Abstract

Partially coherent light provides attractive benefits in imaging, beam shaping, free-space communications, random medium monitoring, among other applications. However, the experimental characterization of the spatial coherence is a difficult problem involving second-order statistics represented by four-dimensional functions that cannot be directly measured and analyzed. In addition, real-world applications usually require quantitative characterization of the local spatial coherence of a beam in the absence of a priori information, together with fast acquisition and processing of the experimental data. Here we propose and experimentally demonstrate a technique that solves this problem. It comprises an optical setup developed for automatized video-rate measurement and a method –phase-space tomographic coherenscopy– allowing parallel data acquisition, processing, and analysis. This technique significantly simplifies the spatial coherence analysis and opens up new perspectives for the development of tools exploiting the degrees of freedom hidden into light coherence.

© 2013 OSA

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2012 (2)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photon.6, 488–496 (2012).
[CrossRef]

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photon.6, 474–479 (2012).
[CrossRef]

2011 (5)

2010 (3)

2009 (3)

2008 (1)

2006 (2)

2005 (1)

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E71, 056607 (2005).
[CrossRef]

2004 (1)

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

2003 (2)

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun.225, 19–30 (2003).
[CrossRef]

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A20, 1094–1102 (2003).
[CrossRef]

2002 (1)

2000 (1)

Z. Zalevsky, D. Medlovic, and H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A2, 83–87 (2000).
[CrossRef]

1999 (1)

1998 (2)

1996 (1)

1995 (2)

T. Alieva and F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun.114, 161–169 (1995).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A12, 1942–1946 (1995).
[CrossRef]

1994 (1)

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

1991 (1)

1989 (1)

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: Experimental results,” Appl. Phys. B49, 409–414 (1989).
[CrossRef]

1986 (1)

J. Radon, “On the determination of functions from their integral values along certain manifolds,” IEEE Trans. Med. Imag.5, 170–176 (1986).
[CrossRef]

1982 (1)

1981 (1)

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta28, 1215–1224 (1981).
[CrossRef]

1979 (1)

M. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta26, 1333–1344 (1979).
[CrossRef]

1921 (1)

A. A. Michelson and F. G. Pease, “Measurement of the diameter of Alpha-Orionis by the interferometer,” Astrophys. J.53, 249–259 (1921).
[CrossRef]

Agarwal, G. S.

Agullo-Lopez, F.

T. Alieva and F. Agullo-Lopez, “Reconstruction of the optical correlation function in a quadratic refractive index medium,” Opt. Commun.114, 161–169 (1995).
[CrossRef]

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photon.6, 488–496 (2012).
[CrossRef]

Alieva, T.

Alonso, M.

Arce, G.

X. Ma and G. Arce, Computational Lithography, Wiley Series in Pure and Applied Optics (Wiley, 2011).

Arce, G. R.

Baldis, H. A.

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

Bandulet, H. C.

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

Bastiaans, M.

M. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta26, 1333–1344 (1979).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta28, 1215–1224 (1981).
[CrossRef]

J. A. Rodrigo, T. Alieva, and M. J. Bastiaans, Optical and Digital Image Processing (Wiley-VCH Verlag, 2011), chap. 12.

Baykal, Y.

Beck, M.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

Bonnaud, G.

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 2006).

Brenner, K. H.

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun.225, 19–30 (2003).
[CrossRef]

Cai, Y.

Calvo, M. L.

Cámara, A.

Cheben, P.

Chmelík, R.

Davidson, F. M.

Depierreux, S.

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

Dogariu, A.

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photon.6, 488–496 (2012).
[CrossRef]

Dong, B.-Z.

Dorsch, R. G.

Dubois, F.

Erkmen, B. I.

Eyyuboglu, H. T.

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photon.6, 488–496 (2012).
[CrossRef]

Ferreira, C.

Fleischer, J. W.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photon.6, 474–479 (2012).
[CrossRef]

Friberg, A. T.

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: Experimental results,” Appl. Phys. B49, 409–414 (1989).
[CrossRef]

Gbur, G.

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics (Elsevier, 2010), 55, 285–341.
[CrossRef]

González, A. I.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, 2000).

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, Colorado, USA, 2005).

Gori, F.

Gu, B.-Y.

Guattari, G.

Gureyev, T. E.

Huang, H.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photon.6, 488–496 (2012).
[CrossRef]

Hulin, S.

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

Joannes, L.

Kolman, P.

Korotkova, O.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Labaune, C.

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

Legros, J.-C.

Lewis, K.

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

Liu, X.

X. Liu and K. H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun.225, 19–30 (2003).
[CrossRef]

Lohmann, A. W.

