Abstract

In classical optical imaging, the Rayleigh diffraction limit dR is defined as the minimum resolvable separation between two points under incoherent light illumination. In this paper, we analyze the minimum resolvable separation between two points under partially coherent beam illumination. We find that the image resolution of two points can overcome the classic Rayleigh diffraction limit through manipulating the correlation function of a partially coherent source, and the image resolution, which independent of the specified positions of two points in the object plane, can in principle reach the value of 0.17dR. Furthermore, we carry out an experimental demonstration of sub-Rayleigh imaging of a 1951 USAF resolution target via engineering the correlation function of the illuminating beam. Our experimental results are in agreement with our theoretical predictions.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (6)

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Plasmon coherence determination by nanoscattering,” Opt. Lett. 42(17), 3279–3282 (2017).
[PubMed]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Sci. Rep. 7, 39957 (2017).
[PubMed]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).

C. Liang, C. Mi, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Vector optical coherence lattices generating controllable far-field beam profiles,” Opt. Express 25(9), 9872–9885 (2017).
[PubMed]

X. Liu, F. Wang, L. Liu, Y. Chen, Y. Cai, and S. A. Ponomarenko, “Complex degree of coherence measurement for classical statistical fields,” Opt. Lett. 42(1), 77–80 (2017).
[PubMed]

2016 (5)

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).

S. Avramov-Zamurovic, C. Nelson, S. Guth, and O. Korotkova, “Flatness parameter influence on scintillation reduction for multi-Gaussian Schell-model beams propagating in turbulent air,” Appl. Opt. 55(13), 3442–3446 (2016).
[PubMed]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).

M. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).

A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially coherent surface plasmon polaritons,” Europhys. Lett. 116(6), 64001 (2016).

2015 (4)

L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
[PubMed]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).

M. V. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).

D. Xu, X. Song, H. Li, D. Zhang, H. Wang, J. Xiong, and K. Wang, “Experimental observation of sub-Rayleigh quantum imaging with a two-photon entangled source,” Appl. Phys. Lett. 106(17), 171104 (2015).

2014 (6)

2013 (5)

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(15), 57–65 (2013).

J. E. Oh, Y. W. Cho, G. Scarcelli, and Y. H. Kim, “Sub-Rayleigh imaging via speckle illumination,” Opt. Lett. 38(5), 682–684 (2013).
[PubMed]

2012 (7)

E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012).
[PubMed]

Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012).
[PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[PubMed]

S. Cho, M. A. Alonso, and T. G. Brown, “Measurement of spatial coherence through diffraction from a transparent mask with a phase discontinuity,” Opt. Lett. 37(13), 2724–2726 (2012).
[PubMed]

J. N. Clark, X. Huang, R. Harder, and I. K. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[PubMed]

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).

2010 (1)

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

2009 (2)

2008 (2)

2007 (2)

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[PubMed]

C. H. Gan, G. Gbur, and T. D. Visser, “Surface plasmons modulate the spatial coherence of light in Young’s interference experiment,” Phys. Rev. Lett. 98(4), 043908 (2007).
[PubMed]

2006 (2)

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[PubMed]

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[PubMed]

2005 (2)

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86(2), 021112 (2005).

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5 Pt 2), 056607 (2005).
[PubMed]

2004 (1)

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[PubMed]

2003 (1)

2002 (1)

J. Garcia and R. Castandeda, “Spatial partially coherent imaging,” J. Mod. Opt. 49(13), 2093–2104 (2002).

2001 (1)

2000 (2)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(Pt 2), 82–87 (2000).
[PubMed]

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE 3919, 141–150 (2000).

1999 (1)

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: Improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).

1994 (1)

Agard, D. A.

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE 3919, 141–150 (2000).

Alonso, M. A.

Anzolin, G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[PubMed]

Avramov-Zamurovic, S.

Barbieri, C.

E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012).
[PubMed]

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[PubMed]

Basu, S.

M. V. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[PubMed]

Betancur, R.

Bianchini, A.

E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012).
[PubMed]

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[PubMed]

Bose-Pillai, S.

M. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).

Brown, D. P.

Brown, T. G.

Cai, Y.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Sci. Rep. 7, 39957 (2017).
[PubMed]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).

X. Liu, F. Wang, L. Liu, Y. Chen, Y. Cai, and S. A. Ponomarenko, “Complex degree of coherence measurement for classical statistical fields,” Opt. Lett. 42(1), 77–80 (2017).
[PubMed]

C. Liang, C. Mi, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Vector optical coherence lattices generating controllable far-field beam profiles,” Opt. Express 25(9), 9872–9885 (2017).
[PubMed]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(15), 57–65 (2013).

