Abstract

A recursive algorithm, which appears to be new, is presented for estimating the amplitude and phase of a wave field from intensity-only measurements on two or more scan planes at different axial positions. The problem is framed as a nonlinear optimization, in which the angular spectrum of the complex field model is adjusted in order to minimize the relative entropy, or Kullback–Leibler divergence, between the measured and reconstructed intensities. The most common approach to this so-called phase retrieval problem is a variation of the well-known Gerchberg–Saxton algorithm devised by Misell (J. Phys. D 6, L6, 1973) , which is efficient and extremely simple to implement. The new algorithm has a computational structure that is very similar to Misell’s approach, despite the fundamental difference in the optimization criteria used for each. Based upon results from noisy simulated data, the new algorithm appears to be more robust than Misell’s approach and to produce better results from low signal-to-noise ratio data. The convergence of the new algorithm is examined.

© 2007 Optical Society of America

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References

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    [CrossRef]
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  29. P. J. Bones, C. R. Parker, B. L. Satherley, and R. W. Watson, "Deconvolution and phase retrieval with use of zero sheets," J. Opt. Soc. Am. A 12, 1842-1857 (1995).
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    [CrossRef]
  38. D. Titterington, "On the iterative image space reconstruction algorithm for ECT," IEEE Trans. Med. Imaging MI-6, 52-56 (1987).
    [CrossRef]
  39. D. N. Politis, "ARMA models, prewhitening and minimum cross entropy," IEEE Trans. Signal Process. 41, 781-787 (1993).
    [CrossRef]
  40. M. A. Tzannes, D. Politis, and N. S. Tzannes, "A general method of minimum cross-entropy spectral estimation," IEEE Trans. Acoust., Speech, Signal Process. ASSP-33, 748-752 (1985).
    [CrossRef]
  41. N. J. Dusaussoy and I. E. Abdou, "The extended MENT algorithm: amaximum entropy type algorithm using prior knowlege for computerized tomography," IEEE Trans. Signal Process. 39, 1164-1180 (1991).
    [CrossRef]
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    [CrossRef]
  44. C. L. Byrne, "Iterative image reconstruction algorithms based on cross-entropy minimization," IEEE Trans. Image Process. 2, 96-103 (1993).
    [CrossRef] [PubMed]
  45. P. Réfrégier and F. Goudail, "Kullback relative entropy and characterization of partially polarized optical waves," J. Opt. Soc. Am. A 23, 671-678 (2006).
    [CrossRef]
  46. R. Malouf, "A comparison of algorithms for maximum entropy parameter estimation," in International Conference on Computational Linguistics, Proceedings of the 6th Conference on Natural Language in Learning, 20, 49-55 (Association for Computational Linguistics, 2002).
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2006 (3)

2005 (2)

L. I. Perlovsky and R. W. Deming, "A mathematical theory for learning and its application to time-varying computed tomography," New Math. Natural Comput. 1, 147-171 (2005).
[CrossRef]

F. Soldovieri, A. Liseno, G. D'Elia, and R. Pierri, "Global convergence of phase retrieval by quadratic approach," IEEE Trans. Antennas Propag. AP-53, 3135-3141 (2005).
[CrossRef]

2004 (1)

2003 (2)

2002 (1)

W. Chalodhorn and D. DeBoer, "Use of microwave lenses in phase retrieval microwave holography of reflector antennas," IEEE Trans. Antennas Propag. AP-50, 1274-1284 (2002).
[CrossRef]

1999 (2)

R. Pierri, G. D'Elia, and F. Soldovieri, "A two probes scanning phaseless near-field far-field transformation technique," IEEE Trans. Antennas Propag. AP-47, 792-802 (1999).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "The role of support information and zero locations in phase retrieval by a quadratic approach," J. Opt. Soc. Am. A 16, 1845-1856 (1999).
[CrossRef]

1998 (1)

1997 (1)

L. I. Perlovsky, C. P. Plum, P. R. Franchi, E. J. Tichovolsky, D. S. Choi, and B. Weijers, "Einsteinian neural network for spectrum estimation," Neural Networks 10, 1541-1546 (1997).
[CrossRef]

1996 (3)

1995 (2)

1994 (1)

