Abstract

A triple-sensor multiplexed fiber-optic displacement sensor, which can measure the displacements of three different objects or the three-dimensional displacement of a single object, is introduced. The sensor is based on the principles of optical frequency-modulated continuous-wave interference and frequency-division multiplexing. The beat signals from the individual sensors are assigned in the frequency domain and separated with different electrical bandpass filters. The displacements of objects can be determined simultaneously by detecting the phase shifts of the corresponding signals. The cross talk between the individual sensors is evaluated, and an accuracy of 0.08  μm in a dynamic range of 1000  μm is achieved.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
  2. J. M. Senior, G. Murtaza, A. I. Stirling, and G. H. Wainwright, "Referenced intensity-modulated optical fiber sensor system for measuring linear displacement," Proc. SPIE 1795, 353-361 (1993).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  7. S. T. Lee, V. P. N. Nampoori, C. P. G. Vallabhan, and P. Radhakrishnan, "Macrobending in a fiber-optic interferometric sensor for displacement and weight measurement," Proc. SPIE 4946, 126-128 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  20. F. Farahi, A. S. Gerges, J. D. C. Jones, and D. A. Jackson, "Time-division multiplexing of fiber-optic interferometric sensors using a frequency modulated laser diode," Electron. Lett. 24, 54-55 (1988).
    [CrossRef]
  21. F. Farahi, J. D. C. Jones, and D. A. Jackson, "Multiplexed fiber-optic interferometric sensing system: combined frequency and time division," Electron. Lett. 24, 409-410 (1988).
    [CrossRef]
  22. J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, and H. J. Shaw, "Coherence multiplexing of fiber-optic interferometric sensors," J. Lightwave Technol. LT-3, 1062-1072 (1985).
    [CrossRef]
  23. J. Zheng, "Reflectometric fiber-optic frequency-modulated continuous-wave interferometric displacement sensor," Opt. Eng. 44, 124404 (2005).
    [CrossRef]

2005

J. Zheng, "Reflectometric fiber-optic frequency-modulated continuous-wave interferometric displacement sensor," Opt. Eng. 44, 124404 (2005).
[CrossRef]

2004

2003

S. T. Lee, V. P. N. Nampoori, C. P. G. Vallabhan, and P. Radhakrishnan, "Macrobending in a fiber-optic interferometric sensor for displacement and weight measurement," Proc. SPIE 4946, 126-128 (2003).
[CrossRef]

2002

1999

F. Suganuma, A. Shimamoto, and K. Tanaka, "Development of a differential optical-fiber displacement sensor," Appl. Opt. 38, 1103-1109 (1999).
[CrossRef]

J. Zheng and S. Albin, "Self-referenced reflective intensity modulated fiber-optic displacement sensor," Opt. Eng. 38, 227-232 (1999).
[CrossRef]

1998

1996

Y. J. Rao and D. A. Jackson, "Recent progress in fibre optic low-coherence interferometry," Meas. Sci. Technol. 7, 981-999 (1996).
[CrossRef]

A. Shimamoto and K. Tanaka, "Geometrical analysis of an optical fiber bundle displacement sensor," Appl. Opt. 35, 6767-6774 (1996).
[CrossRef] [PubMed]

1995

1993

Y. J. Rao, Y. N. Ning, and D. A. Jackson, "Synthesized source for white-light sensing systems," Opt. Lett. 18, 462-464 (1993).
[CrossRef] [PubMed]

J. M. Senior, G. Murtaza, A. I. Stirling, and G. H. Wainwright, "Referenced intensity-modulated optical fiber sensor system for measuring linear displacement," Proc. SPIE 1795, 353-361 (1993).
[CrossRef]

1990

A. S. Gerges, T. P. Newson, and D. A. Jackson, "Coherence tuned fiber-optic sensing system, with self-initialization, based on a multimode laser diode," Appl. Opt. 29, 4473-4480 (1990).
[CrossRef] [PubMed]

T. A. Berkoff and A. D. Kersey, "Interferometric fiber displacement/strain sensor based on source coherence synthesis," Electron. Lett. 26, 452-453 (1990).
[CrossRef]

1988

F. Farahi, A. S. Gerges, J. D. C. Jones, and D. A. Jackson, "Time-division multiplexing of fiber-optic interferometric sensors using a frequency modulated laser diode," Electron. Lett. 24, 54-55 (1988).
[CrossRef]

