Abstract

We consider the heterodyne efficiency as a measure of quality for a coherent detection system. The heterodyne efficiency reflects the matching between the received beam and the local oscillator beam on the detector surface, and one can use this property for the alignment of the system. In this paper we derive a general expression for the heterodyne efficiency of a detection system for beams at any state of coherence, assuming that the propagation directions for the two signals (the received signal and the locally generated one) are slightly different. We derive an analytical expression for the heterodyne efficiency when mixing coherently two partially coherent Gaussian Schell-model beams on a photodetector surface. Numerical examples are given for the variation in the heterodyne efficiency with the misalignment angle, the detector radius, and the parameters of the overlapping beams. We show that partially coherent beams, although they suffer more than coherent beams from a decrease in the heterodyne efficiency, are less affected than coherent beams by the misalignment of the detection system.

© 2010 Optical Society of America

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  1. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001).
    [CrossRef]
  2. I. Melngails, W. Keicher, C. Freed, S. Marcus, B. Edwards, A. Sanchez, T. Fan, and D. Spears, “Laser radar component technology,” Proc. IEEE 84, 227–267 (1996).
    [CrossRef]
  3. J. McElroy, “Infrared heterodyne solar radiometry,” Appl. Opt. 11, 1619–1622 (1972).
    [CrossRef] [PubMed]
  4. M. Abbas, M. Mumma, T. Kostiuk, and D. Buhl, “Sensitivity limits of an infrared heterodyne spectrometer for astrophysical applications,” Appl. Opt. 15, 427–436 (1976).
    [CrossRef] [PubMed]
  5. D. Sun, Z. Liu, J. Nan, Y. Dai, and M. Pi, “Wavefront matching measurement in coherent CO2 laser-radar,” Appl. Opt. 31, 7647–7649 (1992).
    [CrossRef] [PubMed]
  6. A. Siegman, “The antenna properties of optical heterodyne receivers,” Appl. Opt. 5, 1588–1594 (1966).
    [CrossRef] [PubMed]
  7. G. R. Osche, Optical Detection Theory(Wiley, 2002).
  8. S. Cohen, “Heterodyne detection: phase front alignment, beam spot size and detector uniformity,” Appl. Opt. 14, 1953–1959 (1975).
    [CrossRef] [PubMed]
  9. D. Chambers, “Modeling of heterodyne efficiency for coherent laser radar in the presence of aberrations,” Opt. Express l1, 60–67 (1997).
    [CrossRef]
  10. Y. Zhao, M. J. Post, and R. M. Hardesty, “Receiving efficiency of monostatic pulsed coherent lidars. 1: Theory,” Appl. Opt. 29, 4111–4119 (1990).
    [CrossRef] [PubMed]
  11. R. Frehlich, “Heterodyne efficiency for a coherent laser radar with diffuse or aerosol targets,” J. Mod. Opt. 41, 2115–2129 (1994).
    [CrossRef]
  12. D. Fink, “Coherent detection signal-to-noise,” Appl. Opt. 14, 689–690 (1975).
    [CrossRef] [PubMed]
  13. J. Wang, “Heterodyne laser radar SNR from a diffuse target containing multiple glints,” Appl. Opt. 21, 464–476 (1982).
    [CrossRef] [PubMed]
  14. R. Frehlich and M. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
    [CrossRef] [PubMed]
  15. K. Tanaka and N. Ohta, “Effect of tilt and offset of signal field on heterodyne efficiency,” Appl. Opt. 26, 627–632 (1987).
    [CrossRef] [PubMed]
  16. K. Tanaka and N. Saga, “Maximum heterodyne efficiency of optical heterodyne detection in the presence of background radiation,” Appl. Opt. 23, 3901–3904 (1984).
    [CrossRef] [PubMed]
  17. D. Delautre, S. Breugnot, and V. Laude, “Measurement of the sensitivity of heterodyne detection to aberrations using a programmable liquid-crystal modulator,” Opt. Commun. 160, 61–65 (1999).
    [CrossRef]
  18. J. Salzman and A. Katzir, “Signal-to-noise ratio of heterodyne detection: matrix formalism,” Appl. Opt. 22, 888–890 (1983).
    [CrossRef] [PubMed]
  19. T. Tanaka, M. Taguchi, and K. Tanaka, “Heterodyne efficiency for a partially coherent optical signal,” Appl. Opt. 31, 5391–5394 (1992).
    [CrossRef] [PubMed]
  20. M. Salem and A. Dogariu, “Optical heterodyne detection of random electromagnetic beams,” J. Mod. Opt. 51, 2305–2313 (2004).
  21. H. Yura, “Optical heterodyne signal power obtained from finite sized sources of radiation,” Appl. Opt. 13, 150–157 (1974).
    [CrossRef] [PubMed]
  22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics(Cambridge Univ. Press, 1995).
  23. R. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).
  24. J. Lathi and C. Nagel, “Mixing partially coherent fields with Gaussian irradiance profiles; optimization criteria,” Appl. Opt. 9, 115–123 (1970).
    [CrossRef]
  25. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media(SPIE Optical Engineering Press, 1998).
  26. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, 1962).
  27. I. S. Gradshteyn and I. M. Ryzhik, 1965 Table of Integrals, Series, and Products, 6th ed. (Academic, 1965).

