Abstract

A novel class of partially coherent light sources that can yield stable optical lattice termed hollow array in the far field is introduced. The array dimension, the distance of hollow lobes intensity profile, the size and shape of the inner and outer lobe contours and other features can be flexibly controlled by altering the source parameters. Further, every lobe can be shaped with polar and Cartesian symmetry and even combined to form nested structures. The applications of the work are envisioned in material surface processing and particle trapping.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

2019 (1)

2018 (4)

2017 (2)

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

D. Zou, X. Li, X. Pang, H. Zheng, and Y. Ge, “Propagation properties of hollow sinh-Gaussian beams in quadratic-index medium,” Opt. Commun. 401, 54–58 (2017).
[Crossref]

2016 (1)

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

2015 (3)

2014 (5)

2013 (2)

2012 (1)

2011 (1)

2007 (6)

2006 (1)

2005 (3)

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A: Pure Appl. Opt. 7(3), 147–153 (2005).
[Crossref]

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1(1), 23–30 (2005).
[Crossref]

Y. Xia and J. P. Yin, “Generation of a focused hollow beam by an 2pi-phase plate and its application in atom or molecule optics,” J. Opt. Soc. Am. A 22(3), 529–536 (2005).
[Crossref]

2003 (3)

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Adv. Phys. 426(6965), 421–424 (2003).
[Crossref]

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

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, “Atomic trapping and guiding by quasi-dark hollow beams,” IEEE Wireless Commun. 10(2), 45–53 (2003).
[Crossref]

2000 (2)

M. Yan, J. Yin, and Y. Zhu, “Dark-hollow-beam guiding and splitting of a low-velocity atomic beam,” J. Opt. Soc. Am. B 17(11), 1817–1820 (2000).
[Crossref]

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000).
[Crossref]

1997 (2)

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82(1), 60–64 (1997).
[Crossref]

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

1994 (2)

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994).
[Crossref]

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, “Holographic non-diverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[Crossref]

Ahmad, M. A.

Ahufinger, V.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Atewart, B. W.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, “Holographic non-diverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[Crossref]

Berger, V.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82(1), 60–64 (1997).
[Crossref]

Bloch, I.

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1(1), 23–30 (2005).
[Crossref]

Cai, Y.

Campbell, M.

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000).
[Crossref]

Chen, C.

Y. Cai, C. Chen, and F. Wang, “Modified hollow Gaussian beam and its paraxial propagation,” Opt. Commun. 278(1), 34–41 (2007).
[Crossref]

Chen, Y.

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

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref]

Choi, K.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, “Holographic non-diverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[Crossref]

Costard, E.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82(1), 60–64 (1997).
[Crossref]

Damski, B.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Denning, R.

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000).
[Crossref]

Dholakia, K.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Adv. Phys. 426(6965), 421–424 (2003).
[Crossref]

Dolezal, F.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, “Atomic trapping and guiding by quasi-dark hollow beams,” IEEE Wireless Commun. 10(2), 45–53 (2003).
[Crossref]

Dong, Y.

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A: Pure Appl. Opt. 7(3), 147–153 (2005).
[Crossref]

Elbatt, T.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, “Atomic trapping and guiding by quasi-dark hollow beams,” IEEE Wireless Commun. 10(2), 45–53 (2003).
[Crossref]

Fang, G.

Fenichel, H.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, “Holographic non-diverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[Crossref]

Gauthier-Lafaye, O.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82(1), 60–64 (1997).
[Crossref]

Ge, Y.

D. Zou, X. Li, X. Pang, H. Zheng, and Y. Ge, “Propagation properties of hollow sinh-Gaussian beams in quadratic-index medium,” Opt. Commun. 401, 54–58 (2017).
[Crossref]

Gori, F.

Harrison, M.

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000).
[Crossref]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Izadpanah, H.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, “Atomic trapping and guiding by quasi-dark hollow beams,” IEEE Wireless Commun. 10(2), 45–53 (2003).
[Crossref]

Jiang, C.

B. Tang, S. Jiang, C. Jiang, and H. Zhu, “Propagation properties of hollow sinh-Gaussian beams through fractional Fourier transform optical systems,” Opt. Laser Technol. 59, 116–122 (2014).
[Crossref]

Jiang, S.