F. Farahi, J. D. C. Jones, and D. A. Jackson, "Multiplexed fiber-optic interferometric sensing system: combined frequency and time division," Electron. Lett. 24, 409-410 (1988).
[CrossRef]

1987

1986

I. Sakai, "Frequency-division multiplexing of optical-fiber sensors using a frequency-modulated source," Opt. Quantum. Electron. 18, 279-289 (1986).
[CrossRef]

1985

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, and H. J. Shaw, "Coherence multiplexing of fiber-optic interferometric sensors," J. Lightwave Technol. LT-3, 1062-1072 (1985).
[CrossRef]

M. Johnson and G. Goodman, "One- and two-dimensional, differential, reflective fiber displacement sensors," Appl. Opt. 24, 2369-2372 (1985).
[CrossRef] [PubMed]

1982

B. Culshaw and I. P. Giles, "Frequency modulated heterodyne optical Sagnac interferometer," IEEE J. Quantum. Electron. QE-18, 690-693 (1982).
[CrossRef]

D. A. Jackson, A. D. Kersey, M. Corke, and J. D. C. Jones, "Pseudoheterodyne detection scheme for optical interferometers," Electron. Lett. 18, 1081-1083 (1982).
[CrossRef]

Albin, S.

J. Zheng and S. Albin, "Self-referenced reflective intensity modulated fiber-optic displacement sensor," Opt. Eng. 38, 227-232 (1999).
[CrossRef]

Berkoff, T. A.

T. A. Berkoff and A. D. Kersey, "Interferometric fiber displacement/strain sensor based on source coherence synthesis," Electron. Lett. 26, 452-453 (1990).
[CrossRef]

Brooks, J. L.

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, and H. J. Shaw, "Coherence multiplexing of fiber-optic interferometric sensors," J. Lightwave Technol. LT-3, 1062-1072 (1985).
[CrossRef]

Carr, S.

Chen, Y.

Claus, R.

Corke, M.

D. A. Jackson, A. D. Kersey, M. Corke, and J. D. C. Jones, "Pseudoheterodyne detection scheme for optical interferometers," Electron. Lett. 18, 1081-1083 (1982).
[CrossRef]

Culshaw, B.

B. Culshaw and I. P. Giles, "Frequency modulated heterodyne optical Sagnac interferometer," IEEE J. Quantum. Electron. QE-18, 690-693 (1982).
[CrossRef]

Davies, D. E. N.

Farahi, F.

F. Farahi, A. S. Gerges, J. D. C. Jones, and D. A. Jackson, "Time-division multiplexing of fiber-optic interferometric sensors using a frequency modulated laser diode," Electron. Lett. 24, 54-55 (1988).
[CrossRef]

F. Farahi, J. D. C. Jones, and D. A. Jackson, "Multiplexed fiber-optic interferometric sensing system: combined frequency and time division," Electron. Lett. 24, 409-410 (1988).
[CrossRef]

Gerges, A. S.

A. S. Gerges, T. P. Newson, and D. A. Jackson, "Coherence tuned fiber-optic sensing system, with self-initialization, based on a multimode laser diode," Appl. Opt. 29, 4473-4480 (1990).
[CrossRef] [PubMed]

F. Farahi, A. S. Gerges, J. D. C. Jones, and D. A. Jackson, "Time-division multiplexing of fiber-optic interferometric sensors using a frequency modulated laser diode," Electron. Lett. 24, 54-55 (1988).
[CrossRef]

Giles, I. P.

B. Culshaw and I. P. Giles, "Frequency modulated heterodyne optical Sagnac interferometer," IEEE J. Quantum. Electron. QE-18, 690-693 (1982).
[CrossRef]

Goodman, G.

Jackson, D. A.

Y. J. Rao and D. A. Jackson, "Recent progress in fibre optic low-coherence interferometry," Meas. Sci. Technol. 7, 981-999 (1996).
[CrossRef]

Y. J. Rao, Y. N. Ning, and D. A. Jackson, "Synthesized source for white-light sensing systems," Opt. Lett. 18, 462-464 (1993).
[CrossRef] [PubMed]

A. S. Gerges, T. P. Newson, and D. A. Jackson, "Coherence tuned fiber-optic sensing system, with self-initialization, based on a multimode laser diode," Appl. Opt. 29, 4473-4480 (1990).
[CrossRef] [PubMed]

F. Farahi, A. S. Gerges, J. D. C. Jones, and D. A. Jackson, "Time-division multiplexing of fiber-optic interferometric sensors using a frequency modulated laser diode," Electron. Lett. 24, 54-55 (1988).
[CrossRef]

F. Farahi, J. D. C. Jones, and D. A. Jackson, "Multiplexed fiber-optic interferometric sensing system: combined frequency and time division," Electron. Lett. 24, 409-410 (1988).
[CrossRef]

D. A. Jackson, A. D. Kersey, M. Corke, and J. D. C. Jones, "Pseudoheterodyne detection scheme for optical interferometers," Electron. Lett. 18, 1081-1083 (1982).
[CrossRef]

Johnson, M.