2004

M. Salem and A. Dogariu, “Optical heterodyne detection of random electromagnetic beams,” J. Mod. Opt. 51, 2305–2313 (2004).

2002

G. R. Osche, Optical Detection Theory(Wiley, 2002).

2001

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001).
[CrossRef]

1999

D. Delautre, S. Breugnot, and V. Laude, “Measurement of the sensitivity of heterodyne detection to aberrations using a programmable liquid-crystal modulator,” Opt. Commun. 160, 61–65 (1999).
[CrossRef]

1998

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media(SPIE Optical Engineering Press, 1998).

1997

D. Chambers, “Modeling of heterodyne efficiency for coherent laser radar in the presence of aberrations,” Opt. Express l1, 60–67 (1997).
[CrossRef]

1996

I. Melngails, W. Keicher, C. Freed, S. Marcus, B. Edwards, A. Sanchez, T. Fan, and D. Spears, “Laser radar component technology,” Proc. IEEE 84, 227–267 (1996).
[CrossRef]

1995

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics(Cambridge Univ. Press, 1995).

1994

R. Frehlich, “Heterodyne efficiency for a coherent laser radar with diffuse or aerosol targets,” J. Mod. Opt. 41, 2115–2129 (1994).
[CrossRef]

1992

1991

1990

1987

1984

1983

1982

1976

1975

1974

1972

1970

1966

1965

I. S. Gradshteyn and I. M. Ryzhik, 1965 Table of Integrals, Series, and Products, 6th ed. (Academic, 1965).

1962

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, 1962).

Abbas, M.

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media(SPIE Optical Engineering Press, 1998).

Boyd, R.

R. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).

Breugnot, S.

D. Delautre, S. Breugnot, and V. Laude, “Measurement of the sensitivity of heterodyne detection to aberrations using a programmable liquid-crystal modulator,” Opt. Commun. 160, 61–65 (1999).
[CrossRef]

Buhl, D.

Chambers, D.

D. Chambers, “Modeling of heterodyne efficiency for coherent laser radar in the presence of aberrations,” Opt. Express l1, 60–67 (1997).
[CrossRef]

Cohen, S.

Dai, Y.

Delautre, D.

D. Delautre, S. Breugnot, and V. Laude, “Measurement of the sensitivity of heterodyne detection to aberrations using a programmable liquid-crystal modulator,” Opt. Commun. 160, 61–65 (1999).
[CrossRef]

Dogariu, A.

M. Salem and A. Dogariu, “Optical heterodyne detection of random electromagnetic beams,” J. Mod. Opt. 51, 2305–2313 (2004).

Edwards, B.

I. Melngails, W. Keicher, C. Freed, S. Marcus, B. Edwards, A. Sanchez, T. Fan, and D. Spears, “Laser radar component technology,” Proc. IEEE 84, 227–267 (1996).
[CrossRef]

Fan, T.

I. Melngails, W. Keicher, C. Freed, S. Marcus, B. Edwards, A. Sanchez, T. Fan, and D. Spears, “Laser radar component technology,” Proc. IEEE 84, 227–267 (1996).
[CrossRef]

Fink, D.

Freed, C.