1993 (4)

C. L. Byrne, "Iterative image reconstruction algorithms based on cross-entropy minimization," IEEE Trans. Image Process. 2, 96-103 (1993).
[CrossRef] [PubMed]

D. N. Politis, "ARMA models, prewhitening and minimum cross entropy," IEEE Trans. Signal Process. 41, 781-787 (1993).
[CrossRef]

M. H. Maleki and A. J. Devaney, "Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography," J. Opt. Soc. Am. A 10, 1086-1092 (1993).
[CrossRef]

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

1991 (1)

N. J. Dusaussoy and I. E. Abdou, "The extended MENT algorithm: amaximum entropy type algorithm using prior knowlege for computerized tomography," IEEE Trans. Signal Process. 39, 1164-1180 (1991).
[CrossRef]

1990 (1)

1987 (3)

1986 (2)

1985 (1)

M. A. Tzannes, D. Politis, and N. S. Tzannes, "A general method of minimum cross-entropy spectral estimation," IEEE Trans. Acoust., Speech, Signal Process. ASSP-33, 748-752 (1985).
[CrossRef]

1982 (1)

1981 (1)

J. E. Shore, "Minimum cross-entropy spectral analysis," IEEE Trans. Acoust., Speech, Signal Process. ASSP-29, 230-237 (1981).
[CrossRef]

1980 (1)

R. H. Boucher, "Convergence of algorithms for phase retrieval from two intensity distributions," Proc. SPIE 231, 130-141 (1980).

1978 (1)

1976 (1)

1975 (1)

J. P. Burg, "The relationship between maximum entropy spectra and maximum likelihood spectra," Geophysics 37, 375-376 (1975).
[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 and W. O. Saxton, "A practical algorithm for the determination of phase from image diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Appl. Opt. (4)

Geophysics (1)

J. P. Burg, "The relationship between maximum entropy spectra and maximum likelihood spectra," Geophysics 37, 375-376 (1975).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process. (2)

J. E. Shore, "Minimum cross-entropy spectral analysis," IEEE Trans. Acoust., Speech, Signal Process. ASSP-29, 230-237 (1981).
[CrossRef]

M. A. Tzannes, D. Politis, and N. S. Tzannes, "A general method of minimum cross-entropy spectral estimation," IEEE Trans. Acoust., Speech, Signal Process. ASSP-33, 748-752 (1985).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

W. Chalodhorn and D. DeBoer, "Use of microwave lenses in phase retrieval microwave holography of reflector antennas," IEEE Trans. Antennas Propag. AP-50, 1274-1284 (2002).
[CrossRef]

T. Isernia, G. Leone, and R. Pierri, "Radiation pattern evaluation from near-field intensities on planes," IEEE Trans. Antennas Propag. AP-44, 701-710 (1996).
[CrossRef]

R. Pierri, G. D'Elia, and F. Soldovieri, "A two probes scanning phaseless near-field far-field transformation technique," IEEE Trans. Antennas Propag. AP-47, 792-802 (1999).
[CrossRef]

F. Soldovieri, A. Liseno, G. D'Elia, and R. Pierri, "Global convergence of phase retrieval by quadratic approach," IEEE Trans. Antennas Propag. AP-53, 3135-3141 (2005).
[CrossRef]

IEEE Trans. Image Process. (1)

C. L. Byrne, "Iterative image reconstruction algorithms based on cross-entropy minimization," IEEE Trans. Image Process. 2, 96-103 (1993).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging (1)

D. Titterington, "On the iterative image space reconstruction algorithm for ECT," IEEE Trans. Med. Imaging MI-6, 52-56 (1987).
[CrossRef]

IEEE Trans. Signal Process. (2)

D. N. Politis, "ARMA models, prewhitening and minimum cross entropy," IEEE Trans. Signal Process. 41, 781-787 (1993).
[CrossRef]

N. J. Dusaussoy and I. E. Abdou, "The extended MENT algorithm: amaximum entropy type algorithm using prior knowlege for computerized tomography," IEEE Trans. Signal Process. 39, 1164-1180 (1991).
[CrossRef]

Inverse Probl. (1)

T. Isernia, G. Leone, and R. Pierri, "Phase retrieval of radiated fields," Inverse Probl. 11, 183-203 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (13)

