Abstract

In this work, the nonlinear parametric interaction of optical radiation in various transverse modes in a Raman-active medium is investigated both experimentally and theoretically. Verification of the orbital angular momentum algebra (OAM-algebra) [Strohaber et al., Opt. Lett. 37,3411 (2012)] was performed for high-order Laguerre Gaussian modes >1. It was found that this same algebra also describes the coherent transfer of OAM when Ince-Gaussian modes were used. New theoretical considerations extend the OAM-algebra to even and odd Laguerre Gaussian, and Hermite Gaussian beam modes through a change of basis. The results of this work provide details in the spatiotemporal synthesis of custom broadband pulses of radiation from Raman sideband generation.

© 2015 Optical Society of America

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References

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    [Crossref]
  2. J. Strohaber, C. Petersen, and C. J. G. J. Uiterwaal, “Efficient angular dispersion compensation in holographic generation of intense ultrashort paraxial beam modes,” Opt. Lett. 32(16), 2387–2389 (2007).
    [Crossref] [PubMed]
  3. A. Muthukrishnan and C. R. Stroud., “Entanglement of internal and external angular momenta of a single atom,” J. Opt. B Quantum Semiclassical Opt. 4(2), S73–S77 (2002).
    [Crossref]
  4. T. Meyrath, F. Schreck, J. Hanssen, C. Chuu, and M. Raizen, “A high frequency optical trap for atoms using Hermite-Gaussian beams,” Opt. Express 13(8), 2843–2851 (2005).
    [Crossref] [PubMed]
  5. K. C. Wright, L. S. Leslie, and N. P. Bigelow, “Optical control of the internal and external angular momentum of a Bose-Einstein condensate,” Phys. Rev. A 77(4), 441601 (2008).
    [Crossref]
  6. J. F. S. Brachmann, W. S. Bakr, J. Gillen, A. Peng, and M. Greiner, “Inducing vortices in a Bose-Einstein condensate using holographically produced light beams,” Opt. Express 19(14), 12984–12991 (2011).
    [Crossref] [PubMed]
  7. M. J. Padgett and L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31(1), 1–12 (1999).
    [Crossref]
  8. K. Ladavac and D. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12(6), 1144–1149 (2004).
    [Crossref] [PubMed]
  9. B. Neupane, F. Chen, W. Sun, D. T. Chiu, and G. Wang, “Tuning donut profile for spatial resolution in stimulated emission depletion microscopy,” Rev. Sci. Instrum. 84(4), 043701 (2013).
    [Crossref] [PubMed]
  10. Z. Wang, N. Zhang, and X.-C. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free-space optical communication,” Opt. Express 19(2), 482–492 (2011).
    [Crossref] [PubMed]
  11. G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
    [Crossref] [PubMed]
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  14. J. Strohaber, G. Kaya, N. Kaya, N. Hart, A. A. Kolomenskii, G. G. Paulus, and H. A. Schuessler, “In situ tomography of femtosecond optical beams with a holographic knife-edge,” Opt. Express 19(15), 14321–14334 (2011).
    [Crossref] [PubMed]
  15. J. Strohaber, M. Zhi, A. V. Sokolov, A. A. Kolomenskii, G. G. Paulus, and H. A. Schuessler, “Coherent transfer of optical orbital angular momentum in multi-order Raman sideband generation,” Opt. Lett. 37(16), 3411–3413 (2012).
    [Crossref] [PubMed]
  16. M. Zhi, K. Wang, X. Hua, H. Schuessler, J. Strohaber, and A. V. Sokolov, “Generation of femtosecond optical vortices by molecular modulation in a Raman-active crystal,” Opt. Express 21(23), 27750–27758 (2013).
    [PubMed]
  17. P. Hansinger, G. Maleshkov, I. L. Garanovich, D. V. Skryabin, D. N. Neshev, A. Dreischuh, and G. G. Paulus, “Vortex algebra by multiply cascaded four-wave mixing of femtosecond optical beams,” Opt. Express 22(9), 11079–11089 (2014).
    [Crossref] [PubMed]
  18. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21(5), 873–880 (2004).
    [Crossref] [PubMed]
  19. F. Ricci, W. Löffler, and M. P. van Exter, “Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer,” Opt. Express 20(20), 22961–22975 (2012).
    [Crossref] [PubMed]
  20. R. W. Boyd, Nonlinear Optics (Academic, 2003).
  21. I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
    [Crossref]

