Abstract

During the last decade, optical memory effects have been explored extensively for various applications. In this letter we propose phase screen models to facilitate the analysis and the simulation of wave propagation through optical media that exhibits memory effects. We show that the classical optical memory effect, which implies tilt wave correlations of the input and the scattered fields, can be readily modeled by a single random phase screen. For the recently discovered generalized optical memory effect, which implies the existence of shift wave correlations in addition to the tilt correlation, we propose an appropriate generalized random phase screen model.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. H. G. Booker, J. Ratcliffe, and D. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. Lond. A 242(856), 579–607 (1950).
    [Crossref]
  2. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007).
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  4. S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
    [Crossref] [PubMed]
  5. S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89(1), 015005 (2017).
    [Crossref]
  6. J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2017 (6)

2016 (3)

2015 (4)

V. Durán, F. Soldevila, E. Irles, P. Clemente, E. Tajahuerce, P. Andrés, and J. Lancis, “Compressive imaging in scattering media,” Opt. Express 23(11), 14424–14433 (2015).
[Crossref] [PubMed]

B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11(8), 684–689 (2015).
[Crossref]

J. H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U.S.A. 112(30), 9236–9241 (2015).
[Crossref] [PubMed]

H. Y. Liu, E. Jonas, L. Tian, J. Zhong, B. Recht, and L. Waller, “3D imaging in volumetric scattering media using phase-space measurements,” Opt. Express 23(11), 14461–14471 (2015).
[Crossref] [PubMed]

2012 (2)

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics 6(8), 549–553 (2012).
[Crossref]

2011 (1)

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photonics 3(4), 272–365 (2011).
[Crossref]

2010 (1)

2007 (1)

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Disp. Technol. 3(3), 315–320 (2007).
[Crossref]

1988 (1)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[Crossref] [PubMed]

1979 (1)

1950 (1)

H. G. Booker, J. Ratcliffe, and D. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. Lond. A 242(856), 579–607 (1950).
[Crossref]

Alonso, M. A.

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photonics 3(4), 272–365 (2011).
[Crossref]

Andrés, P.

Bastiaans, M.

Berto, P.

Bertolotti, J.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Blum, C.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Booker, H. G.

H. G. Booker, J. Ratcliffe, and D. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. Lond. A 242(856), 579–607 (1950).
[Crossref]

Bose-Pillai, S. R.

Clemente, P.

Cui, M.

J. H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U.S.A. 112(30), 9236–9241 (2015).
[Crossref] [PubMed]

Durán, V.

Feng, Q.

Q. Feng and B. Li, “Convolution and correlation theorems for the two-dimensional linear canonical transform and its applications,” IET Signal Process. 10(2), 125–132 (2016).
[Crossref]

Feng, S.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[Crossref] [PubMed]

Gigan, S.

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89(1), 015005 (2017).
[Crossref]

Guillon, M.

Haskel, M.

Horisaki, R.

Horstmeyer, R.

G. Osnabrugge, R. Horstmeyer, I. N. Papadopoulos, B. Judkewitz, and I. M. Vellekoop, “Generalized optical memory effect,” Optica 4(8), 886–892 (2017).
[Crossref]

B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11(8), 684–689 (2015).
[Crossref]

Hyde, M. W.

Irles, E.

Javidi, B.

Jonas, E.

Jouhanneau, J. S.

I. N. Papadopoulos, J. S. Jouhanneau, J. F. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11(2), 116–123 (2017).
[Crossref]

Judkewitz, B.

I. N. Papadopoulos, J. S. Jouhanneau, J. F. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11(2), 116–123 (2017).
[Crossref]

G. Osnabrugge, R. Horstmeyer, I. N. Papadopoulos, B. Judkewitz, and I. M. Vellekoop, “Generalized optical memory effect,” Optica 4(8), 886–892 (2017).
[Crossref]

B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11(8), 684–689 (2015).
[Crossref]

Kane, C.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[Crossref] [PubMed]

Katz, O.

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics 6(8), 549–553 (2012).
[Crossref]

Komatsu, S.

Lagendijk, A.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Lancis, J.

Lee, P. A.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[Crossref] [PubMed]

Li, B.

Q. Feng and B. Li, “Convolution and correlation theorems for the two-dimensional linear canonical transform and its applications,” IET Signal Process. 10(2), 125–132 (2016).
[Crossref]

Liu, H. Y.

