Abstract

An amplitude-sensitive technique associated with a heterodyne interferometer for detecting small differential phase is reported. The excess noise with the amplitude-sensitive technique is reduced by optical subtraction instead of electronic subtraction. The differential phase introduced by the orthogonally polarized laser beams is converted to the amplitudes of two heterodyne interferometric signals, which presents amplitude and phase quadrature simultaneously. Thus the excess noise power and quantum noise power are both differential phase dependent. The advantages of differential and additive operations by optical technique and the real time differential phase determination without phase lock in are demonstrated experimentally. The theoretical signal-to-noise ratio (SNR) and minimum detectable differential phase are derived, which takes quantum noise and excess noise into consideration. The experimental results demonstrated the resolutions of differential phase detection closes to 106rad/Hz(1013m/Hz) level over 100kHz bandwidth and at 108rad/Hz(1015m/Hz) level over 125MHz bandwidth, respectively, under 2.5mW incident power.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2008 (1)

B. K. Spears and N. B. Tufillaro, “A chaotic lock-in amplifier,” Am. J. Phys. 76, 213-217 (2008).
[CrossRef]

2007 (3)

M. B. Gray, J. H. Chow, K. McKenzie, and D. E. McClelland, “Using a passive fiber ring cavity to generate shot-noise-limited laser light for low-power quantum optics applications,” IEEE Photonics Technol. Lett. 19, 1063-1065 (2007).
[CrossRef]

H. K. Teng and K. C. Lang, “Heterodyne interferometer for displacement measurement with amplitude quadrature and noise suppression,” Opt. Commun. 280, 16-22 (2007).
[CrossRef]

K. McKenzie, M. B. Gray, P. K. Lam, and D. E. McClelland, “Technical limitations to homodyne detection at audio frequency,” Appl. Opt. 46, 3389-3395 (2007).
[CrossRef] [PubMed]

2006 (3)

2005 (1)

H. K. Teng, K. C. Lang, and C. C. Yen, “Optical polarization rotating technique for characterizing linear birefringence with full range,” Opt. Eng. 44, 123602(2005).
[CrossRef]

2004 (1)

2003 (1)

2001 (1)

2000 (1)

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum. 71, 2669-2676 (2000).
[CrossRef]

1999 (2)

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

1998 (1)

M. C. Cox, N. J. Copner, and B. Williams, “High sensitivity precision relative intensity noise calibration standard using low noise reference laser source,” IEE. Proc. Sci. Meas. Technol. 145, 163-165 (1998).
[CrossRef]

1997 (2)

1993 (2)

1989 (2)

H. A. Bachor and P. T. H. Fisk, “Quantum noise--a limit in photodetection,” Appl. Phys. B 49, 291-300 (1989).
[CrossRef]

G. Mana, “Diffraction effects in optical interferometers illuminated by laser sources,” Metrologia 26, 87-93 (1989).
[CrossRef]

1987 (1)

1985 (2)

G. L. Abbas, V. W. S. Chan, and T. K. Yee, “A dual-detector optical heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. 3, 1110-1122 (1985).
[CrossRef]

J. H. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. 21, 237-250 (1985).
[CrossRef]

1983 (1)

1975 (1)

Abbas, G. L.

G. L. Abbas, V. W. S. Chan, and T. K. Yee, “A dual-detector optical heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. 3, 1110-1122 (1985).
[CrossRef]

G. L. Abbas, V. W. S. Chan, and T. K. Yee, “Local-oscillator excess-noise suppression for homodyne and heterodyne detection,” Opt. Lett. 8, 419-421 (1983).
[CrossRef] [PubMed]

Bachor, H. A.

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

H. A. Bachor and P. T. H. Fisk, “Quantum noise--a limit in photodetection,” Appl. Phys. B 49, 291-300 (1989).
[CrossRef]

Bachor, H.-A.

Buchler, B. C.

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

Byer, B. L.

Chan, V. W. S.

G. L. Abbas, V. W. S. Chan, and T. K. Yee, “A dual-detector optical heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. 3, 1110-1122 (1985).
[CrossRef]

G. L. Abbas, V. W. S. Chan, and T. K. Yee, “Local-oscillator excess-noise suppression for homodyne and heterodyne detection,” Opt. Lett. 8, 419-421 (1983).
[CrossRef] [PubMed]

Chang, C. N.

Chou, C.

Chow, J. H.

