Abstract

A rigorous method for finding the best-connected orthogonal communication channels, modes, or degrees of freedom for scalar waves between two volumes of arbitrary shape and position is derived explicitly without assuming planar surfaces or paraxial approximations. The communication channels are the solutions of two eigenvalue problems and are identical to the cavity modes of a double phase-conjugate resonator. A sum rule for the connection strengths is also derived, the sum being a simple volume integral. These results are used to analyze rectangular prism volumes, small volumes, thin volumes in different relative orientations, and arbitrary near-field volumes: all situations in which previous planar approaches have failed for one or more reasons. Previous planar results are reproduced explicitly, extending them to finite depth. Depth is shown not to increase the number of communications modes unless the volumes are close when compared with their depths. How to estimate the connection strengths in some cases without a full solution of the eigenvalue problem is discussed so that estimates of the number of usable communications modes can be made from the sum rule. In general, the approach gives a rigorous basis for handling problems related to volume sources and receivers. It may be especially applicable in near-field problems and in situations in which volume is an intrinsic part of the problem.

© 2000 Optical Society of America

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References

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  1. I previously have briefly presented the derivation of the communications modes and the sum rule in D. A. B. Miller, “Spatial channels for communicating with waves between volumes,” Opt. Lett. 23, 1645–1647 (1998), and in summary form in D. A. B. Miller, “Communicating with waves between volumes—how many different spatial channels are there?” in Optics in Computing ’98, P. Chavel, D. A. B. Miller, H. Thienpont, eds., Proc. SPIE3490, 111–114 (1998).
  2. R. Piestun, D. A. B. Miller, “Degrees of freedom of an electromagnetic wave,” in Eighteenth Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 110–111 (1999); R. Piestun, D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A (to be published).
    [CrossRef]
  3. D. Gabor, “Light and information,” in Progress in Optics, Vol. 1, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1961), pp. 109–153.
    [CrossRef]
  4. G. Toraldo di Francia, “Resolving power and information,” J. Opt. Soc. Am. 45, 497–501 (1955).
    [CrossRef]
  5. G. Toraldo di Francia, “Directivity, supergain, and information,” Trans. IRE 4, 473–478 (1956).
  6. A. Walther, “Gabor’s theorem and energy transfer through lenses,” J. Opt. Soc. Am. 57, 639–644 (1967).
    [CrossRef]
  7. I omit here for simplicity the two distinct phases and, in the case of light, the two distinct polarizations, the effects of each of which can be viewed as doubling the number of degrees of freedom.
  8. G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969).
    [CrossRef] [PubMed]
  9. C. W. Barnes, “Object restoration in a diffraction-limited imaging system,” J. Opt. Soc. Am. 56, 575–578 (1966).
    [CrossRef]
  10. B. R. Frieden, “Evaluation, design, and extrapolation methods for optical signals, based on the use of the prolate functions,” in Progress in Optics IX, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1971), pp. 311–407.
    [CrossRef]
  11. F. Gori, G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
    [CrossRef]
  12. F. Gori, G. Guattari, “Degrees of freedom of images from point-like-element pupils,” J. Opt. Soc. Am. 64, 453–458 (1974).
    [CrossRef]
  13. L. Ronchi, A. Consortini, “Degrees of freedom of images in coherent and incoherent illumination,” Alta Frequenza 43, 1034–1036 (1974).
  14. M. Bendinelli, A. Consortini, L. Ronchi, B. R. Frieden, “Degrees of freedom and eigenfunctions for the noisy image,” J. Opt. Soc. Am. 64, 1498–1502 (1974).
    [CrossRef]
  15. D. Slepian, H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
    [CrossRef]
  16. J. H. Shapiro, “Optimal power transfer through atmospheric turbulence using state knowledge,” IEEE Trans. Commun. Technol. COM-19, 410–414 (1971).
    [CrossRef]
  17. J. H. Shapiro, “Normal-mode approach to wave propagation in the turbulent atmosphere,” Appl. Opt. 13, 2614–2619 (1974).
    [CrossRef] [PubMed]
  18. J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 171–222.
    [CrossRef]
  19. D. A. B. Miller, “Huygens’s wave propagation principle corrected,” Opt. Lett. 16, 1370–1372 (1991).
    [CrossRef] [PubMed]
  20. See, for example, D. Porter, D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, 1990), for a discussion of these issues.
  21. R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1989).
  22. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).
  23. T. A. Beu, R. I. Câmpeanu, “Prolate radial spheroidal wave functions,” Comp. Phys. Commun. 30, 177–185 (1983).
    [CrossRef]
  24. T. A. Beu, R. I. Câmpeanu, “Prolate angular spheroidal wave functions,” Comp. Phys. Commun. 30, 187–192 (1983).
    [CrossRef]
  25. The phase-conjugate cavity discussed here has two phase-conjugating mirrors. This configuration differs from the situation with one phase-conjugating mirror and one conventional mirror that is more extensively discussed in the literature. The case with two phase-conjugate mirrors is briefly discussed in, for example, J. F. Lam, W. P. Brown, “Optical resonators with phase-conjugate mirrors,” Opt. Lett. 2, 61–63 (1980).

