Abstract

Geometrical–optical arguments have traditionally been used to explain how a lenslet array measures the distribution of light jointly over space and spatial frequency. Here, we rigorously derive the connection between the intensity measured by a lenslet array and wave–optical representations of such light distributions for partially coherent optical beams by using the Wigner distribution function (WDF). It is shown that the action of the lenslet array is to sample a smoothed version of the beam’s WDF (SWDF). We consider the effect of lenslet geometry and coherence properties of the beam on this measurement, and we derive an expression for cross–talk between lenslets that corrupts the measurement. Conditions for a high fidelity measurement of the SWDF and the discrepancies between the measured SWDF and the WDF are investigated for a Schell–model beam.

© 2013 OSA

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References

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2013 (1)

2012 (2)

L. Waller, G. Situ, and J. Fleischer, “Phase–space measurement and coherence synthesis of optical beams,” Nat. Photonics6, 474–479 (2012).
[CrossRef]

L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express20, 8296–8308 (2012).
[CrossRef] [PubMed]

2011 (1)

2009 (2)

J.-H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt.48, H77–H94 (2009).
[CrossRef] [PubMed]

S. Cho, J. Petruccelli, and M. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Optic.56, 1843–1852 (2009).
[CrossRef]

2006 (1)

2003 (2)

2001 (2)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg.17(2001).
[PubMed]

N. Lindlein, J. Pfund, and J. Schwider, “Algorithm for expanding the dynamic range of a shack-hartmann sensor by using a spatial light modulator array,” Opt. Eng.40, 837–840 (2001).
[CrossRef]

2000 (1)

1999 (3)

1996 (2)

A. Wax and J. E. Thomas, “Optical heterodyne imaging and Wigner phase space distributions,” Opt. Lett.21, 1427–1429 (1996).
[CrossRef] [PubMed]

H. N. Chapman, “Phase-retrieval X-ray microscopy by Wigner–distribution deconvolution,” Ultramicroscopy66, 153–172 (1996).
[CrossRef]

1994 (1)

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

1992 (1)

E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell.14, 99–106 (1992).
[CrossRef]

1986 (1)

1980 (2)

W. Tango and R. Twiss, “Michelson stellar interferometry,” Prog. Optics17, 239–277 (1980).
[CrossRef]

H. Bartelt, K.-H. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun.32, 32–38 (1980).
[CrossRef]

1979 (1)

1968 (1)

1964 (1)

L. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz7, 559–563 (1964).

1957 (1)

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev.40, 0749–0759 (1932).
[CrossRef]

1908 (1)

G. Lippmann, “La photographie integrale,” Comptes-Rendus, Academie des Sciences146 (1908).

Adelson, E. H.

E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell.14, 99–106 (1992).
[CrossRef]

Alonso, M.

S. Cho, J. Petruccelli, and M. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Optic.56, 1843–1852 (2009).
[CrossRef]

Alonso, M. A.

Barbastathis, G.

Bartelt, H.

H. Bartelt, K.-H. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun.32, 32–38 (1980).
[CrossRef]

Beck, M.

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

Brady, D. J.

Bredif, M.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford (2005).

Brenner, K.-H.

H. Bartelt, K.-H. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun.32, 32–38 (1980).
[CrossRef]

Chapman, H. N.

H. N. Chapman, “Phase-retrieval X-ray microscopy by Wigner–distribution deconvolution,” Ultramicroscopy66, 153–172 (1996).
[CrossRef]

Chen, Z.

Cho, S.

S. Cho and M. A. Alonso, “Ambiguity function and phase-space tomography for nonparaxial fields,” J. Opt. Soc. Am. A28, 897–902 (2011).
[CrossRef]

S. Cho, J. Petruccelli, and M. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Optic.56, 1843–1852 (2009).
[CrossRef]

Choi, H.

Dolin, L.

L. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz7, 559–563 (1964).

Duval, G.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford (2005).

Fleischer, J.