M. B. Gray, J. H. Chow, K. McKenzie, and D. E. McClelland, “Using a passive fiber ring cavity to generate shot-noise-limited laser light for low-power quantum optics applications,” IEEE Photonics Technol. Lett. 19, 1063-1065 (2007).
[CrossRef]

Cohen, S.

Copner, N. J.

M. C. Cox, N. J. Copner, and B. Williams, “High sensitivity precision relative intensity noise calibration standard using low noise reference laser source,” IEE. Proc. Sci. Meas. Technol. 145, 163-165 (1998).
[CrossRef]

Cox, M. C.

M. C. Cox, N. J. Copner, and B. Williams, “High sensitivity precision relative intensity noise calibration standard using low noise reference laser source,” IEE. Proc. Sci. Meas. Technol. 145, 163-165 (1998).
[CrossRef]

De Yoreo, J. J.

J. J. De Yoreo and B. W. Woods, “Investigation of strain birefringence and wavefront distortion in 001 plates of KD2PO4,” UCRL-JC-108125 (Lawrence Livermore National Laboratory, 1991).

Fejer, M. M.

Fisk, P. T. H.

H. A. Bachor and P. T. H. Fisk, “Quantum noise--a limit in photodetection,” Appl. Phys. B 49, 291-300 (1989).
[CrossRef]

Gray, M. B.

Gustafson, E. K.

Harb, C. C.

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

Haus, H. A.

H. A. Haus, “Detection,” in Electromagnetic Noise and Quantum Optical Measurements, (Springer-Verlag, 2000), pp. 281-302.

Hobbs, P. C. D.

Hollberg, L.

E. N. Ivanov and L. Hollberg, “Wide-band suppression of laser intensity noise,” Joint Meeting of the European Time and Frequency Forum and the IEEE International Frequency Control Symposium, Geneva, Switzerland (29 May 2007), pp. 1082-1087.

Huntington, E. H.

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

Ivanov, E. N.

E. N. Ivanov and L. Hollberg, “Wide-band suppression of laser intensity noise,” Joint Meeting of the European Time and Frequency Forum and the IEEE International Frequency Control Symposium, Geneva, Switzerland (29 May 2007), pp. 1082-1087.

Johansson, B. S.

M. Schiess and B. S. Johansson, “Excess noise in heterodyne interferometer,” IEE Proc. J. 140, 217-220 (1993).

Katoh, K.

Kessler, E.

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum. 71, 2669-2676 (2000).
[CrossRef]

Kikuchi, K.

Lam, P. K.

K. McKenzie, M. B. Gray, P. K. Lam, and D. E. McClelland, “Technical limitations to homodyne detection at audio frequency,” Appl. Opt. 46, 3389-3395 (2007).
[CrossRef] [PubMed]

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

Lang, K. C.

H. K. Teng and K. C. Lang, “Heterodyne interferometer for displacement measurement with amplitude quadrature and noise suppression,” Opt. Commun. 280, 16-22 (2007).
[CrossRef]

H. K. Teng, K. C. Lang, and C. C. Yen, “Optical polarization rotating technique for characterizing linear birefringence with full range,” Opt. Eng. 44, 123602(2005).
[CrossRef]

Lawall, J.

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum. 71, 2669-2676 (2000).
[CrossRef]

Ly-Gagnon, D. S.

Lyu, C. W.

Mana, G.

G. Mana, “Diffraction effects in optical interferometers illuminated by laser sources,” Metrologia 26, 87-93 (1989).
[CrossRef]

McClelland, D. E.

K. McKenzie, M. B. Gray, P. K. Lam, and D. E. McClelland, “Technical limitations to homodyne detection at audio frequency,” Appl. Opt. 46, 3389-3395 (2007).
[CrossRef] [PubMed]

M. B. Gray, J. H. Chow, K. McKenzie, and D. E. McClelland, “Using a passive fiber ring cavity to generate shot-noise-limited laser light for low-power quantum optics applications,” IEEE Photonics Technol. Lett. 19, 1063-1065 (2007).
[CrossRef]

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

A. J. Stevenson, M. B. Gray, H.-A. Bachor, and D. E. McClelland, “Quantum-noise-limited interferometric phase measurement,” Appl. Opt. 32, 3481-3488 (1993).
[CrossRef] [PubMed]

McKenzie, K.

M. B. Gray, J. H. Chow, K. McKenzie, and D. E. McClelland, “Using a passive fiber ring cavity to generate shot-noise-limited laser light for low-power quantum optics applications,” IEEE Photonics Technol. Lett. 19, 1063-1065 (2007).
[CrossRef]

K. McKenzie, M. B. Gray, P. K. Lam, and D. E. McClelland, “Technical limitations to homodyne detection at audio frequency,” Appl. Opt. 46, 3389-3395 (2007).
[CrossRef] [PubMed]

Mlejnek, M.

