Abstract

Morphology-dependent resonance (MDR) of the optical forces for a particle illuminated by Airy beams is investigated with respect to its internal field distribution. We find the ring structures arising from the resonance transform significantly with the parametric evolution of Airy evanescent wave, and the interference of the internal waves have a great impact on the Q factor and the background of the resonant peak, but it’s not proper for Airy transmitted wave. The multiple reflections of the evanescent wave between the particle and the interface are also investigated, which show significant impacts on the region where the energy concentrate in.

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
    [CrossRef]
  25. S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997).
    [CrossRef]

2012 (1)

2011 (1)

2010 (1)

2009 (2)

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. 34(21), 3430–3432 (2009).
[CrossRef] [PubMed]

2008 (3)

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

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[CrossRef] [PubMed]

J. Ng and C. T. Chan, “Size-selective optical forces for microspheres using evanescent wave excitation of whispering gallery modes,” Appl. Phys. Lett. 92(25), 251109 (2008).
[CrossRef]

2007 (2)

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] [PubMed]

2006 (1)

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[CrossRef]

2002 (2)

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically Locked-In Colloidal Transport in an Array of Optical Tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

P. T. Leung, S. W. Ng, K. M. Pang, and K. M. Lee, “Morphology-dependent resonances in dielectric spheres with many tiny inclusions,” Opt. Lett. 27(20), 1749–1751 (2002).
[CrossRef] [PubMed]

2000 (1)

H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bisphere,” Phys. Rev. B 62(12), 7976–7997 (2000).
[CrossRef]

1999 (1)

1997 (2)

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385(6611), 45–47 (1997).
[CrossRef]

S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997).
[CrossRef]

1993 (1)

1992 (1)

1989 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[CrossRef]

1988 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[CrossRef]

1985 (2)

1978 (1)

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18(5), 2229–2233 (1978).
[CrossRef]

1970 (1)

A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Aabo, T.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[CrossRef] [PubMed]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[CrossRef]

Ashkin, A.

A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[CrossRef]

Baumgartl, J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

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

Bendix, P. M.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[CrossRef] [PubMed]

Bosanac, L.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[CrossRef] [PubMed]

Broky, J.

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

Chan, C. T.

J. Ng and C. T. Chan, “Size-selective optical forces for microspheres using evanescent wave excitation of whispering gallery modes,” Appl. Phys. Lett. 92(25), 251109 (2008).
[CrossRef]

Chang, R. K.

Chang, S.

S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997).
[CrossRef]

Christodoulides, D. N.

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] [PubMed]

Chylek, P.

Day, D.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Dholakia, K.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

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

Dogariu, A.

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

Gorodetsky, M. L.

Grier, D. G.

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically Locked-In Colloidal Transport in an Array of Optical Tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

Gu, M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Hannappel, G. M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Ilchenko, V. S.

Jimba, Y.

H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bisphere,” Phys. Rev. B 62(12), 7976–7997 (2000).
[CrossRef]

Jo, J. H.

S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997).
[CrossRef]

Johnson, B. R.

Jones, P. H.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[CrossRef]

Kawata, S.

Kiehl, J. T.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18(5), 2229–2233 (1978).
[CrossRef]

Kim, J. T.

S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997).
[CrossRef]

Ko, M. K. W.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18(5), 2229–2233 (1978).
[CrossRef]

Korda, P. T.

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically Locked-In Colloidal Transport in an Array of Optical Tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

Lee, K. M.

Lee, S. S.

S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997).
[CrossRef]

Leung, P. T.

Li, Z. J.

Mazilu, M.

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

Miyazaki, H.

H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bisphere,” Phys. Rev. B 62(12), 7976–7997 (2000).
[CrossRef]

Ng, J.

J. Ng and C. T. Chan, “Size-selective optical forces for microspheres using evanescent wave excitation of whispering gallery modes,” Appl. Phys. Lett. 92(25), 251109 (2008).
[CrossRef]

Ng, S. W.

Nieto-Vesperinas, M.

Nöckel, J. U.

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385(6611), 45–47 (1997).
[CrossRef]

Novitsky, A. V.

Novitsky, D. V.

