Abstract

Trajectory control of spatial solitons is an important subject in optical transmission field. Here we investigate the propagation dynamics of Laguerre-Gaussian soliton arrays in nonlinear media with a strong nonlocality and introduce two parameters, which we refer to as initial tangential velocity and displacement, to control the propagation path. The general analytical expression for the evolution of the soliton array is derived and the propagation properties, such as the intensity distribution, the propagation trajectory, the center distance, and the angular velocity are analyzed. It is found that the initial tangential velocity and displacement make the solitons sinusoidally oscillate in the $x$ and $y$ directions, and each constituent soliton undergoes elliptically or circularly spiral trajectory during propagation. A series of numerical examples is exhibited to graphically illustrate these typical propagation properties. Our results may provide a new perspective and stimulate further active investigations of multisoliton interaction and may be applied in optical communication and particle control.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (4)

V. Biloshytskyi, A. Oliinyk, P. Kruglenko, A. Desyatnikov, and A. Yakimenko, “Vortex nucleation in nonlocal nonlinear media,” Phys. Rev. A 99(4), 043835 (2019).
[Crossref]

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[Crossref]

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[Crossref]

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10(1), 1865 (2019).
[Crossref]

2018 (5)

S. Zeng, M. Chen, T. Zhang, W. Hu, and D. Lu, “Analytical modeling of soliton interactions in a nonlocal nonlinear medium analogous to gravitational force,” Phys. Rev. A 97(1), 013817 (2018).
[Crossref]

Z.-J. Yang, S.-M. Zhang, X.-L. Li, and Z.-G. Pang, “Variable sinh-Gaussian solitons in nonlocal nonlinear Schrödinger equation,” Appl. Math. Lett. 82, 64–70 (2018).
[Crossref]

Z.-J. Yang, S.-M. Zhang, X.-L. Li, Z.-G. Pang, and H.-X. Bu, “High-order revivable complex-valued hyperbolic-sine-Gaussian solitons and breathers in nonlinear media with a spatial nonlocality,” Nonlinear Dyn. 94(4), 2563–2573 (2018).
[Crossref]

X. Ma and S. Schumacher, “Vortex multistability and Bessel vortices in polariton condensates,” Phys. Rev. Lett. 121(22), 227404 (2018).
[Crossref]

Y. V. Izdebskaya, V. G. Shvedov, P. S. Jung, and W. Krolikowski, “Stable vortex soliton in nonlocal media with orientational nonlinearity,” Opt. Lett. 43(1), 66–69 (2018).
[Crossref]

2017 (5)

X. Ma, O. A. Egorov, and S. Schumacher, “Creation and manipulation of stable dark solitons and vortices in microcavity polariton condensates,” Phys. Rev. Lett. 118(15), 157401 (2017).
[Crossref]

Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-angular-momentum mode selection by rotationally symmetric superposition of chiral states with application to electron vortex beams,” Phys. Rev. Lett. 119(9), 094802 (2017).
[Crossref]

S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: Beams with orbital angular momentum,” Rev. Mod. Phys. 89(3), 035004 (2017).
[Crossref]

G. Liang, W. Cheng, Z. Dai, T. Jia, M. Wang, and H. Li, “Spiraling elliptic solitons in lossy nonlocal nonlinear media,” Opt. Express 25(10), 11717–11724 (2017).
[Crossref]

S. L. Xu, G. P. Zhao, M. R. Belić, J. R. He, and L. Xue, “Light bullets in coupled nonlinear Schrödinger equations with variable coefficients and a trapping potential,” Opt. Express 25(8), 9094–9104 (2017).
[Crossref]

2016 (3)

2015 (2)

Y. Zhang, M. R. Belić, L. Zhang, W. Zhong, D. Zhu, R. Wang, and Y. Zhang, “Periodic inversion and phase transition of finite energy Airy beams in a medium with parabolic potential,” Opt. Express 23(8), 10467–10480 (2015).
[Crossref]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and X. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref]

2014 (1)

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

2013 (1)

2012 (1)

Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75(8), 086401 (2012).
[Crossref]

2011 (2)

2009 (3)

Z. Xu, N. F. Smyth, A. A. Minzoni, and Y. S. Kivshar, “Vector vortex solitons in nematic liquid crystals,” Opt. Lett. 34(9), 1414–1416 (2009).
[Crossref]

D. Lu, W. Hu, and Q. Guo, “The relation between optical beam propagation in free space and in strongly nonlocal nonlinear media,” Europhys. Lett. 86(4), 44004 (2009).
[Crossref]

D. Lu and W. Hu, “Theory of multibeam interactions in strongly nonlocal nonlinear media,” Phys. Rev. A 80(5), 053818 (2009).
[Crossref]

2008 (2)