Peng, L. C.

Podoleanu, A. G.

Ralph, T. C.

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

Rosa, C. C.

Schiess, M.

M. Schiess and B. S. Johansson, “Excess noise in heterodyne interferometer,” IEE Proc. J. 140, 217-220 (1993).

Shapiro, J. H.

J. H. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. 21, 237-250 (1985).
[CrossRef]

Spears, B. K.

B. K. Spears and N. B. Tufillaro, “A chaotic lock-in amplifier,” Am. J. Phys. 76, 213-217 (2008).
[CrossRef]

Spicer, J. B.

Stevenson, A. J.

Sun, K. X.

Teng, H. K.

H. K. Teng and K. C. Lang, “Heterodyne interferometer for displacement measurement with amplitude quadrature and noise suppression,” Opt. Commun. 280, 16-22 (2007).
[CrossRef]

C. Chou, H. K. Teng, C. C. Tsai, and L. P. Yu, “Balanced detector interferometric ellipsometer,” J. Opt. Soc. Am. A 23, 2871-2879 (2006).
[CrossRef]

H. K. Teng, K. C. Lang, and C. C. Yen, “Optical polarization rotating technique for characterizing linear birefringence with full range,” Opt. Eng. 44, 123602(2005).
[CrossRef]

H. K. Teng, C. Chou, C. N. Chang, and H. T. Wu, “Application of phase-to-amplitude conversion technique to linear birefringence measurements,” Appl. Opt. 42, 1798-1804(2003).
[CrossRef] [PubMed]

Tsai, C. C.

Tsukamoto, S.

Tufillaro, N. B.

B. K. Spears and N. B. Tufillaro, “A chaotic lock-in amplifier,” Am. J. Phys. 76, 213-217 (2008).
[CrossRef]

Wagner, J. W.

Williams, B.

M. C. Cox, N. J. Copner, and B. Williams, “High sensitivity precision relative intensity noise calibration standard using low noise reference laser source,” IEE. Proc. Sci. Meas. Technol. 145, 163-165 (1998).
[CrossRef]

Woods, B. W.

J. J. De Yoreo and B. W. Woods, “Investigation of strain birefringence and wavefront distortion in 001 plates of KD2PO4,” UCRL-JC-108125 (Lawrence Livermore National Laboratory, 1991).

Wu, H. T.

Yee, T. K.

G. L. Abbas, V. W. S. Chan, and T. K. Yee, “A dual-detector optical heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. 3, 1110-1122 (1985).
[CrossRef]

G. L. Abbas, V. W. S. Chan, and T. K. Yee, “Local-oscillator excess-noise suppression for homodyne and heterodyne detection,” Opt. Lett. 8, 419-421 (1983).
[CrossRef] [PubMed]

Yen, C. C.

H. K. Teng, K. C. Lang, and C. C. Yen, “Optical polarization rotating technique for characterizing linear birefringence with full range,” Opt. Eng. 44, 123602(2005).
[CrossRef]

Yu, L. P.

Am. J. Phys. (1)

B. K. Spears and N. B. Tufillaro, “A chaotic lock-in amplifier,” Am. J. Phys. 76, 213-217 (2008).
[CrossRef]

Appl. Opt. (7)

Appl. Phys. B (1)

H. A. Bachor and P. T. H. Fisk, “Quantum noise--a limit in photodetection,” Appl. Phys. B 49, 291-300 (1989).
[CrossRef]

IEE Proc. J. (1)

M. Schiess and B. S. Johansson, “Excess noise in heterodyne interferometer,” IEE Proc. J. 140, 217-220 (1993).

IEE. Proc. Sci. Meas. Technol. (1)

M. C. Cox, N. J. Copner, and B. Williams, “High sensitivity precision relative intensity noise calibration standard using low noise reference laser source,” IEE. Proc. Sci. Meas. Technol. 145, 163-165 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. H. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. 21, 237-250 (1985).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

M. B. Gray, J. H. Chow, K. McKenzie, and D. E. McClelland, “Using a passive fiber ring cavity to generate shot-noise-limited laser light for low-power quantum optics applications,” IEEE Photonics Technol. Lett. 19, 1063-1065 (2007).
[CrossRef]