H. H. Bauschke, P. L. Combettes, and D. R. Luke, "Hybrid projection-reflection method for phase retrieval," J. Opt. Soc. Am. A 20, 1025-1034 (2003).
[CrossRef]

V. Elser, "Phase retrieval by iterated projections," J. Opt. Soc. Am. A 20, 40-55 (2003).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, and G. Schirinzi, "Image reconstruction from Fourier transform magnitude with applications to synthetic aperture radar imaging," J. Opt. Soc. Am. A 13, 922-934 (1996).
[CrossRef]

G. Liu and P. D. Scott, "Phase retrieval and twin-image elimination for in-line Fresnel holograms," J. Opt. Soc. Am. A 4, 159-165 (1987).
[CrossRef]

C. C. Wackerman and A. E. Yagle, "Phase retrieval and estimation with use of real-plane zeros," J. Opt. Soc. Am. A 11, 2016-2026 (1994).
[CrossRef]

P. J. Bones, C. R. Parker, B. L. Satherley, and R. W. Watson, "Deconvolution and phase retrieval with use of zero sheets," J. Opt. Soc. Am. A 12, 1842-1857 (1995).
[CrossRef]

P. Chen, M. A. Fiddey, C. Liao, and D. A. Pommet, "Blind deconvolution and phase retrieval from zero points," J. Opt. Soc. Am. A 13, 1524-1531 (1996).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "The role of support information and zero locations in phase retrieval by a quadratic approach," J. Opt. Soc. Am. A 16, 1845-1856 (1999).
[CrossRef]

M. H. Maleki and A. J. Devaney, "Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography," J. Opt. Soc. Am. A 10, 1086-1092 (1993).
[CrossRef]

J. R. Fienup and C. C. Wackerman, "Phase-retrieval stagnation problems and solutions," J. Opt. Soc. Am. A 3, 1897-1907 (1986).
[CrossRef]

J. H. Seldin and J. R. Fienup, "Numerical investigation of the uniqueness of phase retrieval," J. Opt. Soc. Am. A 7, 412-427 (1990).
[CrossRef]

H. Takajo, T. Takahashi, R. Ueda, and M. Taninaka, "Study on the convergence property of the hybrid input-output algorithm used for phase retrieval," J. Opt. Soc. Am. A 15, 2849-2861 (1998).
[CrossRef]

P. Réfrégier and F. Goudail, "Kullback relative entropy and characterization of partially polarized optical waves," J. Opt. Soc. Am. A 23, 671-678 (2006).
[CrossRef]

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]

Neural Networks (1)

L. I. Perlovsky, C. P. Plum, P. R. Franchi, E. J. Tichovolsky, D. S. Choi, and B. Weijers, "Einsteinian neural network for spectrum estimation," Neural Networks 10, 1541-1546 (1997).
[CrossRef]

New Math. Natural Comput. (1)

L. I. Perlovsky and R. W. Deming, "A mathematical theory for learning and its application to time-varying computed tomography," New Math. Natural Comput. 1, 147-171 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Optik (Stuttgart) (1)

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

Proc. SPIE (2)

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, "Phase retrieval algorithm for JWST flight and testbed telescope," Proc. SPIE 6265, 1-17 (2006).

R. H. Boucher, "Convergence of algorithms for phase retrieval from two intensity distributions," Proc. SPIE 231, 130-141 (1980).

Other (11)

J. C. Dainty and J. R. Fienup, "Phase retrieval and image reconstruction for astronomy," in Image Recovery: Theory and Application, H.Stark, ed. (Academic, 1987), pp. 231-275.

C. Giacovazzo, Direct Phasing in Crystallography (Oxford U. Press, 1998).

G. Leone, R. Pierri, and F. Soldovieri, "On the performances of two algorithms in phaseless antenna measurements," in Tenth International Conference on Antennas and Propagation (IEEE, 1997), Vol. 1, pp. 136-141.

T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. (Wiley, 2006), Chapter 2.

