Abstract

The parameters of the semiconductor laser source are vital for the performances of optical coherent systems. In this paper, a novel method to measure the phase-shift φm between laser central optical-frequency modulation (COFM) and the accompanied optical-intensity modulation (AOIM) is proposed, which is easy to realize and requires no further fiber etalons or high-speed demodulators. An orthogonal test is utilized to measure φm. Experimental results show that the value of φm approaches 1.09π under different COFM conditions. Then the interference model of phase-generated carrier (PGC) demodulation is modified by taking into account the effect of φm, and the influences of φm on the demodulation results using two methods (look-up table and AOIM-factor division) are further analyzed in detail.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. K. Kirkendall and A. Dandridge, “Overview of high-performance fiber-optic sensing,” J. Phys. D 37, R197–R216 (2004).
    [CrossRef]
  2. M. Zhang, X. H. Ma, L. W. Wang, S. R. Lai, H. P. Hong, H. F. Zhao, and Y. B. Liao, “Progress of optical fiber sensors and its application in harsh environment,” Photon. Sens. 1, 84–89 (2011).
  3. C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
    [CrossRef]
  4. Y. Liu, L. W. Wang, C. D. Tian, M. Zhang, and Y. B. Liao, “Analysis and optimization of the PGC method in all digital demodulation systems,” J. Lightwave Technol. 26, 3225–3233 (2008).
    [CrossRef]
  5. H. Y. Zhang, M. Zhang, L. W. Wang, Y. B. Liao, D. N. Wang, and Y. Q. Zhu, “An improved PGC demodulation method to suppress the impact of laser intensity modulation,” Proc. SPIE 8199, 1–8 (2011).
  6. S. C. Huang, Y. F. Huang, and Z. Z. Wu, “Sensitivity normalization technique of PGC demodulation with low harmonic distortion and high stability using laser modulation to generate carrier signal,” Sens. Actuators A 174, 198–206 (2012).
    [CrossRef]
  7. Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).
  8. S. Schilt, L. Thévenaz, and P. Robert, “Wavelength modulation spectroscopy: combined frequency and intensity laser modulation,” Appl. Opt. 42, 6728–6738 (2003).
    [CrossRef]
  9. J. Wang, M. Maiorov, D. S. Baer, D. Z. Garbuzov, J. C. Connolly, and R. K. Hanson, “In situ combustion measurements of CO with diode-laser absorption near 2.3 μm,” Appl. Opt. 39, 5579–5589 (2000).
    [CrossRef]
  10. L. C. Philippe and R. K. Hanson, “Laser diode wavelength-modulation spectroscopy for simultaneous measurement of temperature, pressure, and velocity in shock-heated oxygen flows,” Appl. Opt. 32, 6090–6103 (1993).
    [CrossRef]
  11. H. J. Li, G. B. Rieker, X. Liu, J. B. Jeffries, and R. K. Hanson, “Extension of wavelength-modulation spectroscopy to large modulation depth for diode laser absorption measurements in high-pressure gases,” Appl. Opt. 45, 1052–1061 (2006).
    [CrossRef]
  12. K. Yin, H. P. Zhou, M. Zhang, T. H. Ding, S. R. Lai, L. W. Wang, and Y. B. Liao, “Optimization design of the pressure phase sensitivity of the fiber-optic air-backed mandrel hydrophone,” Proc. SPIE 7004, 700414 (2008).
    [CrossRef]

2012 (1)

S. C. Huang, Y. F. Huang, and Z. Z. Wu, “Sensitivity normalization technique of PGC demodulation with low harmonic distortion and high stability using laser modulation to generate carrier signal,” Sens. Actuators A 174, 198–206 (2012).
[CrossRef]

2011 (2)

M. Zhang, X. H. Ma, L. W. Wang, S. R. Lai, H. P. Hong, H. F. Zhao, and Y. B. Liao, “Progress of optical fiber sensors and its application in harsh environment,” Photon. Sens. 1, 84–89 (2011).

