 (1990).
[CrossRef]

Vali, V.

Walsh, J.

T. J. Kaiser, D. Cardarelli, and J. Walsh, “Experimental developments in the RFOG,” Proc. SPIE 1367, 121–126 (1991).
[CrossRef]

Wang, S.

H. Ma, S. Wang, and Z. Jin, “Silica waveguide ring resonators with multi-turn structure,” Opt. Commun. 281, 2509–2512 (2008).

Werner, J.

M. Oehme, J. Werner, E. Kasper, M. Jutzi, and M. Berroth, “High bandwidth Ge p-i-n photodetector integrated on Si,” Appl. Phys. Lett. 89(7), 071117 (2006).
[CrossRef]

Winick, K. A.

Yamazaki, H.

K. Iwatsuki, M. Saruwatari, M. Kawachi, and H. Yamazaki, “Waveguide-type optical passive ring-resonator gyro using time division detection scheme,” Electron. Lett. 25(11), 688–689 (1989).
[CrossRef]

Zhang, X.

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[CrossRef]

Zhang, X. L.

Adv. Opt. Photon.

Alta Freq., Rivista Di Electron.

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Progress on the GaAlAs ring laser gyroscope,” Alta Freq., Rivista Di Electron. 10, 45 (1998).

Appl. Opt.

Appl. Phys. Lett.

M. Oehme, J. Werner, E. Kasper, M. Jutzi, and M. Berroth, “High bandwidth Ge p-i-n photodetector integrated on Si,” Appl. Phys. Lett. 89(7), 071117 (2006).
[CrossRef]

W. M. Macek and D. T. M. Davis., “Rotation rate sensing with traveling wave ring laser,” Appl. Phys. Lett. 2(3), 67 (1963).
[CrossRef]

H.- Cao, C.-y. Liu, H. Ling, H. Deng, M. Benavidez, V. A. Smagley, R. B. Caldwell, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osiński, “Frequency beating between monolithically integrated semiconductor ring lasers,” Appl. Phys. Lett. 86(4), 041101 (2005).
[CrossRef]

C. R. Acad. Sci.

G. Sagnac, “L’ether lumineux demontre par l’effet du vent relatif d’ether dans un interferometer en rotation uniforme,” C. R. Acad. Sci. 95, 708 (1913).

Electron. Lett.

K. Iwatsuki, M. Saruwatari, M. Kawachi, and H. Yamazaki, “Waveguide-type optical passive ring-resonator gyro using time division detection scheme,” Electron. Lett. 25(11), 688–689 (1989).
[CrossRef]

K. Suzuki, K. Takiguchi, and K. Hotate, “Reduction of backscattering-induced noise by ternary phase shift keying in resonator micro-optic gyro integrated on silica planar lightwave circuit,” Electron. Lett. 35(13), 1076–1077 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photon. Technol. Lett. 2(1), 75–77 (1990).
[CrossRef]

IEEE Trans. Instrum. Meas.

K. Taguchi, K. Fukushima, A. Ishitani, and M. Ikeda, “Experimental investigation of a semiconductor ring laser as an optical gyroscope,” IEEE Trans. Instrum. Meas. 48(6), 1314–1318 (1999).
[CrossRef]

J. Lightwave Technol.

Nature

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Opt. Commun.

H. Ma, S. Wang, and Z. Jin, “Silica waveguide ring resonators with multi-turn structure,” Opt. Commun. 281, 2509–2512 (2008).

Opt. Eng.

M. R. Poulsen, P. I. Borel, J. Fage-Pedersen, J. Hübner, M. Kristensen, J. H. Povlsen, K. Rottwitt, M. Svalgaard, W. Svendsen, J. F Pedersen, J Hubner, M Kristensen, J. H. Povlsen, K Rottwitt, M Svalgaard, and W. Svendsen, “Advances in silica-based integrated optics,” Opt. Eng. 42(10), 2821–2834 (2003).
[CrossRef]

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[CrossRef]

Opt. Express

Proc. Soc. Photo Opt. Instrum. Eng.

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Proc. SPIE

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5μm wavelength,” Proc. SPIE 985, 202–210 (1988).