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86(2), 021112 (2005).

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5 Pt 2), 056607 (2005).
[PubMed]

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28(13), 1084–1086 (2003).
[PubMed]

Castandeda, R.

J. Garcia and R. Castandeda, “Spatial partially coherent imaging,” J. Mod. Opt. 49(13), 2093–2104 (2002).

Castañeda, R.

Chen, Y.

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Plasmon coherence determination by nanoscattering,” Opt. Lett. 42(17), 3279–3282 (2017).
[PubMed]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Sci. Rep. 7, 39957 (2017).
[PubMed]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).

X. Liu, F. Wang, L. Liu, Y. Chen, Y. Cai, and S. A. Ponomarenko, “Complex degree of coherence measurement for classical statistical fields,” Opt. Lett. 42(1), 77–80 (2017).
[PubMed]

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(15), 57–65 (2013).

Cho, S.

Cho, Y. W.

Clark, J. N.

J. N. Clark, X. Huang, R. Harder, and I. K. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[PubMed]

Cremer, C.

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: Improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).

Dong, Y.

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).

Eyyuboglu, H. T.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(15), 57–65 (2013).

Fischer, D. G.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

Friberg, A. T.

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Plasmon coherence determination by nanoscattering,” Opt. Lett. 42(17), 3279–3282 (2017).
[PubMed]

A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially coherent surface plasmon polaritons,” Europhys. Lett. 116(6), 64001 (2016).

Gan, C. H.

C. H. Gan, G. Gbur, and T. D. Visser, “Surface plasmons modulate the spatial coherence of light in Young’s interference experiment,” Phys. Rev. Lett. 98(4), 043908 (2007).
[PubMed]

Garcia, J.

J. Garcia and R. Castandeda, “Spatial partially coherent imaging,” J. Mod. Opt. 49(13), 2093–2104 (2002).

Gbur, G.

G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
[PubMed]

C. H. Gan, G. Gbur, and T. D. Visser, “Surface plasmons modulate the spatial coherence of light in Young’s interference experiment,” Phys. Rev. Lett. 98(4), 043908 (2007).
[PubMed]

Gori, F.

Gu, J.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).

Gureyev, T. E.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[PubMed]

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(Pt 2), 82–87 (2000).
[PubMed]

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE 3919, 141–150 (2000).

Guth, S.

Han, Y.

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).

Harder, R.

J. N. Clark, X. Huang, R. Harder, and I. K. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[PubMed]

Heintzmann, R.

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: Improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).

Hell, S. W.

Hradil, Z.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[PubMed]

Huang, X.

J. N. Clark, X. Huang, R. Harder, and I. K. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[PubMed]

Hyde, M.

M. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).

Hyde, M. V.

M. V. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).

Kim, Y. H.

Korotkova, O.

Li, H.

D. Xu, X. Song, H. Li, D. Zhang, H. Wang, J. Xiong, and K. Wang, “Experimental observation of sub-Rayleigh quantum imaging with a two-photon entangled source,” Appl. Phys. Lett. 106(17), 171104 (2015).

Li, Z.

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).

Liang, C.

Lin, Q.

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86(2), 021112 (2005).

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28(13), 1084–1086 (2003).
[PubMed]

Liu, L.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).

X. Liu, F. Wang, L. Liu, Y. Chen, Y. Cai, and S. A. Ponomarenko, “Complex degree of coherence measurement for classical statistical fields,” Opt. Lett. 42(1), 77–80 (2017).
[PubMed]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

Liu, X.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).

X. Liu, F. Wang, L. Liu, Y. Chen, Y. Cai, and S. A. Ponomarenko, “Complex degree of coherence measurement for classical statistical fields,” Opt. Lett. 42(1), 77–80 (2017).
[PubMed]

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(15), 57–65 (2013).

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[PubMed]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

Lu, X.

Ma, L.

Mari, E.

Martínez-Herrero, R.

Mayo, S. C.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[PubMed]

Mei, Z.

Mejías, P. M.

Mi, C.

Motka, L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[PubMed]

Nelson, C.

Norrman, A.

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Plasmon coherence determination by nanoscattering,” Opt. Lett. 42(17), 3279–3282 (2017).
[PubMed]

A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially coherent surface plasmon polaritons,” Europhys. Lett. 116(6), 64001 (2016).

Oh, J. E.

Paganin, D. M.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[PubMed]

Peng, X.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).

Ponomarenko, S. A.

Qu, J.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(15), 57–65 (2013).