I. Melngails, W. Keicher, C. Freed, S. Marcus, B. Edwards, A. Sanchez, T. Fan, and D. Spears, “Laser radar component technology,” Proc. IEEE 84, 227–267 (1996).
[CrossRef]

Frehlich, R.

R. Frehlich, “Heterodyne efficiency for a coherent laser radar with diffuse or aerosol targets,” J. Mod. Opt. 41, 2115–2129 (1994).
[CrossRef]

R. Frehlich and M. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
[CrossRef] [PubMed]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, 1965 Table of Integrals, Series, and Products, 6th ed. (Academic, 1965).

Hardesty, R. M.

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001).
[CrossRef]

Katzir, A.

Kavaya, M.

Keicher, W.

I. Melngails, W. Keicher, C. Freed, S. Marcus, B. Edwards, A. Sanchez, T. Fan, and D. Spears, “Laser radar component technology,” Proc. IEEE 84, 227–267 (1996).
[CrossRef]

Kostiuk, T.

Lathi, J.

Laude, V.

D. Delautre, S. Breugnot, and V. Laude, “Measurement of the sensitivity of heterodyne detection to aberrations using a programmable liquid-crystal modulator,” Opt. Commun. 160, 61–65 (1999).
[CrossRef]

Liu, Z.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics(Cambridge Univ. Press, 1995).

Marcus, S.

I. Melngails, W. Keicher, C. Freed, S. Marcus, B. Edwards, A. Sanchez, T. Fan, and D. Spears, “Laser radar component technology,” Proc. IEEE 84, 227–267 (1996).
[CrossRef]

McElroy, J.

Melngails, I.

I. Melngails, W. Keicher, C. Freed, S. Marcus, B. Edwards, A. Sanchez, T. Fan, and D. Spears, “Laser radar component technology,” Proc. IEEE 84, 227–267 (1996).
[CrossRef]

Mumma, M.

Nagel, C.

Nan, J.

Ohta, N.

Osche, G. R.

G. R. Osche, Optical Detection Theory(Wiley, 2002).

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media(SPIE Optical Engineering Press, 1998).

Pi, M.

Post, M. J.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, 1965 Table of Integrals, Series, and Products, 6th ed. (Academic, 1965).

Saga, N.

Salem, M.

M. Salem and A. Dogariu, “Optical heterodyne detection of random electromagnetic beams,” J. Mod. Opt. 51, 2305–2313 (2004).

Salzman, J.

Sanchez, A.

I. Melngails, W. Keicher, C. Freed, S. Marcus, B. Edwards, A. Sanchez, T. Fan, and D. Spears, “Laser radar component technology,” Proc. IEEE 84, 227–267 (1996).
[CrossRef]

Siegman, A.

Spears, D.

I. Melngails, W. Keicher, C. Freed, S. Marcus, B. Edwards, A. Sanchez, T. Fan, and D. Spears, “Laser radar component technology,” Proc. IEEE 84, 227–267 (1996).
[CrossRef]

Sun, D.

Taguchi, M.

Tanaka, K.

Tanaka, T.

Wang, J.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, 1962).

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics(Cambridge Univ. Press, 1995).

Yura, H.

Zhao, Y.

Appl. Opt.

A. Siegman, “The antenna properties of optical heterodyne receivers,” Appl. Opt. 5, 1588–1594 (1966).
[CrossRef] [PubMed]

J. Lathi and C. Nagel, “Mixing partially coherent fields with Gaussian irradiance profiles; optimization criteria,” Appl. Opt. 9, 115–123 (1970).
[CrossRef]

J. McElroy, “Infrared heterodyne solar radiometry,” Appl. Opt. 11, 1619–1622 (1972).
[CrossRef] [PubMed]

H. Yura, “Optical heterodyne signal power obtained from finite sized sources of radiation,” Appl. Opt. 13, 150–157 (1974).
[CrossRef] [PubMed]

D. Fink, “Coherent detection signal-to-noise,” Appl. Opt. 14, 689–690 (1975).
[CrossRef] [PubMed]

S. Cohen, “Heterodyne detection: phase front alignment, beam spot size and detector uniformity,” Appl. Opt. 14, 1953–1959 (1975).
[CrossRef] [PubMed]

