Abstract

I show that there is an exact, complete method for finding the orthogonal spatial channels, or communications modes, between two arbitrary volumes, and the associated connection strengths, for the case of scalar waves. I also show that the sum of the squared connection strengths is given exactly by a simple volume integral. The method is illustrated by a calculation for a particular extreme pair of volumes, and the communications modes are interpreted physically as the modes of a double phase-conjugate resonator.

© 1998 Optical Society of America

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References

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  1. D. Gabor, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 109–153.
    [CrossRef]
  2. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
    [CrossRef] [PubMed]
  3. General results have also been derived for arbitary shapes of apertures in the eigenfunction picture in the study of propagation through atmospheric turbulence; see, e.g., J. H. Shapiro, in Laser Beam Propagation in the Atmosphere, J. W. Strohbehm, ed. (Springer-Verlag, Berlin, 1978), pp. 171–222.
    [CrossRef]
  4. It is possible to use one source type rigorously over a closed surface for the case of phase fronts; see D. A. B. Miller, Opt. Lett. 16, 1370 (1991).
    [CrossRef] [PubMed]
  5. Portions of this research were presented in D. A. B. Miller, Proc. SPIE 3490, 111–114 (1998).
    [CrossRef]
  6. See, e.g., D. Porter and D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, 1990), for a discussion of these issues.
    [CrossRef]
  7. R. Courant and D. Hilbert, Methods of Mathematical Physics, Wiley Classics ed. (Wiley, New York, 1989).
  8. This process of finding the communications modes is analogous to singular value decomposition of the matrix gji; to prove the existence and completeness of the sets of eigenfunctions, however, requires results from functional analysis and hence an integral representation of the problem.

1998 (1)

Portions of this research were presented in D. A. B. Miller, Proc. SPIE 3490, 111–114 (1998).
[CrossRef]

1991 (1)

1969 (1)

Courant, R.

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

Gabor, D.

D. Gabor, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 109–153.
[CrossRef]

Hilbert, D.

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

Miller, D. A. B.

Porter, D.

See, e.g., D. Porter and D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, 1990), for a discussion of these issues.
[CrossRef]

Shapiro, J. H.

General results have also been derived for arbitary shapes of apertures in the eigenfunction picture in the study of propagation through atmospheric turbulence; see, e.g., J. H. Shapiro, in Laser Beam Propagation in the Atmosphere, J. W. Strohbehm, ed. (Springer-Verlag, Berlin, 1978), pp. 171–222.
[CrossRef]

Stirling, D. S. G.

See, e.g., D. Porter and D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, 1990), for a discussion of these issues.
[CrossRef]

Toraldo di Francia, G.

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Proc. SPIE (1)

Portions of this research were presented in D. A. B. Miller, Proc. SPIE 3490, 111–114 (1998).
[CrossRef]

Other (5)

See, e.g., D. Porter and D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, 1990), for a discussion of these issues.
[CrossRef]

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

This process of finding the communications modes is analogous to singular value decomposition of the matrix gji; to prove the existence and completeness of the sets of eigenfunctions, however, requires results from functional analysis and hence an integral representation of the problem.

D. Gabor, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 109–153.
[CrossRef]

General results have also been derived for arbitary shapes of apertures in the eigenfunction picture in the study of propagation through atmospheric turbulence; see, e.g., J. H. Shapiro, in Laser Beam Propagation in the Atmosphere, J. W. Strohbehm, ed. (Springer-Verlag, Berlin, 1978), pp. 171–222.
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Illustrations (a) of two thin volumes considered in this example, (b) the strongest communications mode, and (c) the second communications mode. For the transmitting volume the real part of the wave amplitude along the length of the volume is shown for a particular arbitrary phase. For the receiving volume, the real part of the wave is shown in a contour plot that illustrates approximately half a period of the wave and with a horizontal scale such that 2π of phase is the same size as one wavelength on the diagram. With this choice of scale the curvature of the phase fronts corresponds approximately to the actual curvature of the propagating wave. λ is the wavelength.

Equations (14)

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2ϕr+k2ϕr=-ψr.
Gr,rT=exp-ikr-rT/4πr-rT;
ϕr=VTGr,rTψrTd3rT.
gji=VRVTaRj*rRGrR,rTaTirTd3rTd3rR.
g2=VTψ*rTVTKrT,rTψrTd3rTd3rT,
KrT,rT=VRG*rR,rTGrR,rTd3rR.
g2ψrT=VTKrT,rTψrTd3rT
gnϕnrR=VTGrR,rTψnrTd3rT.
gn*ψnrT=VRG*rR,rTϕnrRd3rR.
gn2ϕnrR=VRJrR,rRϕnrRd3rR,
JrR,rR=VTGrR,rTG*rT,rRd3rT,
GrR,rT=i,jgjiaRjrRaTi*rT.
VRVTGrR,rT2 d3rT d3rR=i,jgji2;
γRTi,jgji2=14π2VRVT1rR-rT2d3rTd3rR

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