2014 (2)

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref] [PubMed]

P. Hansinger, G. Maleshkov, I. L. Garanovich, D. V. Skryabin, D. N. Neshev, A. Dreischuh, and G. G. Paulus, “Vortex algebra by multiply cascaded four-wave mixing of femtosecond optical beams,” Opt. Express 22(9), 11079–11089 (2014).
[Crossref] [PubMed]

2013 (3)

M. Zhi, K. Wang, X. Hua, H. Schuessler, J. Strohaber, and A. V. Sokolov, “Generation of femtosecond optical vortices by molecular modulation in a Raman-active crystal,” Opt. Express 21(23), 27750–27758 (2013).
[PubMed]

J. Strohaber, “Frame dragging with optical vortices,” Gen. Relativ. Gravit. 45(12), 2457–2465 (2013).
[Crossref]

B. Neupane, F. Chen, W. Sun, D. T. Chiu, and G. Wang, “Tuning donut profile for spatial resolution in stimulated emission depletion microscopy,” Rev. Sci. Instrum. 84(4), 043701 (2013).
[Crossref] [PubMed]

2012 (2)

2011 (3)

2008 (1)

K. C. Wright, L. S. Leslie, and N. P. Bigelow, “Optical control of the internal and external angular momentum of a Bose-Einstein condensate,” Phys. Rev. A 77(4), 441601 (2008).
[Crossref]

2007 (1)

2005 (2)

2004 (3)

2002 (1)

A. Muthukrishnan and C. R. Stroud., “Entanglement of internal and external angular momenta of a single atom,” J. Opt. B Quantum Semiclassical Opt. 4(2), S73–S77 (2002).
[Crossref]

1999 (1)

M. J. Padgett and L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31(1), 1–12 (1999).
[Crossref]

1993 (1)

I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

Allen, L.

M. J. Padgett and L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31(1), 1–12 (1999).
[Crossref]

Bakr, W. S.

Bandres, M. A.

Bezuhanov, K.

Bigelow, N. P.

K. C. Wright, L. S. Leslie, and N. P. Bigelow, “Optical control of the internal and external angular momentum of a Bose-Einstein condensate,” Phys. Rev. A 77(4), 441601 (2008).
[Crossref]

Brachmann, J. F. S.

Chen, F.

B. Neupane, F. Chen, W. Sun, D. T. Chiu, and G. Wang, “Tuning donut profile for spatial resolution in stimulated emission depletion microscopy,” Rev. Sci. Instrum. 84(4), 043701 (2013).
[Crossref] [PubMed]

Chiu, D. T.

B. Neupane, F. Chen, W. Sun, D. T. Chiu, and G. Wang, “Tuning donut profile for spatial resolution in stimulated emission depletion microscopy,” Rev. Sci. Instrum. 84(4), 043701 (2013).
[Crossref] [PubMed]

Chuu, C.

D’Ambrosio, V.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref] [PubMed]

Dreischuh, A.

Elias, L. R.

I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

Foo, G.

Garanovich, I. L.

Gillen, J.

Greiner, M.

Grier, D.

Gutiérrez-Vega, J. C.

Hansinger, P.

Hanssen, J.

Hart, N.

Hua, X.

Kaya, G.

Kaya, N.

Kimel, I.

I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

Kolomenskii, A. A.

Ladavac, K.

Leslie, L. S.

K. C. Wright, L. S. Leslie, and N. P. Bigelow, “Optical control of the internal and external angular momentum of a Bose-Einstein condensate,” Phys. Rev. A 77(4), 441601 (2008).
[Crossref]

Löffler, W.

Maleshkov, G.

Marrucci, L.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref] [PubMed]

Meyrath, T.

Muthukrishnan, A.

A. Muthukrishnan and C. R. Stroud., “Entanglement of internal and external angular momenta of a single atom,” J. Opt. B Quantum Semiclassical Opt. 4(2), S73–S77 (2002).
[Crossref]

Neshev, D. N.

Neupane, B.

B. Neupane, F. Chen, W. Sun, D. T. Chiu, and G. Wang, “Tuning donut profile for spatial resolution in stimulated emission depletion microscopy,” Rev. Sci. Instrum. 84(4), 043701 (2013).
[Crossref] [PubMed]

Padgett, M. J.

M. J. Padgett and L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31(1), 1–12 (1999).
[Crossref]

Palacios, D. M.

Paulus, G. G.