Oddershede, L. B.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[CrossRef] [PubMed]

Pang, K. M.

Pendleton, J. D.

Pinnick, R. G.

Qian, S. X.

Saenz, J. J.

Saffari, N.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[CrossRef]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[CrossRef]

Shang, Q. C.

Siviloglou, G. A.

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] [PubMed]

Snow, J. B.

Stevenson, D. J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Stone, A. D.

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385(6611), 45–47 (1997).
[CrossRef]

Stride, E.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[CrossRef]

Sugiura, T.

Taylor, M. B.

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically Locked-In Colloidal Transport in an Array of Optical Tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

Tian, J. G.

Wu, Z. S.

Yang, Y.

Zang, W. P.

Zhao, Z. Y.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

J. Ng and C. T. Chan, “Size-selective optical forces for microspheres using evanescent wave excitation of whispering gallery modes,” Appl. Phys. Lett. 92(25), 251109 (2008).
[CrossRef]

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[CrossRef]

J. Appl. Phys. (2)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[CrossRef]

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

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

Lab Chip (1)

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Nano Lett. (1)

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[CrossRef] [PubMed]

Nat. Photonics (1)

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

Nature (1)

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385(6611), 45–47 (1997).
[CrossRef]

Opt. Commun. (1)

S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997).
[CrossRef]

Opt. Express (2)

Opt. Lett. (6)

Phys. Rev. A (1)

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18(5), 2229–2233 (1978).
[CrossRef]

Phys. Rev. B (1)

H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bisphere,” Phys. Rev. B 62(12), 7976–7997 (2000).
[CrossRef]

Phys. Rev. Lett. (3)

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

A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically Locked-In Colloidal Transport in an Array of Optical Tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

An Airy beam is incident from below with incident angle θ1. A spherical particle with radius a is located above the interface ( z>d ) .

Fig. 2
Fig. 2

Optical forces Fx (black line) and Fz (red line) as a function of particle radius: for Airy evanescent wave, θ1 = 0.9 rad, (a) d = a, n3 = 1.59, (b) d = a, n3 = 1.59 + 10−3i, (c) d = 2.1a, n3 = 1.59; (d) Airy transmitted wave, θ1 = 0.3 rad, d = a, n3 = 1.59; (e) enlarged plot of Fx of b18 in (a) with its resonant item l = 18.

Fig. 3
Fig. 3

Distributions of the electric field magnitude |E| of Airy evanescent wave incident upon a polystyrene spherical particle situating on the interface: (a) y-z plane (x = 0), (b) x-y plane (z = 0), (c) x-z plane (y = 0) are for b18 (α = 14.01); (d) x-z plane (y = 0) for the non-resonant case (α = 13.56). While d = a, n3 = 1.59, xc = yc = 0, zc = −(d + λ), θ1 = 0.9 rad. The unit of electric field magnitude is statvolt/cm and 1 statvolt/cm = 3 × 104 V/m.

Fig. 4
Fig. 4

Variations of the ring structures of the resonant peaks: b18, (a)–(c), θ1 = 0.9 rad with transmitted distance: (a) d = 1.3a, (b) d = 1.7a, (c) d = 2.1a; (d)–(f), d = 2a, with incident angle: (d) θ1 = 0.75 rad, (e) θ1 = 0.9 rad, (f) θ1 = 1.05 rad; (g) a17, θ1 = 0.9 rad, d = 2.1a, the above for Airy evanescent wave. (h) θ1 = 0.2 rad, d = a, (i) θ1 = 0.5 rad, d = 2a, for Airy transmitted wave b18.

Fig. 5
Fig. 5

The resonant peaks of the optical forces correspond to the internal wave distributions in Figs. 4(a)-(c), and 4(g), respectively.

Fig. 6
Fig. 6

Effects of multiple scattering of the Airy evanescent wave between the particle and the interface on the optical forces as a function of particle radius. The curves near the left arrow and near the right arrow represent Fx and Fz, respectively: the black lines denote the optical forces without reflection; the red, magenta, yellow, green, and blue lines represent the evanescent wave is reflected for once, twice, three, four and five times, respectively. While xc = yc = 0, n3 = 1.59, d = a, zc = −(d + λ), θ1 = 0.9 rad. The inset is the variation of the peaks of Fz versus the reflected times.