1998 (1)

1991 (1)

1983 (2)

T. A. Beu, R. I. Câmpeanu, “Prolate radial spheroidal wave functions,” Comp. Phys. Commun. 30, 177–185 (1983).
[CrossRef]

T. A. Beu, R. I. Câmpeanu, “Prolate angular spheroidal wave functions,” Comp. Phys. Commun. 30, 187–192 (1983).
[CrossRef]

1980 (1)

1974 (4)

1973 (1)

F. Gori, G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

1971 (1)

J. H. Shapiro, “Optimal power transfer through atmospheric turbulence using state knowledge,” IEEE Trans. Commun. Technol. COM-19, 410–414 (1971).
[CrossRef]

1969 (1)

1967 (1)

1966 (1)

1961 (1)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

1956 (1)

G. Toraldo di Francia, “Directivity, supergain, and information,” Trans. IRE 4, 473–478 (1956).

1955 (1)

Barnes, C. W.

Bendinelli, M.

Beu, T. A.

T. A. Beu, R. I. Câmpeanu, “Prolate radial spheroidal wave functions,” Comp. Phys. Commun. 30, 177–185 (1983).
[CrossRef]

T. A. Beu, R. I. Câmpeanu, “Prolate angular spheroidal wave functions,” Comp. Phys. Commun. 30, 187–192 (1983).
[CrossRef]

Brown, W. P.

Câmpeanu, R. I.

T. A. Beu, R. I. Câmpeanu, “Prolate angular spheroidal wave functions,” Comp. Phys. Commun. 30, 187–192 (1983).
[CrossRef]

T. A. Beu, R. I. Câmpeanu, “Prolate radial spheroidal wave functions,” Comp. Phys. Commun. 30, 177–185 (1983).
[CrossRef]

Consortini, A.

L. Ronchi, A. Consortini, “Degrees of freedom of images in coherent and incoherent illumination,” Alta Frequenza 43, 1034–1036 (1974).

M. Bendinelli, A. Consortini, L. Ronchi, B. R. Frieden, “Degrees of freedom and eigenfunctions for the noisy image,” J. Opt. Soc. Am. 64, 1498–1502 (1974).
[CrossRef]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1989).

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

Frieden, B. R.

M. Bendinelli, A. Consortini, L. Ronchi, B. R. Frieden, “Degrees of freedom and eigenfunctions for the noisy image,” J. Opt. Soc. Am. 64, 1498–1502 (1974).
[CrossRef]

B. R. Frieden, “Evaluation, design, and extrapolation methods for optical signals, based on the use of the prolate functions,” in Progress in Optics IX, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1971), pp. 311–407.
[CrossRef]

Gabor, D.

D. Gabor, “Light and information,” in Progress in Optics, Vol. 1, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1961), pp. 109–153.
[CrossRef]

Gori, F.

F. Gori, G. Guattari, “Degrees of freedom of images from point-like-element pupils,” J. Opt. Soc. Am. 64, 453–458 (1974).
[CrossRef]

F. Gori, G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, “Degrees of freedom of images from point-like-element pupils,” J. Opt. Soc. Am. 64, 453–458 (1974).
[CrossRef]

F. Gori, G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1989).

Lam, J. F.

Miller, D. A. B.

Piestun, R.

R. Piestun, D. A. B. Miller, “Degrees of freedom of an electromagnetic wave,” in Eighteenth Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 110–111 (1999); R. Piestun, D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A (to be published).
[CrossRef]

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Porter, D.

See, for example, D. Porter, D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, 1990), for a discussion of these issues.