D. Lu, W. Hu, Y. Zheng, Y. Liang, L. Cao, S. Lan, and Q. Guo, “Self-induced fractional Fourier transform and revivable higher-order spatial solitons in strongly nonlocal nonlinear media,” Phys. Rev. A 78(4), 043815 (2008).
[Crossref]

D. Deng and Q. Guo, “Propagation of Laguerre-Gaussian beams in nonlocal nonlinear media,” J. Opt. A: Pure Appl. Opt. 10(3), 035101 (2008).
[Crossref]

2007 (2)

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. 98(21), 213901 (2007).
[Crossref]

D. Buccoliero, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98(5), 053901 (2007).
[Crossref]

2006 (2)

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2(11), 769–774 (2006).
[Crossref]

C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, and O. Cohen, “Two-dimensional multipole solitons in nonlocal nonlinear media,” Opt. Lett. 31(22), 3312–3314 (2006).
[Crossref]

2005 (1)

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref]

2004 (1)

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69(1), 016602 (2004).
[Crossref]

2003 (2)

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
[Crossref]

N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68(3), 036614 (2003).
[Crossref]

2002 (2)

J. Wyller, W. Krolikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66(6), 066615 (2002).
[Crossref]

M. Peccianti, K. A. Brzda’kiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27(16), 1460–1462 (2002).
[Crossref]

2001 (1)

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64(1), 016612 (2001).
[Crossref]

2000 (1)

W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: exact solutions,” Phys. Rev. E 63(1), 016610 (2000).
[Crossref]

1999 (1)

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286(5444), 1518–1523 (1999).
[Crossref]

1998 (1)

1997 (2)

M. Shih and M. Segev, “Three-dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. 78(13), 2551–2554 (1997).
[Crossref]

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276(5318), 1538–1541 (1997).
[Crossref]

1994 (1)

1987 (1)

M. Newstein and K. Lin, “Laguerre-Gaussian periodically focusing beams in a quadratic index medium,” IEEE J. Quantum Electron. 23(5), 481–482 (1987).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013).

Ahmed, N.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

Alfassi, B.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. 98(21), 213901 (2007).
[Crossref]

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2(11), 769–774 (2006).
[Crossref]

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
[Crossref]

M. Peccianti, K. A. Brzda’kiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27(16), 1460–1462 (2002).
[Crossref]

Babiker, M.

Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-angular-momentum mode selection by rotationally symmetric superposition of chiral states with application to electron vortex beams,” Phys. Rev. Lett. 119(9), 094802 (2017).
[Crossref]

S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: Beams with orbital angular momentum,” Rev. Mod. Phys. 89(3), 035004 (2017).
[Crossref]

Bang, O.

N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68(3), 036614 (2003).
[Crossref]

J. Wyller, W. Krolikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66(6), 066615 (2002).
[Crossref]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64(1), 016612 (2001).
[Crossref]

W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: exact solutions,” Phys. Rev. E 63(1), 016610 (2000).
[Crossref]

Bao, C.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

Belic, M. R.

Bian, K.

J. Chen, F. Zhang, K. Bian, C. Jiang, W. Hu, and D. Lu, “Dynamics of shape-invariant rotating beams in linear media with harmonic potentials,” Phys. Rev. A 99(3), 033808 (2019).
[Crossref]

Biloshytskyi, V.

V. Biloshytskyi, A. Oliinyk, P. Kruglenko, A. Desyatnikov, and A. Yakimenko, “Vortex nucleation in nonlocal nonlinear media,” Phys. Rev. A 99(4), 043835 (2019).
[Crossref]

Brzda’kiewicz, K. A.

Bu, H.-X.

Z.-J. Yang, S.-M. Zhang, X.-L. Li, Z.-G. Pang, and H.-X. Bu, “High-order revivable complex-valued hyperbolic-sine-Gaussian solitons and breathers in nonlinear media with a spatial nonlocality,” Nonlinear Dyn. 94(4), 2563–2573 (2018).
[Crossref]

Buccoliero, D.

D. Buccoliero, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98(5), 053901 (2007).
[Crossref]

Buryak, A. V.

Cai, Y.

Cao, L.

D. Lu, W. Hu, Y. Zheng, Y. Liang, L. Cao, S. Lan, and Q. Guo, “Self-induced fractional Fourier transform and revivable higher-order spatial solitons in strongly nonlocal nonlinear media,” Phys. Rev. A 78(4), 043815 (2008).
[Crossref]

Cao, Y.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

Carmon, T.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref]

Carpenter, J.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10(1), 1865 (2019).
[Crossref]

Chen, H.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10(1), 1865 (2019).
[Crossref]

Chen, J.

