Abstract

A phase-diversity wave-front sensor has been developed and tested at the Lockheed Palo Alto Research Labs (LPARL). The sensor consists of two CCD-array focal planes that record the best-focus image of an adaptive imaging system and an image that is defocused. This information is used to generate an object-independent function that is the input to a LPARL-developed neural network algorithm known as the General Regression Neural Network (GRNN). The GRNN algorithm calculates the wave-front errors that are present in the adaptive optics system. A control algorithm uses the calculated values to correct the errors in the optical system. Simulation studies and closed-loop experimental results are presented.

© 1994 Optical Society of America

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References

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  1. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
  2. R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
    [CrossRef]
  3. J. A. Hogbom, “On the intensity distribution over the focal volume,” in Proceedings of the Tenth Sacramento Peak Summer Workshop, O. von der Lühe, ed. (National Solar Observatory, Sunspot, N.M., 1988), pp. 166–176.
  4. S. R. Restaino, “Wave-front sensing and image deconvolution of solar data,” Appl. Opt. 31, 7442–7449 (1992).
    [CrossRef] [PubMed]
  5. M. G. Löfdahl, G. B. Scharmer, “Phase-diversity restoration of solar images,” in Proceedings of the 13th Sacramento Peak Summer Workshop, R. R. Radick, ed. (National Solar Observatory, Sunspot, N.M.1992), pp. 89–104.
  6. M. G. Löfdahl, G. B. Scharmer, “Wave front sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. (to be published).
  7. D. F. Specht, “A general regression neural network,” IEEE Trans. Neural Networks 2, 568–576 (1991).
    [CrossRef]
  8. D. F. Specht, “Enhancements to probabilistic neural networks,” in Proceedings of the IEEE International Joint Conference on Neural Networks, F. Salam, ed. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1761–1768.
  9. D. S. Acton, R. C. Smithson, “Solar imaging with a segmented adaptive mirror,” Appl. Opt. 31, 3161–3169 (1992).
    [CrossRef] [PubMed]

1992 (2)

1991 (1)

D. F. Specht, “A general regression neural network,” IEEE Trans. Neural Networks 2, 568–576 (1991).
[CrossRef]

1988 (1)

1982 (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

Acton, D. S.

Fienup, J. R.

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

Hogbom, J. A.

J. A. Hogbom, “On the intensity distribution over the focal volume,” in Proceedings of the Tenth Sacramento Peak Summer Workshop, O. von der Lühe, ed. (National Solar Observatory, Sunspot, N.M., 1988), pp. 166–176.

Löfdahl, M. G.

M. G. Löfdahl, G. B. Scharmer, “Phase-diversity restoration of solar images,” in Proceedings of the 13th Sacramento Peak Summer Workshop, R. R. Radick, ed. (National Solar Observatory, Sunspot, N.M.1992), pp. 89–104.

M. G. Löfdahl, G. B. Scharmer, “Wave front sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. (to be published).

Paxman, R. G.

Restaino, S. R.

Scharmer, G. B.

M. G. Löfdahl, G. B. Scharmer, “Wave front sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. (to be published).

M. G. Löfdahl, G. B. Scharmer, “Phase-diversity restoration of solar images,” in Proceedings of the 13th Sacramento Peak Summer Workshop, R. R. Radick, ed. (National Solar Observatory, Sunspot, N.M.1992), pp. 89–104.

Smithson, R. C.

Specht, D. F.

D. F. Specht, “A general regression neural network,” IEEE Trans. Neural Networks 2, 568–576 (1991).
[CrossRef]

D. F. Specht, “Enhancements to probabilistic neural networks,” in Proceedings of the IEEE International Joint Conference on Neural Networks, F. Salam, ed. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1761–1768.

