Abstract

We describe several results characterizing the Hubble Space Telescope from measured point spread functions by using phase-retrieval algorithms. The Cramer–Rao lower bounds show that point spread functions taken well out of focus result in smaller errors when aberrations are estimated and that, for those images, photon noise is not a limiting factor. Reconstruction experiments with both simulated and real data show that the calculation of wave-front propagation by the retrieval algorithms must be performed with a multiple-plane propagation rather than a simple fast Fourier transform to ensure the high accuracy ruired. Pupil reconstruction was performed and indicates a misalignment of the optical axis of a camera relay telescope relative to the main telescope. After we accounted for measured spherical aberration in the relay telescope, our estimate of the conic constant of the primary mirror of the HST was −1.0144.

© 1993 Optical Society of America

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References

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  1. A. G. Tescher, ed., Applications of Digital Image Processing XTV, Proc. Soc. Photo-Opt. Instrum. Eng.1567 (1991).
  2. Space Optics for Astrophysics and Earth and Planetary Remote Sensing Vol. 19 of 1991 Technical Digest Series (Optical Society of America, Washington, D.C., 1991).
  3. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  4. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), Chap. 7, pp. 231–275.
  5. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  6. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  7. D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
    [CrossRef]
  8. R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. 66, 961–964 (1976).
    [CrossRef]
  9. J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front phase estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
    [CrossRef]
  10. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 15 and 20.
  11. J. N. Cederquist, C. C. Wackerman, “Phase-retrieval error: a lower bound,” J. Opt. Soc. Am. A 4, 1788–1792 (1987).
    [CrossRef]
  12. J. M. Mendel, Lessons in Digital Estimation Theory (Prentice-Hall, Englewood Cliffs, N.J., 1987), p. 51.
  13. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [CrossRef] [PubMed]
  14. S. Brewer, “WFPC generic prescriptions,” Rep. 92-1 (Jet Propulsion Laboratory, Pasadena, Calif., March1991).
  15. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  17. M. Born, E. Wolf, Progress in Optics (Macmillan, New York, 1964).
  18. L. Adams, Jet Propulsion Laboratory, Pasadena, Calif. 91109, “WF/PC relay optics spider clocking on OTA primary” (personal communication, 17August1990).
  19. C. Roddier, F. Roddier, “Reconstruction of the Hubble Space Telescope wave-front distortion from stellar images taken at various focus positions,” Final Rep., Contract 958893 (Jet Propulsion Laboratory, Pasadena, Calif., May1991).
  20. R. G. Lyon, P. E. Miller, A. Grusczak, “Hubble Space Telescope assembly project report, phase retrieval special studies task,” Final Rep. PR J14-0013, Contract NAS 8-38494 (Goddard Space Flight Center, NASA, Greenbelt, Md., 4June1991).
  21. G. R. Ayers, J. C. Dainty, “An iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
    [CrossRef] [PubMed]
  22. B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
    [CrossRef]
  23. L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
    [CrossRef]
  24. J. R. Fienup, “Phase retrieval for the Hubble Space Telescope using iterative propagation algorithms,” in Applications of Digital Image Processing XIV, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1567, 327–332 (1991).
  25. J. R. Fienup, “Hubble Space Telescope aberrations and alignment determined by phase retrieval algorithms,” in Space Optics for Astrophysics and Earth and Planetary Remote Sensing, Vol. 19 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 19–21.
  26. C. Burrows, Hubble Space Telescope Optical Telescope Assembly Handbook (Space Telescope Science Institute, Baltimore, Md., May1990).

1993 (1)

1989 (2)

1988 (1)

1987 (1)

1982 (1)

1976 (1)

1974 (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

1973 (1)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[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, 237–246 (1972).

Adams, L.

L. Adams, Jet Propulsion Laboratory, Pasadena, Calif. 91109, “WF/PC relay optics spider clocking on OTA primary” (personal communication, 17August1990).

Ayers, G. R.

Bates, R. H. T.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Born, M.

M. Born, E. Wolf, Progress in Optics (Macmillan, New York, 1964).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Brewer, S.

S. Brewer, “WFPC generic prescriptions,” Rep. 92-1 (Jet Propulsion Laboratory, Pasadena, Calif., March1991).

Burrows, C.