S. Kullback, Information Theory and Statistics (Wiley, 1959).

L. I. Perlovsky, Neural Networks and Intellect (Oxford U. Press, 2001), Chapters 4 and 6.

R. W. Deming, "Reconstruction of time-varying objects in computerized tomography using a model-based neural network," in Proceedings of the IEEE International Symposium on Intelligent Control (ISIC '98) (IEEE, 1998), pp. 422-427.

R. Malouf, "A comparison of algorithms for maximum entropy parameter estimation," in International Conference on Computational Linguistics, Proceedings of the 6th Conference on Natural Language in Learning, 20, 49-55 (Association for Computational Linguistics, 2002).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chapter 3.

S. B. Howell, Handbook of CCD Astronomy (Cambridge U. Press, 2000).

G. Holst, "Noise in imaging: the good, the bad and the right, in Photonics Spectra, 40, 88-92 (Laurin, 2006), Vol. 12.

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Figures (16)

Fig. 1
Fig. 1

Cut along the line y = 0 from the intensity distribution shown in Fig. 5b.

Fig. 2
Fig. 2

Log-likelihood versus iterations for the MRE phase retrieval method described in this paper.

Fig. 3
Fig. 3

Optical schematic for the phase retrieval simulations.

Fig. 4
Fig. 4

Simulated wave field at the aperture plane z = 0 for the first experiment. The (a) magnitude is constant over the aperture, however a set of transparent lines spanning the aperture introduce local phase shifts of 14 ° , 8 ° , 4 ° , and 2 ° degrees, visible in the (b) phase distribution.

Fig. 5
Fig. 5

Noisy intensity data associated with the aperture field shown Fig. 4. The data are acquired on two scan planes at axial positions (a) z 1 = 7 λ and (b) z 2 = 16 λ . The units of intensity are photoelectrons ( e ) .

Fig. 6
Fig. 6

SNR versus signal intensity for a read noise level of σ R = 16 e . The units of intensity are photoelectrons ( e ) .

Fig. 7
Fig. 7

Magnitude of the field reconstructed at the aperture plane using (a) GS and (b) MRE (the approach introduced in this paper). The data had relatively high SNR ( 13 ) , and a scan plane separation of Δ z = 9 λ . These images can be compared with the true field shown in Fig. 4a. The optimum number of iterations were used for each method, as determined from the curves in Fig. 11, i.e., ten iterations for GS and eight iterations for MRE.

Fig. 8
Fig. 8

Phase (degrees) of the field reconstructed at the aperture plane using (a) GS and (b) MRE (the approach introduced in this paper). The data had relatively high SNR ( 13 ) , and a scan plane separation of Δ z = 9 λ . These images can be compared with the true phase shown in Fig. 4b.

Fig. 9
Fig. 9

Cut along the line y = 0 from the real part of the reconstructed fields shown in Figs. 7a, 7b, 8a, 8b. GS denotes the Gerchberg–Saxton method while MRE denotes the minimum relative entropy method introduced in this paper.

Fig. 10
Fig. 10

Phase (degrees) of the field reconstructed at the aperture plane using (a) GS and (b) MRE. The data had relatively low SNR ( 7 ) , and a scan plane separation of Δ z = 9 λ . These images can be compared with the true phase shown in Fig. 4b as well as the reconstructions from higher SNR data shown in Figs. 8a, 8b. The optimum number of iterations were used for each method, as determined from the curves in Fig. 11, i.e., ten iterations for GS and four iterations for MRE.

Fig. 11
Fig. 11

Error versus iterations for GS and MRE methods, for various SNR levels and separations between scan planes Δ z . The error metric, defined in Eq. (32), evaluates the quality of the field reconstruction on the aperture plane z = 0 . The optimum number of iterations for each scenario is marked by the diamond symbol ( ) . Beyond the optimum number of iterations, the reconstructions become degraded due to overfitting the noise in the data.

Fig. 12
Fig. 12

Phase (degrees) of the field reconstructed at the aperture plane using 100 iterations of (a) GS and (b) MRE, for SNR 13 and Δ z = 9 λ . These images show the effects of overfitting the noise in the data, and are choppier than the optimum reconstructions shown in Figs. 8a, 8b.