Peng, A.

Petersen, C.

Raizen, M.

Ricci, F.

Schätzel, M. G.

Schreck, F.

Schuessler, H.

Schuessler, H. A.

Sciarrino, F.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref] [PubMed]

Skryabin, D. V.

Slussarenko, S.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref] [PubMed]

Sokolov, A. V.

Sponselli, A.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref] [PubMed]

Strohaber, J.

Stroud, C. R.

A. Muthukrishnan and C. R. Stroud., “Entanglement of internal and external angular momenta of a single atom,” J. Opt. B Quantum Semiclassical Opt. 4(2), S73–S77 (2002).
[Crossref]

Sun, W.

B. Neupane, F. Chen, W. Sun, D. T. Chiu, and G. Wang, “Tuning donut profile for spatial resolution in stimulated emission depletion microscopy,” Rev. Sci. Instrum. 84(4), 043701 (2013).
[Crossref] [PubMed]

Swartzlander, G. A.

Uiterwaal, C. J. G. J.

Vallone, G.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref] [PubMed]

van Exter, M. P.

Villoresi, P.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref] [PubMed]

Walther, H.

Wang, G.

B. Neupane, F. Chen, W. Sun, D. T. Chiu, and G. Wang, “Tuning donut profile for spatial resolution in stimulated emission depletion microscopy,” Rev. Sci. Instrum. 84(4), 043701 (2013).
[Crossref] [PubMed]

Wang, K.

Wang, Z.

Wright, K. C.

K. C. Wright, L. S. Leslie, and N. P. Bigelow, “Optical control of the internal and external angular momentum of a Bose-Einstein condensate,” Phys. Rev. A 77(4), 441601 (2008).
[Crossref]

Yuan, X.-C.

Zhang, N.

Zhi, M.

Gen. Relativ. Gravit. (1)

J. Strohaber, “Frame dragging with optical vortices,” Gen. Relativ. Gravit. 45(12), 2457–2465 (2013).
[Crossref]

IEEE J. Quantum Electron. (1)

I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

J. Opt. B Quantum Semiclassical Opt. (1)

A. Muthukrishnan and C. R. Stroud., “Entanglement of internal and external angular momenta of a single atom,” J. Opt. B Quantum Semiclassical Opt. 4(2), S73–S77 (2002).
[Crossref]

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

Opt. Express (8)

F. Ricci, W. Löffler, and M. P. van Exter, “Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer,” Opt. Express 20(20), 22961–22975 (2012).
[Crossref] [PubMed]

T. Meyrath, F. Schreck, J. Hanssen, C. Chuu, and M. Raizen, “A high frequency optical trap for atoms using Hermite-Gaussian beams,” Opt. Express 13(8), 2843–2851 (2005).
[Crossref] [PubMed]

J. F. S. Brachmann, W. S. Bakr, J. Gillen, A. Peng, and M. Greiner, “Inducing vortices in a Bose-Einstein condensate using holographically produced light beams,” Opt. Express 19(14), 12984–12991 (2011).
[Crossref] [PubMed]

K. Ladavac and D. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12(6), 1144–1149 (2004).
[Crossref] [PubMed]

Z. Wang, N. Zhang, and X.-C. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free-space optical communication,” Opt. Express 19(2), 482–492 (2011).
[Crossref] [PubMed]

J. Strohaber, G. Kaya, N. Kaya, N. Hart, A. A. Kolomenskii, G. G. Paulus, and H. A. Schuessler, “In situ tomography of femtosecond optical beams with a holographic knife-edge,” Opt. Express 19(15), 14321–14334 (2011).
[Crossref] [PubMed]

M. Zhi, K. Wang, X. Hua, H. Schuessler, J. Strohaber, and A. V. Sokolov, “Generation of femtosecond optical vortices by molecular modulation in a Raman-active crystal,” Opt. Express 21(23), 27750–27758 (2013).
[PubMed]

P. Hansinger, G. Maleshkov, I. L. Garanovich, D. V. Skryabin, D. N. Neshev, A. Dreischuh, and G. G. Paulus, “Vortex algebra by multiply cascaded four-wave mixing of femtosecond optical beams,” Opt. Express 22(9), 11079–11089 (2014).
[Crossref] [PubMed]

Opt. Lett. (4)

Opt. Quantum Electron. (1)

M. J. Padgett and L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31(1), 1–12 (1999).
[Crossref]

Phys. Rev. A (1)