Markman, A.

Mosk, A. P.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Osnabrugge, G.

Papadopoulos, I. N.

G. Osnabrugge, R. Horstmeyer, I. N. Papadopoulos, B. Judkewitz, and I. M. Vellekoop, “Generalized optical memory effect,” Optica 4(8), 886–892 (2017).
[Crossref]

I. N. Papadopoulos, J. S. Jouhanneau, J. F. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11(2), 116–123 (2017).
[Crossref]

B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11(8), 684–689 (2015).
[Crossref]

Park, J. H.

J. H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U.S.A. 112(30), 9236–9241 (2015).
[Crossref] [PubMed]

Poulet, J. F.

I. N. Papadopoulos, J. S. Jouhanneau, J. F. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11(2), 116–123 (2017).
[Crossref]

Ratcliffe, J.

H. G. Booker, J. Ratcliffe, and D. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. Lond. A 242(856), 579–607 (1950).
[Crossref]

Rawat, S.

Recht, B.

Rigneault, H.

Rivenson, Y.

Rotter, S.

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89(1), 015005 (2017).
[Crossref]

Shinn, D.

H. G. Booker, J. Ratcliffe, and D. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. Lond. A 242(856), 579–607 (1950).
[Crossref]

Silberberg, Y.

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics 6(8), 549–553 (2012).
[Crossref]

Small, E.

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics 6(8), 549–553 (2012).
[Crossref]

Soldevila, F.

Stern, A.

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[Crossref] [PubMed]

Sun, W.

J. H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U.S.A. 112(30), 9236–9241 (2015).
[Crossref] [PubMed]

Tajahuerce, E.

Takagi, R.

Tanida, J.

Tian, L.

van Putten, E. G.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Vellekoop, I. M.

G. Osnabrugge, R. Horstmeyer, I. N. Papadopoulos, B. Judkewitz, and I. M. Vellekoop, “Generalized optical memory effect,” Optica 4(8), 886–892 (2017).
[Crossref]

B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11(8), 684–689 (2015).
[Crossref]

Voelz, D. G.

Vos, W. L.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Waller, L.

Xiao, X.

Yang, C.

B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11(8), 684–689 (2015).
[Crossref]

Zhong, J.

Adv. Opt. Photonics (1)

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photonics 3(4), 272–365 (2011).
[Crossref]

IET Signal Process. (1)

Q. Feng and B. Li, “Convolution and correlation theorems for the two-dimensional linear canonical transform and its applications,” IET Signal Process. 10(2), 125–132 (2016).
[Crossref]

J. Disp. Technol. (1)

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Disp. Technol. 3(3), 315–320 (2007).
[Crossref]

J. Opt. Soc. Am. (1)

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

Nat. Photonics (2)

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics 6(8), 549–553 (2012).
[Crossref]

I. N. Papadopoulos, J. S. Jouhanneau, J. F. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11(2), 116–123 (2017).
[Crossref]

Nat. Phys. (1)

B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11(8), 684–689 (2015).
[Crossref]

Nature (1)

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (2)

Optica (1)

Philos. Trans. R. Soc. Lond. A (1)

H. G. Booker, J. Ratcliffe, and D. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. Lond. A 242(856), 579–607 (1950).
[Crossref]

Phys. Rev. Lett. (1)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

J. H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U.S.A. 112(30), 9236–9241 (2015).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89(1), 015005 (2017).
[Crossref]

Other (6)

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007).

E. Jakeman and K. D. Ridley, Modeling Fluctuations in Scattered Waves (CRC, 2006).

T. Markus, H. Bryan, and O. Jorge, Phase-Space Optics Fundamentals and Applications (McGraw-Hill, 2010).

M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution—Theory and Applications in Signal Processing (Elsevier, 1997), pp. 375–426.

J. W. Goodman, Statistical Optics (John Wiley & Sons, 2015).

J. J. Healy, M. A. Kutay, H. M. Ozaktas, and J. T. Sheridan, Linear Canonical Transforms: Theory and Applications (Springer, 2015).