Jones, J. D. C.

F. Farahi, J. D. C. Jones, and D. A. Jackson, "Multiplexed fiber-optic interferometric sensing system: combined frequency and time division," Electron. Lett. 24, 409-410 (1988).
[CrossRef]

F. Farahi, A. S. Gerges, J. D. C. Jones, and D. A. Jackson, "Time-division multiplexing of fiber-optic interferometric sensors using a frequency modulated laser diode," Electron. Lett. 24, 54-55 (1988).
[CrossRef]

D. A. Jackson, A. D. Kersey, M. Corke, and J. D. C. Jones, "Pseudoheterodyne detection scheme for optical interferometers," Electron. Lett. 18, 1081-1083 (1982).
[CrossRef]

Kersey, A. D.

T. A. Berkoff and A. D. Kersey, "Interferometric fiber displacement/strain sensor based on source coherence synthesis," Electron. Lett. 26, 452-453 (1990).
[CrossRef]

D. A. Jackson, A. D. Kersey, M. Corke, and J. D. C. Jones, "Pseudoheterodyne detection scheme for optical interferometers," Electron. Lett. 18, 1081-1083 (1982).
[CrossRef]

Kim, B. Y.

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, and H. J. Shaw, "Coherence multiplexing of fiber-optic interferometric sensors," J. Lightwave Technol. LT-3, 1062-1072 (1985).
[CrossRef]

Lee, S. T.

S. T. Lee, V. P. N. Nampoori, C. P. G. Vallabhan, and P. Radhakrishnan, "Macrobending in a fiber-optic interferometric sensor for displacement and weight measurement," Proc. SPIE 4946, 126-128 (2003).
[CrossRef]

Li, T.

Murphy, K.

Murtaza, G.

J. M. Senior, G. Murtaza, A. I. Stirling, and G. H. Wainwright, "Referenced intensity-modulated optical fiber sensor system for measuring linear displacement," Proc. SPIE 1795, 353-361 (1993).
[CrossRef]

Nampoori, V. P. N.

S. T. Lee, V. P. N. Nampoori, C. P. G. Vallabhan, and P. Radhakrishnan, "Macrobending in a fiber-optic interferometric sensor for displacement and weight measurement," Proc. SPIE 4946, 126-128 (2003).
[CrossRef]

Newson, T. P.

Ning, Y. N.

Radhakrishnan, P.

S. T. Lee, V. P. N. Nampoori, C. P. G. Vallabhan, and P. Radhakrishnan, "Macrobending in a fiber-optic interferometric sensor for displacement and weight measurement," Proc. SPIE 4946, 126-128 (2003).
[CrossRef]

Rao, Y. J.

Y. J. Rao and D. A. Jackson, "Recent progress in fibre optic low-coherence interferometry," Meas. Sci. Technol. 7, 981-999 (1996).
[CrossRef]

Y. J. Rao, Y. N. Ning, and D. A. Jackson, "Synthesized source for white-light sensing systems," Opt. Lett. 18, 462-464 (1993).
[CrossRef] [PubMed]

Sakai, I.

I. Sakai, "Frequency-division multiplexing of optical-fiber sensors using a frequency-modulated source," Opt. Quantum. Electron. 18, 279-289 (1986).
[CrossRef]

Senior, J. M.

J. M. Senior, G. Murtaza, A. I. Stirling, and G. H. Wainwright, "Referenced intensity-modulated optical fiber sensor system for measuring linear displacement," Proc. SPIE 1795, 353-361 (1993).
[CrossRef]

Shaw, H. J.

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, and H. J. Shaw, "Coherence multiplexing of fiber-optic interferometric sensors," J. Lightwave Technol. LT-3, 1062-1072 (1985).
[CrossRef]

Shimamoto, A.

Stirling, A. I.

J. M. Senior, G. Murtaza, A. I. Stirling, and G. H. Wainwright, "Referenced intensity-modulated optical fiber sensor system for measuring linear displacement," Proc. SPIE 1795, 353-361 (1993).
[CrossRef]

Suganuma, F.