M. Abbas, M. Mumma, T. Kostiuk, and D. Buhl, “Sensitivity limits of an infrared heterodyne spectrometer for astrophysical applications,” Appl. Opt. 15, 427–436 (1976).
[CrossRef] [PubMed]

J. Wang, “Heterodyne laser radar SNR from a diffuse target containing multiple glints,” Appl. Opt. 21, 464–476 (1982).
[CrossRef] [PubMed]

J. Salzman and A. Katzir, “Signal-to-noise ratio of heterodyne detection: matrix formalism,” Appl. Opt. 22, 888–890 (1983).
[CrossRef] [PubMed]

K. Tanaka and N. Saga, “Maximum heterodyne efficiency of optical heterodyne detection in the presence of background radiation,” Appl. Opt. 23, 3901–3904 (1984).
[CrossRef] [PubMed]

K. Tanaka and N. Ohta, “Effect of tilt and offset of signal field on heterodyne efficiency,” Appl. Opt. 26, 627–632 (1987).
[CrossRef] [PubMed]

Y. Zhao, M. J. Post, and R. M. Hardesty, “Receiving efficiency of monostatic pulsed coherent lidars. 1: Theory,” Appl. Opt. 29, 4111–4119 (1990).
[CrossRef] [PubMed]

R. Frehlich and M. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
[CrossRef] [PubMed]

D. Sun, Z. Liu, J. Nan, Y. Dai, and M. Pi, “Wavefront matching measurement in coherent CO2 laser-radar,” Appl. Opt. 31, 7647–7649 (1992).
[CrossRef] [PubMed]

T. Tanaka, M. Taguchi, and K. Tanaka, “Heterodyne efficiency for a partially coherent optical signal,” Appl. Opt. 31, 5391–5394 (1992).
[CrossRef] [PubMed]

J. Mod. Opt.

M. Salem and A. Dogariu, “Optical heterodyne detection of random electromagnetic beams,” J. Mod. Opt. 51, 2305–2313 (2004).

R. Frehlich, “Heterodyne efficiency for a coherent laser radar with diffuse or aerosol targets,” J. Mod. Opt. 41, 2115–2129 (1994).
[CrossRef]

Opt. Commun.

D. Delautre, S. Breugnot, and V. Laude, “Measurement of the sensitivity of heterodyne detection to aberrations using a programmable liquid-crystal modulator,” Opt. Commun. 160, 61–65 (1999).
[CrossRef]

Opt. Express

D. Chambers, “Modeling of heterodyne efficiency for coherent laser radar in the presence of aberrations,” Opt. Express l1, 60–67 (1997).
[CrossRef]

Proc. IEEE

I. Melngails, W. Keicher, C. Freed, S. Marcus, B. Edwards, A. Sanchez, T. Fan, and D. Spears, “Laser radar component technology,” Proc. IEEE 84, 227–267 (1996).
[CrossRef]

Other

G. R. Osche, Optical Detection Theory(Wiley, 2002).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics(Cambridge Univ. Press, 1995).

R. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media(SPIE Optical Engineering Press, 1998).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, 1962).

I. S. Gradshteyn and I. M. Ryzhik, 1965 Table of Integrals, Series, and Products, 6th ed. (Academic, 1965).

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Figures (5)

Fig. 1
Fig. 1

Illustration of the notation of mixing two beams, with a shift θ between their phase fronts, on a detector in the z = 0 plane.

Fig. 2
Fig. 2

Variation of the heterodyne efficiency versus the detector radius for several values of the intensity widths of the locally generated coherent beams. The received signal was assumed to be coherent ( δ s = ) and has an intensity width σ s = 1 mm . The misalignment angle θ was (a) 0 rad , (b) 0.0002 rad .

Fig. 3
Fig. 3

Same as in Fig. 2, but the received beam was assumed to be partially coherent with a coherence width δ s = 0.5 mm .

Fig. 4
Fig. 4

Variation of the HER versus the detector radius for several values of the coherence widths of the received signal ( δ s = and δ s = 0.5 mm ) that has an intensity width of σ s = 1 mm . The locally generated beam is assumed to be coherent ( δ o = ) and has an intensity width of (a) σ o = 0.5 mm , (b) σ o = 5 mm .