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Figures (4)

Fig. 1
Fig. 1 (a) The incident and the output wavefronts from a thin diffusive medium layer, (b) system block diagram of the RPS model.
Fig. 2
Fig. 2 The tilt and shift effects, (a) Illustration of the classical optical (tilt) memory effect where a tilt of the input wavefront yields a similar tilt in the output plane - a phenomenon common to disordered media at small angles [4]; (b) shift memory effect, where a shift of the input wavefront yields a shift in the output plane [14] These two effects and their combination are described by the “generalized memory effect model” [15].
Fig. 3
Fig. 3 The proposed generalized phase screen model.
Fig. 4
Fig. 4 A first order optical realization of the generalized RPS model in Fig. 3.

Equations (38)

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

U o ( r o )=m( r o ) U i ( r i )= e jϕ( r o ) U i ( r i )
m( r o )( e j k t r i U i ( r i ) )=( e jϕ( r o ) U i ( r i ) ) e j k t r i = U o ( r o ) e j k t r i
W o ( r o , k o )= K( r o , k o , r i , k i ) W i ( r i , k i )d r i d k i
W( r,k )= U( r+ 1 2 r' ) U * ( r- 1 2 r' )exp[ jkr' ]dr'
W ¯ o ( r o , k o )= K ¯ ( r o , k o , r i , k i ) W i ( r i , k i )d r i d k i
K ¯ ( r o , k o , r i , k i )=δ( r o r i ) Γ m ( | r o , | ) e j r o , ( k o k i ) d r o ,
K ¯ ( r o , k o , r i , k i )=δ( r o r i ) e 1 2 σ ϕ' 2 | k o k i | 2
W ¯ o ( r o , k o )= δ( r o r i ) e 1 2 σ ϕ' 2 | k o k i | 2 W i ( r i , k i )d r i d k i .
W ¯ o ( r o , k o - k t )= δ( r o r i ) e 1 2 σ ϕ' 2 | k o k i | 2 W i ( r i , k o - k t )d r i d k i .
K ¯ ( r ˜ , k ˜ )= e 6 l tr L ( | r ˜ | 2 L 2 k ˜ r ˜ k 0 L + | k ˜ | 2 3 k 0 2 ) ,
W ¯ o ( r o , k o )= e 6 l tr L ( | r o r i L k i / k 0 | 2 L 2 ( k o k i )( r o r i L k i / k 0 ) k 0 L + | k o k i | 2 3 k 0 2 ) W i ( r i , k i )d r i d k i
U o ( r o )=L( M ) U i ( r i )= j 2 | B | 1/2 exp[ j 2 ( r o t D B 1 r o 2 r i t B 1 r o + r i t B 1 A r i ) ] U i ( r i )d r i
M 1 =[ I 1 k 0 Z 0 I ], M 2 =[ 0 f k 0 I - k 0 f I 0 ], M 3 =[ 1 f Z - f k 0 I k 0 f I 0 ],
K ¯ ( r o , k o , r i , k i )= e 1 2 σ 1 2 | k o k i | 2 e 1 2 f 2 σ 2 2 | k 0 ( r o r i L k 0 k i )+z( k o - k i ) | 2 = e k 0 2 2 σ 2 2 f 2 | r o r i L k 0 k i | 2 + k 0 z σ 2 2 f 2 [ k o k i ][ r o r i L k 0 k i ] e ( z 2 2 σ 2 2 f 2 + 1 2 σ 1 2 ) | k o k i | 2
σ 1 2 = L k 0 2 l tr , σ 2 2 f 2 = L 3 k 0 2 12 l tr ,z= L 2 ,
U o ( r o )= h rr ( r o , r i ) U i ( r i )d r i