C. Burrows, Hubble Space Telescope Optical Telescope Assembly Handbook (Space Telescope Science Institute, Baltimore, Md., May1990).

Cederquist, J. N.

Dainty, J. C.

G. R. Ayers, J. C. Dainty, “An iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
[CrossRef] [PubMed]

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), Chap. 7, pp. 231–275.

Davey, B. L. K.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Fienup, J. R.

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[CrossRef] [PubMed]

J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front phase estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
[CrossRef]

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

J. R. Fienup, “Phase retrieval for the Hubble Space Telescope using iterative propagation algorithms,” in Applications of Digital Image Processing XIV, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1567, 327–332 (1991).

J. R. Fienup, “Hubble Space Telescope aberrations and alignment determined by phase retrieval algorithms,” in Space Optics for Astrophysics and Earth and Planetary Remote Sensing, Vol. 19 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 19–21.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), Chap. 7, pp. 231–275.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gonsalves, R. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Grusczak, A.

R. G. Lyon, P. E. Miller, A. Grusczak, “Hubble Space Telescope assembly project report, phase retrieval special studies task,” Final Rep. PR J14-0013, Contract NAS 8-38494 (Goddard Space Flight Center, NASA, Greenbelt, Md., 4June1991).

Kryskowski, D.

Lane, R. G.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Lyon, R. G.

R. G. Lyon, P. E. Miller, A. Grusczak, “Hubble Space Telescope assembly project report, phase retrieval special studies task,” Final Rep. PR J14-0013, Contract NAS 8-38494 (Goddard Space Flight Center, NASA, Greenbelt, Md., 4June1991).

Mendel, J. M.

J. M. Mendel, Lessons in Digital Estimation Theory (Prentice-Hall, Englewood Cliffs, N.J., 1987), p. 51.

Miller, P. E.

R. G. Lyon, P. E. Miller, A. Grusczak, “Hubble Space Telescope assembly project report, phase retrieval special studies task,” Final Rep. PR J14-0013, Contract NAS 8-38494 (Goddard Space Flight Center, NASA, Greenbelt, Md., 4June1991).

Misell, D. L.

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Robinson, S. R.

Roddier, C.

C. Roddier, F. Roddier, “Reconstruction of the Hubble Space Telescope wave-front distortion from stellar images taken at various focus positions,” Final Rep., Contract 958893 (Jet Propulsion Laboratory, Pasadena, Calif., May1991).

Roddier, F.

C. Roddier, F. Roddier, “Reconstruction of the Hubble Space Telescope wave-front distortion from stellar images taken at various focus positions,” Final Rep., Contract 958893 (Jet Propulsion Laboratory, Pasadena, Calif., May1991).

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, 237–246 (1972).

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 15 and 20.

Wackerman, C. C.

Wolf, E.

M. Born, E. Wolf, Progress in Optics (Macmillan, New York, 1964).

Appl. Opt. (2)

Astron. J. (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Opt. Commun. (1)

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Other (16)

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), Chap. 7, pp. 231–275.

J. R. Fienup, “Phase retrieval for the Hubble Space Telescope using iterative propagation algorithms,” in Applications of Digital Image Processing XIV, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1567, 327–332 (1991).

J. R. Fienup, “Hubble Space Telescope aberrations and alignment determined by phase retrieval algorithms,” in Space Optics for Astrophysics and Earth and Planetary Remote Sensing, Vol. 19 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 19–21.

C. Burrows, Hubble Space Telescope Optical Telescope Assembly Handbook (Space Telescope Science Institute, Baltimore, Md., May1990).

J. M. Mendel, Lessons in Digital Estimation Theory (Prentice-Hall, Englewood Cliffs, N.J., 1987), p. 51.

A. G. Tescher, ed., Applications of Digital Image Processing XTV, Proc. Soc. Photo-Opt. Instrum. Eng.1567 (1991).

Space Optics for Astrophysics and Earth and Planetary Remote Sensing Vol. 19 of 1991 Technical Digest Series (Optical Society of America, Washington, D.C., 1991).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 15 and 20.

S. Brewer, “WFPC generic prescriptions,” Rep. 92-1 (Jet Propulsion Laboratory, Pasadena, Calif., March1991).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born, E. Wolf, Progress in Optics (Macmillan, New York, 1964).