Rehacek, J.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[PubMed]

Robinson, I. K.

J. N. Clark, X. Huang, R. Harder, and I. K. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[PubMed]

Romanato, F.

Rust, M. J.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[PubMed]

Sahin, S.

Sánchez-Soto, L. L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[PubMed]

Santarsiero, M.

Scarcelli, G.

Sedat, J. W.

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE 3919, 141–150 (2000).

Sharma, K. A.

Song, X.

D. Xu, X. Song, H. Li, D. Zhang, H. Wang, J. Xiong, and K. Wang, “Experimental observation of sub-Rayleigh quantum imaging with a two-photon entangled source,” Appl. Phys. Lett. 106(17), 171104 (2015).

Stevenson, A. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[PubMed]

Stoklasa, B.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[PubMed]

Swartzlander, G. A.

Tamburini, F.

E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012).
[PubMed]

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[PubMed]

Thidé, B.

Tong, Z.

Umbriaco, G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[PubMed]

van Dijk, T.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

Visser, T. D.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

C. H. Gan, G. Gbur, and T. D. Visser, “Surface plasmons modulate the spatial coherence of light in Young’s interference experiment,” Phys. Rev. Lett. 98(4), 043908 (2007).
[PubMed]

Voelz, D. G.

M. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).

M. V. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).

Wang, F.

X. Liu, F. Wang, L. Liu, Y. Chen, Y. Cai, and S. A. Ponomarenko, “Complex degree of coherence measurement for classical statistical fields,” Opt. Lett. 42(1), 77–80 (2017).
[PubMed]

C. Liang, C. Mi, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Vector optical coherence lattices generating controllable far-field beam profiles,” Opt. Express 25(9), 9872–9885 (2017).
[PubMed]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(15), 57–65 (2013).

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).

Wang, H.

D. Xu, X. Song, H. Li, D. Zhang, H. Wang, J. Xiong, and K. Wang, “Experimental observation of sub-Rayleigh quantum imaging with a two-photon entangled source,” Appl. Phys. Lett. 106(17), 171104 (2015).

Wang, J.

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).

Wang, K.

D. Xu, X. Song, H. Li, D. Zhang, H. Wang, J. Xiong, and K. Wang, “Experimental observation of sub-Rayleigh quantum imaging with a two-photon entangled source,” Appl. Phys. Lett. 106(17), 171104 (2015).

Wichmann, J.

Wilkins, S. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004).
[PubMed]

Wolf, E.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[PubMed]

Wood, J. K.

Wu, G.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).

Xiao, X.

M. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).

M. V. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).

Xiong, J.

D. Xu, X. Song, H. Li, D. Zhang, H. Wang, J. Xiong, and K. Wang, “Experimental observation of sub-Rayleigh quantum imaging with a two-photon entangled source,” Appl. Phys. Lett. 106(17), 171104 (2015).

Xu, D.

D. Xu, X. Song, H. Li, D. Zhang, H. Wang, J. Xiong, and K. Wang, “Experimental observation of sub-Rayleigh quantum imaging with a two-photon entangled source,” Appl. Phys. Lett. 106(17), 171104 (2015).

Yu, J.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).

Yuan, Y.

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(15), 57–65 (2013).

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

Zhang, D.

D. Xu, X. Song, H. Li, D. Zhang, H. Wang, J. Xiong, and K. Wang, “Experimental observation of sub-Rayleigh quantum imaging with a two-photon entangled source,” Appl. Phys. Lett. 106(17), 171104 (2015).

Zhao, C.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).

C. Liang, C. Mi, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Vector optical coherence lattices generating controllable far-field beam profiles,” Opt. Express 25(9), 9872–9885 (2017).
[PubMed]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).

Zhu, S.

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86(2), 021112 (2005).

Zhu, S. Y.

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5 Pt 2), 056607 (2005).
[PubMed]

Zhuang, X.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (7)

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86(2), 021112 (2005).

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).

D. Xu, X. Song, H. Li, D. Zhang, H. Wang, J. Xiong, and K. Wang, “Experimental observation of sub-Rayleigh quantum imaging with a two-photon entangled source,” Appl. Phys. Lett. 106(17), 171104 (2015).

Europhys. Lett. (1)

A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially coherent surface plasmon polaritons,” Europhys. Lett. 116(6), 64001 (2016).