T. J. Kaiser, D. Cardarelli, and J. Walsh, “Experimental developments in the RFOG,” Proc. SPIE 1367, 121–126 (1991).
[CrossRef]

C. Ciminelli, F. Peluso, and M. N. Armenise, “A new integrated optical angular velocity sensor,” Proc. SPIE 5728, 93–100 (2005).
[CrossRef]

C. Monovoukas, A. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).
[CrossRef]

M. Armenise and P. J. R. Laybourn, “Design and Simulation of a Ring Laser for Miniaturised Gyroscopes,” Proc. SPIE 3464, 81–90 (1998).
[CrossRef]

Other

S. Ezekiel, “Optical Gyroscope Options: Principles and Challenges,” in Optical Fiber Sensors, OSA Technical Digest (CD) (Optical Society of America, 2006), paper MC1.

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A. Lawrence, “Providing an inexpensive gyro for the navigation mass market,” Institute of Navigation, National Technical Meeting, (Academic, San Diego, Calif. 1990), 161–166.

H. Mao, H. Ma, and Z. Jin, “Resonator Micro-Optic Gyroscope Based on the Double Phase Modulation Technique,” in Proc. CLEO-2010, JWA52 (2010).

C. Vannahme, H. Suche, S. Reza, R. Ricken, V. Quiring, and W. Sohler, “Integrated optical Ti:LiNbO3 ring resonator for rotation rate sensing,” 13th Eur. Conf. Integrated Optics, (The Technical University of Denmark, Building 116, Copenhagen, Denmark, 2007), http://www.ecio-conference.org/2007/index.html .

C. Ciminelli, F. Dell’Olio, V. M. N. Passaro, and M. N. Armenise, “Low loss InP-based ring resonators for integrated optical gyroscopes,” presented at Caneus 2009 Workshop, NASA Ames Research Center, Moffett Field, Calif, March1–6, 2009.

H. Ma, Z. He, and K. Hotate, “Sensitivity improvement of waveguide-type optical passive ring resonator gyroscope by carrier suppression,” in Proc. OFS-20, 750353–1-750353–4 (2009).

T. Imai, Y. Miki, S. Maeda, and K. Nishide, “Development of resonator fiber optic gyros, Eleventh International Conference on Optical Fiber Sensors,” Advanced Sensing Photonics, Japan Society of Applied Physics, Special Exhibition Ex2–1 (1996).

W. X. Zhang, Fiber Optic Gyroscope and its Application, 1st ed., (National Defense Industry Press, Beijing, China, 2008).

Y. Chen, H. Ma, and Z. Jin, “New method to measure the half-wave voltage of the phase modulator,” The 2nd Asia-Pacific Optical Sensors Conference (APOS), 28–30 June, Guangzhou, China, paper TU6 (2010).

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Figures (7)

Fig. 1
Fig. 1

Experimental setup of the RMOG based on the DPMT. FL: fiber laser; ISO: isolator; C1, C2: couplers; PM1, PM2, PM3, PM4: phase modulators; SG1, SG2, SG3, SG4: signal generators; CIR1, CIR2: circulators; Sync: synchronized signal; PLC: planar lightwave circuit; WRR: waveguide ring resonator; PD1, PD2: photodetectors; LIA1, LIA2: lock-in amplifiers; FBC: feedback circuit.

Fig. 2
Fig. 2

(a) Basic model of the DPMT with two phase modulators (PM), (b) Basic model of the DPMT with one PM and an adder, (c) Analysis model of the WRR with the DPMT.

Fig. 3
Fig. 3

Calculated backscattering error in the RMOG. (a) relationship between the modulation amplitude accuracy and the backscattering noise with the DPMT (bottom curve) and the SPMT (upper curve), respectively. (b) relationship between the temperature drift and the backscattering error in the RMOG with the DPMT (bottom curve) and the SPMT (upper curve) applied, respectively. The half-wave voltage of the phase modulator is 3.0 V and its temperature coefficient is 500 ppm/°C.

Fig. 4
Fig. 4

Simulation results of the RMOG with the DPMT. (a) influence of modulation frequencies on the slope of the demodulation curve with the unit gain of the LIA when the phase modulation indices for both M1 and M2 are 2.38. The incident light power of the photodetector is about 200 µW. The responsivity of the photodetector Rv is 0.848 V/mW; (b) normalized demodulation curve with the modulation frequencies f1 of 9 MHz, f2 of 1 kHz, and the modulation indices for both M1 and M2 of 2.38.