J. Chen, F. Zhang, K. Bian, C. Jiang, W. Hu, and D. Lu, “Dynamics of shape-invariant rotating beams in linear media with harmonic potentials,” Phys. Rev. A 99(3), 033808 (2019).
[Crossref]

Chen, M.

S. Zeng, M. Chen, T. Zhang, W. Hu, and D. Lu, “Analytical modeling of soliton interactions in a nonlocal nonlinear medium analogous to gravitational force,” Phys. Rev. A 97(1), 013817 (2018).
[Crossref]

Chen, Y.

Y. Chen and G. Liang, “Rotating vortex clusters nested in Gaussian envelope in nonlocal nonlinear media,” Opt. Commun. 449, 69–72 (2019).
[Crossref]

Chen, Z.

Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75(8), 086401 (2012).
[Crossref]

Cheng, W.

Chi, S.

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69(1), 016602 (2004).
[Crossref]

Christodoulides, D. N.

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Figures (8)

Fig. 1.
Fig. 1. (a)–(e) The evolution of intensity pattern of LG soliton array with tangential velocity parameter $\xi =0$. (f) and (g) The normalized intensity distribution in the transverse plane and $x$-direction, respectively, at $z=0.5T$. The total number of the constituent solitons is $N_t=6$. Parameters: $P_0=P_c$, $r=9w_0$, $n=1$, $m=2$.
Fig. 2.
Fig. 2. The evolution of intensity pattern of LG soliton array with different tangential velocity parameters. $\xi =0.5$ for (a)–(e), $\xi =1$ for (f)–(j), $\xi =2$ for (k)–(o). The beginning of each arrow points to the position of soliton $A$, the end of each arrow indicates the direction of rotation. Common parameters: $P_0=P_c$, $N_t=6$, $r=9w_0$, $n=1$, $m=2$.
Fig. 3.
Fig. 3. $C$ (solid line) and $C'$ (dashed line) are taken as examples to illustrate the dynamic changes in the propagation process of multiple solitons. (a) Three-dimensional propagation trajectories of $C$ and $C'$. (b) Projection trajectories of $C$ and $C'$ in the $x$-$z$ plane. (c) Projection trajectories of $C$ and $C'$ in the $y$-$z$ plane. (d) Projection trajectories of $C$ and $C'$ in the $x$-$y$ plane. Parameters: $P_0=P_c$, $\xi =0.5$, $r=9w_0$.
Fig. 4.
Fig. 4. (a)–(c) The projection trajectories of the rotating LG solitons with velocity parameter $\xi =0.5$. (a) $x$-$z$ plane, (b) $y$-$z$ plane, (c) $x$-$y$ plan. Solid red, green, blue lines represent $A, B, C$, respectively. Dashed red, green, blue lines represent $A', B', C'$, respectively. (d)–(f) and (g)–(i) are the same as (a)–(c) except that the velocity parameter are taken as $\xi =1$ and $\xi =2$, respectively. Parameters: $P_0=P_c$, $r=9w_0$.
Fig. 5.
Fig. 5. (a)–(c) Evolution of the rotating angle versus propagation distance without taking initial azimuth angle into account. (d)–(f) Evolution of the angular velocity $\omega (z)$ (green dashed line, left ordinate) and the center distance $d(z)$ (blue dash-dotted line, right ordinate) for different tangential velocity parameters.
Fig. 6.
Fig. 6. The examples of rotating LG soliton arrays with different negative tangential velocity parameters. (a) $\xi =-0.5$, (b) $\xi =-1$, (c) $\xi =-2$. Surrounded by the dashed circles represent the same soliton at different propagation positions in each row, the arrows indicate the direction of rotation. Parameters: $P_0=P_c$, $N_t=9$, $n=0$, $m=2$.
Fig. 7.
Fig. 7. The examples of rotating LG soliton arrays with the same tangential velocity parameter $\xi =1$. Surrounded by the dashed circles represent the same soliton at different propagation positions in each row, the arrows indicate the direction of rotation. (a) Ring-like soliton array with $n=2$, $m=1$, $N_t=10$; (b) rhombus-like soliton array with $n=4$, $m=2$, $N_t=8$; (c) pentagram-like soliton array with $n=2$, $m=2$, $N_t=10$. The input power is taken as $P_0=P_c$.
Fig. 8.
Fig. 8. The examples of rotating LG soliton arrays with different tangential velocity parameters. $\xi =0.7$ for (a), $\xi =1$ for (b), $\xi =1.5$ for (c). Parameters: $P_0=P_c$, $N=6$, $r=12w_0$.