B. Tang, S. Jiang, C. Jiang, and H. Zhu, “Propagation properties of hollow sinh-Gaussian beams through fractional Fourier transform optical systems,” Opt. Laser Technol. 59, 116–122 (2014).
[Crossref]

Karimi, E.

Khonina, S. N.

Korotkova, O.

Kotlyar, V. V.

Kozawa, Y.

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Kukshya, V.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, “Atomic trapping and guiding by quasi-dark hollow beams,” IEEE Wireless Commun. 10(2), 45–53 (2003).
[Crossref]

Lajunen, H.

Lee, H. S.

H. S. Lee, B. W. Atewart, K. Choi, and H. Fenichel, “Holographic non-diverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[Crossref]

Lewenstein, M.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Li, X.

D. Zou, X. Li, X. Pang, H. Zheng, and Y. Ge, “Propagation properties of hollow sinh-Gaussian beams in quadratic-index medium,” Opt. Commun. 401, 54–58 (2017).
[Crossref]

Liang, C.

Lin, J.

Lin, Q.

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A: Pure Appl. Opt. 7(3), 147–153 (2005).
[Crossref]

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

Liu, J.

Liu, L.

Liu, S.

Liu, Z.

Lu, X.

Ma, L.

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Adv. Phys. 426(6965), 421–424 (2003).
[Crossref]

Mandel, L.

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

Mao, Y.

Marksteiner, S.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994).
[Crossref]

Marrucci, L.

Mei, Z.

Mi, C.

Pang, X.

D. Zou, X. Li, X. Pang, H. Zheng, and Y. Ge, “Propagation properties of hollow sinh-Gaussian beams in quadratic-index medium,” Opt. Commun. 401, 54–58 (2017).
[Crossref]

Peng, X.

Piccirillo, B.

Ponomarenko, S. A.

Rolston, S. L.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994).
[Crossref]

Ryu, B. K.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, “Atomic trapping and guiding by quasi-dark hollow beams,” IEEE Wireless Commun. 10(2), 45–53 (2003).
[Crossref]

Saastamoinen, T.

Sanpera, A.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Santamato, E.

Santarsiero, M.

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Sato, S.

Savage, C. M.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994).
[Crossref]

Sen, A.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Sen, U.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Sharp, D.

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000).
[Crossref]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Skidanov, R. V.

Soifer, V. A.

Spalding, G. C.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Adv. Phys. 426(6965), 421–424 (2003).
[Crossref]

Sun, Q.

Tang, B.

B. Tang, S. Jiang, C. Jiang, and H. Zhu, “Propagation properties of hollow sinh-Gaussian beams through fractional Fourier transform optical systems,” Opt. Laser Technol. 59, 116–122 (2014).
[Crossref]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Turberfield, A.

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000).
[Crossref]

Vyas, S.

Wan, L.

Wang, F.

Wang, Z.

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A: Pure Appl. Opt. 7(3), 147–153 (2005).
[Crossref]

Wolf, E.

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

Xia, Y.

Yan, M.

Yin, J.

Yin, J. P.

Zhang, G.

Zhao, C.

Zhao, D.

Zhao, H.

Zheng, H.

D. Zou, X. Li, X. Pang, H. Zheng, and Y. Ge, “Propagation properties of hollow sinh-Gaussian beams in quadratic-index medium,” Opt. Commun. 401, 54–58 (2017).
[Crossref]

Zhou, K.

Zhou, Y.

Zhu, H.

B. Tang, S. Jiang, C. Jiang, and H. Zhu, “Propagation properties of hollow sinh-Gaussian beams through fractional Fourier transform optical systems,” Opt. Laser Technol. 59, 116–122 (2014).
[Crossref]

Zhu, X.

Zhu, Y.

Zito, G.

Zoller, P.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994).
[Crossref]

Zou, D.