Fig. 5
Fig. 5

Measured resonance curve and the demodulation curve for the CCW lightwave. (a) resonance curve for the CCW lightwave. (b) demodulation curve for the CCW lightwave with a gain of 80. The modulation frequencies f1 and f2 are 9.0 MHz and 1.0 KHz, respectively. The modulation indices M1 and M2 are 2.38. The optical power at the photodetector is about 200 µW. The responsivity of the photodetector Rv is 0.848 V/mW.

Fig. 6
Fig. 6

Experimental results of the RMOG compared with the DPMT and the SPMT. (a) Influence of the modulation amplitude applied with PM3 on the bias stability of the RMOG with the SPMT (f1 = 9.0 MHz, f3 = 9.3 MHz, M1 = 2.38) and the DPMT (f1 = 9.0 MHz, f2 = 1 KHz, f3 = 9.3 MHz, f4 = 1.1 KHz, M1 = M2 = M4 = 2.38). The half-wave voltage of the phase modulator is 3.0 V. (b) Typical outputs of the RMOG over 60 seconds with an integration time of 10 s. M1 = M2 = M3 = M4 = 2.38; f1 = 9.0 MHz, f2 = 1 KHz, f3 = 9.3 MHz, f4 = 1.1 KHz.

Fig. 7
Fig. 7

Experimental result of the RMOG based on the DPMT. Gyro output when the rotary table swings from CW to CCW directions with a rotation rate of 1.75 × 10−3 rad/s.

Equations (9)

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E R i n ( t ) = E 0 n = + m = + J n ( M 1 ) J m ( M 2 ) exp j ( ω 0 + n ω 1 + m ω 2 ) t ,
E R o u t ( t ) = E 0 n = + m = + J n ( M 1 ) J m ( M 2 ) exp [ j ω ( n , m ) t ] H ( ω ( n , m ) ) exp [ j Φ ( ω ( n , m ) ) ] ,
V P D o u t ( t ) = 1 2 c ε 0 E 0 2 R v n = m = n = m ' = J n ( M 1 ) J m ( M 2 ) J n ( M 1 ) J m ' ( M 2 ) exp j ( 2 π ( ( n n ) f 1 + ( m m ' ) f 2 ) t ) H ( ω ( n , m ) ) H ( ω ( n ' , m ' ) ) exp j ( Φ ( ω ( n , m ) ) Φ ( ω ( n ' , m ' ) ) ) ,
V P D f 1 ( t ) = 1 2 c ε 0 E 0 2 R v n = m = J n ( M 1 ) J n + 1 ( M 1 ) J m 2 ( M 2 ) H ( ω ( n , m ) ) H ( ω ( n + 1 , m ) ) [ cos ( 2 π f 1 t + Φ ( ω ( n + 1 , m ) ) Φ ( ω ( n , m ) ) ) ] .
V LIA = A 1 sin φ 1 B 1 cos φ 1 ,
A 1 = 1 2 c ε 0 E 0 2 R v m = n = 0 J n ( M 1 ) J n + 1 ( M 1 ) J m 2 ( M 2 ) { H ( ω ( n , m ) ) H ( ω ( n + 1 , m ) ) cos ( Φ ( ω ( n + 1 , m ) ) Φ ( ω ( n , m ) ) ) H ( ω ( n , m ) ) H ( ω ( n 1 , m ) ) cos ( Φ ( ω ( n , m ) ) Φ ( ω ( n 1 , m ) ) ) } ,
B 1 = 1 2 c ε 0 E 0 2 R v m = n = 0 J n ( M 1 ) J n + 1 ( M 1 ) J m 2 ( M 2 ) { H ( ω ( n , m ) ) H ( ω ( n + 1 , m ) ) sin ( Φ ( ω ( n + 1 , m ) ) Φ ( ω ( n , m ) ) ) H ( ω ( n , m ) ) H ( ω ( n 1 , m ) ) sin ( Φ ( ω ( n , m ) ) Φ ( ω ( n 1 , m ) ) ) } .
Ω min = ( 2 c λ 0 4 A η D τ ) ( 1 F P i / ( h f 0 ) ) ,
B I A S b s 1 = c λ 0 σ R ( 2 π ) 2 D L ( Δ V V ) N ,

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