Fig. 7
Fig. 7

Effects of multiple scattering on the optical forces: as a function of beam center’s displacements xc, while yc = 0, a = 0.8 μm, other parameters are the same as in Fig. 6.

Equations (20)

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A= y ^ C i k 0 d s x d s y Φ( s x , s y )exp[ i f i ( x,y,z ) ],
Φ( s x , s y )exp[ i 3 ( x ¯ 0 3 s x 3 + y ¯ 0 3 s y 3 3 a 0 2 x ¯ 0 s x 3 a 0 2 y ¯ 0 s y ) a 0 ( x ¯ 0 2 s x 2 + y ¯ 0 2 s y 2 ) ].
s x = n x cos θ 1 n z sin θ 1 , s y = n y , s z = n x sin θ 1 + n z cos θ 1 , x ¯ 0 = n 1 k 0 x 0 , y ¯ 0 = n 1 k 0 y 0 .
E= i n 1 2 k 0 ××A,H=×A,
E ( i ) =C d s x d s y n x s ^ i + n y n z p ^ i 1 n z 2 Φ( s x , s y )exp[ i f i ( x,y,z ) ],
H ( i ) = n 1 C d s x d s y n y n z s ^ i n x p ^ i 1 n z 2 Φ( s x , s y )exp[ i f i ( x,y,z ) ],
s ^ i = n y x ^ + n x y ^ , p ^ i =( n x x ^ + n y y ^ ) n z ( 1 n z 2 ) z ^ .
T p = 2 n 1 n z n 1 ξ+ n 2 n z , T s = 2 n 1 n z n 1 n z + n 2 ξ .
E ( t ) =C d s x d s y n x T s s ^ t + n y n z T p p ^ t 1 n z 2 Φ( s x , s y )exp[ i f t ( x,y,z ) ],
H ( t ) = n 2 C d s x d s y n y n z T p s ^ t n x T s p ^ t 1 n z 2 Φ( s x , s y )exp[ i f t ( x,y,z ) ],
s ^ t = n y x ^ + n x y ^ , p ^ t =( n x x ^ + n y y ^ )ξ( n 1 / n 2 )( 1 n z 2 ) z ^ .
f t ( x,y,z )= n 1 k 0 [ n x ( x x c )+ n y ( y y c ) n z ( z c +d ) ]+ n 2 k 0 ( z+d )ξ.
A lm = A c d s x d s y Φ( s x , s y )exp[ ig( r 0 ) ] ( n x i n y ) m ( 1 n z 2 ) ( m+1 )/2 [ i n x β lm ( ξ ) T s + n y n z α lm ( ξ ) T p ] ,
B lm = B c d s x d s y Φ( s x , s y )exp[ ig( r 0 ) ] ( n x i n y ) m ( 1 n z 2 ) ( m+1 )/2 [ i n x α lm ( ξ ) T s + n y n z β lm ( ξ ) T p ] ,
A c =C i l+1 l( l+1 ) α 2 π( 2l+1 ) ( lm )! ( l+m )! , B c = n 2 C i l l( l+1 ) α 2 π( 2l+1 ) ( lm )! ( l+m )! ,
g( r 0 )= n 1 k 0 [ n x x c + n y y c + n z ( z c +d ) ]+ n 2 k 0 ξd.
α lm ( ξ )=ξ[ P l m+1 ( ξ )+( l+m )( lm+1 ) P l m1 ( ξ ) ]2m ( 1 ξ 2 ) 1/2 P l m ( ξ ),
β lm ( ξ )= P l m+1 ( ξ )( l+m )( lm+1 ) P l m1 ( ξ ).
a l = ψ l ( n ˜ α ) ψ l ( α ) n ˜ ψ l ( n ˜ α ) ψ l ( α ) n ˜ ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) ,
b l = n ˜ ψ l ( n ˜ α ) ψ l ( α ) ψ l ( n ˜ α ) ψ l ( α ) ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) n ˜ ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) .

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