D. Zou, X. Li, X. Pang, H. Zheng, and Y. Ge, “Propagation properties of hollow sinh-Gaussian beams in quadratic-index medium,” Opt. Commun. 401, 54–58 (2017).
[Crossref]

Adv. Phys. (2)

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Adv. Phys. 426(6965), 421–424 (2003).
[Crossref]

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Appl. Phys. Lett. (1)

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

IEEE Wireless Commun. (1)

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, “Atomic trapping and guiding by quasi-dark hollow beams,” IEEE Wireless Commun. 10(2), 45–53 (2003).
[Crossref]

J. Appl. Phys. (1)

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82(1), 60–64 (1997).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A: Pure Appl. Opt. 7(3), 147–153 (2005).
[Crossref]

J. Opt. Soc. Am. A (2)

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

Nat. Phys. (1)

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1(1), 23–30 (2005).
[Crossref]

Nature (1)

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Figures (7)

Fig. 1.
Fig. 1. Illustration of the degree of coherence (a) for a the hollow multi-Gaussian Schell-model array source calculated from Eq. (10) and function $p(\mathbf {v})$ (b) calculated from Eq. (12). Calculated parameters are set as follows: $M=5$ , $N=5$ , $L_{1}=2$ , $L_{2}=1$ , $\delta =1mm$ , $R=0.9m$ , $\lambda =632.8nm$ .
Fig. 2.
Fig. 2. Influence of parameter $M$ , $N$ and $R$ on distribution of normalized spectral intensity of far field: $L_{1}=2$ , $L_{2}=1$ , $\delta =1mm$ , $\sigma _{0}=1mm$ ; (a) $M=3$ , $N=2$ , $R_{x}=R_{y}=1.2m$ ; (b) $M=2$ , $N=2$ , $R_{x}=R_{y}=1.2m$ ; (c) $M=2$ , $N=2$ , $R_{x}=R_{y}=2m$ ; (d) $M=2$ , $N=2$ , $R_{x}=1.2m$ , $R_{y}=2m$ .
Fig. 3.
Fig. 3. Influence of parameter $L$ and $\delta$ on distribution of normalized spectral intensity of far field: $M=2$ , $N=2$ , $R_{x}=R_{y}=1.5m$ , $\sigma _{0}=1mm$ ; (a) $L_{1}=40$ , $L_{2}=1$ , $\delta _{x}=\delta _{y}=1mm$ ; (b) $L_{1}=40$ , $L_{2}=5$ , $\delta _{x}=\delta _{y}=1mm$ ; (c) $L_{1}=40$ , $L_{2}=20$ , $\delta _{x}=\delta _{y}=1mm$ ; (d) $L_{1}=40$ , $L_{2}=20$ , $\delta _{x}=1.2mm$ , $\delta _{y}=0.8mm$ .
Fig. 4.
Fig. 4. Illustration of the degree of coherence (a) and its Fourier transform $p(\mathbf {v})$ (b) for the rectangular Hollow multi-Gaussian Schell-model array source corresponding to Eq. (24). Calculated parameters are set as follows: $M=5$ , $N=5$ , $L_{1}=3$ , $L_{2}=2$ , $\delta =1mm$ , $R=0.82m$ .
Fig. 5.
Fig. 5. Influence of parameter $M$ , $N$ and $R$ on distribution of normalized spectral intensity of far field corresponding to Eq. (26): $L_{1}=10$ , $L_{2}=8$ and rest parameters as in Fig. 2.
Fig. 6.
Fig. 6. Influence of parameter $L$ and $\delta$ on distribution of normalized spectral intensity of far field corresponding to Eq. (26): (a) $L_{1}=20$ , $L_{2}=1$ ; (b) $L_{1}=20$ , $L_{2}=4$ ; (c) $L_{1}=20$ , $L_{2}=18$ ; (d) $L_{1}=20$ , $L_{2}=18$ and the paramenter $\delta$ as in Fig. 3.
Fig. 7.
Fig. 7. Normalized spectral intensity distribution of the far field corresponding to Eq. (28): (a) $L_{1}=40$ , $L_{2}=3$ , $\delta _{x}=\delta _{y}=1mm$ ; (b) $L_{1}=40$ , $L_{2}=3$ , $\delta _{x}=1.2mm$ , $\delta _{y}=0.8mm$ ; (c) $L_{1}=40$ , $L_{2}=8$ , $\delta _{x}=\delta _{y}=1mm$ ; (d) $L_{2}=25$ , $L_{1}=2$ , $\delta _{x}=\delta _{y}=1mm$ ; (e) $L_{2}=25$ , $L_{1}=2$ , $\delta _{x}=1.2mm$ , $\delta _{y}=0.8mm$ ; (f) $L_{2}=25$ , $L_{1}=20$ , $\delta _{x}=\delta _{y}=1mm$ .