H. Y. Zhang, M. Zhang, L. W. Wang, Y. B. Liao, D. N. Wang, and Y. Q. Zhu, “An improved PGC demodulation method to suppress the impact of laser intensity modulation,” Proc. SPIE 8199, 1–8 (2011).

2010 (1)

Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).

2009 (1)

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[CrossRef]

2008 (2)

Y. Liu, L. W. Wang, C. D. Tian, M. Zhang, and Y. B. Liao, “Analysis and optimization of the PGC method in all digital demodulation systems,” J. Lightwave Technol. 26, 3225–3233 (2008).
[CrossRef]

K. Yin, H. P. Zhou, M. Zhang, T. H. Ding, S. R. Lai, L. W. Wang, and Y. B. Liao, “Optimization design of the pressure phase sensitivity of the fiber-optic air-backed mandrel hydrophone,” Proc. SPIE 7004, 700414 (2008).
[CrossRef]

2006 (1)

2004 (1)

C. K. Kirkendall and A. Dandridge, “Overview of high-performance fiber-optic sensing,” J. Phys. D 37, R197–R216 (2004).
[CrossRef]

2003 (1)

2000 (1)

1993 (1)

Baer, D. S.

Chu, X. H.

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[CrossRef]

Connolly, J. C.

Dandridge, A.

C. K. Kirkendall and A. Dandridge, “Overview of high-performance fiber-optic sensing,” J. Phys. D 37, R197–R216 (2004).
[CrossRef]

Ding, T. H.

K. Yin, H. P. Zhou, M. Zhang, T. H. Ding, S. R. Lai, L. W. Wang, and Y. B. Liao, “Optimization design of the pressure phase sensitivity of the fiber-optic air-backed mandrel hydrophone,” Proc. SPIE 7004, 700414 (2008).
[CrossRef]

Garbuzov, D. Z.

Hanson, R. K.

Hong, H. P.

M. Zhang, X. H. Ma, L. W. Wang, S. R. Lai, H. P. Hong, H. F. Zhao, and Y. B. Liao, “Progress of optical fiber sensors and its application in harsh environment,” Photon. Sens. 1, 84–89 (2011).

Huang, L. J.

Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).

Huang, S. C.

S. C. Huang, Y. F. Huang, and Z. Z. Wu, “Sensitivity normalization technique of PGC demodulation with low harmonic distortion and high stability using laser modulation to generate carrier signal,” Sens. Actuators A 174, 198–206 (2012).
[CrossRef]

Huang, Y. F.

S. C. Huang, Y. F. Huang, and Z. Z. Wu, “Sensitivity normalization technique of PGC demodulation with low harmonic distortion and high stability using laser modulation to generate carrier signal,” Sens. Actuators A 174, 198–206 (2012).
[CrossRef]

Jeffries, J. B.

Kirkendall, C. K.

C. K. Kirkendall and A. Dandridge, “Overview of high-performance fiber-optic sensing,” J. Phys. D 37, R197–R216 (2004).
[CrossRef]

Lai, S. R.

M. Zhang, X. H. Ma, L. W. Wang, S. R. Lai, H. P. Hong, H. F. Zhao, and Y. B. Liao, “Progress of optical fiber sensors and its application in harsh environment,” Photon. Sens. 1, 84–89 (2011).

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[CrossRef]

K. Yin, H. P. Zhou, M. Zhang, T. H. Ding, S. R. Lai, L. W. Wang, and Y. B. Liao, “Optimization design of the pressure phase sensitivity of the fiber-optic air-backed mandrel hydrophone,” Proc. SPIE 7004, 700414 (2008).
[CrossRef]

Li, H. J.

Liao, Y. B.

M. Zhang, X. H. Ma, L. W. Wang, S. R. Lai, H. P. Hong, H. F. Zhao, and Y. B. Liao, “Progress of optical fiber sensors and its application in harsh environment,” Photon. Sens. 1, 84–89 (2011).