K. C. Wright, L. S. Leslie, and N. P. Bigelow, “Optical control of the internal and external angular momentum of a Bose-Einstein condensate,” Phys. Rev. A 77(4), 441601 (2008).
[Crossref]

Phys. Rev. Lett. (1)

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

B. Neupane, F. Chen, W. Sun, D. T. Chiu, and G. Wang, “Tuning donut profile for spatial resolution in stimulated emission depletion microscopy,” Rev. Sci. Instrum. 84(4), 043701 (2013).
[Crossref] [PubMed]

Other (1)

R. W. Boyd, Nonlinear Optics (Academic, 2003).

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Figures (7)

Fig. 1
Fig. 1 Illustration of (a) the four-port Michelson and (b) the beam crossing setup with the following elements: beam splitter (BS), Mirror (M), Lens (L), translation stage (T), spatial light modulator (SLM) and lead tungsten crystal (PbWO4). The SLM, used to produce the desired beam mode, is positioned in the stationary arm of the four-port Michelson. Radiation from the variable arm was used as a reference.
Fig. 2
Fig. 2 Cascaded generation of sidebands: (a). Schematic representation of the time-delayed chirped pulse scheme employed to access the Raman transitions of the Raman-active crystal (PbWO4). Shown are the spectral density distribution versus time plots of two positively b > 0 , linearly chirped pulses delayed in time by t d . At all times, the frequency difference between the pulses is Δ ω = b t d . In order to access the Raman transitions, this difference must equal the Raman frequency ω R = ω t d . (b). Diagram of cascade generation of sideband frequencies.
Fig. 3
Fig. 3 Generation of Raman sidebands with LG beams having the same helicity P = S . Columns 1—5: S1, S, P, AS1 and AS2; rows 1 and 3 are sidebands generated with = 1 and 2 respectively. Rows 2 and 4 are the interferograms of those beams in rows 1 and 3 with the reference beam from the 4-port Michelson. The measured topological charges are consistent with that predicted by the OAM-algebra. Columns 6—9: S1, S, P, and AS1; row 1 are sidebands produced with = 3 and row 2 shows the interferograms. Panels (a) and (b) show the intensity distribution and the interferogram of the AS1 order beam produced with = 5 . As can be seen, the multifurcation has broken up into 5 single bifurcations [denoted by the red dots in panel (b)].
Fig. 4
Fig. 4 Generation of Raman sidebands with LG beam of the same order but opposite helicity P = S (unbalanced arms). Columns 1—5: S1, S, P, AS1 and AS2; rows 1 and 2, and rows 3 and 4 show sidebands generated with = 1 and = 2 respectively. Rows 2 and 4 are the interference of those beams in rows 1 and 3 with the reference beam for the Michelson. The measured topological charges are consistent with those predicted by the OAM-algebra n A S = ( n + 1 ) P n S . Column (a) shows images of the AS1 beams generated with = 4, 5, 6 and 7 (rows 1—d respectively); and column (b) shows the interferograms. In all data, the topological charges were determined by counting multifurcation and fringes.
Fig. 5
Fig. 5 Ellipticity-dependence of the generation Raman sidebands with Ince-Gaussian beams of the same order and helicity P = S (balanced arms). All images are of the AS1 order. Columns 1—5: ε = ( 0.0 , 0.2 , 0.4 , 0.6 , 1.0 ) [columns 6—10: ε = ( 0.0 , 0.1 , 0.2 , 0.3 , 1.0 ) ] rows 1 and 3 (columns 1—5) are sidebands generated with = 2 , and 3 . Rows 1 and 3, columns 6—10 are sidebands generated with = 4 , and 5 . The interferograms appear under the images of the orders. The measured topological charges are determined by counting multifurcation and fringes and are consistent with those predicted by the OAM-algebra found for the LG beams. The general trend is for the multifurcations (bifurcations) to split up and move along a line as the ellipticity increases. An interesting consequence of the control of the bifurcation is that the initial instability of the higher order IG beams can be partially corrected for by changing the ellipticity parameter (beams in columns 2 and 7 have a more circular shape than in column 1 and 6).
Fig. 6
Fig. 6 Generation of Raman sidebands with e LG beams. Columns a—e: S1, S, P, AS1 and AS2; row 1 are sidebands generated with = 2 and Row 2 are sidebands generated with = 3 . For the even (odd) LG beams, the number of angular nodes is 2 . From the OAM-algebra (see text, Eq. (10)), the AS1 and S1 orders are expected to have modal contributions from and 3 . So for LG 0 , 2 e in the first row, the S1 and AS1 order are expected to have similarities with LG 0 , 2 e and LG 0 , 6 e modes having 4 and 12 nodes. In the bottom row = 3 the S1 and AS1 orders are expected to have contributions from the LG 0 , 3 e and LG 0 , 9 e modes. These modes have 6 and 18 nodes respectively. For AS2 sideband, the OAM-algebra results in angular mode numbers of , 3 and 5 . For LG 0 , 2 e (top row) the AS2 order is expected to have contributions from LG 0 , 2 e , LG 0 , 6 e and LG 0 , 10 e (4, 12 and 20 angular nodes) and for LG 0 , 2 e (bottom row) the AS2 order is expected to have contributions from LG 0 , 3 e , LG 0 , 9 e and LG 0 , 15 e (6, 18 and 30 angular nodes).
Fig. 7
Fig. 7 Generation of Raman sidebands with even e LG beams having a blocked lobe. Columns 1—3: S, P, and AS1. Row 1 is generated with = 2 . Row 2 is generation with even e LG beams having = 2 . In the last row, a modal lobe was blocked to investigate the generation of the lowest order contribution to the OAM-algebra which in this case is LG 0 , 2 e . The AS1 orders for both scenarios (blocked and unblocked) are nearly identical demonstrating the generation of the predicted modes and that the entire modes structures of the pump and Stokes are participating in the generation process.

