Abstract

A type of chirped Airy Gaussian vortex (CAiGV) localized wave packets in a quadratic index medium are studied by solving the paraxial differential equation. For the first time, the propagation properties of spatiotemporal CAiGV light bullets in the quadratic index medium are demonstrated. Some typical examples of the obtained solutions are based on the temporal and spatial chirp parameters, the initial velocity, the distribution factor, and the topological charge. The radiation force of the spatial CAiGV wave packet on a Rayleigh dielectric particle has the periodically reversion and recovery abilities due to the quadratic potential. What we report here can obtain different radiation force trajectory and may have potential application in optical tweezing and bio-medical field.

© 2017 Optical Society of America

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2017 (2)

2016 (4)

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy-Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

B. Malomed, L. Torner, F. Wise, and D. Mihalache, “On multidimensional solitons and their legacy in contemporary atomic, molecular and optical physics,” J. Phys. B 49(17), 170502 (2016).
[Crossref]

Y. Peng, B. Chen, X. Peng, M. Zhou, L. Zhang, D. Li, and D. Deng, “Self-accelerating Airy-Ince-Gaussian and Airy-Helical-Ince-Gaussian light bullets in free space,” Opt. Express 24(17), 18973–18985 (2016).
[Crossref] [PubMed]

D. Bongiovanni, B. Wetzel, Y. Hu, Z. Chen, and R. Morandotti, “Optimal compression and energy confinement of optical Airy bullets,” Opt. Express 24(23), 26454–26463 (2016).
[Crossref] [PubMed]

2015 (4)

2014 (1)

2013 (1)

W. P. Zhong, M. R. Belić, and T. Huang, “Three-dimensional finite-energy Airy self-accelerating parabolic-cylinder light bullets,” Phys. Rev. A 88(3), 033824 (2013).
[Crossref]

2012 (2)

D. Mihalache, “Linear and nonlinear light bullets: Recent theoretical and experimental studies,” Rom. J. Phys. 57(1–2), 352–371 (2012).

D. Deng and H. Li, “Propagation of Airy-Gaussian beams,” Appl. Phys. B 106, (3)677–681 (2012).
[Crossref]

2011 (4)

2010 (3)

T. J. Eichelkraut, G. A. Siviloglou, I. M. Besieris, and D. N. Christodoulides, “Oblique Airy wave packets in bidispersive optical media,” Opt. Lett. 35(21), 3655–3657 (2010).
[Crossref] [PubMed]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4(2), 103–106 (2010).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

2009 (2)

D. Deng and Q. Guo, “Airy complex variable function Gaussian beams,” New. J. Phys. 11(10) 103029 (2009).
[Crossref]

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

2008 (3)

2007 (3)

2005 (3)

2003 (1)

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

1999 (1)

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82(23), 4631–4634 (1999).
[Crossref]

1996 (1)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

1995 (1)

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, (4)3254–3278 (1995).
[Crossref] [PubMed]

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]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Abdollahpour, D.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon. 2(11), 675–678 (2008).
[Crossref]

Belic, M.

Belic, M. R.

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Besieris, I. M.

Blair, S.

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, (4)3254–3278 (1995).
[Crossref] [PubMed]

Bongiovanni, D.

Brambilla, G.

Broky, J.

Chen, B.

Chen, C.

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy-Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

B. Chen, C. Chen, X. Peng, and D. Deng, “Propagation of Airy Gaussian vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc. Am. B 32, (1)173–178 (2015).
[Crossref]

Chen, H.

Chen, Z.

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4(2), 103–106 (2010).
[Crossref]

Christodoulides, D. N.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4(2), 103–106 (2010).
[Crossref]

T. J. Eichelkraut, G. A. Siviloglou, I. M. Besieris, and D. N. Christodoulides, “Oblique Airy wave packets in bidispersive optical media,” Opt. Lett. 35(21), 3655–3657 (2010).
[Crossref] [PubMed]

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008).
[Crossref] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Chu, R.

Conti, C.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

Deng, D.