Fig. 5
Fig. 5

Variation of the heterodyne efficiency with the misalignment angle ( θ ) for several values of the coherence widths of the received signal ( δ s = and δ s = 0.5 mm ) that has an intensity width of σ s = 1 mm . The locally generated signal is assumed to be coherent ( δ o = ) and has an intensity width of σ s = 1 mm . The detector radius is assumed as (a) R = 0.5 mm , (b) R = 2 mm .

Equations (57)

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U o ( r , t ) = U o ( r ) e j ω o t ,
U s ( r , t ) = U s ( r ) e j ω s t e j k . r ,
d i ( r , t ) = R ( r ) [ U ( r , t ) . U * ( r , t ) ] ,
d i ( r , t ) = R ( r ) [ U o ( r ) . U o * ( r ) + U s ( r ) . U s * ( r ) + U o ( r ) . U s * ( r ) e j ( ω o ω s ) t e j k . r + U s ( r ) . U o * ( r ) e j ( ω o ω s ) t e j k . r ] .
d i ( r , t ) = R ( r ) [ U o ( r ) . U s * ( r ) e j ( ω o ω s ) t e j k . r + U s ( r ) . U o * ( r ) e j ( ω o ω s ) t e j k . r ] .
i I F ( t ) = R ( r ) [ U o ( r ) . U s * ( r ) e j ( ω o ω s ) t e j k . r + U s ( r ) . U o * ( r ) e j ( ω o ω s ) t e j k . r ] d 2 r .
P total = i I F ( t ) 2 ¯ = R ( r 1 ) R ( r 2 ) [ U o ( r 1 ) . U s * ( r 1 ) . exp ( j Δ ω t + j k . r 1 ) ¯ + C.C . ] . [ U o ( r 2 ) . U s * ( r 2 ) . exp ( j Δ ω t + j k . r 2 ) ¯ + C.C ] d 2 r 1 d 2 r 2 ,
P total = 2 Re R ( r 1 ) R ( r 2 ) [ U o ( r 1 ) . U s * ( r 1 ) . U o * ( r 2 ) . U s ( r 2 ) exp ( j ( k . r 1 k . r 2 ) ) ] d 2 r 1 d 2 r 2 .
P total = 2 Re R ( r 1 ) R ( r 2 ) [ ( U o ( r 1 ) . U o * ( r 2 ) U s * ( r 1 ) . U s ( r 2 ) + U o ( r 1 ) . U s * ( r 1 ) U o * ( r 2 ) . U s ( r 2 ) ) exp ( j ( k . r 1 k . r 2 ) ) ] d 2 r 1 d 2 r 2 .
P total = 2 Re R ( r 1 ) R ( r 2 ) [ Γ o ( r 1 , r 2 ) . Γ s * ( r 1 , r 2 ) exp ( j ( k . r 1 k . r 2 ) ) ] d 2 r 1 d 2 r 2 ,
NEP = 2 e B R ( r ) Γ o ( r , r ) d r 2 ,
SNR = 2 Re R ( r 1 ) R ( r 2 ) [ Γ o ( r 1 , r 2 ) . Γ s * ( r 1 , r 2 ) exp ( j ( k . r 1 k . r 2 ) ) ] d 2 r 1 d 2 r 2 2 e B R ( r ) Γ o ( r , r ) d r 2 .
SNR * = Re [ Γ o ( r 1 , r 2 ) . Γ s * ( r 1 , r 2 ) exp ( j ( k . r 1 k . r 2 ) ) ] d 2 r 1 d 2 r 2 Γ o ( r , r ) d r 2 .