Ronchi, L.

L. Ronchi, A. Consortini, “Degrees of freedom of images in coherent and incoherent illumination,” Alta Frequenza 43, 1034–1036 (1974).

M. Bendinelli, A. Consortini, L. Ronchi, B. R. Frieden, “Degrees of freedom and eigenfunctions for the noisy image,” J. Opt. Soc. Am. 64, 1498–1502 (1974).
[CrossRef]

Shapiro, J. H.

J. H. Shapiro, “Normal-mode approach to wave propagation in the turbulent atmosphere,” Appl. Opt. 13, 2614–2619 (1974).
[CrossRef] [PubMed]

J. H. Shapiro, “Optimal power transfer through atmospheric turbulence using state knowledge,” IEEE Trans. Commun. Technol. COM-19, 410–414 (1971).
[CrossRef]

J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 171–222.
[CrossRef]

Slepian, D.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Stirling, D. S. G.

See, for example, D. Porter, D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, 1990), for a discussion of these issues.

Toraldo di Francia, G.

Walther, A.

Alta Frequenza (1)

L. Ronchi, A. Consortini, “Degrees of freedom of images in coherent and incoherent illumination,” Alta Frequenza 43, 1034–1036 (1974).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Comp. Phys. Commun. (2)

T. A. Beu, R. I. Câmpeanu, “Prolate radial spheroidal wave functions,” Comp. Phys. Commun. 30, 177–185 (1983).
[CrossRef]

T. A. Beu, R. I. Câmpeanu, “Prolate angular spheroidal wave functions,” Comp. Phys. Commun. 30, 187–192 (1983).
[CrossRef]

IEEE Trans. Commun. Technol. (1)

J. H. Shapiro, “Optimal power transfer through atmospheric turbulence using state knowledge,” IEEE Trans. Commun. Technol. COM-19, 410–414 (1971).
[CrossRef]

J. Opt. Soc. Am. (6)

Opt. Commun. (1)

F. Gori, G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

Opt. Lett. (3)

Trans. IRE (1)

G. Toraldo di Francia, “Directivity, supergain, and information,” Trans. IRE 4, 473–478 (1956).

Other (8)

R. Piestun, D. A. B. Miller, “Degrees of freedom of an electromagnetic wave,” in Eighteenth Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 110–111 (1999); R. Piestun, D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A (to be published).
[CrossRef]

D. Gabor, “Light and information,” in Progress in Optics, Vol. 1, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1961), pp. 109–153.
[CrossRef]

B. R. Frieden, “Evaluation, design, and extrapolation methods for optical signals, based on the use of the prolate functions,” in Progress in Optics IX, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1971), pp. 311–407.
[CrossRef]

I omit here for simplicity the two distinct phases and, in the case of light, the two distinct polarizations, the effects of each of which can be viewed as doubling the number of degrees of freedom.

J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 171–222.
[CrossRef]

See, for example, D. Porter, D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, 1990), for a discussion of these issues.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1989).

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

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Figures (8)

Fig. 1
Fig. 1

Illustration of the diffraction approach to estimating the number of independent channels or resolvable spots for communicating between two surfaces. N ∼ ΩA2, where N is the number of channels and λ is the wavelength.

Fig. 2
Fig. 2

Schematic diagram of the transmitting volume V T , which contains sources Ψ(r T ) of waves, and the receiving volume V R , which contains a wave field ϕ(r R ) that arises from the sources in the transmitting volume.

Fig. 3
Fig. 3

Transmitting volume V T , receiving volume V R , rectangular prism volumes, and their coordinates and dimensions.

Fig. 4
Fig. 4

Illustration of the focusing functions F T (r T ) and F R (r R ) that arise in the analysis of communications between rectangular prism volumes.

Fig. 5
Fig. 5

Normalized prolate spheroidal functions for various parameter values: (a) 1 degree of freedom (c = π/2), (b) 2 degrees of freedom (c = π), (c) 3 degrees of freedom (c = 3π/2).

Fig. 6
Fig. 6

Most extreme well-connected communications mode with nx = 0 and ny = 2 between volumes with dimensions of 9λ × 27λ × 9λ (left-hand image) and 9λ × 9λ × 18λ (right-hand image). The two volumes are separated by 81λ (center to center). The volumes are shown to scale in an isometric projection.