Tanaka, K.

Taylor, H. F.

Tur, M.

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, and H. J. Shaw, "Coherence multiplexing of fiber-optic interferometric sensors," J. Lightwave Technol. LT-3, 1062-1072 (1985).
[CrossRef]

Vallabhan, C. P. G.

S. T. Lee, V. P. N. Nampoori, C. P. G. Vallabhan, and P. Radhakrishnan, "Macrobending in a fiber-optic interferometric sensor for displacement and weight measurement," Proc. SPIE 4946, 126-128 (2003).
[CrossRef]

Wainwright, G. H.

J. M. Senior, G. Murtaza, A. I. Stirling, and G. H. Wainwright, "Referenced intensity-modulated optical fiber sensor system for measuring linear displacement," Proc. SPIE 1795, 353-361 (1993).
[CrossRef]

Wang, A.

Wentworth, R. H.

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, and H. J. Shaw, "Coherence multiplexing of fiber-optic interferometric sensors," J. Lightwave Technol. LT-3, 1062-1072 (1985).
[CrossRef]

Youngquist, R. C.

R. C. Youngquist, S. Carr, and D. E. N. Davies, "Optical coherence-domain reflectometry: a new optical evaluation technique," Opt. Lett. 12, 158-160 (1987).
[CrossRef] [PubMed]

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, and H. J. Shaw, "Coherence multiplexing of fiber-optic interferometric sensors," J. Lightwave Technol. LT-3, 1062-1072 (1985).
[CrossRef]

Yuan, L.

Zheng, J.

J. Zheng, "Reflectometric fiber-optic frequency-modulated continuous-wave interferometric displacement sensor," Opt. Eng. 44, 124404 (2005).
[CrossRef]

J. Zheng, "Analysis of optical frequency-modulated continuous-wave interference," Appl. Opt. 43, 4189-4198 (2004).
[CrossRef] [PubMed]

J. Zheng and S. Albin, "Self-referenced reflective intensity modulated fiber-optic displacement sensor," Opt. Eng. 38, 227-232 (1999).
[CrossRef]

Zhou, L.

Appl. Opt.

Electron. Lett.

D. A. Jackson, A. D. Kersey, M. Corke, and J. D. C. Jones, "Pseudoheterodyne detection scheme for optical interferometers," Electron. Lett. 18, 1081-1083 (1982).
[CrossRef]

T. A. Berkoff and A. D. Kersey, "Interferometric fiber displacement/strain sensor based on source coherence synthesis," Electron. Lett. 26, 452-453 (1990).
[CrossRef]

F. Farahi, A. S. Gerges, J. D. C. Jones, and D. A. Jackson, "Time-division multiplexing of fiber-optic interferometric sensors using a frequency modulated laser diode," Electron. Lett. 24, 54-55 (1988).
[CrossRef]

F. Farahi, J. D. C. Jones, and D. A. Jackson, "Multiplexed fiber-optic interferometric sensing system: combined frequency and time division," Electron. Lett. 24, 409-410 (1988).
[CrossRef]

IEEE J. Quantum. Electron.

B. Culshaw and I. P. Giles, "Frequency modulated heterodyne optical Sagnac interferometer," IEEE J. Quantum. Electron. QE-18, 690-693 (1982).
[CrossRef]

J. Lightwave Technol.

J. L. Brooks, R. H. Wentworth, R. C. Youngquist, M. Tur, B. Y. Kim, and H. J. Shaw, "Coherence multiplexing of fiber-optic interferometric sensors," J. Lightwave Technol. LT-3, 1062-1072 (1985).
[CrossRef]

Meas. Sci. Technol.

Y. J. Rao and D. A. Jackson, "Recent progress in fibre optic low-coherence interferometry," Meas. Sci. Technol. 7, 981-999 (1996).
[CrossRef]

Opt. Eng.

J. Zheng and S. Albin, "Self-referenced reflective intensity modulated fiber-optic displacement sensor," Opt. Eng. 38, 227-232 (1999).
[CrossRef]

J. Zheng, "Reflectometric fiber-optic frequency-modulated continuous-wave interferometric displacement sensor," Opt. Eng. 44, 124404 (2005).
[CrossRef]

Opt. Lett.

Opt. Quantum. Electron.