h rr ( r o , r i )= e jϕ( r o ) δ( r o r i )
K( r o , k o , r i , k i )= h rr ( r o + 1 2 r o , r i 1 2 r i , ) h rr * ( r o 1 2 r o , r i + 1 2 r i , ) e j k o r o , +j k i r i , d r o , d r i ,
K( r o , k o , r i , k i )= e jϕ( r o + 1 2 r o , ) e +jϕ( r o 1 2 r o , ) δ( r o + 1 2 r o , r i 1 2 r i , ) δ( r o 1 2 r o , r i + 1 2 r i , ) e j k o r o , +j k i r i , d r o , d r i , = = e jϕ( r o + 1 2 r o , ) e +jϕ( r o 1 2 r o , ) e j k o r o , d r o , δ( r o + 1 2 r o , r i 1 2 r i , ) δ( r o 1 2 r o , r i + 1 2 r i , ) e +j k i r i , d r i , = = e jϕ( r o + 1 2 r o , ) e jϕ( r o 1 2 r o , ) e j k o r o , δ( 2 r o 2 r i ) e j k i ( 2 r o + r o , 2 r i ) d r o , = =δ( r o r i ) e j2 k i ( r o r i ) e jϕ( r o + 1 2 r o , ) e +jϕ( r o 1 2 r o , ) e j k o r o , e j k i r o , d r o , = =δ( r o r i ) e jϕ( r o + 1 2 r o , ) e +jϕ( r o 1 2 r o , ) e j r o , ( k o k i ) d r o ,
K ¯ ( r o , k o , r i , k i )E[ K( r o , k o , r i , k i ) ]=E[ δ( r o r i ) e jϕ( r o + 1 2 r o , ) e +jϕ( r o 1 2 r o , ) e j r o , ( k o k i ) d r o , ]= δ( r o r i ) E[ e jϕ( r o + 1 2 r o , ) e +jϕ( r o 1 2 r o , ) ] e j r o , ( k o k i ) d r o , =δ( r o r i ) Γ m ( | r o , | ) e j r o , ( k o k i ) d r o ,
Γ m ( | r o , | )= e 1 2 D ϕ ( r o , )
Γ m ( | r o , | )= e σ ϕ' 2 2 | r o , | 2
K ¯ ( r o , k o , r i , k i )=δ( r o r i ) e σ ϕ' 2 2 | r o , | 2 e j r o , ( k o k i ) d r o , =δ( r o r i ) e | ( k o k i ) | 2 2 σ ϕ' 2 .
W ¯ o ( r o , k o )= K ¯ ( r o , k o , r i , k i ) W i ( r i , k i )d r i d k i = δ( r o r i ) e | k o k i | 2 2 σ ϕ' 2 W i ( r i , k i )d r i d k i .
[ r o k o ]=[ aI bI cI dI ][ r i k i ]=M[ r i k i ]
U o ( r o )=L( M ) U i ( r i )= j 2b exp[ j 2b ( r o 2 d2 r o r i + r i 2 a ) ] U i ( r i )d r i
K( r o , k o , r i , k i )=δ( r i A r o B k o )δ( k i C r o D k o ),
[ AI BI CI DI ]= M 1 = [ aI bI cI dI ] 1
K 1 ( r',k', r i , k i )=δ( r i A 1 r' B 1 k' )δ( k i C 1 r' D 1 k' )
K ¯ 2 ( r'',k'',r',k' )=δ( r''r' ) e 1 2 σ 1 2 | k''k' | 2
K 3 ( r''',k''',r'',k'' )=δ( r'' A 2 r''' B 2 k''' )δ( k'' C 2 r''' D 2 k''' )
K ¯ 4 ( r'''',k'''',r''',k''' )=δ( r''''r''' ) e 1 2 σ 2 2 | k''''k''' | 2
K 5 ( r o , k o ,r'''',k'''' )=δ( r'''' A 3 r o B 3 k o )δ( k'''' C 3 r o D 3 k o )
K αβ ( r o , k 0 , r i , k i )= K α ( r',k', r i , k i ) K β ( r o , k 0 ,r',k' ) dr'dk'
K ¯ 12 ( r'',k'', r i , k i )= = δ( r i A 1 r' B 1 k' )δ( k i C 1 r' D 1 k' ) 2π σ 1 δ( r''r' ) e σ 1 2 2 | k''k' | 2 dr'dk' = e 1 2 σ 1 2 | k''k' | 2 δ( r i A 1 r'' B 1 k' )δ( k i C 1 r'' D 1 k' )dk' = e 1 2 σ 1 2 | k'' 1 D 1 k i + C 1 D 1 r'' | 2 δ( r i r''( A 1 C 1 D 1 ) B 1 D 1 k i ).
