Abstract

We present a general definition of the radiation efficiency of stationary electromagnetic fields and prove that it is bounded between zero and unity for beams of any state of coherence and polarization. The radiation efficiency may be interpreted as a measure of how directed the radiated fields are, and therefore it can be used to assess the allowed spatial coherence and intensity variations across a beam. We consider a class of partially coherent electromagnetic fields that were recently introduced in the literature and evaluate the radiation efficiencies for two particular examples, namely, the azimuthally polarized symmetric beams and the dipolar beams that are nearly linearly polarized in the central region. The results show that the radiation efficiency is fairly insensitive to the state of polarization and that it differs appreciably from unity for only small values of source and correlation widths.

© 2001 Optical Society of America

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References

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  1. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
  2. N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
    [CrossRef]
  3. P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
    [CrossRef]
  4. R. Martı́nez-Herrero, P. M. Mejı́as, J. M. Movilla, “Spatial characterization of general partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
    [CrossRef] [PubMed]
  5. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
    [CrossRef]
  6. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
    [CrossRef]
  7. N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
    [CrossRef]
  8. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [CrossRef]
  9. S. R. Seshadri, “Average characteristics of partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 17, 780–789 (2000).
    [CrossRef]
  10. P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
    [CrossRef]
  11. D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
    [CrossRef]
  12. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
    [CrossRef]
  13. See, for example, Ref. 1or A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, Vol. 69 of the SPIE Milestone Series (SPIE, Bellingham, Wash., 1993).
  14. T. Shirai, T. Asakura, “Radiation efficiency of partially coherent planar sources with various intensity distributions,” J. Mod. Opt. 40, 1143–1159 (1993).
    [CrossRef]
  15. T. Shirai, T. Asakura, “Radiation efficiency of partially coherent three-dimensional sources. A comparison with planar sources,” J. Mod. Opt. 40, 2451–2465 (1993).
    [CrossRef]
  16. A. Gamliel, “Radiation efficiency of planar Gaussian Schell-model sources,” Opt. Commun. 60, 333–338 (1986).
    [CrossRef]
  17. A. D. Ostrowsky, “Paraxial approximation of modern radiometry for beamlike wave propagation,” Opt. Rev. 3, 83–88 (1996).
    [CrossRef]
  18. D. K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, Reading, Mass., 1989).
  19. We emphasize that this is not a rigorous free-space representation because it ignores the divergency condition of the field. However, from the exact far-field formulas, one can show that Eq. (5) remains valid for reasonably directional electromagnetic fields, or, more specifically, when s⋅E(r′)is negligible for each realization in the source plane.
  20. S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
    [CrossRef]
  21. J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
    [CrossRef]
  22. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), p. 319.
  23. A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927–936 (1982).
    [CrossRef]

2000 (1)

1999 (1)

1998 (3)

1997 (2)

1996 (2)

P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
[CrossRef]

A. D. Ostrowsky, “Paraxial approximation of modern radiometry for beamlike wave propagation,” Opt. Rev. 3, 83–88 (1996).
[CrossRef]

1994 (1)

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
[CrossRef]

1993 (2)

T. Shirai, T. Asakura, “Radiation efficiency of partially coherent planar sources with various intensity distributions,” J. Mod. Opt. 40, 1143–1159 (1993).
[CrossRef]

T. Shirai, T. Asakura, “Radiation efficiency of partially coherent three-dimensional sources. A comparison with planar sources,” J. Mod. Opt. 40, 2451–2465 (1993).
[CrossRef]

1992 (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

1986 (1)

A. Gamliel, “Radiation efficiency of planar Gaussian Schell-model sources,” Opt. Commun. 60, 333–338 (1986).
[CrossRef]

1985 (1)

1983 (1)

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

1982 (1)

A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927–936 (1982).
[CrossRef]

1939 (1)

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Anderson, E. H.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Asakura, T.

T. Shirai, T. Asakura, “Radiation efficiency of partially coherent three-dimensional sources. A comparison with planar sources,” J. Mod. Opt. 40, 2451–2465 (1993).
[CrossRef]

T. Shirai, T. Asakura, “Radiation efficiency of partially coherent planar sources with various intensity distributions,” J. Mod. Opt. 40, 1143–1159 (1993).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Cheng, D. K.

D. K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, Reading, Mass., 1989).

Chu, L. J.

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Friberg, A. T.

Gamliel, A.

A. Gamliel, “Radiation efficiency of planar Gaussian Schell-model sources,” Opt. Commun. 60, 333–338 (1986).
[CrossRef]

Gori, F.

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Greene, P. L.

Guattari, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Hall, D. G.

P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

James, D. F. V.

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
[CrossRef]

Kettunen, V.

King, O.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Kuittinen, M.

Mandel, L.

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

Marti´nez-Herrero, R.