Appl. Opt. (2)

IEEE Trans. Neural Networks (1)

D. F. Specht, “A general regression neural network,” IEEE Trans. Neural Networks 2, 568–576 (1991).
[CrossRef]

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

Opt. Eng. (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

Other (4)

J. A. Hogbom, “On the intensity distribution over the focal volume,” in Proceedings of the Tenth Sacramento Peak Summer Workshop, O. von der Lühe, ed. (National Solar Observatory, Sunspot, N.M., 1988), pp. 166–176.

M. G. Löfdahl, G. B. Scharmer, “Phase-diversity restoration of solar images,” in Proceedings of the 13th Sacramento Peak Summer Workshop, R. R. Radick, ed. (National Solar Observatory, Sunspot, N.M.1992), pp. 89–104.

M. G. Löfdahl, G. B. Scharmer, “Wave front sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. (to be published).

D. F. Specht, “Enhancements to probabilistic neural networks,” in Proceedings of the IEEE International Joint Conference on Neural Networks, F. Salam, ed. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1761–1768.

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Figures (19)

Fig. 1
Fig. 1

Phase-diversity sensor in a staring imager.

Fig. 2
Fig. 2

Phase-diversity sensor in a scanning imager.

Fig. 3
Fig. 3

Comparison between the simulation and the experiment for the sharpness metric M 2 and the power metric M 4.

Fig. 4
Fig. 4

Simulation results for two different objects: (a) Panama City, Florida, and (b) a computer-generated target. M 4 with the center segment dephased 0.5 waves is shown for (c) the Panama City object and (d) the target object. M 4 with 100:1 multiplicative noise added with 8-bit digitization is shown for (e) the Panama City object and (f) the target object. M 4 with 100:1 multiplicative noise added with 12-bit digitization is shown for (g) the Panama City object and (h) the target object.

Fig. 5
Fig. 5

Phase-diversity test facility at LPARL.

Fig. 6
Fig. 6

LPARL 19-segment mirror developed for solar astronomy.

Fig. 7
Fig. 7

Seven segments of the 19-segment mirror were used to simulate a segmented primary mirror. Each segment is controllable in tilt and piston.

Fig. 8
Fig. 8

Phase-diversity wave-front-sensor focal plane consists of a beam splitter and two CCD arrays.

Fig. 9
Fig. 9

Fringe pattern from the white-light interferometer after the segmented mirror has been aligned.

Fig. 10
Fig. 10

(a) Best-focus image and (b) the defocused (or diversity) image are shown for the phase-diversity test facility.

Fig. 11
Fig. 11

Two test objects that were used in the open-loop tests are shown: (a) high-contrast scene, (b) lower-contrast scene.

Fig. 12
Fig. 12

100 elements are selected from the metric M 4. The same points are used for the metric M 2.

Fig. 13
Fig. 13

M 4 metric for the pattern and the house object with piston errors on the center segment. Computer simulations are shown for comparison.

Fig. 14
Fig. 14

Open-loop test results. Plots of the GRNN-estimated position versus the actual position for each segment are shown. Computer-simulation results are shown for (a) segment 1, (b) segment 2, and (c) the center segment (segment 4).

Fig. 15
Fig. 15

Open-loop test results. Plots of the GRNN-estimated position versus the actual position for each segment are shown. Measured data results are shown for (a) segment 1, (b) segment 2, and (c) the center segment (segment 4).

Fig. 16
Fig. 16

Four-segment closed-loop test results. The initial, aberrated, and corrected images are shown for each of the three test objects.

Fig. 17
Fig. 17

Four-segment test. The percent modulation is for the vertical bars on the bar target shown in Fig. 16: —, initial; ***, aberrated; +++, corrected.

Fig. 18
Fig. 18

Six-segment closed-loop test results. The initial, aberrated, and corrected images are shown for the U.S. Air Force bar target and the pattern test objects.

Fig. 19
Fig. 19

Six-segment test. The percent modulation for the vertical bars on the target shown in Fig. 18: —, initial; ***, aberrated; +++, corrected.