J. Lightwave Technol. (2)

G. L. Abbas, V. W. S. Chan, and T. K. Yee, “A dual-detector optical heterodyne receiver for local oscillator noise suppression,” J. Lightwave Technol. 3, 1110-1122 (1985).
[CrossRef]

D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying sensing with carrier phase estimation,” J. Lightwave Technol. 24, 12-21 (2006).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Metrologia (1)

G. Mana, “Diffraction effects in optical interferometers illuminated by laser sources,” Metrologia 26, 87-93 (1989).
[CrossRef]

Opt. Commun. (1)

H. K. Teng and K. C. Lang, “Heterodyne interferometer for displacement measurement with amplitude quadrature and noise suppression,” Opt. Commun. 280, 16-22 (2007).
[CrossRef]

Opt. Eng. (1)

H. K. Teng, K. C. Lang, and C. C. Yen, “Optical polarization rotating technique for characterizing linear birefringence with full range,” Opt. Eng. 44, 123602(2005).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (2)

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

T. C. Ralph, E. H. Huntington, C. C. Harb, B. C. Buchler, P. K. Lam, D. E. McClelland, and H. A. Bachor, “Understanding and controlling the laser intensity noise,” Opt. Quantum Electron. 31, 583-598 (1999).
[CrossRef]

Rev. Sci. Instrum. (1)

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum. 71, 2669-2676 (2000).
[CrossRef]

Other (4)

H. A. Haus, “Detection,” in Electromagnetic Noise and Quantum Optical Measurements, (Springer-Verlag, 2000), pp. 281-302.

Waveform Generation, Measurements and Analysis Committee (TC-10), “IEEE standard for digitizing waveform recordings,” IEEE-STD-1057-2007 (2007).

E. N. Ivanov and L. Hollberg, “Wide-band suppression of laser intensity noise,” Joint Meeting of the European Time and Frequency Forum and the IEEE International Frequency Control Symposium, Geneva, Switzerland (29 May 2007), pp. 1082-1087.

J. J. De Yoreo and B. W. Woods, “Investigation of strain birefringence and wavefront distortion in 001 plates of KD2PO4,” UCRL-JC-108125 (Lawrence Livermore National Laboratory, 1991).

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Figures (9)

Fig. 1
Fig. 1

(a) Optical setup of proposed OVDPD. ISO, isolator; L, focus lens, DAQ; PC, data acquisition board and personal computer. (b) The orientation of linearly polarized sensing beam before incident into sensing part and that of local oscillator beam are described by points A and D. The orthogonally polarized components of sensing beam before H2 are described by point B and then point C after passing through H2. The dished lines of point C are the vertically and horizontally polarized components of P and S waves.

Fig. 2
Fig. 2

Normalized power distribution of signal, shot noise, and excess noise over the detection bandwidth as a function of differential phase are shown. The excess noise power at RIN = 10 13 is 67 dB enlarged, and the quantum noise power is 87 dB enlarged for clear observation. The solid curves are the results from i P , and dotted curves are that from i S .

Fig. 3
Fig. 3

Normalized excess noise powers resulted from ODM (solid curves) and OAM (dotted curves) show functions of phase bias with respect to different power-splitting ratio ε. The excess noise power from ODM is smaller with smaller phase bias, where that from OAM shows the opposite trend. The curves also show the excess noise power is power-splitting-ratio-ε dependent.

Fig. 4
Fig. 4

Theoretical minimum detectable phase Δ ϕ diff min as function of power-splitting ratio ε with respect to different RIN. It can be concluded from the curves that Δ ϕ diff min highly depends on ε at larger RIN and less depends on ε at smaller RIN.

Fig. 5
Fig. 5

(a) Sampled sinusoidal variations of output voltages that are proportional to the photocurrents i S , AC (circles) and i P , AC (stars), respectively, where the mirror Ms is driven slowly by a PZT. Each circle and star is the instantaneous amplitude calculated from 16 sampled data. (b) The observed phase quadrature signals of photocurrents i S , AC (OAM) and i P , AC (ODM), where the mirrors Mr and Ms remain still.

Fig. 6
Fig. 6

Spectral distributions of output voltage with differential mode at Δ ϕ 1 mrad for Curve III, 25 mrad for Curve II and close to π for Curve I. Curve IV shows the noise voltage floor by blocking the laser source where all the electronics remain power on.

