Abstract

We study the spatial coherence of a partially coherent beam before and after being transmitted through a spiral phase plate that changes the overall orbital angular momentum of the field. The two-point coherence function is measured and directly visualized on a CCD through interference in a Mach–Zehnder interferometer equipped with an image rotator. We show, in particular, how the coherence singularities associated with Airy rings are strongly affected by the spiral phase plate.

© 2010 Optical Society of America

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  36. B. Smith, B. Killett, M. Raymer, I. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]

2010 (1)

2009 (3)

2008 (3)

2007 (2)

G. A. Swartzlander, Jr. and R. I. Hernandez-Aranda, “The optical Rankine vortex and the anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[Crossref] [PubMed]

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, “Experimental measurement of the four-dimensional coherence function for an undulator x ray source,” Phys. Rev. Lett. 98, 224801 (2007).
[Crossref] [PubMed]

2006 (3)

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: Local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[Crossref] [PubMed]

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
[Crossref]

M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 31, 861–863 (2006).
[Crossref] [PubMed]

2005 (2)

2004 (3)

I. Maleev, D. Palacios, A. Marathay, and G. Swartzlander, Jr., “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21, 1895–1900 (2004).
[Crossref]

I. Maleev, “Partial coherence and optical vortices,” Ph.D. dissertation (Worcester Polytechnic Institute, 2004), http://www.wpi.edu/Pubs/ETD/Available/etd-0713104-021808/unrestricted/Maleev_PhD.pdf.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref] [PubMed]

2003 (5)

2002 (1)

2001 (2)

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150–156 (2001).
[Crossref]

M. S. Soskin and M. V. Vasnetsov, “Singular optics” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, 2001), pp. 219–276.
[Crossref]

2000 (1)

C. C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence of light,” J. Mod. Opt. 47, 1237–1246 (2000).

1999 (2)

1998 (3)

1997 (1)

1995 (2)

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

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

1982 (1)

1977 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[Crossref]

1964 (1)

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[Crossref]

Agarwal, G. S.

Banaszek, K.

Basistiy, I. V.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

Beijersbergen, M. W.

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[Crossref]

Bogatyryova, G.

Borghi, R.

M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 31, 861–863 (2006).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

Burvall, A.

Carter, W.

Cheng, C. C.

C. C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence of light,” J. Mod. Opt. 47, 1237–1246 (2000).

Dainty, C.

de Jonge, M. D.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, “Experimental measurement of the four-dimensional coherence function for an undulator x ray source,” Phys. Rev. Lett. 98, 224801 (2007).
[Crossref] [PubMed]

Di Lorenzo Pires, H.

Dorrer, C.

Duan, Z.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: Local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[Crossref] [PubMed]

Duncan, D.

Fel’de, C.

Fischer, D.

Flewett, S.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, “Experimental measurement of the four-dimensional coherence function for an undulator x ray source,” Phys. Rev. Lett. 98, 224801 (2007).
[Crossref] [PubMed]

Gbur, G.

Golay, M. J. E.

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[Crossref]

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[Crossref]

Hanson, S. G.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: Local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[Crossref] [PubMed]

Heier, H.

C. C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence of light,” J. Mod. Opt. 47, 1237–1246 (2000).

Heintzmann, R.

Hernandez-Aranda, R. I.

G. A. Swartzlander, Jr. and R. I. Hernandez-Aranda, “The optical Rankine vortex and the anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[Crossref] [PubMed]

Karman, G. P.

Killett, B.

Kivshar, Y.

Konforti, N.

Lohmann, A.

Maleev, I.

Maleev, I. D.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref] [PubMed]

Mandel, L.

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

Marathay, A.

Marathay, A. S.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref] [PubMed]

Mendlovic, D.

Miyamoto, Y.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: Local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[Crossref] [PubMed]

Motzek, K.

Mukamel, E.

Nugent, K. A.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, “Experimental measurement of the four-dimensional coherence function for an undulator x ray source,” Phys. Rev. Lett. 98, 224801 (2007).
[Crossref] [PubMed]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[Crossref]

Palacios, D.

Palacios, D. M.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref] [PubMed]

Paterson, D.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, “Experimental measurement of the four-dimensional coherence function for an undulator x ray source,” Phys. Rev. Lett. 98, 224801 (2007).
[Crossref] [PubMed]

Peele, A. G.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, “Experimental measurement of the four-dimensional coherence function for an undulator x ray source,” Phys. Rev. Lett. 98, 224801 (2007).
[Crossref] [PubMed]

Polyanskii, P.

Ponomarenko, S.

Ponomarenko, S. A.

Prahl, S.

Rabbani, M.

Raymer, M.

Raymer, M. G.