L. Adams, Jet Propulsion Laboratory, Pasadena, Calif. 91109, “WF/PC relay optics spider clocking on OTA primary” (personal communication, 17August1990).

C. Roddier, F. Roddier, “Reconstruction of the Hubble Space Telescope wave-front distortion from stellar images taken at various focus positions,” Final Rep., Contract 958893 (Jet Propulsion Laboratory, Pasadena, Calif., May1991).

R. G. Lyon, P. E. Miller, A. Grusczak, “Hubble Space Telescope assembly project report, phase retrieval special studies task,” Final Rep. PR J14-0013, Contract NAS 8-38494 (Goddard Space Flight Center, NASA, Greenbelt, Md., 4June1991).

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Figures (10)

Fig. 1
Fig. 1

Pupil functions used for the numeric computation of CR lower bounds on aberration estimates: (a) aperture 1, (b) aperture 2.

Fig. 2
Fig. 2

Simplified thin-lens model of the PC mode of the HST. Plane x3 contains the obscurations in the PC. A pair of imaginary thin lenses is inserted just before and just after plane x3 to reduce the size of the FFT required for the digital propagation of a wave front through the system.

Fig. 3
Fig. 3

Transmittance functions for OTA and PC for simulations: (a) OTA, (b) PC, (c) composite for the single-FFT model.

Fig. 4
Fig. 4

(a) Simulated PSF. (b) The difference in the PSF's computed using single-FFT and multiple-plane propagation models of HST.

Fig. 5
Fig. 5

Estimated spherical aberration a11 as a function of assumed spatial scale s for simulated data. The solid curve shows the quadratic phase factor BQ optimized for the given spatial scale; the dashed curve indicates when the true value of BQ is used.

Fig. 6
Fig. 6

Quadratic curve fit through the plate scale versus the CCD position. The plate scale is in arc seconds per pixel, and the CCD position is in millimeters. (The data points are from Ref. 14.)

Fig. 7
Fig. 7

Measured image PC-6F889N_P2 from HARP1A and the images computed from it. Measured PSF (upper left), the PSF deconvolved with the Ayers/Dainty algorithm (upper right), the PSF deconvolved with the Wiener filter by using jitter data (lower left), and the PSF computed from the model by using a polynomial phase estimate (lower right).

Fig. 8
Fig. 8

Optimization of the fitting error as a function of the plate scale. The spatial scale parameter s is proportional to the plate scale.

Fig. 9
Fig. 9

Pupil function reconstructed by one iteration of the iterative transform algorithm.

Fig. 10
Fig. 10

Model of the pupil function inferred from the reconstructed pupil shown in Fig. 9.

Tables (17)

Tables Icon

Table 1 Normalized Lower Bounds on E[(âjaj)2] for Aperture 1

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Table 2 Normalized Lower Bounds on E[(âjaj)2] for Aperture 2

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Table 3 Normalized Lower Bounds on E[(âjaj)2] for Aperture 1 when a4 is Known

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Table 4 Normalized Lower Bounds on E[(âjaj)2] for Aperture 2 when a4 is Known

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Table 5 Parameters of the HST (PC) for Simulations

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Table 6 ABCD Values for Simulationsa

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Table 7 Sample Spacings and Array Widths for Simulations (N = 256, λ = 0.889 μm)

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Table 8 Effect of the Plate Scale on Retrieved Zernike Coefficients with Optimized Quadratic Coefficient BQ

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Table 9 Zernike Phase Coefficients Estimated in a Blind Test

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Table 10 Sample Spacings for HST Cameras and Wavelengths for Nyquist Sampling

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Table 11 Scale Factors and Aperture Sizes (in Pixels) for Various HST Cameras and Wavelengths, for N = 256a

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Table 12 Ray Heights and Slopes from Paraxial Ray Tracinga

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Table 13 ABCD Matrix Valuesa

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Table 14 Example Evaluation of Spatial Scale Factors for λ = 889 nm and N = 256

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Table 15 Zernike Coefficients (Micrometer rms Wave-Front Error) for HARP1A images PC-6 F889N_P2 and PC-6 F889N_Q2

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Table 16 Effect of the Plate Scale on Zernike Coefficients for Image PC-6 F889N_P2 (Micrometer rms Wave-Front Error)a

