Abstract

We report on a novel class of higher-order Bessel-Gauss beams in which the well-known Bessel-Gauss beam is the fundamental mode and the azimuthally symmetric Laguerre-Gaussian beams are special cases. We find these higher-order Bessel-Gauss beams by superimposing decentered Hermite-Gaussian beams. We show analytically and experimentally that these higher-order Bessel-Gauss beams resemble higher-order eigenmodes of optical resonators consisting of aspheric mirrors. This work is relevant for the many applications of Bessel-Gauss beams in particular the more recently proposed high-intensity Bessel-Gauss enhancement cavities for strong-field physics applications.

© 2012 Optical Society of America

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References

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  1. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microw. Opt. Acoust. 2, 105–112 (1978).
    [Crossref]
  2. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491– 495 (1987).
    [Crossref]
  3. F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
    [Crossref]
  4. T. A Planchon, L. Gao, D. E Milkie, M. W Davidson, J. A Galbraith, C. G Galbraith, and Eric Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
    [Crossref] [PubMed]
  5. M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
    [Crossref]
  6. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
    [Crossref] [PubMed]
  7. D. Li and K. Imasaki, “Vacuum laser-driven acceleration by two slits-truncated Bessel beams,” Appl. Phys. Lett. 87, 091106 (2005).
    [Crossref]
  8. J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis,” J. Opt. Soc. Am. A 20, 2113–2122 (2003).
    [Crossref]
  9. A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: analytical description and experiment,” J. Opt. Soc. Am. A 18, 1986–1992 (2001).
    [Crossref]
  10. B. Ma, F. Wu, W. Lu, and J. Pu, “Nanosecond zero-order pulsed Bessel beam generated from unstable resonator based on an axicon,” Opt. Laser Technol. 42, 941–944 (2010).
    [Crossref]
  11. W. P. Putnam, D. N. Schimpf, G. Abram, and F. X. Kärtner, “Bessel-Gauss beam enhancement cavities for high-intensity applications,” Opt. Express 20, 24429–24443 (2012).
    [Crossref]
  12. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43(6), 1155–1166 (1996).
  13. R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higherorder Bessel beams,” Opt. Express 17, 23389–23395 (2009).
    [Crossref]
  14. C. Palma, “Decentered Gaussian beams, ray bundles, and Bessel-Gauss beams,” Appl. Opt. 36, 1116–1120 (1997).
    [Crossref] [PubMed]
  15. A. R. Al-Rashed and B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt. 34, 6819–6825 (1995).
    [Crossref] [PubMed]
  16. A. R. Al-Rashed, “Spatial and temporal modes of resonators with dispersive phase-conjugate mirrors,” PhD Thesis (1997).
  17. S. A. Collins, “Lens-System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [Crossref]
  18. G. Ryshik, Tables of Series, Products and Integrals (Verlag Harri Deutsch, 1981).
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    [Crossref]
  20. L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett. 23, 409–411 (1998).
    [Crossref]
  21. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21, 53–58 (2004).
    [Crossref]
  22. A. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron. 4, 460–465 (1968).
    [Crossref]
  23. A. E. Siegman, Lasers (University Science Books, 1986).
  24. O. Brzobohaty, T. Cizmar, and P. Zemanek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express 16, 12688–12700 (2008).
    [Crossref] [PubMed]

2012 (2)

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[Crossref]

W. P. Putnam, D. N. Schimpf, G. Abram, and F. X. Kärtner, “Bessel-Gauss beam enhancement cavities for high-intensity applications,” Opt. Express 20, 24429–24443 (2012).
[Crossref]

2011 (1)

T. A Planchon, L. Gao, D. E Milkie, M. W Davidson, J. A Galbraith, C. G Galbraith, and Eric Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref] [PubMed]

2010 (2)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

B. Ma, F. Wu, W. Lu, and J. Pu, “Nanosecond zero-order pulsed Bessel beam generated from unstable resonator based on an axicon,” Opt. Laser Technol. 42, 941–944 (2010).
[Crossref]

2009 (1)

2008 (1)

2005 (1)

D. Li and K. Imasaki, “Vacuum laser-driven acceleration by two slits-truncated Bessel beams,” Appl. Phys. Lett. 87, 091106 (2005).
[Crossref]

2004 (1)

2003 (1)

2002 (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

2001 (1)

1998 (1)

1997 (1)

1996 (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43(6), 1155–1166 (1996).

1995 (1)

1987 (2)

H. F. Johnson, “An improved method for computing a discrete hankel transform,” Comp. Phys. Comm. 43, 181–202 (1987).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491– 495 (1987).
[Crossref]

1978 (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microw. Opt. Acoust. 2, 105–112 (1978).
[Crossref]

1970 (1)

1968 (1)

A. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron. 4, 460–465 (1968).
[Crossref]

Abram, G.