L. Waller, G. Situ, and J. Fleischer, “Phase–space measurement and coherence synthesis of optical beams,” Nat. Photonics6, 474–479 (2012).
[CrossRef]

Forbes, G. W.

Friberg, A. T.

Gehm, M. E.

Hanrahan, P.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford (2005).

Hong, K.

Horowitz, M.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford (2005).

Itoh, K.

Javidi, B.

Jung, S.

Larkin, K. G.

Lee, B.

Lee, J.

Levoy, M.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford (2005).

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in “IEEE International Conference on Computational Photography (ICCP),” (IEEE, 2009), pp. 1–10.
[CrossRef]

Lindlein, N.

N. Lindlein, J. Pfund, and J. Schwider, “Algorithm for expanding the dynamic range of a shack-hartmann sensor by using a spatial light modulator array,” Opt. Eng.40, 837–840 (2001).
[CrossRef]

Lippmann, G.

G. Lippmann, “La photographie integrale,” Comptes-Rendus, Academie des Sciences146 (1908).

Lohmann, A.

H. Bartelt, K.-H. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun.32, 32–38 (1980).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Marks, D. L.

Marks, D. M.

McAlister, D.

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

McCain, S. T.

Min, S.-W.

Ng, R.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford (2005).

Oh, S. B.

Ohtsuka, Y.

Park, J.-H.

Petruccelli, J.

S. Cho, J. Petruccelli, and M. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Optic.56, 1843–1852 (2009).
[CrossRef]

Pfund, J.

N. Lindlein, J. Pfund, and J. Schwider, “Algorithm for expanding the dynamic range of a shack-hartmann sensor by using a spatial light modulator array,” Opt. Eng.40, 837–840 (2001).
[CrossRef]

Pitsianis, N. P.

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg.17(2001).
[PubMed]

Potuluri, P.

Raymer, M. G.

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

Rehman, S.

Z. Zhang, Z. Chen, S. Rehman, and G. Barbastathis, “Factored form descent: a practical algorithm for coherence retrieval,” Opt. Express21, 5759–5780 (2013).
[CrossRef] [PubMed]

L. Tian, S. Rehman, and G. Barbastathis, “Experimental 4D compressive phase space tomography,” in “Frontiers in Optics,” (Optical Society of America, 2012), p. FM4C.4.

Schwider, J.

N. Lindlein, J. Pfund, and J. Schwider, “Algorithm for expanding the dynamic range of a shack-hartmann sensor by using a spatial light modulator array,” Opt. Eng.40, 837–840 (2001).
[CrossRef]

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg.17(2001).
[PubMed]

Sheppard, C. J. R.

Situ, G.

L. Waller, G. Situ, and J. Fleischer, “Phase–space measurement and coherence synthesis of optical beams,” Nat. Photonics6, 474–479 (2012).
[CrossRef]

Stack, R. A.

Stern, A.

Sullivan, M. E.

Tango, W.

W. Tango and R. Twiss, “Michelson stellar interferometry,” Prog. Optics17, 239–277 (1980).
[CrossRef]

Thomas, J. E.

Thompson, B. J.

Tian, L.

L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express20, 8296–8308 (2012).
[CrossRef] [PubMed]

L. Tian, S. Rehman, and G. Barbastathis, “Experimental 4D compressive phase space tomography,” in “Frontiers in Optics,” (Optical Society of America, 2012), p. FM4C.4.

Twiss, R.

W. Tango and R. Twiss, “Michelson stellar interferometry,” Prog. Optics17, 239–277 (1980).
[CrossRef]

Waller, L.

L. Waller, G. Situ, and J. Fleischer, “Phase–space measurement and coherence synthesis of optical beams,” Nat. Photonics6, 474–479 (2012).
[CrossRef]

Walther, A.

Wang, J. Y. A.

E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell.14, 99–106 (1992).
[CrossRef]

Wax, A.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev.40, 0749–0759 (1932).
[CrossRef]

Wolf, E.