I. Sakai, "Frequency-division multiplexing of optical-fiber sensors using a frequency-modulated source," Opt. Quantum. Electron. 18, 279-289 (1986).
[CrossRef]

Proc. SPIE

J. M. Senior, G. Murtaza, A. I. Stirling, and G. H. Wainwright, "Referenced intensity-modulated optical fiber sensor system for measuring linear displacement," Proc. SPIE 1795, 353-361 (1993).
[CrossRef]

S. T. Lee, V. P. N. Nampoori, C. P. G. Vallabhan, and P. Radhakrishnan, "Macrobending in a fiber-optic interferometric sensor for displacement and weight measurement," Proc. SPIE 4946, 126-128 (2003).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Triple-sensor multiplexed reflectometric fiber-optic FMCW interferometric displacement sensor.

Fig. 2
Fig. 2

Another version of the triple-sensor multiplexed reflectometric fiber-optic FMCW interferometric displacement sensor.

Fig. 3
Fig. 3

Signal spectrum of the triple-sensor multiplexed fiber-optic FMCW interferometric displacement sensor.

Fig. 4
Fig. 4

Phasor diagrams for calculating the cross talk: (a) The sum of P 1M , P 2M , and P 3M equals P M . (b) When P M P 2M and P 2M P 3M , the phase error δϕ b01 reaches an extreme.

Fig. 5
Fig. 5

Waveforms of the beat signals from the individual sensors: (a) Signal from the sensor with a short air cavity. (b) Signal from the sensor with a medium cavity. (c) Signal from the sensor with a long air cavity.

Fig. 6
Fig. 6

Relationship between the phase shift of the beat signal and displacement of the object.

Tables (1)

Tables Icon

Table 1 Maximum Displacement Error in the Triple-Sensor Multiplexed Fiber-Optic FMCW Interferometric Displacement Sensor a

Equations (28)

