Abstract

We address the problem of designing phase gratings that generate prescribed light intensities in the far field. The method of generalized projections is used to find a solution consistent with prior constraints and available data. While the method of generalized projections is known to converge to erroneous solutions occasionally, such solutions were not observed in this problem.

© 1991 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
    [Crossref]
  2. C. Warde, A. D. Fisher, “Spatial light modulators: applications and functional capabilities,” in Optical Signal Processing, J. Horner, ed. (Academic, San Diego, Calif., 1988), pp. 478–523.
  3. R. E. Collin, Antenna and Radiowave Propagation (McGraw-Hill, New York, 1985), pp. 107–151.
  4. M. King, “Fourier optics and radar signal processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 209–251.
    [Crossref]
  5. R. J. Mailloux, “Phased array theory and technology,” Proc. IEEE 70, 246–291 (1982).
    [Crossref]
  6. R. T. Compton, Adaptive Antennas (Prentice-Hall, Engle-wood Cliffs, N.J., 1988).
  7. L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
    [Crossref]
  8. R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [Crossref]
  9. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [Crossref]
  10. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I, theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
    [Crossref]
  11. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [Crossref]
  12. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237 (1972).
  13. J. R. Fineup, “Phase retrieval algorithm: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref]
  14. A. Levi, “Image restoration by the method of projections with applications to the phase and magnitude retrieval problems,” Ph.D. dissertation (Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, N.Y., 1983).
  15. A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987).

1989 (1)

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[Crossref]

1984 (1)

1982 (3)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I, theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[Crossref]

J. R. Fineup, “Phase retrieval algorithm: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref]

R. J. Mailloux, “Phased array theory and technology,” Proc. IEEE 70, 246–291 (1982).
[Crossref]

1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

1974 (1)

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237 (1972).

1951 (1)

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
[Crossref]

Collin, R. E.

R. E. Collin, Antenna and Radiowave Propagation (McGraw-Hill, New York, 1985), pp. 107–151.

Compton, R. T.

R. T. Compton, Adaptive Antennas (Prentice-Hall, Engle-wood Cliffs, N.J., 1988).

Fineup, J. R.

Fisher, A. D.

C. Warde, A. D. Fisher, “Spatial light modulators: applications and functional capabilities,” in Optical Signal Processing, J. Horner, ed. (Academic, San Diego, Calif., 1988), pp. 478–523.

Flannery, D. L.

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237 (1972).

Horner, J. L.

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[Crossref]

King, M.

M. King, “Fourier optics and radar signal processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 209–251.
[Crossref]

Landweber, L.

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
[Crossref]

Levi, A.

A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[Crossref]

A. Levi, “Image restoration by the method of projections with applications to the phase and magnitude retrieval problems,” Ph.D. dissertation (Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, N.Y., 1983).

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987).

Mailloux, R. J.

R. J. Mailloux, “Phased array theory and technology,” Proc. IEEE 70, 246–291 (1982).
[Crossref]

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237 (1972).

Stark, H.

A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[Crossref]

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987).

Warde, C.

C. Warde, A. D. Fisher, “Spatial light modulators: applications and functional capabilities,” in Optical Signal Processing, J. Horner, ed. (Academic, San Diego, Calif., 1988), pp. 478–523.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I, theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[Crossref]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I, theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[Crossref]

Am. J. Math. (1)

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Circuits Syst. (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

IEEE Trans. Med. Imag. (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I, theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237 (1972).

Proc. IEEE (2)

R. J. Mailloux, “Phased array theory and technology,” Proc. IEEE 70, 246–291 (1982).
[Crossref]

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[Crossref]

Other (6)

C. Warde, A. D. Fisher, “Spatial light modulators: applications and functional capabilities,” in Optical Signal Processing, J. Horner, ed. (Academic, San Diego, Calif., 1988), pp. 478–523.