B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am.47, 895 (1957).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wolf, K. B.

Zhang, Z.

Z. Zhang, Z. Chen, S. Rehman, and G. Barbastathis, “Factored form descent: a practical algorithm for coherence retrieval,” Opt. Express21, 5759–5780 (2013).
[CrossRef] [PubMed]

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in “IEEE International Conference on Computational Photography (ICCP),” (IEEE, 2009), pp. 1–10.
[CrossRef]

Appl. Opt. (4)

Comptes-Rendus, Academie des Sciences (1)

G. Lippmann, “La photographie integrale,” Comptes-Rendus, Academie des Sciences146 (1908).

IEEE Trans. Pattern Anal. Mach. Intell. (1)

E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell.14, 99–106 (1992).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz (1)

L. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz7, 559–563 (1964).

J. Mod. Optic. (1)

S. Cho, J. Petruccelli, and M. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Optic.56, 1843–1852 (2009).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

J. Refract. Surg. (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg.17(2001).
[PubMed]

Nat. Photonics (1)

L. Waller, G. Situ, and J. Fleischer, “Phase–space measurement and coherence synthesis of optical beams,” Nat. Photonics6, 474–479 (2012).
[CrossRef]

Opt. Commun. (1)

H. Bartelt, K.-H. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun.32, 32–38 (1980).
[CrossRef]

Opt. Eng. (1)

N. Lindlein, J. Pfund, and J. Schwider, “Algorithm for expanding the dynamic range of a shack-hartmann sensor by using a spatial light modulator array,” Opt. Eng.40, 837–840 (2001).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev.40, 0749–0759 (1932).
[CrossRef]

Phys. Rev. Lett. (1)

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

Prog. Optics (1)

W. Tango and R. Twiss, “Michelson stellar interferometry,” Prog. Optics17, 239–277 (1980).
[CrossRef]

Ultramicroscopy (1)

H. N. Chapman, “Phase-retrieval X-ray microscopy by Wigner–distribution deconvolution,” Ultramicroscopy66, 153–172 (1996).
[CrossRef]

Other (4)

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in “IEEE International Conference on Computational Photography (ICCP),” (IEEE, 2009), pp. 1–10.
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

L. Tian, S. Rehman, and G. Barbastathis, “Experimental 4D compressive phase space tomography,” in “Frontiers in Optics,” (Optical Society of America, 2012), p. FM4C.4.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford (2005).

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Figures (6)

Fig. 1
Fig. 1

Illustration of the measurement under a single lenslet of the SWDF in one spatial dimension (r0 = x0 and u = ux). (a) A lenslet centered at x0 maps points (x0, ux) in the SWDF domain to positions x0 + λfux at the lenslet’s Fourier plane. (b) A convolution of the aperture WDF with the incident WDF forms the SWDF.

Fig. 2
Fig. 2

Lenslet array geometry.

Fig. 3
Fig. 3

Sampling of the SWDF using an array of three lenslets. (a) One–to–one mapping from the SWDF to the detector coordinate according to u = (xolw)/(λf) as the angular spread of the SWDF is narrower than the numerical aperture of a lenslet. (b) Multiple points in the SWDF domain contribute to detector pixels in the cross–talk region as the angular spread of the incident field is wider than the numerical aperture of a lenslet, which produces the 0th order cross–talk.

Fig. 4
Fig. 4

Left: highly incoherent; middle: highly coherent; and right: partially coherent case. (a) Total output intensity is composed of (b) SWDF term and (c) total contribution from cross–talk terms. The total cross–talk is composed of (d) 0th order cross–talk and (e) total of higher order cross–talk. All the intensities are normalized to the maximum value in the total output. The horizontal axis is the spatial coordinate normalized by the width of a lenslet.

Fig. 5
Fig. 5

Comparison of WDF (solid red line), SWDF (dashed blue lines) and total output intensity (dotted green lines) for (a) highly incoherent (σc = 0.01w), (b) highly coherent (σc = 20w), and (c) partially coherent (σc = 0.1w) incident light.