Meji´as, P. M.

Movilla, J. M.

Mukunda, N.

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Ostrowsky, A. D.

A. D. Ostrowsky, “Paraxial approximation of modern radiometry for beamlike wave propagation,” Opt. Rev. 3, 83–88 (1996).
[CrossRef]

Rooks, M. J.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Seshadri, S. R.

Shirai, T.

T. Shirai, T. Asakura, “Radiation efficiency of partially coherent three-dimensional sources. A comparison with planar sources,” J. Mod. Opt. 40, 2451–2465 (1993).
[CrossRef]

T. Shirai, T. Asakura, “Radiation efficiency of partially coherent planar sources with various intensity distributions,” J. Mod. Opt. 40, 1143–1159 (1993).
[CrossRef]

Simon, R.

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Stratton, J. A.

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Sudarshan, E. C. G.

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Turunen, J.

Vahimaa, P.

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Wicks, G. W.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Wolf, E.

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

Appl. Phys. Lett. (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

J. Mod. Opt. (2)

T. Shirai, T. Asakura, “Radiation efficiency of partially coherent planar sources with various intensity distributions,” J. Mod. Opt. 40, 1143–1159 (1993).
[CrossRef]

T. Shirai, T. Asakura, “Radiation efficiency of partially coherent three-dimensional sources. A comparison with planar sources,” J. Mod. Opt. 40, 2451–2465 (1993).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Commun. (2)

A. Gamliel, “Radiation efficiency of planar Gaussian Schell-model sources,” Opt. Commun. 60, 333–338 (1986).
[CrossRef]

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
[CrossRef]

Opt. Eng. (1)

A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927–936 (1982).
[CrossRef]

Opt. Lett. (2)

Opt. Rev. (1)

A. D. Ostrowsky, “Paraxial approximation of modern radiometry for beamlike wave propagation,” Opt. Rev. 3, 83–88 (1996).
[CrossRef]

Phys. Rev. (1)

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Phys. Rev. A (1)

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence–polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Other (5)

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

D. K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, Reading, Mass., 1989).

We emphasize that this is not a rigorous free-space representation because it ignores the divergency condition of the field. However, from the exact far-field formulas, one can show that Eq. (5) remains valid for reasonably directional electromagnetic fields, or, more specifically, when s⋅E(r′)is negligible for each realization in the source plane.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), p. 319.

See, for example, Ref. 1or A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, Vol. 69 of the SPIE Milestone Series (SPIE, Bellingham, Wash., 1993).

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Figures (9)

Fig. 1
Fig. 1

Illustration of the source in the coordinate system and the unit vector nˆ pointing out from the hemisphere at the position r in the far zone.

Fig. 2
Fig. 2

Arrows that represent the strength and the direction of the electric field of an azimuthally polarized beam in the source plane.

Fig. 3
Fig. 3

Radiation efficiency as a function of kσS for kσg=1, 2, 8 for a Gaussian correlated, azimuthally polarized field.

Fig. 4
Fig. 4

Radiation efficiency as a function of kσg for kσS=1, 2, 8 for a Gaussian correlated, azimuthally polarized field.

Fig. 5
Fig. 5

Three-dimensional plot of the radiation efficiency as a function of kσS and kσg for a Gaussian correlated, azimuthally polarized field.

Fig. 6
Fig. 6

Visualization of the magnitude and the direction of the electric field vector for a dipolar beam across the source plane.

Fig. 7
Fig. 7

Radiation efficiency plotted as a function of kσS for kσg=1, 2, 8 for a Gaussian correlated, dipolar field.

Fig. 8
Fig. 8

Radiation efficiency as a function of kσg for kσS=1, 2, 8 for a Gaussian correlated, dipolar field.

Fig. 9
Fig. 9

Three-dimensional plot of the radiation efficiency as a function of kσS and kσg for a Gaussian correlated, dipolar field.

Equations (47)