R. E. Collin, Antenna and Radiowave Propagation (McGraw-Hill, New York, 1985), pp. 107–151.

M. King, “Fourier optics and radar signal processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 209–251.
[Crossref]

R. T. Compton, Adaptive Antennas (Prentice-Hall, Engle-wood Cliffs, N.J., 1988).

A. Levi, “Image restoration by the method of projections with applications to the phase and magnitude retrieval problems,” Ph.D. dissertation (Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, N.Y., 1983).

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Configuration in which light beams are switched to preselected locations by the application of the proper phase function on the grating.

Fig. 2
Fig. 2

Block diagram of the phase-design algorithm for the method of generalized projections.

Fig. 3
Fig. 3

(a) Flow chart for realizing the phase-design algorithm (DFT, discrete Fourier transform); (b) subroutines for realizing the operators P1 and P2.

Fig. 4
Fig. 4

Trap points in the set geometry of nonconvex sets. (a) Stable trap point at T. Slight deviations in the trajectory do not avoid T. (b) No stable trap points occur in the geometry associated with nonconvex circular sets. Point T is an unstable trap point. A slight deviation in the trajectory leading to T will avoid the trap point.

Fig. 5
Fig. 5

Reconstructed light intensity in the Fourier plane for a phase grating designed by the method of generalized projections: (a) the desired light intensity pattern, (b) the realized light intensity pattern after 50 iterations.

Fig. 6
Fig. 6

Error history of reconstructed Fourier magnitude for the prescribed pattern of Fig. 5(a).

Fig. 7
Fig. 7

Reconstruction of a uniform-intensity spot pattern for the method of generalized projections: (a) the desired impulse distribution, (b) the realized intensity distribution after 10 iterations.

Fig. 8
Fig. 8

Efficiency history of the realized uniform-intensity spot-array pattern for the method of generalized projections.

Fig. 9
Fig. 9

Normalized variance history of the realized uniform-intensity spot-array pattern for the method of generalized projections.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

I ( u ) = | Ω exp [ - j ϕ ( x ) ] exp ( - j 2 π u · x ) d x | 2 ,
exp [ j ϕ ( x ) ] = 1.
M ( u ) = | Ω exp [ - j ϕ ( x ) ] exp ( - j 2 π u · x ) d x | .
g - h = Min all y C i y - h
f n + 1 = T 1 T 2 T m f n ,
T i 1 + λ i ( P i - 1 )             ( i = 1 , 2 , m ) .
f n + 1 = T 1 T 2 f n             ( f 0 arbitrary ) ,
J ( g ) = P 1 g - g + P 2 g - g .
J ( f n + 1 ) J ( T 2 f n ) J ( f n ) ,
0 λ i Γ ( f n )             ( i = 1 , 2 ) ,
C 1 = { h ( x ) H ( u ) : H ( u ) = M ( u ) }
C 2 = { h ( x ) : h ( x ) = 0 , for all x Ω c ; h ( x ) = 1 , for all x Ω } .
P 1 g M ( u ) exp [ j θ G ( u ) ] .
P 2 g = 0 ( x Ω c ) = exp [ j θ g ( x ) ] ( x Ω ) ,
f k + 1 = P 2 P 1 f k .
f k + 1 = P 1 P 2 f k .
f k + 1 = T 2 T 1 f k ,
T i 1 + λ i , k ( P i - 1 )             ( i = 1 , 2 ) .
0 λ i , k A i 2 ( k ) + A i ( k ) A i 2 ( k ) + A i ( k ) - ½ [ A i ( k ) + B i ( k ) ] 1 ,
A 1 P 2 T 1 f k - T 1 f k P 1 T 1 f k - T 1 f k ,
A 2 P 1 f k - f k P 2 f k - f k ,
B 1 ( P 1 T 1 f k - T 1 f k , P 2 T 1 f k - T 1 f k ) P 1 T 1 f k - T 1 f k 2 ,
B 2 ( P 2 f k - f k , P 1 f k - f k ) P 2 f k - f k 2 .
η = i I D F ( u i ) 2 all i F ( u i ) 2 ,
σ n 2 = 1 N i I D F ( u i ) - F ¯ 2 F ¯ 2 ,
F ¯ = 1 N i I D F ( u i ) .

Metrics