Al-Rashed, A. R.

A. R. Al-Rashed and B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt. 34, 6819–6825 (1995).
[Crossref] [PubMed]

A. R. Al-Rashed, “Spatial and temporal modes of resonators with dispersive phase-conjugate mirrors,” PhD Thesis (1997).

Arnold, C. B.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[Crossref]

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43(6), 1155–1166 (1996).

Betzig, Eric

T. A Planchon, L. Gao, D. E Milkie, M. W Davidson, J. A Galbraith, C. G Galbraith, and Eric Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref] [PubMed]

Brzobohaty, O.

Chávez-Cerda, S.

Chen, M.

Chen, W.

Cizmar, T.

Collins, S. A.

Davidson, M. W

T. A Planchon, L. Gao, D. E Milkie, M. W Davidson, J. A Galbraith, C. G Galbraith, and Eric Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref] [PubMed]

Dholakia, K.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Dudley, A.

Duocastella, M.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[Crossref]

Fahrbach, F. O.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

Forbes, A.

Fox, A.

A. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron. 4, 460–465 (1968).
[Crossref]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43(6), 1155–1166 (1996).

Galbraith, C. G

T. A Planchon, L. Gao, D. E Milkie, M. W Davidson, J. A Galbraith, C. G Galbraith, and Eric Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref] [PubMed]

Galbraith, J. A

T. A Planchon, L. Gao, D. E Milkie, M. W Davidson, J. A Galbraith, C. G Galbraith, and Eric Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref] [PubMed]

Gao, L.

T. A Planchon, L. Gao, D. E Milkie, M. W Davidson, J. A Galbraith, C. G Galbraith, and Eric Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref] [PubMed]

Garcés-Chávez, V.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491– 495 (1987).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491– 495 (1987).
[Crossref]

Guizar-Sicairos, M.

Gutiérrez-Vega, J. C.

Huang, M.

Huang, W.

Imasaki, K.

D. Li and K. Imasaki, “Vacuum laser-driven acceleration by two slits-truncated Bessel beams,” Appl. Phys. Lett. 87, 091106 (2005).
[Crossref]

Johnson, H. F.

H. F. Johnson, “An improved method for computing a discrete hankel transform,” Comp. Phys. Comm. 43, 181–202 (1987).
[Crossref]

Kärtner, F. X.

Katranji, E. G.

Khilo, A. N.

Khilo, N.

Li, D.

D. Li and K. Imasaki, “Vacuum laser-driven acceleration by two slits-truncated Bessel beams,” Appl. Phys. Lett. 87, 091106 (2005).
[Crossref]

Li, T.

A. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron. 4, 460–465 (1968).
[Crossref]

Lu, W.

B. Ma, F. Wu, W. Lu, and J. Pu, “Nanosecond zero-order pulsed Bessel beam generated from unstable resonator based on an axicon,” Opt. Laser Technol. 42, 941–944 (2010).
[Crossref]

Ma, B.

B. Ma, F. Wu, W. Lu, and J. Pu, “Nanosecond zero-order pulsed Bessel beam generated from unstable resonator based on an axicon,” Opt. Laser Technol. 42, 941–944 (2010).
[Crossref]

McGloin, D.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Milkie, D. E

T. A Planchon, L. Gao, D. E Milkie, M. W Davidson, J. A Galbraith, C. G Galbraith, and Eric Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref] [PubMed]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491– 495 (1987).
[Crossref]

Palma, C.

Planchon, T. A

T. A Planchon, L. Gao, D. E Milkie, M. W Davidson, J. A Galbraith, C. G Galbraith, and Eric Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref] [PubMed]

Pu, J.

B. Ma, F. Wu, W. Lu, and J. Pu, “Nanosecond zero-order pulsed Bessel beam generated from unstable resonator based on an axicon,” Opt. Laser Technol. 42, 941–944 (2010).
[Crossref]

Putnam, W. P.

Rodríguez-Masegosa, R.

Rohrbach, A.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

Ryshik, G.

G. Ryshik, Tables of Series, Products and Integrals (Verlag Harri Deutsch, 1981).

Ryzhevich, A. A.

Saleh, B. E. A.

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43(6), 1155–1166 (1996).

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43(6), 1155–1166 (1996).

Schimpf, D. N.

Sheppard, C. J. R.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microw. Opt. Acoust. 2, 105–112 (1978).
[Crossref]

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

Spagnolo Schirripa, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43(6), 1155–1166 (1996).

Vasilyeu, R.

Wilson, T.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microw. Opt. Acoust. 2, 105–112 (1978).
[Crossref]

Wu, F.