Ma, X.

Martínez-Matos, Ó.

McAlister, D. F.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

Medlovic, D.

Z. Zalevsky, D. Medlovic, and H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A2, 83–87 (2000).
[CrossRef]

Mejía, Y.

Mendlovic, D.

Michard, A.

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

Michel, P.

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

Michelson, A. A.

A. A. Michelson and F. G. Pease, “Measurement of the diameter of Alpha-Orionis by the interferometer,” Astrophys. J.53, 249–259 (1921).
[CrossRef]

Nugent, K. A.

Ozaktas, H. M.

Z. Zalevsky, D. Medlovic, and H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A2, 83–87 (2000).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Pease, F. G.

A. A. Michelson and F. G. Pease, “Measurement of the diameter of Alpha-Orionis by the interferometer,” Astrophys. J.53, 249–259 (1921).
[CrossRef]

Qu, J.

Radon, J.

J. Radon, “On the determination of functions from their integral values along certain manifolds,” IEEE Trans. Med. Imag.5, 170–176 (1986).
[CrossRef]

Raymer, M. G.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994).
[CrossRef] [PubMed]

Ren, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photon.6, 488–496 (2012).
[CrossRef]

Riazuelo, G.

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

Ricklin, J. C.

Roberts, A.

Rodrigo, J. A.

Santarsiero, M.

Shapiro, J. H.

Shirai, T.

Simon, R.

Situ, G.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photon.6, 474–479 (2012).
[CrossRef]

Tervonen, E.

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: Experimental results,” Appl. Phys. B49, 409–414 (1989).
[CrossRef]

Tikhonchuk, V. T.

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

Tran, C. Q.

Tur, M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photon.6, 488–496 (2012).
[CrossRef]

Turunen, J.

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: Experimental results,” Appl. Phys. B49, 409–414 (1989).
[CrossRef]

Visser, T. D.

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics (Elsevier, 2010), 55, 285–341.
[CrossRef]

Waller, L.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photon.6, 474–479 (2012).
[CrossRef]

Wang, J.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photon.6, 488–496 (2012).
[CrossRef]

Weber, S.

P. Michel, C. Labaune, H. C. Bandulet, K. Lewis, S. Depierreux, S. Hulin, G. Bonnaud, V. T. Tikhonchuk, S. Weber, G. Riazuelo, H. A. Baldis, and A. Michard, “Strong reduction of the degree of spatial coherence of a laser beam propagating through a preformed plasma,” Phys. Rev. Lett.92, 175001 (2004).
[CrossRef] [PubMed]

Willner, A. E.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photon.6, 488–496 (2012).
[CrossRef]

Wolf, E.

Yan, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photon.6, 488–496 (2012).
[CrossRef]

Yang, G.-Z.

Yang, J.-Y.

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Supplementary Material (1)

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Figures (6)

Fig. 1
Fig. 1

Tomographic data acquisition and processing schemes. (a–b), Examples illustrating different projection choices available in conventional tomography of 3D objects. The vertical projection set drastically simplifies the reconstruction of the horizontal 2D sections of the 3D object. (c), Scheme of the proposed tomographic coherenscopy technique: the projection subset is measured for a fixed β0 and then fed into the reconstruction algorithm providing parallel processing. The Wigner distribution Wx0(y, ky) is reconstructed for each projection subset slice defined by both β0 and x0, for instance see the blue slice. From the Wx0(y, ky), the corresponding amplitude and phase profiles of the MI are retrieved. This calculation is performed in parallel for the rest of the considered slices, for example the orange and purple ones.

Fig. 2
Fig. 2

Sketch of the experimental setups. (a), Setup for generation of test beams. While the degree of coherence of the test beam is controlled by the position d of the RGG diffuser, the spatial shaping of its amplitude and phase is achieved by a phase hologram addressed into the SLM. The diffuser is placed between the focusing (FL) and the collimating (CL) lenses. (b), Coherenscope setup. The test beam is projected by the relay lenses (RL) into the coherenscope setup which comprises two SLMs and a CCD camera placed at a fixed distance z. SLM 1 and SLM 2 address digital lenses to measure the required WD projections at video-rate, see Methods. As an example, the insets show the case of the WD projection acquisition corresponding to α = 5π/4 and β0 = 0.

Fig. 3
Fig. 3

Experiment 1: Analysis of the coherent beam. (a–b), Theoretical and experimentally obtained intensity and phase distributions of the input test beam LG0,3 + LG4,1. The experimental phase is reconstructed via an iterative phase retrieval (IPR) algorithm. (c), Amplitude and phase of the reconstructed MI obtained via IPR and phase-space tomography (PST). (d), Amplitude and phase profiles of the MI, Γβ0 (r0, s), corresponding to β0 = 0 and β0 = π/4, both sharing the reference point r0 = [0.6, 0]t mm as indicated in (a).