Equations (28)

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W ( 0 ) ( ρ 1 , ρ 2 ; ω ) = E ( ρ 1 ; ω ) E ( ρ 2 ; ω ) ,
W ( 0 ) ( ρ 1 , ρ 2 ) = S ( ρ 1 ) S ( ρ 2 ) μ ( ρ 2 ρ 1 ) ,
W ( 0 ) ( ρ 1 , ρ 2 ) = p ( v ) H 0 ( ρ 1 , v ) H 0 ( ρ 2 , v ) d 2 v ,
H 0 ( ρ , v ) = τ ( ρ ) exp ( 2 π i v ρ ) ,
W ( 0 ) ( ρ 1 , ρ 2 ) = τ ( ρ 1 ) τ ( ρ 2 ) μ ( ρ 2 ρ 1 ) ,
μ ( ρ 2 ρ 1 ) = p ( v ) exp [ 2 π i v ( ρ 2 ρ 1 ) ] d 2 v .
τ ( ρ ) = exp [ ρ 2 / ( 4 σ 0 2 ) ] .
μ ( ρ 2 ρ 1 ) = 1 C 0 l = 1 L ( 1 ) l 1 l ( L l ) exp [ ( x 2 x 1 ) 2 2 l δ x 2 ( y 2 y 1 ) 2 2 l δ y 2 ] ,
μ c ( ρ 2 ρ 1 ) = 1 C 0 N M l = 1 L ( 1 ) l 1 l ( L l ) exp [ ( x 2 x 1 ) 2 2 l δ x 2 ( y 2 y 1 ) 2 2 l δ y 2 ] × n = P P cos [ 2 π n R x ( x 2 x 1 ) δ x ] m = Q Q cos [ 2 π m R y ( y 2 y 1 ) δ y ] ,
μ C ( ρ 2 ρ 1 ) = μ c 1 ( ρ 2 ρ 1 ) μ c 2 ( ρ 2 ρ 1 ) ,
μ c t ( ρ 2 ρ 1 ) = 1 C 0 t N M l = 1 L t ( 1 ) l 1 l ( L t l ) exp [ ( x 2 x 1 ) 2 2 l δ t x 2 ( y 2 y 1 ) 2 2 l δ t y 2 ] × n = P P cos [ 2 π n R t x ( x 2 x 1 ) δ t x ] m = Q Q cos [ 2 π m R t y ( y 2 y 1 ) δ t y ] ( t = 1 , 2 ) .
p c ( v ) = p 1 ( v ) p 2 ( v ) ,
p t ( v ) = δ t x 2 δ t y 2 C 0 t N M l = 1 L t ( 1 ) l 1 l ( L t l ) exp ( l δ t x 2 v x 2 2 l δ t y 2 v y 2 2 ) × n = P P cosh ( 2 l δ t x π n R t x v x ) exp ( 2 l π 2 n 2 R t x 2 ) × m = Q Q cosh ( 2 l δ t y π m R t y v y ) exp ( 2 l π 2 m 2 R t y 2 ) ( t = 1 , 2 ) ,
W ( 0 ) ( ρ 1 , ρ 2 ) = exp ( ρ 1 2 + ρ 2 2 4 σ 0 2 ) [ μ c 1 ( ρ 2 ρ 1 ) μ c 2 ( ρ 2 ρ 1 ) ] .
W ( ) ( r 1 s 1 , r 2 s 2 ) = ( 2 π k ) 2 cos θ 1 cos θ 2 W ~ ( 0 ) ( k s 1 , k s 2 ) exp [ i k ( r 2 r 1 ) ] / ( r 1 r 2 ) ,
W ~ ( 0 ) ( f 1 , f 2 ) = ( 2 π ) 4 W ( 0 ) ( ρ 1 , ρ 2 ) exp [ i ( f 1 ρ 1 + f 2 ρ 2 ) ] d 2 ρ 1 d 2 ρ 2
W C ( ) ( r 1 s 1 , r 2 s 2 ) = W 1 ( ) ( r 1 s 1 , r 2 s 2 ) W 2 ( ) ( r 1 s 1 , r 2 s 2 ) ,