B. Ma, F. Wu, W. Lu, and J. Pu, “Nanosecond zero-order pulsed Bessel beam generated from unstable resonator based on an axicon,” Opt. Laser Technol. 42, 941–944 (2010).
[Crossref]

Yu, L.

Zemanek, P.

Zhu, Z.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

D. Li and K. Imasaki, “Vacuum laser-driven acceleration by two slits-truncated Bessel beams,” Appl. Phys. Lett. 87, 091106 (2005).
[Crossref]

Comp. Phys. Comm. (1)

H. F. Johnson, “An improved method for computing a discrete hankel transform,” Comp. Phys. Comm. 43, 181–202 (1987).
[Crossref]

IEEE J. Microw. Opt. Acoust. (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microw. Opt. Acoust. 2, 105–112 (1978).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron. 4, 460–465 (1968).
[Crossref]

J. Mod. Opt. (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43(6), 1155–1166 (1996).

J. Opt. Soc. Am. (1)

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

Laser Photon. Rev. (1)

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[Crossref]

Nat. Methods (1)

T. A Planchon, L. Gao, D. E Milkie, M. W Davidson, J. A Galbraith, C. G Galbraith, and Eric Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref] [PubMed]

Nat. Photonics (1)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

Nature (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Opt. Commun. (1)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491– 495 (1987).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

B. Ma, F. Wu, W. Lu, and J. Pu, “Nanosecond zero-order pulsed Bessel beam generated from unstable resonator based on an axicon,” Opt. Laser Technol. 42, 941–944 (2010).
[Crossref]

Opt. Lett. (1)

Other (3)

A. R. Al-Rashed, “Spatial and temporal modes of resonators with dispersive phase-conjugate mirrors,” PhD Thesis (1997).

G. Ryshik, Tables of Series, Products and Integrals (Verlag Harri Deutsch, 1981).

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (10)

Fig. 1
Fig. 1 Geometry for the superposition of generalized decentered Hermite-Gaussian beams.

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Fig. 2
Fig. 2 Absolute value of the azimuthally symmetric Bessel-Gauss beam at z=0 for l=0 and (a): m=0, (b): m=1, (c): m=2, and in the far-field (z= 4 · zR); (d): m=0, (e): m=1, (f): m=2.

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Fig. 3
Fig. 3 Absolute value of the superposition of higher-order Bessel-Gauss beams in the far-field, complementary to Fig. 2 (at z= 4 · zR) (a): (u0,1 + u0,−1); (b): (u1,1 + u1,−1); (c): (u2,1 + u2,−1); (d): (u0,2 + u0,−2); (e): (u1,2 + u1,−2); (f): (u2,2 + u2,−2).

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Fig. 4
Fig. 4 Beam propagation and transverse pattern (at z=0) for: (a) and (b) generalized higher-order (m=2, l=0) Bessel-Gauss beam; (c) and (d) ordinary higher-order (m=2, l=0) Bessel-Gauss beam; (e) and (f) modified higher-order (m=2, l=0) Bessel-Gauss beam; (g) and (h) azimuthally symmetric Laguerre-Gaussian beam of order p=1, respectively.

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Fig. 5
Fig. 5 Schematic of the optical resonator.

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Fig. 6
Fig. 6 (a–c) Absolute value of the amplitude of the fundamental (m=0) Bessel-Gauss beam, (d–f) m=1 higher-order Bessel-Gauss mode, and (d–f) m=2 higher-order Bessel-Gauss mode at distance 0, −(0.9 · R − L), and −(R − L) from the axicon.

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Fig. 7
Fig. 7 (a) Absolute value of the amplitude of the generalized Bessel-Gauss beam vs distance from the minimum Gaussian waist point, (b) and (c): patterns for the (m=1) and (m=2) azimuthally symmetric higher-order Bessel-Gauss beams, respectively.

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Fig. 8
Fig. 8 Schematic of the experimental setup.

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Fig. 9
Fig. 9 Signal of the photodiode. The labels correspond to the mode images in Fig. 10.

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Fig. 10
Fig. 10 (a)–(d): Experimentally obtained images of higher-order modes m= 0, 1, 2, 3, respectively. (e)–(h): corresponding numerical simulations.