H. Y. Zhang, M. Zhang, L. W. Wang, Y. B. Liao, D. N. Wang, and Y. Q. Zhu, “An improved PGC demodulation method to suppress the impact of laser intensity modulation,” Proc. SPIE 8199, 1–8 (2011).

Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[CrossRef]

Y. Liu, L. W. Wang, C. D. Tian, M. Zhang, and Y. B. Liao, “Analysis and optimization of the PGC method in all digital demodulation systems,” J. Lightwave Technol. 26, 3225–3233 (2008).
[CrossRef]

K. Yin, H. P. Zhou, M. Zhang, T. H. Ding, S. R. Lai, L. W. Wang, and Y. B. Liao, “Optimization design of the pressure phase sensitivity of the fiber-optic air-backed mandrel hydrophone,” Proc. SPIE 7004, 700414 (2008).
[CrossRef]

Liu, X.

Liu, Y.

Ma, X. H.

M. Zhang, X. H. Ma, L. W. Wang, S. R. Lai, H. P. Hong, H. F. Zhao, and Y. B. Liao, “Progress of optical fiber sensors and its application in harsh environment,” Photon. Sens. 1, 84–89 (2011).

Maiorov, M.

Philippe, L. C.

Rieker, G. B.

Robert, P.

Schilt, S.

Shi, Q. P.

Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).

Thévenaz, L.

Tian, C. D.

Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[CrossRef]

Y. Liu, L. W. Wang, C. D. Tian, M. Zhang, and Y. B. Liao, “Analysis and optimization of the PGC method in all digital demodulation systems,” J. Lightwave Technol. 26, 3225–3233 (2008).
[CrossRef]

Tian, Q.

Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).

Wang, D. N.

H. Y. Zhang, M. Zhang, L. W. Wang, Y. B. Liao, D. N. Wang, and Y. Q. Zhu, “An improved PGC demodulation method to suppress the impact of laser intensity modulation,” Proc. SPIE 8199, 1–8 (2011).

Wang, H. H.

Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).

Wang, J.

Wang, L. W.

H. Y. Zhang, M. Zhang, L. W. Wang, Y. B. Liao, D. N. Wang, and Y. Q. Zhu, “An improved PGC demodulation method to suppress the impact of laser intensity modulation,” Proc. SPIE 8199, 1–8 (2011).

M. Zhang, X. H. Ma, L. W. Wang, S. R. Lai, H. P. Hong, H. F. Zhao, and Y. B. Liao, “Progress of optical fiber sensors and its application in harsh environment,” Photon. Sens. 1, 84–89 (2011).

Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[CrossRef]

Y. Liu, L. W. Wang, C. D. Tian, M. Zhang, and Y. B. Liao, “Analysis and optimization of the PGC method in all digital demodulation systems,” J. Lightwave Technol. 26, 3225–3233 (2008).
[CrossRef]

K. Yin, H. P. Zhou, M. Zhang, T. H. Ding, S. R. Lai, L. W. Wang, and Y. B. Liao, “Optimization design of the pressure phase sensitivity of the fiber-optic air-backed mandrel hydrophone,” Proc. SPIE 7004, 700414 (2008).
[CrossRef]

Wu, Z. Z.

S. C. Huang, Y. F. Huang, and Z. Z. Wu, “Sensitivity normalization technique of PGC demodulation with low harmonic distortion and high stability using laser modulation to generate carrier signal,” Sens. Actuators A 174, 198–206 (2012).
[CrossRef]

Yin, K.

K. Yin, H. P. Zhou, M. Zhang, T. H. Ding, S. R. Lai, L. W. Wang, and Y. B. Liao, “Optimization design of the pressure phase sensitivity of the fiber-optic air-backed mandrel hydrophone,” Proc. SPIE 7004, 700414 (2008).
[CrossRef]

Zeng, X.

Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).

Zhang, H. Y.