C. C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence of light,” J. Mod. Opt. 47, 1237–1246 (2000).

Roberts, A.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, “Experimental measurement of the four-dimensional coherence function for an undulator x ray source,” Phys. Rev. Lett. 98, 224801 (2007).
[Crossref] [PubMed]

Rozas, D.

Sacks, Z.

Saleh, B.

Santarsiero, M.

M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 31, 861–863 (2006).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[Crossref]

Savitzky, A.

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[Crossref]

Schouten, H. F.

T. van Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79, 033805 (2009).
[Crossref]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[Crossref] [PubMed]

Shabtay, G.

Shih, M.

Sindbert, S.

Smith, A.

Smith, B.

Soskin, M.

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, 2001), pp. 219–276.
[Crossref]

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

Swartzlander, G.

Swartzlander, G. A.

G. Gbur and G. A. Swartzlander, Jr., “Complete transverse representation of a correlation singularity of a partially coherent field,” J. Opt. Soc. Am. B 25, 1422–1429 (2008).
[Crossref]

G. A. Swartzlander, Jr. and R. I. Hernandez-Aranda, “The optical Rankine vortex and the anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[Crossref] [PubMed]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref] [PubMed]

Takeda, M.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: Local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[Crossref] [PubMed]

Tran, C. Q.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, “Experimental measurement of the four-dimensional coherence function for an undulator x ray source,” Phys. Rev. Lett. 98, 224801 (2007).
[Crossref] [PubMed]

van Dijk, T.

T. van Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79, 033805 (2009).
[Crossref]

van Duijl, A.

van Exter, M. P.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, 2001), pp. 219–276.
[Crossref]

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

Vicalvi, S.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[Crossref]

Visser, T. D.

T. van Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79, 033805 (2009).
[Crossref]

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
[Crossref]

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[Crossref]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[Crossref] [PubMed]

Walmsley, I.

Wang, W.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: Local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[Crossref] [PubMed]

Wicker, K.

Williams, G. J.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, “Experimental measurement of the four-dimensional coherence function for an undulator x ray source,” Phys. Rev. Lett. 98, 224801 (2007).
[Crossref] [PubMed]

Woerdman, J. P.

Wolf, E.

Woudenberg, J.

Anal. Chem. (1)

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[Crossref]

Appl. Opt. (1)

J. Mod. Opt. (2)

C. C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence of light,” J. Mod. Opt. 47, 1237–1246 (2000).

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (5)

Opt. Commun. (3)

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
[Crossref]

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[Crossref]

Opt. Express (1)

Opt. Lett. (9)

D. Mendlovic, G. Shabtay, A. Lohmann, and N. Konforti, “Display of spatial coherence,” Opt. Lett. 23, 1084–1086 (1998).
[Crossref]

B. Smith, B. Killett, M. Raymer, I. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
[Crossref]

E. Mukamel, K. Banaszek, I. Walmsley, and C. Dorrer, “Direct measurement of the spatial Wigner function with area-integrated detection,” Opt. Lett. 28, 1317–1319 (2003).
[Crossref] [PubMed]

M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 31, 861–863 (2006).
[Crossref] [PubMed]

G. Bogatyryova, C. Fel’de, P. Polyanskii, S. Ponomarenko, M. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
[Crossref] [PubMed]

H. Di Lorenzo Pires, J. Woudenberg, and M. P. van Exter, “Measurement of the orbital angular momentum spectrum of partially coherent beams,” Opt. Lett. 35, 889–891 (2010).
[Crossref] [PubMed]

G. P. Karman, M. W. Beijersbergen, A. van Duijl, and J. P. Woerdman, “Creation and annihilation of phase singularities in a focal field,” Opt. Lett. 22, 1503–1505 (1997).
[Crossref]

S. A. Ponomarenko and E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27, 1211–1213 (2002).
[Crossref]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[Crossref] [PubMed]

Phys. Rev. A (1)

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Figures (9)

Fig. 1
Fig. 1

Geometry considered for the calculations. A circular aperture of diameter d 1 is illuminated by fully incoherent light. A second aperture of diameter d 2 is placed at a distance L from d 1 , at z = 0 . A SPP can be placed just after aperture 2. The field correlations between the transverse positions ρ 1 and ρ 2 at the plane z are studied.

Fig. 2
Fig. 2

Cross sections of the theoretical (normalized) intensity distributions in the plane z = 45   mm for different values of the coherence length L c at the SPP and open second aperture. The curves are vertically displaced by 0.25 from each other.

Fig. 3
Fig. 3

Cross sections of the theoretical coherence X ( ρ ) of the beam in the plane z = 45   mm after the (open) second aperture. The dashed (red) curve is the prediction without phase plate; the continuous (blue) curve corresponds to a SPP placed at aperture 2. The calculations are made for different values of the coherence length: (a) L c = 0.68   mm , (b) L c = 0.40   mm , (c) L c = 0.25   mm , (d) L c = 0.20   mm , (e) L c = 0.16   mm , and (f) L c = 0.13   mm .