Fig. 4
Fig. 4

Experiment 2: Analysis of beams with Gaussian-envelope amplitude of DoC. (a), Intensity distributions of the test beams with different degrees of coherence (first row). The WDs of the beams Wx0(y, ky) for x0 = 0.6 mm exhibit hidden differences associated with their coherence state (second row). (b), Amplitude and phase profiles of the MI and the DoC for the RGG positions d = 0, 14, 28, and 52 mm. These profiles correspond to Γβ0 (r0, s) and γβ0 (r0, s) for β0 = 0 and r0 = [0.6, 0]t mm. (c), Comparison between the profiles |Γβ0 (r0, s)| and |γβ0 (r0, s)| for d = 14 mm corresponding to β0 = 0 and β0 = π/4, sharing the same reference point r0 = [0.6, 0]t mm. Solid curves: theoretically expected amplitudes of the DoC.

Fig. 5
Fig. 5

Experiment 3: Analysis of beams with non-homogeneous amplitude of DoC. (a), Intensity distributions of the two modes ψ1 (r) and ψ2 (r) (first column) used for the generation of the partially coherent beams A and B (second column). (b), Comparison between the experimentally obtained (points) and numerically simulated (solid lines) profiles |γ0 (r1, s)| with r1 = [x1, 0]t = [0.12, 0]t mm for the beams A and B. The profiles correspond to the white dashed lines drawn in (a) at the second column. (c), The profiles |γ0 (r1, s)| and |γ0 (r2, s)| for the beam A, with the previously defined r1 (second panel, red points) and r2 = [x2, 0]t = [0.35, 0]t mm (second panel, green points), demonstrate the non-homogeneity of the amplitude of DoC. The profile associated with the reference point r2 corresponds to the white dashed line drawn in (a). The first and third panels present the intensity of the modes ψ1 (r) (solid line) and ψ2 (r) (dashed line), which form the beam A, along the line associated with x1 and x2, correspondingly. They are used to help the interpretation of the DoC profiles.

Fig. 6
Fig. 6

Experimental results. (a–d), Intensity distributions of the the beam scattered from the RGG diffuser placed at different positions d. These signals were imaged just at the output plane of the diffuser. Their intensity profiles along the x and y directions are shown in the second and third row (red color, scatter plot), correspondingly. These profiles are fitted to a theoretical Gaussian curve (blue color, solid curve) to estimate the beam with w for each case.

Equations (15)

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W ( r , k ) = 1 λ 2 d r Γ ( r r / 2 , r + r / 2 ) exp ( i k t r ) ,
Γ 0 ( r 0 , s ) = λ 2 π d k y W x 0 ( y 0 + s / 2 , k y ) exp ( i s k y ) .
LG p , l ( r ; w ) = A L p | l | ( 2 π w 2 r 2 ) exp ( π w 2 r 2 ) exp ( i l ϕ ) ,
W f ( r , p ) = 1 σ 2 d r Γ ( r r / 2 , r + r / 2 ) exp ( i 2 π p t r σ 2 ) ,
W f T ( r , p ) = W f ( D t r B t p , C t r + A t p ) ,
1 σ 2 d p W f T ( r , p ) = | f T ( r ) | 2 = P T ( r ) .
A FRFT = [ cos α x 0 0 cos α y ] and B FRFT = [ sin α x 0 0 sin α y ] .
A = [ cos β sin β cos α sin β cos α cos β ] and B = [ cos β sin β sin α sin β sin α cos β ] .
L j ( x , y ) = exp [ i π λ g j ( x cos β y sin β ) 2 ] exp [ i π λ f j ( x sin β + y cos β ) 2 ] ,
f 1 = 2 z / ( 2 cot ( α / 2 ) ) ,
f 2 = z / ( 2 2 sin α ) ,
Γ d ( r 1 , r 2 ) d r I ( r ) h ( r 1 , r ) h * ( r 2 , r ) .
h ( r , r ) = 1 i λ f exp ( i π d λ f 2 r 2 i 2 π λ f r r ) .
Γ d ( r 1 , r 2 ) = I 0 exp [ π 1 w c 2 ( r 1 r 2 ) 2 + i π 1 ρ 2 ( r 1 2 r 2 2 ) ] .
Γ ( r 1 , r 2 ) = f ( r 1 ) Γ d ( r 1 , r 2 ) f * ( r 2 ) = Γ c ( r 1 , r 2 ) Γ d ( r 1 , r 2 ) ,

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