Fig. 6
Fig. 6

Rerror in solid blue curve, cross–talk intensity fraction Rcross–talk in dashed green curve, and convolution error Rconv in red dotted curve as functions of the normalized coherence length of incident light σc/w.

Equations (43)

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J ( r 1 , r 2 ; z ) = U ( r 1 ; z ) U * ( r 2 ; z ) ,
B ( r , p ; z ) = B ( r z p , p ; 0 )
I ( r ; z ) = B ( r , p ; z ) d 2 p = B ( r z p , p ; 0 ) d 2 p .
1 λ 2 B ( r , λ u ) = 𝒲 ( r , u ) = J ( r + r / 2 , r r / 2 ) e i 2 π u r d 2 r ,
I ( r ) = 1 λ f S [ 𝒲 i , 𝒲 p ] ( r 0 , u ) ,
S [ 𝒲 1 , 𝒲 2 ] ( r , u ) = 𝒲 1 ( r , u ) 𝒲 2 ( r r , u u ) d 2 r d 2 u .
T ( x ) = l = N N rect ( x l w w ) exp [ i π λ f ( x l w ) 2 ] ,
J 1 ( x ¯ + x 2 , x ¯ x 2 ) = J i ( x ¯ + x 2 , x ¯ x 2 ) T ( x ¯ + x 2 ) T * ( x ¯ x 2 ) ,
𝒲 p ( x , u ) = sin [ 2 π u ( w 2 | x | ) ] π u rect ( x w )
I 0 ( x o ) = 1 λ f l = N N S [ 𝒲 i , 𝒲 p ] ( l w , x o l w λ f ) .
I SWDF ( x o ) = 1 λ f l = N N S [ 𝒲 i , 𝒲 p ] ( l w , x o l w λ f ) rect ( x o l w w ) .
I c ( 0 ) ( x o ) = 1 λ f l = N N S [ 𝒲 i , 𝒲 p ] ( l w , x o l w λ f ) [ 1 rect ( x o l w w ) ] .
I c ( n ) ( x o ) = 2 λ f l = N + n 2 N n 2 𝒲 i ( x ¯ , u ) 𝒲 p ( x ¯ l w , u x o l w λ f ) cos [ 2 π ( x x o λ f + u ) n w ] d x ¯ d u .
I ( x o ) = I SWDF ( x o ) + I c ( 0 ) ( x o ) + n = 1 2 N I c ( n ) ( x o ) .
J i ( x 1 , x 2 ) = exp [ ( x 2 x 1 ) 2 2 σ c 2 ] .
𝒲 i ( x ¯ , u ) = 1 2 π σ u exp [ u 2 2 σ u 2 ] ,
S [ 𝒲 i , 𝒲 p ] ( x ¯ , u ) = w 2 2 π σ u exp [ ( u u ) 2 2 σ u 2 ] [ sin ( π w u ) π w u ] 2 d u .
I SWDF ( x o ) = w 2 2 π σ u λ f l = N N rect ( x o l w w ) exp { [ ( x o l w ) / λ f u ] 2 2 σ u 2 } [ sin ( π w u ) π w u ] d u ,
I c ( 0 ) ( x o ) = w 2 2 π σ u λ f l = N N [ 1 rect ( x o l w w ) ] exp { [ ( x o l w ) / λ f u ] 2 2 σ u 2 } [ sin ( π w u ) π w u ] 2 d u .
I c ( n ) ( x o ) = 2 w 2 2 π σ u λ f l = N + n 2 N n 2 exp { [ ( x o l w ) / λ f u ] 2 2 σ u 2 } cos ( 2 π n w u ) sin [ π w ( n u 0 + 2 u ) / 2 ] π w ( n u 0 + 2 u ) / 2 sin [ π w ( n u 0 2 u ) / 2 ] π w ( n u 0 2 u ) / 2 d u ,
R error = | total output intensity incident WDF | 2 | total output intensity | 2 .