Fig. 4
Fig. 4

Experimental setup used to generate a partially coherent beam and to measure its mutual coherence function (see text for details).

Fig. 5
Fig. 5

Visualization of coherence singularities in X ( ρ ) as measured from interference fringes. The SPP is absent and the aperture 2 is imaged at the ICCD. We set L c = 0.15   mm and d 2 = 0.59   mm . The arctangent of X is shown in order to enhance contrast.

Fig. 6
Fig. 6

Cross sections of the interference pattern for different values of the angle of rotation θ. The coherence length of the source is L c = 0.15   mm , the diameter of the second aperture is d 2 = 0.59   mm , and the imaged plane is z = 0   mm . The SPP is not in the setup.

Fig. 7
Fig. 7

Rotationally averaged cross sections of the intensity profiles in the plane z = 45   mm after the SPP. The (red) dashed curve shows the intensity profile in the absence of the SPP. All curves are normalized and are displaced by 0.25 from each other.

Fig. 8
Fig. 8

Measured coherence X ( ρ ) for (a) L c = 0.40   mm , (b) L c = 0.20   mm , and (c) L c = 0.13   mm . Left column: rotationally averaged cross sections. Continuous (blue) curves are measured with the SPP; dashed (red) curves are without SPP. Right column: Interference patterns measured with beam splitter BS 2 misaligned. Upper figures correspond to the case without SPP, and lower figures correspond to the case with SPP.

Fig. 9
Fig. 9

Dependence of the diameter of the first dislocation ring on the coherence length L c of the input beam. The left (blue) line is defined as D = L c . The right (red) curve is a curve fit of the last eight points and has the form D = ( 0.027 ± 0.001 ) / L c .

Equations (21)

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I = I 1 + I 2 + 2   Re [ W ( ρ 1 , ρ 2 , z ; ω ) e i δ ] ,
W 0 ( ρ 1 , ρ 2 ) = T ( ρ 1 ) T ( ρ 2 ) J 1 ( α | ρ 1 ρ 2 | ) α | ρ 1 ρ 2 | e ( i π / λ L ) ( ρ 1 2 ρ 2 2 ) ,
L c = 1.22 λ L d 1 .
W ( ρ 1 , ρ 2 ) = W 0 ( ρ 1 , ρ 2 ) e i ( ϕ 1 ϕ 2 ) ,
W ( ρ 1 , ρ 2 ; z ) = + W ( ρ 1 , ρ 2 ; 0 ) e i ( k / 2 z ) | ρ 1 ρ 1 | 2 e i ( k / 2 z ) | ρ 2 ρ 2 | 2 d ρ 1 d ρ 2 ,
W ( ρ 1 , ρ 2 ; 0 ) = A ( ρ 1 ) A ( ρ 2 ) f ( ρ 1 ρ 2 ) ,
A ( ρ ) = T ( ρ ) e ( i π / λ L ) ρ 2 e i ϕ ,
f ( ρ ) = J 1 ( α ρ ) α ρ .
ρ + = ρ 1 + ρ 2 2 ,     Δ ρ = ρ 1 ρ 2 ,
σ = ρ 1 + ρ 2 2 ,     δ = ρ 1 ρ 2 ,
W ( ρ 1 , ρ 2 ; z ) = f ( Δ ρ ) e i ( k / z ) Δ ρ σ [ g ( ρ + + Δ ρ 2 ) g ( ρ + Δ ρ 2 ) e i ( k / z ) ρ + δ d ρ + ] d Δ ρ ,
g ( ρ ) A ( ρ ) e i ( k / 2 z ) ρ 2 .
W ( σ , δ ; z ) = G ( u k 2 z δ ) F ( u + k z σ ) G ( u + k 2 z δ ) d u .
I ( ρ ; z ) = | G ( u ) | 2 F ( u + k z ρ ) d u ,
X ( ρ ; z ) = G ( u k z ρ ) F ( u ) G ( u + k z ρ ) d u .
F ( u ) = F [ f ] = J 1 ( α | ρ | ) α | ρ | e i ρ u d ρ = { 1 , u k d 1 2 L 0 , u > k d 1 2 L . }
g ( ρ ) = T ( ρ ) e ( i k / 2 ) ( 1 / z + 1 / L ) ρ 2 e i ϕ ,
G ( u ) = F [ g ] = 0 0 2 π ρ d ρ d ϕ e i a ρ 2 e i ( ϕ + ρ u ) ,
G ( u ) = e i ϕ u e i u 2 / 8 a [ J 1 ( u 2 8 a ) + i J 0 ( u 2 8 a ) ] ,
W 0 ( ρ , θ ) = | T ( ρ ) | 2 J 1 [ α ( θ ) ρ ] α ( θ ) ρ ,
D = 2 π z λ L c .

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