Fig. 7
Fig. 7

Illustration of (a) two thin volumes, (b) the strongest communications mode, (c) the second-strongest communications mode. For the transmitting volume V T [the thin horizontal volume shown in (a)] the real part of the wave amplitude along the length of the volume is shown for a particular arbitrary phase. For the receiving volume V R [the thin vertical volume shown in (a)] the real part of the wave is shown in a contour plot that illustrates approximately half of a period of the wave with a horizontal scale such that 2π of the phase is the same size as one wavelength on the diagram. With this choice of scale the curvatures of the phase fronts correspond approximately to the actual curvature of the propagating waves. Dimensions are in wavelengths λ. Note that the second communications mode changes sign between the peak in the center and those in the upper and the lower lobes. Note also that these upper and lower lobes are more intense than the center peak. At least 86% of the available communications strength is in the first mode, and at least 11% of the strength is in the second mode.

Fig. 8
Fig. 8

Illustration in cross section of the source points r T1 and r T2 in volume V T and the volume V R , which contains the resultant waves of interest. Also shown is a volume ΔV T near r T1. Other sources in ΔV T , with their phases appropriately chosen, are expected to produce waves in V R that are substantially similar (not orthogonal) to the wave from the source at point r T1, whereas sources outside ΔV T are expected to produce waves in V R that are substantially orthogonal to those from the source at point r T1.

Tables (1)

Tables Icon

Table 1 Eigenvalues ν n That Correspond to the Various Functions Shown in Fig. 3a

Equations (96)