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

I ( t ) = I 1 + I 2 + 2 I 1 I 2 cos ( ατ t + ω 0 τ ) = I 0 [ 1 + V cos ( ω b t + ϕ b 0 ) ] ,
I ( t ) = I 0 [ 1 + V cos ( 2 π Δ νν m O P D c t + 2 π λ 0 O P D ) ] = I 0 [ 1 + V cos ( 2 πν b t + ϕ b 0 ) ] ,
I ( t ) = ( I 01 + I 02 + I 03 ) + [ I 01 V 1 cos ( ω b 1 t + ϕ b 01 ) + I 02 V 2 cos ( ω b 2 t + ϕ b 02 ) + I 03 V 3 cos ( ω b 3 t + ϕ b 03 ) ] ,
I ( t ) = ( I 01 + I 02 + I 03 ) + [ I 01 V 1 cos ( 2 π Δ νν m O P D 1 c t + 2 π λ 0 O P D 1 ) + I 02 V 2 cos ( 2 π Δ νν m O P D 2 c t + 2 π λ 0 O P D 2 ) + I 03 V 3 cos ( 2 π Δ νν m O P D 3 c t + 2 π λ 0 O P D 3 ) ] = ( I 01 + I 02 + I 03 ) + [ I 01 V 1 cos ( 2 πν b 1 t + ϕ b 01 ) + I 02 V 2 cos ( 2 πν b 2 t + ϕ b 02 ) + I 03 V 3 cos ( 2 πν b 3 t + ϕ b 03 ) ] ,
O P D 1 = 2 n d 1 ,
O P D 2 = 2 n d 2 ,
O P D 3 = 2 n d 3 ,
ϕ b 01 = 4 π n d 1 λ 0 ,
ϕ b 02 = 4 π n d 2 λ 0 ,
ϕ b 03 = 4 π n d 3 λ 0 .
Δ d 1 = λ 0 4 π n Δ ϕ b 01 ,
Δ d 2 = λ 0 4 π n Δ ϕ b 02 ,
Δ d 3 = λ 0 4 π n Δ ϕ b 03 ,
I ( t ) = ( I 01 + I 02 + I 03 ) + [ I 01 V 1 cos ( ω b 1 t + ϕ b 01 ) + I 02 V 2 cos ( ω b 2 t + ϕ b 02 ) + I 03 V 3 cos ( ω b 3 t + ϕ b 03 ) ] × r e c t T m ( t ) n = δ ( t n T m ) ,
r e c t T ( t ) = { 1 | t | T / 2 0 | t | T / 2 .
I ( ω ) = 2 π ( I 01 + I 02 + I 03 ) δ ( ω ) + 2 π 2 { I 01 V 1 sin [ ( ω + ω b 1 ) T m 2 ] ( ω + ω b 1 ) T m 2 e j ϕ b 01 + I 02 V 2 sin [ ( ω + ω b 2 ) T m 2 ] ( ω + ω b 2 ) T m 2 e j ϕ b 02 + I 03 V 3 sin [ ( ω + ω b 3 ) T m 2 ] ( ω + ω b 3 ) T m 2 e j ϕ b 03 + I 01 V 1 sin [ ( ω ω b 1 ) T m 2 ] ( ω ω b 1 ) T m 2 e j ϕ b 01 + I 02 V 2 sin [ ( ω ω b 2 ) T m 2 ] ( ω ω b 2 ) T m 2 e j ϕ b 02 + I 03 V 3 sin [ ( ω ω b 3 ) T m 2 ] ( ω ω b 3 ) T m 2 e j ϕ b 03 } m = δ ( ω m ω m ) ,
I ( ω ) = 2 π ( I 01 + I 02 + I 03 ) δ ( ω ) + 2 π 2 { I 01 V 1  sinc [ ( ω + ω b 1 ) T m 2 ] e j ϕ b 01 + I 02 V 2  sinc [ ( ω + ω b 2 ) T m 2 ] e j ϕ b 02 + I 03 V 3  sinc [ ( ω + ω b 3 ) T m 2 ] e j ϕ b 03 + I 01 V 1  sinc [ ( ω ω b 1 ) T m 2 ] e j ϕ b 01 + I 02 V 2  sinc [ ( ω ω b 2 ) T m 2 ] e j ϕ b 02 + I 03 V 3  sinc [ ( ω ω b 3 ) T m 2 ] e j ϕ b 03 } m = δ ( ω m ω m ) .
i M ( t ) = C M cos ( M ω m t + arg { I 01 V 1  sinc [ ( M ω m ω b 1 ) T m 2 ] e j ϕ b 01 + I 02 V 2  sinc [ ( M ω m ω b 2 ) T m 2 ] e j ϕ b 02 + I 03 V 3  sinc [ ( M ω m ω b 3 ) T m 2 ] e j ϕ b 03 } ) ,
C M = | I 01 V 1  sinc [ ( M ω m ω b 1 ) T m 2 ] e j ϕ b 01 + I 02 V 2  sinc [ ( M ω m ω b 2 ) T m 2 ] e j ϕ b 02 + I 03 V 3  sinc [ ( M ω m ω b 3 ) T m 2 ] e j ϕ b 03 | ,
δ ϕ b 01 = arg { I 01 V 1  sinc [ ( M ω m ω b 1 ) T m 2 ] e j ϕ b 01 + I 02 V 2  sinc [ ( M ω m ω b 2 ) T m 2 ] e j ϕ b 02 + I 03 V 3  sinc [ ( M ω m ω b 3 ) T m 2 ] e j ϕ b 03 } ϕ b 01 .
P 1 M = I 01 V 1  sinc [ ( M ω m ω b 1 ) T m 2 ] e j ϕ b 01 ,
P 2 M = I 02 V 2  sinc [ ( M ω m ω b 2 ) T m 2 ] e j ϕ b 02 ,
P 3 M = I 03 V 3  sinc [ ( M ω m ω b 3 ) T m 2 ] e j ϕ b 03 ,
P M = P 1 M + P 2 M + P 3 M .
( δ ϕ b 01 ) e x t r = arcsin ( P 2 M + P 3 M P 1 M ) = arcsin { I 02 V 2  sinc [ ( M ω m ω b 2 ) T m 2 ] + I 03 V 3  sinc [ ( M ω m ω b 3 ) T m 2 ] I 01 V 1  sinc [ ( M ω m ω b 1 ) T m 2 ] } ,
( δ ϕ b 01 ) extr P 2 M + P 3 M P 1 M = I 02 V 2  sinc [ ( M ω m ω b 2 ) T m 2 ] + I 03 V 3  sinc [ ( M ω m ω b 3 ) T m 2 ] I 01 V 1  sinc [ ( M ω m ω b 1 ) T m 2 ] .
( δ ϕ b 01 ) max = I 02 V 2 I 01 V 1 [ 2 ( N M ) + 1 ] + I 03 V 3 I 01 V 1 [ 2 ( P M ) + 1 ] ,
( δ Δ d 1 ) max = λ 0 4 π n [ I 02 V 2 I 01 V 1 [ 2 ( N M ) + 1 ] + I 03 V 3 I 01 V 1 [ 2 ( P M ) + 1 ] ] .

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