Equations (29)

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HG n m = N n m H n ( 2 x w ) e ξ 2 / 2 H m ( 2 y w ) e r 2 / w 2 e i k r 2 / 2 R e i ( n + m + 1 ) ψ G ,
LG ρ , e ( o ) = N ρ ( 2 r w ) | | L ρ | | ( 2 r 2 w 2 ) ( cos ( θ ) sin ( θ ) ) e r 2 / w 2 e i k r 2 / 2 R e i ( 2 ρ + | | + 1 ) ψ G ,
IG p , m e ( o ) = N p m ( C p m ( i ξ , ε ) C p m ( η , ε ) S p m ( i ξ , ε ) S p m ( η , ε ) ) e r 2 / w 2 e i k r 2 / 2 R e i ( p + 1 ) ψ G .
LG 0 , = N 0 w 2 2 ( 1 ) | | 2 | | k = 0 | | | | ! k ! ( | | k ) ! ( i sgn ( ) ) k H | | k ( ξ ) H k ( η ) e i k z .
H | | k ( ξ ) = H | | k ( ξ ) H k ( η ) = ( 1 ) k H k ( η ) .
`LG 0 , = N 0 | | w 2 2 ( 1 ) | | 2 | | k = 0 | | | | ! k ! ( | | k ) ! ( i sgn ( ) ) k ( 1 ) k H | | k ( ξ ) H k ( η ) e i k z .
2 E n 2 c 2 2 E t 2 = 1 ε 0 c 2 2 P N L t 2 .
1 r 2 2 θ 2 E A S N ( θ ) e i ( k z w t ) = 1 ε 0 c 2 2 P N L t 2 .
P N L cos N + 1 ( P θ ) cos N ( S θ ) e i [ ( N + 1 ) k P N k S ] z e i [ ( N + 1 ) ω P N ω S ] t .
1 r 2 2 E A S N ( θ ) θ 2 e i ( k z w t ) = 1 ε 0 c 2 [ ( N + 1 ) ω P N ω S ] cos N + 1 ( P θ ) cos N ( S θ ) e i [ ( ( N + 1 ) k P N k S ) z ( ( N + 1 ) ω P N ω S ) t ]
1 r 2 n 2 a n cos ( n θ ) = B cos N + 1 ( P θ ) cos N ( S θ ) .
E A S N ( θ ) = k 1 N + 1 j = N N a j k cos [ ( k P + j S ) θ ] + ( 1 + ( 1 ) N + 1 ) 1 2 j = 1 N a j 0 cos ( j S θ )
E A S N ( θ ) = k 1 N + 1 j = N N a j k cos [ ( k + j ) θ ] + ( 1 + ( 1 ) N + 1 ) 1 2 j = 1 N a j 0 cos ( j θ ) ,
E A S N ( θ ) = k 1 N + 1 j = N N a j k cos [ ( k j ) θ ] + ( 1 + ( 1 ) N + 1 ) 1 2 j = 1 N a j 0 cos ( j θ )
LG 0 , = N 0 | | ( 2 r w ) | | e i θ e r 2 / w 2
HG n m = N n m H n ( 2 x w ) e x 2 / w 2 H m ( 2 y w ) e y 2 / w 2
LG 0 , = n , m a n m HG n , m
I L H S = N 0 | | N n m ( 2 r w ) | | e i θ H n ( ξ ) H m ( η ) e 2 r 2 / w 2 d x d y .
e i θ = k = 0 | | ( | | k ) [ i sgn ( ) sin ( θ ) ] k [ cos ( θ ) ] | | k .