R cross talk = | total intensity in all terms of cross talk | 2 | total output intensity | 2 .
R conv = | SWDF incident WDF | 2 | total output intensity | 2 .
J ( r + r 2 , r r 2 ) = J i ( r + r 2 , r r 2 ) T ( r + r 2 ) T * ( r r 2 ) ,
𝒲 o ( r , u ) = 𝒲 i ( r , u ) 𝒲 T ( r , u u ) d 2 u ,
𝒲 d ( r , u ) = 𝒲 o ( r λ f u , u ) = 𝒲 i ( r λ f u , u ) 𝒲 T ( r λ f u , u u ) d 2 u .
𝒲 T ( r , u ) = 𝒲 p ( r r 0 , r r 0 λ f + u ) ,
𝒲 d ( r , u ) = 𝒲 i ( r λ f u , u ) 𝒲 p ( r λ f u r 0 , r r 0 λ f u ) d 2 u .
I ( r ) = 𝒲 i ( r λ f u , u ) 𝒲 p ( r λ f u r 0 , r r 0 λ f u ) d 2 u d 2 u ,
I ( x o ) = 1 λ f J 1 ( x 1 , x 2 ) exp { i π λ f [ ( x o x 1 ) 2 ( x o x 2 ) 2 ] } d x 1 d x 2 .
I ( x o ) = 1 λ f l 1 = N N l 2 = N N J i ( x 1 , x 2 ) rect ( x 1 l 1 w w ) rect ( x 2 l 2 w w ) exp { i 2 π λ f [ ( l 1 2 l 2 2 ) w 2 2 + ( x 1 x 2 ) x o l 1 w x 1 + l 2 w x 2 ] } d x 1 d x 2 .
x ¯ = x 1 + x 2 2 , x = x 1 x 2 ,
m = l 1 + l 2 , n = l 1 l 2 .
n = 2 q , where q = N , N + 1 , , N 1 , N ;
l 1 = l + q , l 2 = l q ,
m = 2 l , where l = N + | q | , N + | q | + 1 , , N | q | 1 , N | q | .
n = 2 q , where q = N + 1 2 , N + 3 2 , , N 3 2 , N 1 2 ,
l 1 = l + q , l 2 = l q ,
m = 2 l , where l = N + | q | , N + | q | + 1 , , N | q | 1 , N | q | .
I ( x o ) = 1 λ f n = 2 N even 2 N l = N + | n | 2 N | n | 2 rect ( x ¯ l w + ( x n w ) / 2 w ) rect ( x ¯ l w + ( x n w ) / 2 w ) J i ( x ¯ + x 2 , x ¯ x 2 ) exp [ i 2 π λ f ( x o l w ) x + i 2 π λ f ( x ¯ l w ) n w ] d x ¯ d x + 1 λ f n = 2 N + 1 odd 2 N 1 l = N + | n | 2 N | n | 2 rect ( x ¯ l w + ( x n w ) / 2 w ) rect ( x ¯ l w ( x n w ) / 2 w ) J i ( x ¯ + x 2 , x ¯ x 2 ) exp [ i 2 π λ f ( x o l w ) x + i 2 π λ f ( x ¯ l w ) n w ] d x ¯ d x .
| x ¯ l w | < w / 4 ,
( n 1 ) w + 2 | x ¯ l w | < x < ( n + 1 ) w 2 | x ¯ l w | .
I ( x o ) = 1 λ f { n = 2 N + 1 odd 2 N 1 l = N + | n | 2 N | n | 2 𝒲 i ( x ¯ , u ) 𝒲 p ( x ¯ l w , u x o l w λ f ) exp [ i 2 π ( x ¯ x o λ f + u ) n w ] d x ¯ d u + n = 2 N even 2 N l = N + | n | 2 N | n | 2 𝒲 i ( x ¯ , u ) 𝒲 p ( x ¯ , l w , u x o l w λ f ) exp [ i 2 π ( x ¯ x o λ f + u ) n w ] d x ¯ d u . }

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