Abstract

The problem of maximizing the integrated–weighted intensity of a transmitted beam in a receiver plane is equivalent to the problem of finding the largest eigenvalue’s eigenfunction for a particular Hermitian operator. Application of the power method for the determination of this eigenfunction, along with its associated eigenvalue, results in an iterative transform algorithm that can be applied to arbitrary apertures, nonnegative windows, and propagation media. The computational complexity of each iteration of this algorithm is equivalent to the numerical propagation of an arbitrary beam through the transmission medium.

© 2004 Optical Society of America

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References

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  1. D. Slepian, “Analytic solution of two apodization problems,” J. Opt. Soc. Am. 55, 1110–1115 (1965).
    [CrossRef]
  2. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  3. G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovich, Orlando, Fla, 1988).
  4. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holgrams,” Opt. Eng. (Bellingham) 19, 297–305 (1980).
    [CrossRef]
  5. J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Proc. SPIE373, 147–160 (1981).
    [CrossRef]
  6. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  7. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  8. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: extensions to many dimensions; generalized prolate spheroidal functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
    [CrossRef]
  9. B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on the use of the prolate functions,” in Progress in Optics, Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Chap. VIII, pp. 311–407.

1987

1982

1980

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holgrams,” Opt. Eng. (Bellingham) 19, 297–305 (1980).
[CrossRef]

1965

1964

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: extensions to many dimensions; generalized prolate spheroidal functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Fienup, J. R.

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holgrams,” Opt. Eng. (Bellingham) 19, 297–305 (1980).
[CrossRef]

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Proc. SPIE373, 147–160 (1981).
[CrossRef]

Frieden, B. R.

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

Slepian, D.

D. Slepian, “Analytic solution of two apodization problems,” J. Opt. Soc. Am. 55, 1110–1115 (1965).
[CrossRef]

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: extensions to many dimensions; generalized prolate spheroidal functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
[CrossRef]

Strang, G.

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovich, Orlando, Fla, 1988).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Appl. Opt.

Bell Syst. Tech. J.

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: extensions to many dimensions; generalized prolate spheroidal functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng. (Bellingham)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holgrams,” Opt. Eng. (Bellingham) 19, 297–305 (1980).
[CrossRef]

Other

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Proc. SPIE373, 147–160 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovich, Orlando, Fla, 1988).

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

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Figures (8)

Fig. 1
Fig. 1

Propagation model.

Fig. 2
Fig. 2

Fractional intensity as a function of weighting-function diameter for an Airy beam.

Fig. 3
Fig. 3

Iterative transform algorithm for the computation of optimal beams.

Fig. 4
Fig. 4

Fractional intensity for optimal and Airy beams with circular aperture and weighting function.

Fig. 5
Fig. 5

Simulation scenario for determining optimal beams for propagation through a phase screen that is distant from both the pupil plane and destination plane. The phase screen is 5 km from the pupil plane and 5 km from the destination plane. The aperture diameter is 1 m and the operating wavelength is 1 μm.

Fig. 6
Fig. 6

Fractional intensity for propagation through a distant phase screen for both a compensated and uncompensated (optimal) beam with circular aperture and weighting functions. The fractional intensities for a uniform-amplitude and optimal beam through free space are also shown for reference.

Fig. 7
Fig. 7

(a) Normalized intensity for a diffraction-limited beam propagated a distance 10 km through free space, (b) phase screen placed 5 km from the aperture, (c) intensity pattern for an uncompensated beam propagated through the distant phase screen, (d) intensity for a compensated beam to maximize the energy encircled in a region of diamter 0.1 m. The intensity values are normalized to the peak intensity for the diffraction-limited beam, and the phase-screen values have units of radians. The height and width for each figure is approximately 1.2 m.

Fig. 8
Fig. 8

Optimal pupil field to maximize the encircled energy within a radius of 0.1 m for the simulation example: (a) field amplitude, (b) field phase.

Equations (39)

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g(x)=hd(x, u)a(u)f(u)du,
Iaf=a(u)|f(u)|2du,
Ig=|g(x)|2dx,
Iwg=w(x)|g(x)|2dx,
a(u)=1,uD/20,u>D/2,
w(x)=1,xr0,x>r.
hd(x, u)1jλdexpj 2πλexpj πλd x-u2,
f(u)=exp-j πλd u2,
|g(x)|2=1λd2a(u)exp-j2πu xλddu2=1λd2Axλd2,
A(ν)=D22J1[2π(D/2)|ν|](D/2)|ν|
|g(x)|2=1λd2D22J1[π(D/λd)|x|](D/2λd)|x|2=D22J12[π(D/λd)|x|]|x|2,
Iwg=w(x)|g(x)|2dx=D2202π0rJ12[π(D/λd)ρ]ρ2 ρdρdϕ=πD2220π(Dr/λd)J12(z)zdz.
Ig=If=π(D/2)2,
IwgIg=20π(Dr/λd)J12(z)zdz=1-J02π Drλd-J12π Drλd,
fˆ=argmaxIaf=I0,Iwg,=argmaxIaf=I0,f*(u)a*(u)×w(x)hd*(x, u)hd(x, u)dxa(u)f(u)dudu=argmaxIaf=I0,AAf*(u)Hd(u, u)f(u)dudu,
Hd(u, u)=hd*(x, u)hd(x, u)w(x)dx=n=1λnψn(u)ψn*(u),(u, u)A2
fˆ(u)=(I0)1/2a(u)ψ1(u)/aψ1,
aψ1=[a(u)ψ1(u)]2du1/2.
Hd(u, u)=[hd(x0, u)]*[hd(x0, u)]
fˆ(u)=(I0)1/2ahd(x0) a(u)hd(x0, u),
ahd(x0)2=|a(u)hd(x0, u)|2du.
hd(x0, u)1jλdexpj 2πλexpj πλd x0-u2,
zk(u)=AHd(u, u)fk-1(u)du,uA,
fk(u)=I0zk(u)/zk(u),uA.
f0(u)=n=1αnψn(u),
z1(u)=AHd(u, u)f0(u)du=n=1αnλnψn(u),
f1(u)=I0n=1αnλnψn(u)n=1αn2λn21/2.
z2(u)=n=1αnλn2ψn(u)n=1αn2λn21/2,
f2(u)=I0n=1αnλn2ψn(u)n=1αn2λn41/2,
fk(u)=I0n=1αnλnkψn(u)n=1αn2λn2k1/2.
n=1αn2(λn/λ1)2kI01/2fk(u)=ψ1(u)+α2α1λ2λ1kψ2(u)+α3α1λ3λ1kψ3(u)+ ,
f0*(u)ψ1(u)du=0,
zk(u)=w(x)Afk-1(u)hd(x, u)duhd*(x, u)dx,
uA,
gk-1(x)=Afk-1(u)hd(x, u)du.
zk(u)=w(x)gk-1(x)hd*(x, u)dx,uA.
fk(u)=I0zk(u)/a(u)zk(u).
a(u)=1,uD/20,u>D/2,
w(x)=1,xr0,x>r,

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