P max = 2 I o I s .
I o = R ( r ) Γ o ( r , r ) d r 2 ,
I s = R ( r ) Γ s ( r , r ) d r 2 .
η h = Re R ( r 1 ) R ( r 2 ) [ Γ o ( r 1 , r 2 ) . Γ s * ( r 1 , r 2 ) exp ( j ( k . r 1 k . r 2 ) ) ] d 2 r 1 d 2 r 2 R ( r ) Γ o ( r , r ) d r 2 R ( r ) Γ s ( r , r ) d r 2 .
η h = Re [ Γ o ( r 1 , r 2 ) . Γ s * ( r 1 , r 2 ) exp ( j ( k . r 1 k . r 2 ) ) ] d 2 r 1 d 2 r 2 Γ o ( r , r ) d 2 r Γ s ( r , r ) d 2 r .
η h = SNR * Γ s ( r , r ) d 2 r .
Γ α ( r 1 , r 2 ) = I α exp [ ( r 1 2 + r 2 2 ) 4 σ α 2 ( r 1 r 2 ) 2 2 δ α 2 ] .
η h = ϕ 2 = 0 2 π ϕ 1 = 0 2 π r 2 = 0 R r 1 = 0 R Re [ Γ o ( r 1 , r 2 ) . Γ s * ( r 1 , r 2 ) ] exp ( j k r 1 cos ϕ 1 θ j k r 2 cos ϕ 2 θ ) r 1 r 2 d r 1 d r 2 d ϕ 1 d ϕ 2 ϕ = 0 2 π r = 0 R Γ o ( r , r ) r d r d ϕ ϕ = 0 2 π r = 0 R Γ s ( r , r ) r d r d ϕ .
η h = ϕ 2 = 0 2 π ϕ 1 = 0 2 π r 2 = 0 r 1 = 0 Re [ Γ o ( r 1 , r 2 ) . Γ s * ( r 1 , r 2 ) e ( r 1 2 + r 2 2 ) W 2 exp ( j ( k r 1 cos ϕ 1 θ k r 2 cos ϕ 2 θ ) ) ] r 1 r 2 d r 1 d r 2 d ϕ 1 d ϕ 2 ϕ = 0 2 π r = 0 Γ o ( r , r ) e ( r 2 ) W 2 r d r d ϕ ϕ = 0 2 π r = 0 Γ s ( r , r ) e ( r 2 ) W 2 r d r d ϕ .
η h = ( R 2 + 2 σ s 2 ) ( R 2 + 2 σ o 2 ) ( 2 σ s σ o R 2 ) 2 γ 1 γ 2 ( e F 2 4 γ 2 ) ( e F 2 + β 1 2 4 γ 1 ) e F β 1 γ 1 ,
γ 1 = γ 2 + η 2 4 γ 2 ,
β 1 = η F 2 γ 2 ,
γ 2 = 1 4 σ s 2 + 1 2 δ s 2 + 1 4 σ o 2 + 1 2 δ o 2 + 2 R 2 ,
η = 1 δ s 2 + 1 δ o 2 .
I = ϕ 2 = 0 2 π ϕ 1 = 0 2 π r 2 = 0 r 1 = 0 Re [ Γ o ( r 1 , r 2 ) . Γ s * ( r 1 , r 2 ) e ( r 1 2 + r 2 2 ) W 2 exp ( j ( k r 1 cos ϕ 1 θ k r 2 cos ϕ 2 θ ) ) ] r 1 r 2 d r 1 d r 2 d ϕ 1 d ϕ 2 .
I = ϕ 2 = 0 2 π ϕ 1 = 0 2 π r 2 = 0 r 1 = 0 Re [ I s I o e A ( ( r 1 2 + r 2 2 ) ) e 2 B r 1 r 2 cos ( ϕ 1 ϕ 2 ) e ( r 1 2 + r 2 2 ) W 2 exp ( j ( k r 1 cos ϕ 1 θ k r 2 cos ϕ 2 θ ) ) ] r 1 r 2 d r 1 d r 2 d ϕ 1 d ϕ 2 ,
A = 1 4 σ s 2 + 1 2 δ s 2 + 1 4 σ o 2 + 1 2 δ o 2 + 2 R 2 ,
B = 1 2 δ s 2 + 1 2 δ o 2 .
I = r 2 = 0 r 2 ϕ 2 = 0 2 π ϕ 1 = 0 2 π Re [ I s I o e A ( ( r 1 2 + r 2 2 ) ) e C cos ( ϕ 1 ϕ 2 ) exp ( F ( r 1 cos ϕ 1 r 2 cos ϕ 2 ) ) ] r 1 r 2 d ϕ 1 d ϕ 2 d r 2 d r 1 .