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Table 17 Modified Zernike Polynomials

Equations (74)

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f ( x ) = m ( x ) exp [ i θ ( x ) ] ,
θ ( x ) = j = 1 J a j Z j ( x ) .
E [ D ( u ) ] = I a ( u ) .
I a ( u ) = α ( u ) [ F a ( u ) F a * ( u ) + I B ( u ) ] + σ 2 ,
Pr [ D ( u ) ] = [ I a ( u ) ] D ( u ) exp [ I a ( u ) ] D ( u ) ! .
Pr [ { D ( u ) } ] = Π u Pr [ D ( u ) ] ,
L ( a ) = ln Pr [ { D ( u ) } u ] = u ln Pr [ D ( u ) ] = u I a ( u ) + u D ( u ) ln I a ( u ) u ln D ( u ) ! .
L ( a ) = u I a ( u ) + u D ( u ) ln I a ( u ) .
a = E { ( â a ) ( â a ) T } .
J ( a ) = E { J o ( a ) } ,
[ J o ( a ) ] jk = 2 L ( a ) a j a k .
E [ ( â j a j ) 2 ] [ J 1 ( a ) ] jj .
L ( a ) a j = u I a ( u ) a j [ D ( u ) I a ( u ) 1 ] .
[ J o ( a ) ] jk = 2 L ( a ) a j a k = u 2 I a ( u ) a j a k [ D ( u ) I a ( u ) 1 ] u I a ( u ) a j I a ( u ) a k D ( u ) [ I a ( u ) ] 2 ,
[ J ( a ) ] jk = E { [ J o ( a ) ] jk } = u I a ( u ) a j I a ( u ) a k 1 I a ( u ) ,
I a ( u ) a j = α ( u ) [ F a * ( u ) a j F a ( u ) + F a ( u ) a j F a * ( u ) ] ,
F a ( u ) a j = i x Z j ( x ) f a ( x ) exp ( i 2 π ux ) = def i u [ Z j f a ] ,
I a ( u ) a j = α ( u ) i ( { u [ Z j f a ] } F a ( u ) + u [ Z j f a ] F a * ( u ) ) = 2 α ( u ) Im { u [ Z j f a ] F a * ( u ) } ,
[ J ( a ) ] jk = 4 u α 2 ( u ) I a ( u ) × Im { u [ Z j f a ] F a * ( u ) } Im { u [ Z k f a ] F a * ( u ) } ,
σ j = { E [ ( â j a j ) 2 ] / N max } 1 / 2 .
U 2 ( x 2 ) = 1 i λ B U 1 ( x 1 ) exp [ i π λ B ( A x 1 2 2 x 1 x 2 + D x 1 2 ) ] d x 1 = 1 i λ B exp ( i π D λ B x 2 2 ) [ U 1 ( x 1 ) exp ( i π A λ B x 1 2 ) ] × exp ( i 2 π λ B x 1 x 2 ) d x 1
U 2 ( x 2 ) = exp ( i α 1 x 2 2 ) 1 N x 1 = 0 N 1 U 1 ( x 1 ) exp ( i β 1 x 1 2 ) × exp ( i 2 π x 1 x 2 / N ) = exp ( i α 1 x 2 2 ) { U 1 ( x 1 ) exp ( i β 1 x 1 2 ) } ,
1 N = Δ x 1 Δ x 2 λ B or Δ x 2 = λ B N Δ x 1 ,
α 1 = π D Δ x 2 2 λ B ,
β 1 = π A Δ x 1 2 λ B ,
α 1 [ ( N 2 ) 2 ( N 2 1 ) 2 ] = α 1 ( N 1 ) π ,
BQ = 1 2 π ( α 1 π Δ x 2 2 λ f py + β 2 ) = ( D 12 B 12 1 f py + A 23 B 23 ) Δ x 2 2 2 λ ,
E = [ min C x 4 [ C ( PSF both ) PSF OTA ] 2 x 4 [ PSF OTA ] 2 ] 1 / 2 = 0.085 ,
E = u W ( u ) [ | G ( u ) | | F ( u ) | ] 2 ,