Equations (37)

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2 i k A z + Δ A + 2 k 2 δ n n 0 A = 0 ,
δ n n 2 P 0 ( R 0 + 1 2 R 0 r 2 ) ,
2 i k A z + Δ A k 2 γ 2 r 2 P 0 A + 2 n 2 k 2 R 0 P 0 n 0 A = 0 ,
A ( r , z ) = Φ ( r , z ) exp ( i k n 2 R 0 P 0 n 0 z ) ,
2 i k Φ z + Δ Φ k 2 γ 2 r 2 P 0 Φ = 0.
Φ n m = C n m w ( z ) [ r w ( z ) ] m L n m [ r 2 w 2 ( z ) ] exp { r 2 2 w 2 ( z ) + i [ c ( z ) r 2 + ( 2 n + m + 1 ) θ ( z ) + m φ ] }
w ( z ) = w 0 ( cos 2 z s + P c P 0 sin 2 z s ) 1 / 2
w n m ( z ) = 2 n + m + 1 w ( z )
c ( z ) = k ( P c / P 0 1 ) sin ( 2 z s ) 4 z p [ cos 2 z s + ( P c / P 0 ) sin 2 z s ] ,
θ ( z ) = arctan ( P c P 0 tan z s ) ,
Φ n m e , o = C n m e , o w ( z ) [ r w ( z ) ] m L n m [ r 2 w 2 ( z ) ] exp { r 2 2 w 2 ( z ) + i [ c ( z ) r 2 + ( 2 n + m + 1 ) θ ( z ) ] } [ cos ( m φ ) sin ( m φ ) ] ,
d 2 r c ( z ) d z 2 + r c ( z ) z p 2 = 0.
r c ( z ) = r c ( 0 ) cos z s + z p r c ( 0 ) sin z s .
Φ ± ( r , z ) = Φ ( r ± r c ( z ) , z ) exp [ i u ( z ) r + i ϕ ( z ) ] ,
u ( z ) = k r c ( z ) ,
ϕ ( z ) = k 2 [ r c 2 ( z ) z p 2 r c 2 ( z ) ] .
Φ n m ( r , z ) = C 0 j = 1 N Φ n m ( j ) ( r , z ) ,
Φ n m ( j ) ( r , z ) = Φ n m ( j ) [ r r c ( z ) , z ] exp { i k ( r c ( 0 ) z p sin z s + r c ( 0 ) cos z s ) r + i [ k 4 ( r c 2 ( 0 ) z p z p r c 2 ( 0 ) ) sin ( 2 z s ) k 2 r c ( 0 ) r c ( 0 ) cos ( 2 z s ) ] } ,
x c j ( z ) = c x j cos z s + t x j z p sin z s ,
y c j ( z ) = c y j cos z s + t y j z p sin z s ,
u x j ( z ) = k c x j z p sin z s + k t x j cos z s ,
u y j ( z ) = k c y j z p sin z s + k t y j cos z s ,
ϕ j ( z ) = k 4 [ c x j 2 + c y j 2 z p z p ( t x j 2 + t y j 2 ) ] sin ( 2 z s ) k 2 ( c x j t x j + c y j t y j ) cos ( 2 z s ) .
c x j = r cos φ 0 j ,     c y j = r sin φ 0 j ,
t x j = ξ r sin φ 0 j z p ,     t y j = ξ r cos φ 0 j z p ,
Φ A , B , C = Φ n m ( j = 1 , 2 , 3 ) ( x , y , z ; r ) ,
Φ A , B , C = Φ n m ( j = 1 , 2 , 3 ) ( x , y , z ; r 2 ) ,
S A B C ( z ) = 3 3 r 2 4 ( cos 2 z s + ξ 2 sin 2 z s ) .
S m i n ( m a x ) = 3 3 ξ 2 r 2 4
x c j ( z ) = r 2 [ ( 1 ξ ) cos ( φ 0 j z s ) + ( 1 + ξ ) cos ( φ 0 j + z s ) ]
y c j ( z ) = r 2 [ ( 1 ξ ) sin ( φ 0 j z s ) + ( 1 + ξ ) sin ( φ 0 j + z s ) ]
( ξ 2 cos 2 φ 0 j + sin 2 φ 0 j ) x 2 + ( ξ 2 sin 2 φ 0 j + cos 2 φ 0 j ) y 2 + [ ( ξ 2 1 ) sin ( 2 φ 0 j ) ] x y = ξ 2 r 2 .
x 2 + y 2 = r 2
Ω j ( z ) = arctan ( sin φ 0 j cos z s + ξ cos φ 0 j sin z s cos φ 0 j cos z s ξ sin φ 0 j sin z s )
ω j ( z ) = ξ z p ( cos 2 z s + ξ 2 sin 2 z s ) .
d A , B , C ( z ) = r ( cos 2 z s + ξ 2 sin 2 z s ) 1 / 2 ,
S ( z ) = 3 3 r 2 2 ( cos 2 z s + ξ 2 sin 2 z s ) .

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