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Cω=ΦωNω.
S(r)=12R[E*(r)×H(r)],
E(r)=E(r)(r)+iE(i)(r),
H(r)=H(r)(r)+iH(i)(r),
S(r)=12RE*(r)×0μ01/2s×E(r)=120μ01/2E*(r)  E(r)s=120μ01/2tr Wjk(e)(r, r)s=120μ01/2tr Wjk(e)(r, r)n,
Wjk(e)(r, r)=ksz2πr2- Wjk(e)(r1, r2)×exp[iks  (r1-r2)]d2r1d2r2,
Φω=(2π)S(r)  nr2dΩ=120μ01/2(2π)tr Wjk(e)(r, r)r2dΩ,
Nω=c0(z=0)[w(e)(r)+w(h)(r)]d2r=c0(z=0)04 E*(r)  E(r)+μ04 H*(r)  H(r)d2r=c04(z=0)[0tr Wjk(e)(r, r)+μ0tr Wjk(h)(r, r)]d2r,
Cω=20(2π)tr Wjk(e)(r, r)r2dΩ(z=0)[0tr Wjk(e)(r, r)+μ0tr Wjk(h)(r, r)]d2r.
E(r)=10 zˆ×Fz(r),
Ez(r)=0,
H(r)=-1ikμ0 z Fz(r),
Hz(r)=-1ikμ02z2+k2Fz(r),
Ex(r)=-y G(r),
Ey(r)=x G(r),
Ez(r)=0,
tr Wjk(e)(r1, r2)=G*(r1)y1G(r2)y2+G*(r1)x1G(r2)x2g(r1, r2).
G(r)=Gaexp-r24σS2,
Ex(x, y)=-Kyexp-x2+y24σS2,
Ey(x, y)=Kxexp-x2+y24σS2,
g(r1, r2)=exp-(r1-r2)22σg2.
tr Wjk(e)(r1, r2)=K2(x1x2+y1y2)×exp-r12+r224σS2-(r1-r2)22σg2.
S(r)=c008K2sz2σg2σS8r2(σg2+4σS2)2×2+(sx2+sy2) k2σg4σg2+4σS2×exp-(sx2+sy2) 2k2σg2σS2σg2+4σS2n,
Φω=2πc00K2σS41+(kσg)24(kσS)2-1×1+3(kσg)28(kσS)2-1+3(kσg)28(kσS)2+(kσg)42[(kσg)2+4(kσS)2]D(ξ)ξ,
ξ2=2k2σS2σg2σg2+4σS2
D(ξ)=exp(-ξ2)0ξexp(t2)dt
Nω=2πc00K2σS4×1+1+1(kσS)2exp[-2(kσS)2].
G(r)=Gdxexp-r24σS2,
 Ex(x, y)=L xy2σS2exp-x2+y24σS2,
Ey(x, y)=L1-x22σS2exp-x2+y24σS4,
tr Wjk(e)(r1, r2)=L21-(x12+x22) 12σS2+x1x2y1y2+x12x224σS4×exp-r12+r224σS2-(r1-r2)22σg2.
S(r)=8c00(kszLσgσS4)2r2(σg2+4σS2)5 [(4σg4+64σS4+32σg2σS2)+sx2(6k2σg6+24k2σg4σS2)+(sx2+sy2)(k2σg6+4k2σg4σS2)+(sx2sy2+sx4)k4σg8]exp-(sx2+sy2) 2k2σg2σS2σg2+4σS2n,
Φω=πc00L2σS28(σg2+4σS2)44k2σg8σS2+16k2σg6σS4+2048σS8+15σg8+1136σg4σS4+2560σg2σS6+216σg6σS2-(224k2σg6σS4+512k2σg4σS6+16k4σg8σS4+216σg6σS2+2560σg2σS6+1136σg4σS4+2048σS8+15σg8+24k2σg8σS2) D(ξ)ξ,
Nω=2πc00L2σS212+exp(-2k2σS2)×1+k2σS22+1516k2σS2.
S(r)=120μ01/2ksz2πr2 - tr Wjk(e)(r1, r2)×exp[iks  (r1-r2)]d2r1d2r2n.
tr W˜jk(e)(-ks, ks)
= - tr Wjk(e)(r1, r2)×exp[iks(r1-r2)]d2r1d2r2,
Φω=120μ01/2(2π)ksz2πr2tr W˜jk(e)(-ks, ks)r2dΩ=120μ01/2k2π2sx2+sy21sz×tr W˜jk(e)(-ks, ks)dsxdsy,
Φω120μ01/2k2π2sx2+sy21×tr W˜jk(e)(-ks, ks)dsxdsy.
Φω120μ01/2 - tr Wjk(e)(r1, r2)δ(r1-r2)dr1dr2=120μ01/2(z=0)tr Wjk(e)(r, r)d2r.
Φω12μ001/2(z=0)tr Wjk(h)(r, r)d2r,
Φωc04(z=0)[0tr Wjk(e)(r, r)
+μ0tr Wjk(h)(r, r)]d2r=Nω,
tr Wjk(e)(r1, r2)=[Gx*(r1)Gx(r2)+Gy*(r1)Gy(r2)]g(r1, r2),
tr Wjk(e)*(r2, r1)=tr Wjk(e)(r1, r2).
f*(r1)f(r2)tr Wjk(e)(r1, r2)d2r1d2r2
=f*(r1)f(r2)[Gx*(r1)Gx(r2)+Gy*(r1)Gy(r2)]g(r1, r2)d2r1d2r2=f*(r1)Gx*(r1)f(r2)Gx(r2)g(r1, r2)d2r1d2r2+f*(r1)Gy*(r1)f(r2)Gy(r2)g(r1, r2)d2r1d2r20.

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