H. Y. Zhang, M. Zhang, L. W. Wang, Y. B. Liao, D. N. Wang, and Y. Q. Zhu, “An improved PGC demodulation method to suppress the impact of laser intensity modulation,” Proc. SPIE 8199, 1–8 (2011).

Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[CrossRef]

Zhang, M.

H. Y. Zhang, M. Zhang, L. W. Wang, Y. B. Liao, D. N. Wang, and Y. Q. Zhu, “An improved PGC demodulation method to suppress the impact of laser intensity modulation,” Proc. SPIE 8199, 1–8 (2011).

M. Zhang, X. H. Ma, L. W. Wang, S. R. Lai, H. P. Hong, H. F. Zhao, and Y. B. Liao, “Progress of optical fiber sensors and its application in harsh environment,” Photon. Sens. 1, 84–89 (2011).

Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[CrossRef]

Y. Liu, L. W. Wang, C. D. Tian, M. Zhang, and Y. B. Liao, “Analysis and optimization of the PGC method in all digital demodulation systems,” J. Lightwave Technol. 26, 3225–3233 (2008).
[CrossRef]

K. Yin, H. P. Zhou, M. Zhang, T. H. Ding, S. R. Lai, L. W. Wang, and Y. B. Liao, “Optimization design of the pressure phase sensitivity of the fiber-optic air-backed mandrel hydrophone,” Proc. SPIE 7004, 700414 (2008).
[CrossRef]

Zhao, H. F.

M. Zhang, X. H. Ma, L. W. Wang, S. R. Lai, H. P. Hong, H. F. Zhao, and Y. B. Liao, “Progress of optical fiber sensors and its application in harsh environment,” Photon. Sens. 1, 84–89 (2011).

Zhou, H. P.

K. Yin, H. P. Zhou, M. Zhang, T. H. Ding, S. R. Lai, L. W. Wang, and Y. B. Liao, “Optimization design of the pressure phase sensitivity of the fiber-optic air-backed mandrel hydrophone,” Proc. SPIE 7004, 700414 (2008).
[CrossRef]

Zhu, Y. Q.

H. Y. Zhang, M. Zhang, L. W. Wang, Y. B. Liao, D. N. Wang, and Y. Q. Zhu, “An improved PGC demodulation method to suppress the impact of laser intensity modulation,” Proc. SPIE 8199, 1–8 (2011).

Appl. Opt. (4)

J. Lightwave Technol. (1)

J. Phys. D (1)

C. K. Kirkendall and A. Dandridge, “Overview of high-performance fiber-optic sensing,” J. Phys. D 37, R197–R216 (2004).
[CrossRef]

Opt. Eng. (1)

Q. P. Shi, Q. Tian, L. W. Wang, C. D. Tian, H. Y. Zhang, M. Zhang, Y. B. Liao, H. H. Wang, X. Zeng, and L. J. Huang, “Performance improvement of phase-generated carrier method by eliminating laser-intensity modulation for optical seismometer,” Opt. Eng. 49, 1–6 (2010).

Photon. Sens. (1)

M. Zhang, X. H. Ma, L. W. Wang, S. R. Lai, H. P. Hong, H. F. Zhao, and Y. B. Liao, “Progress of optical fiber sensors and its application in harsh environment,” Photon. Sens. 1, 84–89 (2011).

Proc. SPIE (3)

C. D. Tian, L. W. Wang, M. Zhang, H. Y. Zhang, X. H. Chu, S. R. Lai, and Y. B. Liao, “Performance improvement of PGC method by using lookup table for optical seismometer,” Proc. SPIE 7503, 750348 (2009).
[CrossRef]

H. Y. Zhang, M. Zhang, L. W. Wang, Y. B. Liao, D. N. Wang, and Y. Q. Zhu, “An improved PGC demodulation method to suppress the impact of laser intensity modulation,” Proc. SPIE 8199, 1–8 (2011).