J. Appl. Phys. (1)

M. V. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).

J. Microsc. (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(Pt 2), 82–87 (2000).
[PubMed]

J. Mod. Opt. (1)

J. Garcia and R. Castandeda, “Spatial partially coherent imaging,” J. Mod. Opt. 49(13), 2093–2104 (2002).

J. Opt. Soc. Am. A (3)

Nat. Commun. (2)

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[PubMed]

J. N. Clark, X. Huang, R. Harder, and I. K. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[PubMed]

Nat. Methods (1)

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[PubMed]

Opt. Commun. (1)

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboglu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(15), 57–65 (2013).

Opt. Express (5)

Opt. Lett. (15)

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Plasmon coherence determination by nanoscattering,” Opt. Lett. 42(17), 3279–3282 (2017).
[PubMed]

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
[PubMed]

X. Liu, F. Wang, L. Liu, Y. Chen, Y. Cai, and S. A. Ponomarenko, “Complex degree of coherence measurement for classical statistical fields,” Opt. Lett. 42(1), 77–80 (2017).
[PubMed]

L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
[PubMed]

Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012).
[PubMed]

S. Cho, M. A. Alonso, and T. G. Brown, “Measurement of spatial coherence through diffraction from a transparent mask with a phase discontinuity,” Opt. Lett. 37(13), 2724–2726 (2012).
[PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[PubMed]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
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Figures (7)

Fig. 1
Fig. 1 Telecentric imaging system. Lenses L1 and L2 have the same focal length f.
Fig. 2
Fig. 2 Variation of the μ ( x ) of an LGCSM beam with x / δ 0 for different values of n.
Fig. 3
Fig. 3 Images of two pinholes with separation d = d R under the illumination of (a) an incoherent source; (b) an LGCSM source with n = 1 and δ 0 = 0.5 d R ; (c) an LGCSM source with n = 6 and δ 0 = 0.98 d R . The corresponding cross lines ( ρ y = 0 ) of the images are shown in Fig. 3(d).
Fig. 4
Fig. 4 (a) Resolvable separation between two pinholes versus δ 0 / d R ; (b) Minimum resolvable separation between two pinholes versus radial index n.
Fig. 5
Fig. 5 Experimental setup for generating an LGCSM beam and demonstrating sub-Rayleigh imaging. Laser, ND: YAG laser; BE, beam expander; BS, beam splitter; SLM, spatial light modulator; CA1, CA2, circular apertures; L1, L2, L3, L4, thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; OB, 1951 USAF resolution target; CCD, charge-coupled device.
Fig. 6
Fig. 6 Image of the target under the illumination of (a) a GSM beam with δ0 = 55um; (b) a GSM beam with δ0 = 10um; (c) an LGCSM beam with n = 1, δ0 = 28um; (d) an LGCSM beam with n = 6, δ0 = 55um.
Fig. 7
Fig. 7 (a)-(c) Simulation results of the image of the area in rectangular box shown in Fig. 6 under the illumination of different beams; (d)-(f) the corresponding cross lines of simulation results (solid line) and experimental results (circular dots) of the right side triple-slit at ρy = 0.

Equations (10)

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W ( r 1 , r 2 ) = exp ( r 1 2 + r 2 2 4 σ 0 2 ) μ ( Δ r ) ,
W ( ρ 1 , ρ 2 ) = W ( r 1 , r 2 ) O * ( r 1 ) O ( r 2 ) h * ( r 1 , ρ 1 ) h ( r 2 , ρ 2 ) d 2 r 1 d 2 r 2 ,
h ( r , ρ ) = 1 λ 2 f 2 P ( ξ ) exp ( i k f ξ ( r + ρ ) ) d 2 ξ .
h ( r , ρ ) = 2 π R 2 λ 2 f 2 J 1 ( 2 π R | r + ρ | / λ f ) 2 π R | r + ρ | / λ f .
O ( r ) = δ ( x d / 2 , 0 ) + δ ( x + d / 2 , 0 ) .
S ( ρ ) = 1 16 λ 2 f 2 ( π R 2 λ f ) 2 exp ( d 2 8 σ 0 2 ) [ S + 2 + S 2 + 2 Re [ μ ( d ) ] S + S ] ,
S ± = 2 J 1 ( 2 π R ( ρ x ± d / 2 ) 2 + ρ y 2 / λ f ) 2 π R ( ρ x ± d / 2 ) 2 + ρ y 2 / λ f .
μ ( Δ r ) = L n 0 ( Δ r 2 2 δ 0 2 ) exp ( Δ r 2 2 δ 0 2 ) ,
E ( v ) = ( | v | ω 0 ) n exp ( v 2 ω 0 2 ) ,
μ ( Δ r ) I ( v 1 ) exp [ i 2 π v 1 Δ r λ f 2 ] d 2 v 1 ,

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