Tables (4)

Tables Icon

Table 1 Wave-Front Sensor Categories

Tables Icon

Table 2 RMS Difference between the Estimated and the Actual Segment Positions for the Open-Loop Test

Tables Icon

Table 3 Four-Segment Test Results

Tables Icon

Table 4 Six-Segment Test Results

Equations (21)

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P ( x ) = w ( x ) exp [ 2 π i ϕ ( x ) ] ,
h ( ξ ) = | P ( x ) exp ( - 2 π i ξ x ) d x | 2 .
H o ( f ) = N o P ( x ) exp ( - 2 π i ξ x ) d x × P ( x ) exp ( 2 π i ξ x ) d x exp ( - 2 π i f ξ ) d ξ = N o w ( x ) w ( x ) exp { - 2 π i [ ϕ ( x ) - ϕ ( x ) - ξ ( x - x ) + f ξ ] } d x d x d ξ .
H d ( f ) = N o w ( x ) w ( x ) exp { - 2 π i [ ϕ ( x ) - ϕ ( x ) + ϕ d ( x ) - ϕ d ( x ) - ξ ( x - x ) + f ξ ] } d x d x d ξ ,
S o = H o ( f ) Q ( f ) ,
S o = H d ( f ) Q ( f ) .
ϕ ( x ) = k = 1 N C k Ψ ( x ) ,
E = f S o ( f ) H ^ d ( f ) - S d ( f ) H ^ o ( f ) 2 H ^ o ( f ) 2 + H ^ d ( f ) 2
M 1 ( f ) = S o ( f ) S d ( f ) = H o ( f ) H d ( f ) .
M 1 ( f ) = A o ( f ) A d ( f ) exp { i [ θ o ( f ) - θ d ( f ) ] } = A o ( f ) A d ( f ) { cos [ θ o ( f ) - θ d ( f ) ] + i sin [ θ o ( f ) - θ d ( f ) ] } .
M 2 ( f ) = S o ( f ) S d * ( f ) - S o * ( f ) S d ( f ) S o * ( f ) S o ( f ) + S d * ( f ) S d ( f ) = 2 i A o ( f ) A d ( f ) A o ( f ) 2 + A d ( f ) 2 sin [ θ o ( f ) - θ d ( f ) ] ,
M 3 ( f ) = S o ( f ) S d * ( f ) + S o * ( f ) S d ( f ) S o * ( f ) S o ( f ) + S d * ( f ) S d ( f ) = 2 A o ( f ) A d ( f ) A o ( f ) 2 + A d ( f ) 2 cos [ θ o ( f ) - θ d ( f ) ] .
M 4 ( f ) = S o ( f ) S o * ( f ) - S d ( f ) S d * ( f ) S o ( f ) S o * ( f ) + S d ( f ) S d * ( f ) = A o ( f ) 2 - A d ( f ) 2 A o ( f ) 2 + A d ( f ) 2 .
H r ( f ) = N o w ( - x ) w ( - x ) exp { - 2 π i [ ϕ ( - x ) - ϕ ( - x ) + ξ ( x - x ) + f ξ ] } d x d x d ξ .
H r ( - f ) = H r * ( f ) ,
H r ( f ) = N o w ( x ) w ( x ) exp { 2 π i [ ϕ ( x ) - ϕ ( x ) - ξ ( x - x ) + f ξ ] } d x d x d ξ ,
H o ( f ) - H r * ( f ) .
Y o = i = 1 N Y i exp ( - R i 2 σ 2 ) i = 1 N exp ( - R i 2 σ 2 ) ,
R i 2 = ( X i - X o ) T ( X i - X o ) .
Δ z = 4 λ ( f # ) 2 .
[ - 0.4 , 0.4 ] , [ - 0.25 , .025 ] , [ - 0.15 , 0.15 ] , [ - 0.07 , 0.07 ] ,

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