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Equations (26)

Equations on this page are rendered with MathJax. Learn more.

h m n ( x , y , z = 0 ) = 1 w 0 H m ( 2 ( x − x d 0 ) w 0 ) H n ( 2 ( y − y d 0 ) w 0 ) × exp ( i k 2 q 0 [ ( x − x d 0 ) 2 + ( y − y d 0 ) 2 ] ) exp ( i k [ ε x 0 x + ε y 0 y ] ) ,
h m n ( x , y , L ) = 1 w exp ( i k L ) exp ( − i ϕ ⋅ ( m + n + 1 ) ) × H m ( 2 ( x − x d ) w ) H n ( 2 ( y − y d ) w ) × exp ( i k 2 q [ ( x − x d ) 2 + ( y − y d ) 2 ] ) exp ( i k [ ε x x + ε y y ] ) exp ( i φ )
( x d ε x ) = ( A B C D ) ( x d 0 ε x 0 ) ;                   ( y d ε y ) = ( A B C D ) ( y d 0 ε y 0 ) ;
φ = − k 2 [ C ( x d 0 x d + y d 0 y d ) + B ( ε x 0 ε x + ε y 0 ε y ) ] ;
q = A q 0 + B C q 0 + D ;
ϕ = arg ( A + B / q 0 ) .
v γ ( r , θ , L ) = 1 w exp ( i k L ) exp ( − i ϕ ( m + 1 ) ) H m ( 2 ( r cos ( θ − γ ) − r d ) w ) × exp ( i k 2 q ( r 2 + r d 2 − 2 r r d cos ( θ − γ ) ) ) exp ( i k ε r cos ( θ − γ ) ) exp ( i φ ) .
r d = x d 2 + y d 2 = A r d 0 + B ε 0
ε = ε x 2 + ε y 2 = C r d 0 + D ε 0
φ = − k 2 [ C r d 0 r d + B ε 0 ε ]
u m l ( r , θ , L ) = 2 π w exp ( i k L ) exp ( − i ϕ ( m + 1 ) ) exp ( i φ ) exp [ i k 2 q ( r 2 + r d 2 ) ] ℐ m l .
ℐ m l = 1 2 π ∫ 0 2 π d γ H m ( a cos ( θ − γ ) − b ) ⋅ exp ( i α cos ( θ − γ ) ) ⋅ exp ( i l γ ) ,
α = k r ( ε − ( r d / q ) ) ;                 a = 2 ⋅ r / w ;               b = 2 ⋅ r d / w .
ℐ 0 l = e i l θ ⋅ i l ⋅ J l ( α )
ℐ 1 l = e i l θ 2 ⋅ i l ⋅ [ 2 i a ⋅ ( ( − 1 ) l J 1 − l ( α ) + J 1 + l ( α ) ) − 4 b ⋅ J l ( α ) ]
ℐ 2 l = e i l θ 2 ⋅ i l ⋅ [ − 2 a 2 ⋅ ( ( − 1 ) l J 2 − l ( α ) + J 2 + l ( α ) ) − i 8 a b ( ( − 1 ) l J 1 − l ( α ) + J 1 + l ( α ) )
+ ( 4 a 2 + 8 b 2 − 4 ) J l ( α ) ]
ℐ 3 l = e i l θ 2 ⋅ i l ⋅ [ − i 2 a 3 ⋅ ( ( − 1 ) l J 3 − l ( α ) + J 3 + l ( α ) ) + 12 a 2 b ⋅ ( ( − 1 ) l J 2 − l ( α ) J 2 + l ( α ) ) + i ( 6 a 3 + 24 a b 2 − 12 a ) ⋅ ( ( − 1 ) l J 1 − l ( α ) + J 1 + l ( α ) ) + 2 ⋅ ( − 8 b 3 − 12 a 2 b + 12 b ) J l ( α ) ]
1 2 π ∫ 0 2 π d γ H 2 ⋅ p ( 2 ⋅ ( r / w ) ⋅ cos ( θ − γ ) ) = ( − 1 ) p ( 2 p ) ! p ! L p ( 2 r 2 / w 2 ) ,
∫ − 1 1 d t ( 1 − t 2 ) α − 1 2 H 2 p ( u ⋅ t ) = ( − 1 ) p π ⋅ ( 2 p ) ! Γ ( α + 1 2 ) Γ ( p + α + 1 ) L p α ( u )
( A B C D ) = ( 1 L 0 1 ) ( 1 0 − 2 / R 1 ) ( 1 L 0 1 ) .
L = R − r d 0 / ε .
q = − i L ⋅ R − L 2 .
cos 2 ( β ) = ( 1 + cos ( 2 β ) ) / 2 cos 3 ( β ) = ( 3 cos ( β ) + cos ( 3 β ) ) / 4 cos 4 ( β ) = ( 3 + 4 cos ( 2 β ) + cos ( 4 β ) ) / 8
e i l θ 2 π ∫ 0 2 π d β     e i α cos β cos ( n β ) e − i l β = e i l θ ⋅ i n + l 2 [ ( − 1 ) l J n − l ( α ) + J n + 1 ( α ) ] ,
J k ( t ) = 1 2 π i k ∫ 0 2 π d β           exp { i ( t cos β − k β ) } .

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