K. Yin, H. P. Zhou, M. Zhang, T. H. Ding, S. R. Lai, L. W. Wang, and Y. B. Liao, “Optimization design of the pressure phase sensitivity of the fiber-optic air-backed mandrel hydrophone,” Proc. SPIE 7004, 700414 (2008).
[CrossRef]

Sens. Actuators A (1)

S. C. Huang, Y. F. Huang, and Z. Z. Wu, “Sensitivity normalization technique of PGC demodulation with low harmonic distortion and high stability using laser modulation to generate carrier signal,” Sens. Actuators A 174, 198–206 (2012).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

Time delay between received carrier and local carrier in typical PGC setup.

Fig. 2.
Fig. 2.

Schematic diagram for measuring phase shift φm.

Fig. 3.
Fig. 3.

Imbalanced Michelson interferometer used to measure φm.

Fig. 4.
Fig. 4.

Sampled (a) Vinter, (b) Vpower signal, and (c) their ratio.

Fig. 5.
Fig. 5.

Comparison of the PGC dmodulation results (a) without and (b) with considering φm.

Tables (6)

Tables Icon

Table 1. Three Parameter Sets for Diode1 (Threshold=15.6mA)

Tables Icon

Table 2. Orthogonal Test Results of Diode1

Tables Icon

Table 3. Three Parameter Sets for Diode2 (Threshold=16.2mA)

Tables Icon

Table 4. Orthogonal Test Results of Diode2

Tables Icon

Table 5. Three Parameter Sets for Diode3 (Threshold=17.9mA)

Tables Icon

Table 6. Orthogonal Test Results of Diode3

Equations (15)

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

V=kI0[1+mcos(Ωc(tτ)+φm)]·[1+υcos(C·cos(Ωc(tτ))+φs)],
{Vpower=r3I0{1+mcos[Ωc(tτ)+φm]}=r3I0[1+mcos(Ωct+φmφd1)]Vinter=r4Vpower{1+υcos[C·cos(Ωctφd1)+φs]},
Vratio=VinterVpower=r4{1+υcos[C·cos(Ωctφd1)+φs]}.
[Vratio·sin(Ωct)]*hLPF(t)=r4υJ1(C)sin(φs)·sinφd1,
[Vratio·cos(Ωct)]*hLPF(t)=r4υJ1(C)sin(φs)·cosφd1.
VpowerAC=r3I0mcos(Ωct+φmφd1).
[VpowerAC·sin(Ωct)]*hLPF(t)=12sin(φmφd1),
[VpowerAC·cos(Ωct)]*hLPF(t)=12cos(φmφd1).
V=kI0[1+mcos(Ωc(tτ))]·[1+υcos(C·cos(Ωc(tτ))+φs)],
{V1=kI0υK1{sin(φsθ1)mυ2K1}V2=kI0υK2cos(φs+θ2),
{K1={m2[J0(C)J2(C)]}2+[J1(C)]2;K2={m2[J3(C)J1(C)]}2+[J2(C)]2tanθ1=m[J0(C)J2(C)]2J1(C);tanθ2=m[J3(C)J1(C)]2J2(C).
V1V2=K1K2·{sin(φs+θ2)cos(θ1+θ2)δ0cos(φs+θ2)sin(θ1+θ2)},
{V1e=kI0υK1e{sin(φsθ1e)mcos(φm)υ2K1e}V2e=kI0υK2ecos(φs+θ2e),
{K1e={m2[J0(C)J2(C)]cos(φm)}2+[J1(C)]2;K2e={m2[J3(C)J1(C)]cos(φm)}2+[J2(C)]2tanθ1e=mcos(φm)[J0(C)J2(C)]2J1(C);tanθ2e=mcos(φm)[J3(C)J1(C)]2J2(C).
V1eV2e=K1eK2e·{sin(φs+θ2e)cos(θ1e+θ2e)δ0ecos(φs+θ2e)sin(θ1e+θ2e)},

Metrics