Fig. 7
Fig. 7

Output power spectral distributions of ODM with photoreceiver PDp at power-splitting ratio ε = 0.5 of BS1. The peak of spectral curve denoted by Δ ϕ diff = π is 15 dBm , and the peak of spectral curve denoted by Δ ϕ diff = Δ ϕ diff min is 54 dBm . Thus Δ ϕ diff min 2.3 × 10 6 rad / Hz 1 / 2 over a 100 kHz bandwidth. The noise floor is the spectrum of photoreceiver output where laser is blocked and all the electronics remain power on.

Fig. 8
Fig. 8

Spectral distributions of ODM with single photoreceiver PDp, where the beat frequency Δ ω 80 MHz . In comparison with the spectral distributions depicted in Fig. 9, it can be observed that the excess noise is absent and quantum noise dominated at this spectral region. Δ ϕ diff min 5 × 10 8 rad / Hz 1 / 2 is determined from the peaks of curves denoted by Δ ϕ diff = π and Δ ϕ diff = Δ ϕ diff min over a 125 MHz bandwidth.

Fig. 9
Fig. 9

Ratios of measurement error as function of Δ ϕ meas with respect to the ratio r of unequal amplitudes between P and S polarization states are shown. The ratios of error ( Δ ϕ meas Δ ϕ diff ) / Δ ϕ meas in decibels resulting from detection with single photoreceiver or quadrature detection are overlapped.

Equations (20)

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E Sig , P = a E IN { exp [ i ( ω 1 t + ϕ 1 + α ) ] exp [ i ( ω 1 t + ϕ 2 + α ) ] } + b E IN exp [ i ( ω 2 t + β ) ] ,
E Sig , S = a E IN { exp [ i ( ω 1 t + ϕ 1 + α ) ] exp [ i ( ω 1 t + ϕ 2 + α ) ] } + b E IN exp [ i ( ω 2 t + β ) ] ,
Δ ϕ = ϕ 1 ϕ 1 .
i P = χ P IN { 2 a 2 [ 1 cos ( Δ ϕ ) ] + b 2 4 a b sin ( Δ ϕ 2 ) sin ( Δ ω t + γ ) } = i P , DC + i P , AC ,
i S = χ P IN { 2 a 2 [ 1 + cos ( Δ ϕ ) ] + b 2 + 4 a b cos ( Δ ϕ 2 ) cos ( Δ ω t + γ ) } = i S , DC + i S , AC ,
Δ ϕ = 2 arctan ( | i P , AC | | i S , AC | ) ,
Δ ϕ = 2 arcsin ( | i P , AC | | i P , AC , max | ) ,
Δ ϕ = 2 arccos ( | i S , AC | | i S , AC , max | ) ,
SNR = ( P signal / P noise ) 1 / 2 ,
P signal = i AC 2 Z ,
P qn = i qn 2 Z ,
i qn = ( 2 q B i ) 1 / 2 ,
P ex = RIN BZ i 2 ,
a 2 = 1 ϵ 4 , b 2 = 1 ϵ 4 ,
Δ ϕ diff min = 2 b 2 q B + b 4 RIN P B 2 ( ab ) 2 P 2 a 2 q B 4 ( ab ) 2 RIN P B ,
i P = χ P IN { a 2 ( 1 r ) 2 + 2 a 2 r 2 ( 1 cos ( Δ ϕ ) ) + b 2 ) 2 a b ( 1 + r ) sin ( Δ ϕ 2 ) 1 + ( 1 r 1 + r ) 2 cot 2 ( Δ ϕ 2 ) sin ( Δ ω t ) } ,
i S = χ P IN { a 2 ( 1 r ) 2 + 2 a 2 r 2 ( 1 + cos ( Δ ϕ ) ) + b 2 ) + 2 a b ( 1 + r ) cos ( Δ ϕ 2 ) 1 + ( 1 r 1 + r ) 2 tan 2 ( Δ ϕ 2 ) cos ( Δ ω t ) } ,
Δ ϕ meas = 2 arcsin { sin ( Δ ϕ diff 2 ) 1 + ( 1 r 1 + r ) 2 cot 2 ( Δ ϕ diff 2 ) }
Δ ϕ meas = 2 arccos { cos ( Δ ϕ 2 ) 1 + ( 1 r 1 + r ) 2 tan 2 ( Δ ϕ 2 ) }
Δ ϕ meas = 2 arctan { tan ( Δ ϕ diff 2 ) 1 + [ ( 1 r ) / ( 1 + r ) ] 2 cot 2 ( Δ ϕ diff / 2 ) 1 + [ ( 1 r ) / ( 1 + r ) ] 2 tan 2 ( Δ ϕ diff / 2 ) }

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