Fig. 13
Fig. 13

Phase (degrees) of the field reconstructed at the aperture plane using the MRE method, SNR 13 . Here the separation between the scan planes was decreased, relative to the previous simulations, to a distance Δ z = 1 λ . The optimum number of iterations (16) were used, as determined from the curve in Fig. 11. This image can be compared with the Δ z = 9 λ image in Fig. 8b.

Fig. 14
Fig. 14

In the second experiment, a pattern of square objects within the aperture attenuates the amplitude locally, but leaves the phase unaffected. (a) shows the true magnitude distribution on the aperture plane ( z = 0 ) . (b) shows the noisy, diffracted, intensity data acquired on the first scan plane at axial position z 1 = 7 λ . Note that data was also acquired on a second scan plane at z 2 = 16 λ , but this is not shown. Here, the SNR is quite low, i.e., SNR 5 .

Fig. 15
Fig. 15

Magnitude of the field reconstructed at the aperture plane z = 0 using (a) GS and (b) MRE, from the noisy intensity data in shown Fig. 14b. These images can be compared with the true field shown in Fig. 14a. The optimum number of iterations were used for each method, i.e., ten iterations for GS and four iterations for MRE.

Fig. 16
Fig. 16

Cut along the line y λ = 9.5 along the bottom row of small squares in the magnitude reconstructions shown in Figs. 15a, 15b.

Equations (43)