I = r 2 = 0 r 2 = 0 ϕ 2 = 0 2 π Re [ I s I o e A ( ( r 1 2 + r 2 2 ) ) e F r 2 cos ϕ 2 ] r 1 r 2 d ϕ 2 d r 2 d r 1 ϕ 1 = 0 2 π exp ( [ cos ϕ 1 ( C cos ϕ 2 + F r 1 ) sin ϕ 1 ( C sin ϕ 2 ) ] ) d ϕ 1 .
ϕ = 0 2 π e [ α cos ϕ + β sin ϕ ] d ϕ = 2 π I 0 ( α 2 + β 2 ) ,
I 0 ( N 2 + M 2 2 M N cos ( ϕ 2 ψ ) ) = ( 1 ) m I m ( N ) I m ( M ) cos m ( ϕ 2 ψ ) ,
I 1 = 2 π ( 1 ) m r 2 = 0 r 2 = 0 Re [ I s I o e A ( ( r 1 2 + r 2 2 ) ) ] I m ( C ) I m ( F r 1 ) r 1 r 2 d r 2 d r 1 ϕ 2 = 0 2 π exp ( [ F r 2 cos ϕ 2 ] ) cos m ( ϕ 2 ψ ) d ϕ 2 .
ϕ = 0 2 π e [ α cos ϕ + β sin ϕ ] cos m ( ϕ ψ ) d ϕ = 2 π I m ( α 2 + β 2 ) cos m ( ϕ ξ ) ,
I = ( 2 π ) 2 ( 1 ) m cos ( m ψ ) r 2 = 0 r 2 = 0 Re [ I s I o e A ( ( r 1 2 + r 2 2 ) ) ] I m ( C ) I m ( F r 1 ) I m ( F r 2 ) r 1 r 2 d r 2 d r 1 .
I m ( F r 1 ) = j m J m ( F r 1 ) ,
I m ( F r 2 ) = j m J m ( F r 2 ) ,
I = ( 2 π ) 2 I s I o ( 1 ) m cos ( m ψ ) r 1 = 0 Re e A ( r 1 2 ) J m ( F r 1 ) r 1 d r 1 r 2 = 0 r 2 e A ( r 2 2 ) J m ( F r 2 ) I m ( 2 B r 1 r 2 ) d r 2 .
x = 0 x e α x 2 J l ( γ x ) I l ( β x ) d x = 1 2 α e ( β 2 γ 2 ) 4 α J l ( β γ 2 α ) .
I = ( 2 π ) 2 I s I o ( e F 2 4 γ 2 2 γ 2 ) m = ( 1 ) m cos ( m ψ ) r 1 = 0 e ( A + η 2 4 γ 2 ) r 1 2 J m ( F r 1 ) J m ( η F 2 γ 2 ) r 1 d r 1 ,
γ 2 = A ,
η = 2 B .
x = 0 x e ρ 2 x 2 J l ( α x ) J l ( β x ) d x = 1 2 ρ 2 e ( α 2 + β 2 ) 4 ρ 2 I l ( β γ 2 ρ 2 ) .
I = ( 2 π ) 2 I s I o ( e F 2 4 γ 2 2 γ 2 ) ( e F 2 + β 1 2 4 γ 1 2 γ 1 ) m = ( 1 ) m cos ( m ψ ) I m ( F β 1 γ 1 ) ,
γ 1 = A + η 2 4 γ 2 ,
β 1 = η F 2 γ 2 .
m = ( 1 ) m I m ( F β 1 γ 1 ) cos ( m ψ ) = e F β 1 γ 1 cos ( ψ ) ,
I = ( 2 π ) 2 I s I o ( e F 2 4 γ 2 2 γ 2 ) ( e F 2 + β 1 2 4 γ 1 2 γ 1 ) e ( F β 1 γ 1 ) .
I d = ϕ = 0 2 π r = 0 Γ ( r , r ) e r 2 W 2 r d r d ϕ .
I d = ϕ = 0 2 π r = 0 I e ( r 2 2 σ 2 ) e r 2 W 2 r d r d ϕ .
I d = 2 π I r = 0 e ( r 2 2 σ 2 ) e r 2 W 2 r d r d ϕ .
x = 0 x e ρ 2 x 2 d x = 1 2 ρ 2 .
I d = 2 π I 2 [ 1 2 σ 2 + 1 W 2 ] .
η h = ( R 2 + 2 σ s 2 ) ( R 2 + 2 σ o 2 ) ( 2 σ s σ o R 2 ) 2 γ 1 γ 2 ( e F 2 4 γ 2 ) ( e F 2 + β 1 2 4 γ 1 ) e F β 1 γ 1 .

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