W t ( ) ( r 1 s 1 , r 2 s 2 ) = k 2 σ 0 2 cos θ 1 cos θ 2 exp [ i k ( r 2 r 1 ) ] 2 C 0 t N M r 1 r 2 l = 1 L t ( 1 ) l 1 l ( L t l ) × 1 a t x a t y exp ( b t x ) exp ( b t y ) × n = P P cosh [ n c t x k 2 a t x ( s 1 x + s 2 x ) ] exp ( n 2 c t x 2 a t x ) × m = Q Q cosh [ m c t y k 2 a t y ( s 1 y + s 2 y ) ] exp ( m 2 c t y 2 a t y ) ( t = 1 , 2 ) ,
a t j = 1 8 σ 0 2 + 1 2 l t δ t j 2 , b t j = k 2 2 [ σ 0 2 ( s 2 j s 1 j ) 2 + ( s 2 j + s 1 j ) 2 8 a t j ] , c t j = π R t j δ t j , ( j = x , y ; t = 1 , 2 ) .
S C ( ) ( r s ) = S 1 ( ) ( r s ) S 2 ( ) ( r s ) ,
S t ( ) ( r s ) = k 2 σ 0 2 cos 2 θ 2 C 0 t N M r 2 l = 1 L t ( 1 ) l 1 l ( L t l ) × 1 a t x a t y exp ( k 2 s x 2 4 a t x ) exp ( k 2 s y 2 4 a t y ) × n = P P cosh ( n c t x a t x k s x ) exp ( n 2 c t x 2 a t x ) × m = Q Q cosh ( m c t y a t y k s y ) exp ( m 2 c t y 2 a t y ) ( t = 1 , 2 ) ,
exp ( k 2 s j 2 4 a t j ) 0 , j = x , y ; t = 1 , 2.
1 4 σ 0 2 + 1 δ t j 2 2 π 2 λ 2 , j = x , y ; t = 1 , 2.
μ R ( ρ 2 ρ 1 ) = μ r 1 ( ρ 2 ρ 1 ) μ r 2 ( ρ 2 ρ 1 ) ,
μ r t ( ρ 2 ρ 1 ) = 1 C 0 t N M l = 1 L t ( 1 ) l 1 l ( L t l ) exp [ ( x 2 x 1 ) 2 2 l δ t x 2 ] × l = 1 L t ( 1 ) l 1 l ( L t l ) exp [ ( y 2 y 1 ) 2 2 l δ t y 2 ] × n = P P cos [ 2 π n R t x ( x 2 x 1 ) δ t x ] m = Q Q cos [ 2 π m R t y ( y 2 y 1 ) δ t y ] ( t = 1 , 2 ) ,
S R ( ) ( r s ) = S r 1 ( ) ( r s ) S r 2 ( ) ( r s ) ,
S r t ( ) ( r s ) = k 2 σ 0 2 cos 2 θ 2 C 0 t N M r 2 l = 1 L t ( 1 ) l 1 l ( L t l ) 1 a t x exp ( k 2 s x 2 4 a t x ) × l = 1 L t ( 1 ) l 1 l ( L t l ) 1 a t y exp ( k 2 s y 2 4 a t y ) × n = P P cosh ( n c t x a t x k s x ) exp ( n 2 c t x 2 a t x ) × m = Q Q cosh ( m c t y a t y k s y ) exp ( m 2 c t y 2 a t y ) ( t = 1 , 2 ) .
μ C R ( ρ 2 ρ 1 ) = a μ c 1 ( ρ 2 ρ 1 ) + b μ r 2 ( ρ 2 ρ 1 ) ,

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