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NΩA/λ2.
2Φr, t-1c22Φr, tt2=-Ψr, t,
Ψr, t=ψrexpiωt+c.c.
Φr, t=ϕrexpiωt+c.c.,
2ϕr+k2ϕr=-ψr.
Gr, r1=exp±ik|r-r1|4π|r-r1|.
Gr, r1=exp-ik|r-r1|4π|r-r1|.
ϕr=VT Gr, rTψrTd3rT.
VT aTmrTaTn*rTd3rT=δmn,
VR aRmrRaRn*rRd3rR=δmn,
ψrT=i biaTirT,
ϕrR=j djaRjrR.
dj=i gjibi,
gji=VRVT aRj*rRGrR, rTaTirTd3rTd3rR.
d1d2d3=g11g12g13g21g22g23g31g32g33b1b2b3,
ϕ=ΓRTψ.
W=ΨRrR, tΦrR, tt.
W=2ωImψRrRϕ*rR.
P=-VR 2ωImψRrRϕ*rRd3rR.
P=-2ωC VRImaRj*rRϕrRd3rR=-2ωCImgji.
P=-2ωC VRImiaRj*rRϕrRd3rR=-2ωCRegji.
M=VR aj*rRϕrRd3rR=gji.
γRTi,j |gji|2=14π2VRVT1|rR-rT|2d3rTd3rR.
GrR, rT=iνirRaTi*rT,
νirR=VT GrR, rTaTirTd3rT.
νirR=j gjiaRjrR,
GrR, rT=i,j gjiaRjrRaTi*rT.
|GrR, rT|2=i,j gjiaRjrRaTi*rT×p,q gqp*aRq*rRaTprT.
VRVT |GrR, rT|2d3rTd3rR=i,j |gji|2.
|GrR, rT|2=14π2|rR-rT|2,
|g|2=VR ϕ*rRϕrRd3rR.
|g|2=VRVTVT G*rR, rTψ*rTGrR, rT×ψrTd3rTd3rTd3rR,
|g|2=VT ψ*rTVT KrT, rTψrTd3rTd3rT,
KrT, rT=VR G*rR, rTGrR, rTd3rR.
KrT, rT=K*rT, rT.
VTVT |KrT, rT|2d3rTd3rT<.
KrT, rT=j,i kjiaTjrTaTi*rT.
|g|2ψrT=VT KrT, rTψrTd3rT
gnϕnrR=ϕnUrR=VT GrR, rTψnrTd3rT.
VR G*rR, rTVT GrR, rTψnrTd3rTd3rR=|gn|2ψnrT=VR G*rR, rTgnϕnrRd3rR,
gn*ψnrT=VR G*rR, rTϕnrRd3rR.
|gn|2ϕnrR=VR JrR, rRϕnrRd3rR,
JrR, rR=VT GrR, rTG*rR, rTd3rT.
gnψn*rT=VR GrR, rTϕn*rRd3rR,
r2ΔxT, 2ΔyT, 2ΔzT, 2ΔxR, 2ΔyR, 2ΔzR.
|g|2ψrT=VTVRexpik|rR-rT|exp-ik|rR-rT|4π2|rR-rTrR-rT|d3rRψrTd3rT.
|rR-rT|=r+zR-zT2+xR-xT2+yR-yT21/2r+zR-zT+xR-xT22r+yR-yT22r,
expik|rR-rT|exp-ikrR-rTexp-ikzT-12rxT2+yT2-zT+12rxT2+yT2+1rxRxT-xT+yRyT-yT.
ψrT=FTrTβTrT,
FTrT=exp-ikzT-12rxT2+yT2.
|g|2βTrT=14πr2VTVRexp-ikrxRxT-xT+yRyT-yTβrTd3rRd3rT.
βTrT=αTxxTαTyyTαTzzT.
|g|2αTxxTαTyyTαTzzT=14πr2VTVR×exp-ikrxRxT-xT+yRyT-yT×αTxxTαTyyTαTzzTd3rRd3rT,
ηTxαTxxT=-ΔxTΔxT-ΔxRΔxRexp-ikr xRxT-xTdxR×αTxxTdxT.
ηTzαTzzT=-ΔzTΔzT-ΔzRΔzR αTzzTdzRdzT,
ηTxηTyηTz=4πr2|g|2.
ηTz=2ΔzR2ΔzT.
-ΔxRΔxRexp-ikr xRxT-xTdxR=rikxT-xTexpikr ΔxRxT-xT-exp-ikr ΔxRxT-xT=2πΔxRΩTxsinΩTxxT-xTπxT-xT,
ΩTx=kΔxRr.
2πΔxRΩTx=2πrk=λr,
νTxαTxxT=-ΔxTΔxTsinΩTxxT-xTπxT-xT αTxxTdxT,
νTx=ηTxλr.
ψnxnyrT=FTrTαnxxTαnyyT,
νTnxS0nxcx, ξT=-11sincxξT-ξTπξT-ξT S0nxcx, ξTdξT,
ξT=xTΔxT,
cx=ΔxTΩTx,
1ν0>ν1>ν2> 0,
nxcrit=2ΔxTΩxπ=2π cx=2ΔxT2ΔxRλr,
ηTxλr,
|g|2λ22ΔzT2ΔzR4π2.
 |g|2=VTVR4πr2.
Nmax=VRVT4πr24π2λ22ΔzT2ΔzR=nxcritnycrit,
ϕrR=FRrRβRrR,
FRrR=expikzR-12rxR2+yR2
βRrR=αRxxRαRyyRαRzzR
ηRz=2ΔzR2ΔzT.
ϕnxnyrR=FRrRαnxxRαnyyR,
ΔxTMΔxRMλ0r  1,
expik|rR-rT|exp-ik|rR-rT|exp-ikzT-zT,
|g|2ψrT=14πr2VTVRexpikzT-zTd3rRψrTd3rT,
|g|2ϕrR=14πr2VRVTexpikzR-zRd3rTϕrRd3rR,
ψrT=βTrTexp-ikzT,
|g|2βTrT=14πr2VTVR βTrTd3rRd3rT.
ψrT=1VTexp-ikzT,
|g|2=VRVT4πr2.
ϕrR=1VRexp-ikzR
KrT2, rT1=VR G*rR, rT2GrR, rT1d3rR.
G*rR, rT2GrR, rT1=exp-ik|rR-rT1|-|rR-rT2|4πr0214πr02 FTrT2FT*rT1exp-ikxRxT2-xT1r0exp-ikyRyT2-yT1r0,
kΔxRmaxxT2-xT1rπ2,
|xT2-xT1|πr2kΔxRmax=12λr2ΔxRmax.
VTλr2ΔxRmaxλr2ΔyRmax 2ΔzTmax,
KrT2, rT1FT*rT2FTrT1VR4πr2
ψ1rT=1ΔVT1/2 FTrT,
|g|2=VT ψ1*rT2VT KrT2, rT1ψ1rT1d3rT1d3rT2=ΔVT1ΔVT1/2VRΔVT4πr21ΔVT1/2d3rT2=VRΔVT4πr2.
NVTΔVT,
N|g|2=VRVT4πr2,

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