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy-Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

Y. Peng, B. Chen, X. Peng, M. Zhou, L. Zhang, D. Li, and D. Deng, “Self-accelerating Airy-Ince-Gaussian and Airy-Helical-Ince-Gaussian light bullets in free space,” Opt. Express 24(17), 18973–18985 (2016).
[Crossref] [PubMed]

B. Chen, C. Chen, X. Peng, and D. Deng, “Propagation of Airy Gaussian vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc. Am. B 32, (1)173–178 (2015).
[Crossref]

D. Deng and H. Li, “Propagation of Airy-Gaussian beams,” Appl. Phys. B 106, (3)677–681 (2012).
[Crossref]

D. Deng, “Propagation properties of Airy-Gaussian beams in a quadratic-index medium,” Eur. Phys. J. D 65, 553–556 (2011).
[Crossref]

D. Deng and Q. Guo, “Airy complex variable function Gaussian beams,” New. J. Phys. 11(10) 103029 (2009).
[Crossref]

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon. 2(11), 675–678 (2008).
[Crossref]

Dogariu, A.

Efremidis, N. K.

Eichelkraut, T. J.

Fainman, Y.

Guan, C.

Guo, M.

Guo, Q.

D. Deng and Q. Guo, “Airy complex variable function Gaussian beams,” New. J. Phys. 11(10) 103029 (2009).
[Crossref]

Gutiérrez-Vega, J. C.

Harada, Y.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

Hu, Y.

Huang, T.

W. P. Zhong, M. R. Belić, and T. Huang, “Three-dimensional finite-energy Airy self-accelerating parabolic-cylinder light bullets,” Phys. Rev. A 88(3), 033824 (2013).
[Crossref]

Jedrkiewicz, O.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

Ke, Y.

Kim, H. C.

Kivshar, Y. S.

Koleskik, M.

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Levy, U.

Li, C.

Li, D.

Li, H.

D. Deng and H. Li, “Propagation of Airy-Gaussian beams,” Appl. Phys. B 106, (3)677–681 (2012).
[Crossref]

Li, P.

Li, Y.

Lin, K.

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]

Liu, S.

Liu, W.

Liu, X.

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82(23), 4631–4634 (1999).
[Crossref]

Liu, Y.

Lou, C.

Luo, H.

Malomed, B.

B. Malomed, L. Torner, F. Wise, and D. Mihalache, “On multidimensional solitons and their legacy in contemporary atomic, molecular and optical physics,” J. Phys. B 49(17), 170502 (2016).
[Crossref]

Malomed, B. A.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, (5)R53 (2005).
[Crossref]

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon. 2(11), 675–678 (2008).
[Crossref]

McLeod, R.

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, (4)3254–3278 (1995).
[Crossref] [PubMed]

Mihalache, D.

B. Malomed, L. Torner, F. Wise, and D. Mihalache, “On multidimensional solitons and their legacy in contemporary atomic, molecular and optical physics,” J. Phys. B 49(17), 170502 (2016).
[Crossref]

D. Mihalache, “Linear and nonlinear light bullets: Recent theoretical and experimental studies,” Rom. J. Phys. 57(1–2), 352–371 (2012).

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, (5)R53 (2005).
[Crossref]

Miroshnichenko, A. E.

Moloney, J. V.

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Morandotti, R.

Neshev, D. N.

Newstein, M.

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]

Nezhad, M.

Pang, L.

Papazoglou, D. G.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Peng, X.

Peng, Y.

Y. Peng, B. Chen, X. Peng, M. Zhou, L. Zhang, D. Li, and D. Deng, “Self-accelerating Airy-Ince-Gaussian and Airy-Helical-Ince-Gaussian light bullets in free space,” Opt. Express 24(17), 18973–18985 (2016).
[Crossref] [PubMed]

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy-Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

Piskarskas, A.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

Polynkin, P.

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Qian, L. J.

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82(23), 4631–4634 (1999).
[Crossref]

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4(2), 103–106 (2010).
[Crossref]

Shadrivov, I. V.

Shen, Y.

Shi, J.

Siviloglou, G. A.

Song, D.

Suntsov, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Torner, L.