K ¯ 13 ( r''',k''', r i , k i )= e 1 2 σ 1 2 | k'' 1 D 1 k i + C 1 D 1 r'' | 2 δ( r i r''( A 1 C 1 D 1 ) B 1 D 1 k i )× δ( r'' A 2 r''' B 2 k''' )δ( k'' C 2 r''' D 2 k''' )dr''dk'' = = e 1 2 σ 1 2 | ( C 2 r'''+ D 2 k''' ) 1 D 1 k i + C 1 D 1 r'' | 2 δ( r i r''( A 1 C 1 D 1 ) B 1 D 1 k i )δ( r'' A 2 r''' B 2 k''' ) dr'' = e 1 2 σ 1 2 | r'''( C 2 + C 1 D 1 A 2 )+k'''( C 1 D 1 B 2 + D 2 ) 1 D 1 k i | 2 δ( r i B 1 D 1 k i ( A 1 C 1 D 1 )( A 2 r'''+ B 2 k''' ) ).
K ¯ 14 ( r'''',k'''', r i , k i ) e 1 2 σ 1 2 | r'''( C 2 + C 1 D 1 A 2 )+k'''( C 1 D 1 B 2 + D 2 ) 1 D 1 k i | 2 δ( r i B 1 D 1 k i ( A 1 C 1 D 1 )( A 2 r'''+ B 2 k''' ) )× 2π σ 2 δ( r''''r''' ) e σ 2 2 2 | k''''k''' | 2 dr'''dk''' = e 1 2 σ 1 2 | r''''( C 2 + C 1 D 1 A 2 )+k'''( C 1 D 1 B 2 + D 2 ) 1 D 1 k i | 2 e 1 2 σ 2 2 | k''''k''' | 2 × δ( r i B 1 D 1 k i ( A 1 C 1 D 1 )( A 2 r''''+ B 2 k''' ) )dk''' e 1 2 σ 1 2 | r''''( C 2 + C 1 D 1 A 2 )+k'''( C 1 D 1 B 2 + D 2 ) 1 D 1 k i | 2 e 1 2 σ 2 2 | k''''k''' | 2 × δ( 1 ( A 1 C 1 / D 1 ) B 2 r i B 1 ( A 1 C 1 / D 1 ) B 2 D 1 k i A 2 B 2 r''''k''' )dk''' = e 1 2 σ 1 2 | 1 B 2 r''''+ ( C 1 B 2 + D 1 D 2 ) ( A 1 C 1 / D 1 ) B 2 D 1 r i ( B 1 ( C 1 B 2 + D 1 D 2 ) ( A 1 C 1 / D 1 ) B 2 D 1 +1 ) k i D 1 | 2 e 1 2 σ 2 2 | k''''( A 2 B 2 r''''+ 1 ( A 1 C 1 / D 1 ) B 2 r i B 1 ( A 1 C 1 / D 1 ) B 2 D 1 k i ) | 2 .
K ¯ 15 ( r 0 , k 0 , r i , k i ) e 1 2 σ 1 2 | 1 B 2 r''''+ ( C 1 B 2 + D 1 D 2 ) ( A 1 C 1 / D 1 ) B 2 D 1 r i ( B 1 ( C 1 B 2 + D 1 D 2 ) ( A 1 C 1 / D 1 ) B 2 D 1 +1 ) k i D 1 | 2 e 1 2 σ 2 2 | k''''+ A 2 B 2 r'''' 1 ( A 1 C 1 / D 1 ) B 2 r i + B 1 ( A 1 C 1 / D 1 ) B 2 D 1 k i | 2 × δ( r'''' A 3 r o B 3 k o )δ( k'''' C 3 r o D 3 k o )dr''''dk'''' = e 1 2 σ 1 2 | 1 B 2 r''''+ ( C 1 B 2 + D 1 D 2 ) ( A 1 D 1 C 1 ) B 2 r i ( B 1 ( C 1 B 2 + D 1 D 2 ) ( A 1 D 1 C 1 ) B 2 +1 ) k i D 1 | 2 e 1 2 σ 2 2 | ( C 3 x o + D 3 ω o )+ A 2 B 2 r'''' D 1 ( A 1 D 1 C 1 ) B 2 r i + B 1 ( A 1 D 1 C 1 ) B 2 k i | 2 × δ( r'''' A 3 r o B 3 k o )dr'''' = e 1 2 σ 1 2 | A 3 B 2 r o B 3 B 2 k o + ( C 1 B 2 + D 1 D 2 ) ( A 1 D 1 C 1 ) B 2 r i ( B 1 ( C 1 B 2 + D 1 D 2 ) ( A 1 D 1 C 1 ) B 2 +1 ) k i D 1 | 2 e 1 2 σ 2 2 | ( C 3 + A 2 A 3 B 2 ) r o +( D 3 + A 2 B 3 B 2 ) k o D 1 ( A 1 D 1 C 1 ) B 2 r i + B 1 ( A 1 D 1 C 1 ) B 2 k i | 2

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