err = { u W ( u ) [ | G ( u ) | | F ( u ) | ] 2 u W ( u ) | F ( u ) | 2 } 1 / 2 .
ρ s λ / ( 2 D ) ,
ρ s λ / D .
D a = λ / ρ s .
s = N / D a = N ρ s / λ ,
M = [ A B C D ] .
M = M n M 2 , M 1 .
[ y k υ k ] = [ A B C D ] [ y k υ k ]
y k = A y k + B υ k ,
υ k = C y k + D υ k .
M = [ A B C D ] = [ y 1 y 2 υ 1 υ 2 ] [ y 1 y 2 υ 1 υ 2 ] 1
A = y 1 υ 2 y 2 υ 1 y 1 υ 2 y 2 υ 1 ,
B = y 1 y 2 y 2 y 1 y 1 υ 2 y 2 υ 1 ,
C = υ 1 υ 2 υ 2 υ 1 y 1 υ 2 y 2 υ 1 ,
D = y 1 υ 2 y 2 υ 1 y 1 υ 2 y 2 υ 1 .
15.24 μ m / pixel 71,231.5 mm = 0.2156 μ rad / pixel × 1 arcsec 4.848 μ rad = 0.04413 arcsec / pixel .
plate scale ( arcsec ) = 0.044193 0.000016567 x 0.000018200 x 2 + 0.00001690 y + 0.000006469 x y 0.000018315 y 2 ,
BQ = D 12 Δ x 2 2 2 λ B 12 + A 23 Δ x 2 2 2 λ B 23 ,
ϕ ( x , y , z ) = 2 π λ ( x 2 + y 2 + z 2 ) 1 / 2
κ = 1.0023 + 0.043841 a 11 .
λ B 34 N Δ x 4
λ B 34 N Δ x 4
λ B 23 N Δ x 3
B 23 Δ x 4 B 34
λ B 12 N Δ x 2
λ B 12 B 34 N B 23 Δ x 4
λ B 14 N Δ x 4
G ( u ) = P [ U 1 ( x 1 ) ] ,
E = u W ( u ) [ | G ( u ) | | F ( u ) | ] 2
E = u W ( u ) [ | G ( u ) | J | F ( u ) | ] 2 ,
| G ( u ) | J 2 = | G ( u ) | 2 * J ( u ) ,
E p = u W ( u ) [ 1 | F ( u ) | | G ( u ) | J ] | G ( u ) | J 2 p ,
| G ( u ) | J 2 p = u J ( u u ) [ G * ( u ) G ( u ) p + c . c . ] ,
G J ( u ) = G ( u ) u J ( u u ) W ( u ) [ | F ( u ) | | G ( u ) | J 1 ] .
E p = Re [ x 1 U 1 ( x 1 ) p g J * ( x 1 ) ] ,
g J ( x 1 ) = P [ G J ( u ) ]
E a j = 2 Im [ x 1 U 1 ( x 1 ) Z j ( x 1 ) g J * ( x 1 ) ] ,
E θ ( x 1 ) = 2 Im [ U 1 ( x 1 ) g J * ( x 1 ) ]
E q = u G J * ( u ) G ( u ) q + c . c . = x 3 m 3 ( x 3 ) q U 3 ( x 3 ) { P 4 3 [ G J ( u ) ] } * + c . c . ,
E m 3 ( x 3 ) = 2 Re ( U 3 ( x 3 ) { P 4 3 [ G J ( u ) ] } * ) .
m 3 ( x 3 ) = u M 3 ( u ) exp ( i 2 π u x 3 / N ) ,
m 3 ( x 3 x 0 ) x 0 = u ( i 2 π u / N ) M 3 ( u ) × exp [ i 2 π u ( x 3 x 0 ) / N ] ,
E x 0 = x 3 u ( i 2 π u / N ) M 3 ( u ) × exp ( i 2 π u x 3 / N ) U 3 ( x 3 ) { P 4 3 [ G J ( u ) ] } * + c . c . = 2 Im ( x 3 1 [ ( 2 π u / N ) M 3 ( u ) ] × U 3 ( x 3 ) { P 4 3 [ G J ( u ) ] } * ) .
I ( x ) = | exp ( i 2 π d / λ ) + a exp [ i 2 π ( r / λ + c ) ] | 2 = 2 + 2 a cos [ 2 π ( r d ) / λ + 2 π c ] ,
x m = { 2 d λ [ ( m + n 0 ) / 2 c ] } 1 / 2 ,

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