B. Malomed, L. Torner, F. Wise, and D. Mihalache, “On multidimensional solitons and their legacy in contemporary atomic, molecular and optical physics,” J. Phys. B 49(17), 170502 (2016).
[Crossref]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, (5)R53 (2005).
[Crossref]

Trapani, P. Di

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Trillo, S.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

Trull, J.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
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Tsai, C. H.

Tzortzakis, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Valiulis, G.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
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Phys. Rev. A (2)

W. P. Zhong, M. R. Belić, and T. Huang, “Three-dimensional finite-energy Airy self-accelerating parabolic-cylinder light bullets,” Phys. Rev. A 88(3), 033824 (2013).
[Crossref]

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[Crossref] [PubMed]

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X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82(23), 4631–4634 (1999).
[Crossref]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
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Figures (8)

Fig. 1
Fig. 1

(a)–(c) the intensity profiles [I = |A(T,Z)|2] of finite-energy chirp Airy pulses at various propagation distances Z. (d)-(f) the pulse intensity distributions. βt = −1 in (a) and (d), βt = 0 in (b) and (e), and βt = 1 in (c) and (f). at = 0.3.

Fig. 2
Fig. 2

Snapshots describing the propagation evolution of the second order CAiGV light bullets with χ0 = 0.3, ax = ay = at = 0.3, βt = βs = 0.

Fig. 3
Fig. 3

(a) and (d) the snapshots describing the propagation evolution of the second order CAiGV light bullets with βt = 1. (b) and (e) the analytical results of the second order CAiGV distribution in Eq. (5) [i.e. Eq. (12)]. (c) and (f) the numerical simulations of the second order CAiGV distribution in Eq. (5) by using the fast Fourier transform method. ax = ay = at = 0.3, βs = 0, χ0 = 0.3.

Fig. 4
Fig. 4

Snapshots describing the propagation evolution of the second order CAiGV light bullets (a)-(c) and the first order CAiGV light bullets (d)-(f) with βt = −1, βs = 0, χ0 = 0.3, ax = ay = at = 0.3.

Fig. 5
Fig. 5

(a)-(c) the numerical demonstrations of the beam center of the first order CAiGV distributions with different χ0 and different initial velocities with ax = ay = 0.1. (d)-(f) the snapshots of the first order CAiGV light bullets in the quadratic index medium with χ0 = 0.15, ax = ay = at = 0.3, Z = 1 and different initial velocities. βt = βs = 0. (a) cx = cy = 2, (b) cx = cy = 0, (c) cx = cy = −2. (d) cx = 2, cy = −2, and ct = −2. (e) cx = cy = ct = 0. (f) cx = −2, cy = 2, and ct = 2.

Fig. 6
Fig. 6

Snapshots describing the propagation evolution of the first order CAiGV light bullets with βt = −1, χ0 = 0.3, ax = ay = at = 0.3, and different spatial chirp parameters βs = 0.5 in (a)-(c) and βs = −0.5 in (d)-(f).

Fig. 7
Fig. 7

Propagation of the first order spatial CAiGV wave packets with χ0 = 0.3, ax = ay = 0.3. (a) βs = 0.5 and cx = cy = 0, (b) βs = 0.5 and cx = cy = 3, (c) βs = −0.5 and cx = cy = 0, (d) βs = −0.5 and cx = cy = −3.

Fig. 8
Fig. 8

Distributions of the intensity (a1)-(f1), scattering force (a2)-(f2), and transverse gradient force (a3)-(f3) of the first order CAiGV distributions for the particles at different positions with χ0 = 0.15. Other parameters are the same as those in Fig. 7(b).

Equations (21)