I L H S = N 0 | | N n m k = 0 | | ( | | k ) ( i sgn ( ) ) k ξ | | k η k H n ( ξ ) H m ( η ) e 2 r 2 / w 2 d x d y .
η k e η 2 H m ( η ) d η = ( 1 ) m η k e η 2 e η 2 d m d η m ( e η 2 ) d η ,
ξ | | k H n ( ξ ) e ξ 2 d ξ = ( 1 ) n ξ | | k e ξ 2 e ξ 2 d n d ξ n ( e ξ 2 ) d ξ .
η k e η 2 H m ( η ) d η = ( 1 ) k k ! π ,
ξ | | k H n ( ξ ) e ξ 2 d ξ = ( 1 ) | | k ( | | k ) ! π .
LG 0 , = N 0 w 2 2 ( 1 ) | | 2 | | k = 0 | | | | ! k ! ( | | k ) ! ( i sgn ( ) ) k H | | k ( ξ ) H k ( η ) e i k z .
H G 2 k , 2 j = ( 1 ) j 2 j + k [ ( j + k ) ! ] 2 ( 2 j ) ! ( 2 k ) ! L G j + k , 0 e s = 0 j + k ( 1 ) s ( 2 k s ) ( 2 j j + k s ) + ( 1 ) j 2 j + k q = 0 j + k 1 q ! [ 2 ( j + k ) q ] ! 2 ( 2 j ) ! ( 2 k ) ! L G q , 2 ( j + k q ) e × [ q = 0 q ( 1 ) s ( 2 k s ) ( 2 j q s ) + s = 0 2 ( j + k ) q ( 1 ) s ( 2 k + 1 s ) ( 2 j 2 ( j + k ) q s ) ] ,
HG 2 k + 1 , 2 j = ( 1 ) j q = 0 j + k 1 2 j + k + 1 q ! [ 2 ( j + k ) + 1 q ] ! ( 2 j ) ! ( 2 k + 1 ) ! LG q , 2 ( k + j q ) + 1 e × [ s = 0 q ( 1 ) s ( 2 k + 1 s ) ( 2 j q s ) s = 0 2 ( j + k ) + 1 q ( 1 ) s ( 2 k + 1 s ) ( 2 j 2 ( j + k ) + 1 q s ) ] ,
HG 2 k , 2 j + 1 = ( 1 ) j q = 0 j + k 1 2 j + k + 1 q ! [ 2 ( j + k ) + 1 q ] ! ( 2 j + 1 ) ! ( 2 k ) ! LG q , 2 ( k + j q ) + 1 o × [ s = 0 q ( 1 ) s ( 2 k + 1 s ) ( 2 j q s ) s = 0 2 ( j + k ) + 1 q ( 1 ) s ( 2 k s ) ( 2 j + 1 2 ( j + k ) + 1 q s ) ] ,
HG 2 k + 1 , 2 j + 1 = ( 1 ) j 2 j + k + 1 [ ( j + k + 1 ) ! ] 2 2 ( 2 j + 1 ) ! ( 2 k + 1 ) ! LG j + k + 1 , 0 o s = 0 j + k + 1 ( 1 ) s ( 2 k + 1 s ) ( 2 j + 1 j + k + 1 s ) + ( 1 ) 2 j + k + 1 q = 0 j + k q ! [ 2 ( j + k + 1 ) q ] ! 2 ( 2 j + 1 ) ! ( 2 k + 1 ) ! LG q , 2 ( j + k + 1 q ) o × [ s = 0 q ( 1 ) s ( 2 k + 1 s ) ( 2 j + 1 q s ) s = 0 2 ( j + k + 1 ) q ( 1 ) s ( 2 k + 1 s ) ( 2 j 2 ( j + k + 1 ) q s ) ] .

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