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

E GS = j = 1 2 d 2 x ( U m ( x , z j ) U ( x , z j ) ) 2 ,
E ( A ) = j = 1 2 d 2 x [ I m ( x , z j ) I A ( x , z j ) ] 2 .
D ( I m I A ) = j = 1 J d 2 x I m ( x , z j ) ln I m ( x , z j ) I A ( x , z j ) ,
d 2 x I m ( x , z j ) = d 2 x I A ( x , z j ) .
L ( A ) = j = 1 J d 2 x I m ( x , z j ) ln I A ( x , z j ) ,
U A ( x , z j ) = 1 2 π d 2 k A ( k ) e i γ ( k ) z j e i k x ,
A ( k ) = 1 2 π d 2 x U A ( x , z j ) e i γ ( k ) z j e i k x .
d 2 x U A ( x , z j ) 2 = d 2 k A ( k ) 2 e 2 I { γ ( k ) } z j ,
U A ( x , z j ) = 1 2 π k k 0 A ( k ) e i γ ( k ) z j e i k x d 2 k .
U A * ( x , z j ) = 1 2 π k k 0 A * ( k ) e i γ ( k ) z j e i k x d 2 k ,
A ( k ) = 1 2 π x 0 x 0 U A ( x , z j ) e i γ ( k ) z j e i k x d 2 x ,
x 0 x 0 U A ( x , z j ) 2 d 2 x = k k 0 A ( k ) 2 d 2 k .
L ( A ) = j = 1 J x 0 x 0 d 2 x I m ( x , z j ) ln I A ( x , z j ) ,
x 0 x 0 d 2 x I A ( x , z j ) = x 0 x 0 d 2 x I m ( x , z j ) = R ,
I A ( x , z j ) = U A ( x , z j ) 2 + μ .
k k 0 A ( k ) 2 d 2 k = R 4 μ x 0 2 .
G ( A ) = L ( A ) + α ( R 4 μ x 0 2 k k 0 A ( k ) 2 d 2 k ) , = L ( A ) + α [ R 4 μ x 0 2 k k 0 ( a 2 ( k ) + b 2 ( k ) ) d 2 k ] ,
δ G δ a ( k ) = δ L δ a ( k ) 2 α a ( k ) = 0 , δ G δ b ( k ) = δ L δ b ( k ) 2 α b ( k ) = 0 .
δ L δ a ( k ) = 2 α a ( k ) ,
δ L δ b ( k ) = 2 α b ( k ) .
δ L δ a ( k ) = j = 1 J x 0 x 0 d 2 x I m ( x , z j ) δ δ a ( k ) [ ln I A ( x , z j ) ] , = j = 1 J x 0 x 0 d 2 x I m ( x , z j ) I A ( x , z j ) δ I A ( x , z j ) δ a ( k ) .
δ L δ a ( k ) = 1 2 π j = 1 J x 0 x 0 d 2 x I m ( x , z j ) I A ( x , z j ) [ U A ( x , z j ) e i γ ( k ) z j e i k x + U A * ( x , z j ) e i γ ( k ) z j e i k x ] .
i ( δ L δ b ( k ) ) = 1 2 π j = 1 J x 0 x 0 d 2 x I m ( x , z j ) I A ( x , z j ) [ U A ( x , z j ) e i γ ( k ) z j e i k x U A * ( x , z j ) e i γ ( k ) z j e i k x ] .
α [ a ( k ) + i b ( k ) ] = α A ( k ) = 1 2 π j = 1 J x 0 x 0 d 2 x I m ( x , z j ) I A ( x , z j ) U A ( x , z j ) e i γ ( k ) z j e i k x .
α k k 0 A ( k ) 2 d 2 k = j = 1 J x 0 x 0 d 2 x I m ( x , z j ) I A ( x , z j ) U A ( x , z j ) [ 1 2 π k k 0 d 2 k A * ( k ) e i γ ( k ) z j e i k x ] .
α ( R 4 μ x 0 2 ) = j = 1 J x 0 x 0 d 2 x I m ( x , z j ) I A ( x , z j ) U A ( x , z j ) 2 .
α ( R 4 μ x 0 2 ) = j = 1 J x 0 x 0 d 2 x I m ( x , z j ) μ j = 1 J x 0 x 0 d 2 x I m ( x , z j ) I A ( x , z j ) .
α = J .
A ( k ) = 1 2 π J j = 1 J x 0 x 0 d 2 x I m ( x , z j ) I A ( x , z j ) U A ( x , z j ) e i γ ( k ) z j e i k x .
U A ( M 1 ) ( x , z j ) = 1 2 π k k 0 d 2 k A ( M 1 ) ( k ) e i γ ( k ) z j e i k x ,
I A ( M 1 ) ( x , z j ) = U A ( M 1 ) ( x , z j ) 2 + μ ,
A ( M ) ( k ) = 1 2 π J j = 1 J x 0 x 0 d 2 x I m ( x , z j ) I A ( M 1 ) ( x , z j ) U A ( M 1 ) ( x , z j ) e i γ ( k ) z j e i k x .
U A ( x , z s ) = P r s { U A ( x , z r ) } , = ( 1 2 π ) 2 k k 0 d 2 k e i k x e i γ ( k ) ( z s z r ) x 0 x 0 d 2 x e i k x U A ( x , z r ) .
U A ( M ) ( x , z 1 ) = P 21 { [ I m ( x , z 2 ) U A ( M 1 ) ( x , z 2 ) ] U A ( M 1 ) ( x , z 2 ) } .
U A ( M ) ( x , z 2 ) = P 12 { [ I m ( x , z 1 ) U A ( M ) ( x , z 1 ) ] U A ( M ) ( x , z 1 ) } .
U A ( M ) ( x , z j ) = 1 J j = 1 J P j j { [ I m ( x , z j ) I A ( M 1 ) ( x , z j ) ] U A ( M 1 ) ( x , z j ) } .
E ( M ) = x 0 x 0 U A ( M ) ( x , 0 ) U true ( x , 0 ) d 2 x x 0 x 0 U true ( x , 0 ) d 2 x ,
U A ( x , z j ) = 1 2 π k k 0 d 2 k [ a ( k ) + i b ( k ) ] e i γ ( k ) z j e i k x ,
U A * ( x , z j ) = 1 2 π k k 0 d 2 k [ a ( k ) i b ( k ) ] e i γ ( k ) z j e i k x .
δ U A δ a ( k ) = 1 2 π e i γ ( k ) z j e i k x , δ U A δ b ( k ) = i 2 π e i γ ( k ) z j e i k x ,
δ U A * δ a ( k ) = 1 2 π e i γ ( k ) z j e i k x , δ U A * δ b ( k ) = i 2 π e i γ ( k ) z j e i k x .
δ I A δ a ( k ) = 1 2 π [ U A ( x , z j ) e i γ ( k ) z j e i k x + U A * ( x , z j ) e i γ ( k ) z j e i k x ] ,
δ I A δ b ( k ) = i 2 π [ U A ( x , z j ) e i γ ( k ) z j e i k x + U A * ( x , z j ) e i γ ( k ) z j e i k x ] .

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