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2 i ϕ z + 1 k ( 2 ϕ x 2 + 2 ϕ y 2 ) k g 2 ϕ τ 2 k α 2 ( x 2 + y 2 ) ϕ = 0 ,
2 i ϕ z + 2 ϕ X 2 + 2 ϕ Y 2 S i g n [ k g ] L d i f f L d i s p 2 ϕ T 2 L d i f f 2 α 2 ( X 2 + Y 2 ) ϕ = 0 .
2 i ϕ Z + 2 ϕ X 2 + 2 ϕ Y 2 + 2 ϕ T 2 α 2 ( X 2 + Y 2 ) ϕ = 0 .
2 i A ( T , Z ) Z + 2 A ( T , Z ) T 2 = 0 ,
2 i φ ( X , Y , Z ) Z + 2 φ ( X , Y , Z ) X 2 + 2 φ ( X , Y , Z ) Y 2 α 2 ( X 2 + Y 2 ) φ ( X , Y , Z ) = 0 .
A ( T , Z ) = b A i [ b ( T + b 4 Z 2 + c t Z i a t Z ) ] exp [ a t b ( T + b 2 Z 2 + c t Z ) ] × exp [ i b ( β t T 2 c t T b 2 T Z + a t 2 2 Z c t 2 2 Z c t b 2 Z 2 b 2 12 Z 3 ) ] ,
φ m ( X , Y , 0 ) = A i ( X ) A i ( Y ) e a x X + a y Y χ 0 2 ( X 2 + Y 2 ) e i c x X + i c y Y + i β s ( X 2 + Y 2 ) ( X + i Y ) m ,
φ m ( X , Y , Z ) = α csc ( α Z ) 2 ( η + β s + i χ 0 2 ) f ( X , Y ) j = 0 m C m J i j m j a X m j j a Y j × A i [ μ ( X ) ] A i [ μ ( Y ) ] exp [ ν ( X ) + ν ( Y ) ] ,
μ ( S ) = B 2 16 B 2 ( a s + i c s i K S ) ,
ν ( S ) = B 4 ( a s + i c s i K S ) 2 + B 2 8 ( a s + i c s i K S ) B 3 96 .
φ 1 ( X , Y , Z ) = α csc ( α Z ) 2 ( η + β s + i χ 0 2 ) f ( X , Y ) { [ J ( X ) + i J ( Y ) ] A i [ μ ( X ) ] A i [ μ ( Y ) ] B 2 [ A i [ μ ( X ) ] A i [ μ ( Y ) ] + i A i [ μ ( X ) ] A i [ μ ( Y ) ] ] } exp [ ν ( X ) + ν ( Y ) ] ,
φ 2 ( X , Y , Z ) = α csc ( α Z ) 2 ( η + β s + i χ 0 2 ) f ( X , Y ) ( P 1 + P 2 + P 3 ) exp [ ν ( X ) + ν ( Y ) ] ,
J ( S ) = B 2 ( a s + i c s i K S ) + B 2 8 ,
P 1 = [ R 2 ( X ) i B J ( Y ) ] A i [ μ ( X ) ] A i [ μ ( Y ) ] ,
P 2 = [ i B J ( X ) + R 2 ( Y ) ] A i [ μ ( X ) ] A i [ μ ( Y ) ] ,
P 3 = [ R 1 ( X ) R 1 ( Y ) + 2 i J ( X ) J ( Y ) ] A i [ μ ( X ) ] A i [ μ ( Y ) ] + i B 2 2 A i [ μ ( X ) ] A i [ μ ( Y ) ] ,
R 1 ( S ) = B 2 + 1 4 [ B 2 ( a s + i c s i K S ) 2 B 3 ( a s + i c s i K S ) + B 4 8 ] ,
R 2 ( S ) = B 3 8 + B 2 2 ( a s + i c s i K S ) .
ϕ m ( X , Y , T , Z ) = α csc ( α Z ) 2 ( η + β s + i χ 0 2 ) f ( X , Y ) A i [ b ( T + b 4 Z 2 + c t Z i a t Z ) ] × j = 0 m C m j i j m j a x m j j a y j A i [ μ ( X ) ] A i [ μ ( Y ) ] exp [ ν ( X ) + ν ( Y ) ] × exp [ i b ( β t T 2 c t T b 2 T Z + a t 2 2 Z c t 2 2 Z c t b 2 Z 2 b 2 12 Z 3 ) ] + exp [ α t b ( T + b 2 Z 2 + c t Z ) ] .
F s c a t ( X , Y , Z ) = n ( r ) ζ 0 c I ( X , Y , Z ) e z ,
F g r a d ( X , Y , Z ) = 2 π n ( r ) ρ 0 c I ( X , Y , Z ) ,

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