Abstract

An analytical expression for the diffraction of an elliptic Laguerre–Gaussian (LG) beam is derived and analyzed. We show that a beam with even singularity order has nonzero axial intensity for any degree of ellipticity and at any finite distance z from the initial plane, whereas at z=0 and z= the axial intensity is zero. We show that for a beam with a small degree of ellipticity and even order of singularity, two isolated intensity zeroes appear in the Fresnel zone on a straight line at an angle of 45deg or 45deg, depending whether the beam’s spin is right or left. The theoretical conclusions are confirmed by numerical simulation and physical experiments.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. Cai and Q. Lin, "Decentered elliptical Gaussian beam," Appl. Opt. 41, 4336-4340 (2002).
    [CrossRef] [PubMed]
  2. Y. Cai and Q. Lin, "Decentered elliptical Hermite-Gaussian beam," J. Opt. Soc. Am. A 20, 1111-1119 (2003).
    [CrossRef]
  3. Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A, Pure Appl. Opt. 6, 1061-1066 (2004).
    [CrossRef]
  4. Z. Mitreska, "Diffraction of elliptical Gaussian light beams on rectangular profile grating of transmittance," Pure Appl. Opt. 3, 995-1004 (1994).
    [CrossRef]
  5. S. Seshadri, "Basic elliptical Gaussian wave and beam in a uniaxial crystal," J. Opt. Soc. Am. A 20, 1818-1826 (2003).
    [CrossRef]
  6. A. Steinbach, M. Ranner, F. C. Crnz, and J. C. Bergquist, "CW second harmonic generation with elliptical Gaussian beam," Opt. Commun. 123, 207-214 (1996).
    [CrossRef]
  7. Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
    [CrossRef]
  8. Y. Cai and Q. Lin, "Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems," J. Opt. Soc. Am. A 21, 1058-1065 (2004).
    [CrossRef]
  9. M. A. Bandres and J. Gutierrez-Vega, "Ince-Gaussian beams," Opt. Lett. 29, 144-146 (2004).
    [CrossRef] [PubMed]
  10. M. A. Bandres and J. Gutierrez-Vega, "Ince-Gaussian modes of the paraxial wave equation and stable resonators," J. Opt. Soc. Am. A 21, 873-880 (2004).
    [CrossRef]
  11. M. A. Bandres and J. Gutierrez-Vega, "Elegant Ince-Gaussian beams," Opt. Lett. 29, 1724-1726 (2004).
    [CrossRef] [PubMed]
  12. M. A. Bandres and J. Gutierrez-Vega, "Higher-order complex source for elegant Laguerre-Gaussian waves," Opt. Lett. 29, 2213-2215 (2004).
    [CrossRef] [PubMed]
  13. U. T. Schwarz, M. A. Bandres, and J. Gutierrez-Vega, "Observation of Ince-Gaussian modes in stable resonators," Opt. Lett. 29, 1870-1872 (2004).
    [CrossRef] [PubMed]
  14. E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, 5157-5161 (2004).
    [CrossRef]
  15. Z. Bin and L. Zhu, "Diffraction property of an axicon in oblique illumination," Appl. Opt. 37, 2563-2568 (1998).
    [CrossRef]
  16. A. Thaning, Z. Jaroszewicz, and A. T. Friberg, "Diffractive axicons in oblique illumination: Analysis and experiments and comparison with elliptical axicons," Appl. Opt. 42, 9-17 (2003).
    [CrossRef] [PubMed]
  17. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, "Astigmatic Bessel laser beams," J. Mod. Opt. 51, 677-686 (2004).
  18. A. P. Prudnikov, Yu. A. Bychkov, and O. I. Marychev, Integrals and Series: Volume 2: Special Functions (Nauka, 1983).
  19. V.A.Soifer, ed., Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).
  20. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).

2004

2003

2002

2001

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).

1998

1996

A. Steinbach, M. Ranner, F. C. Crnz, and J. C. Bergquist, "CW second harmonic generation with elliptical Gaussian beam," Opt. Commun. 123, 207-214 (1996).
[CrossRef]

1994

Z. Mitreska, "Diffraction of elliptical Gaussian light beams on rectangular profile grating of transmittance," Pure Appl. Opt. 3, 995-1004 (1994).
[CrossRef]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, 5157-5161 (2004).
[CrossRef]

Bandres, M. A.

Bergquist, J. C.

A. Steinbach, M. Ranner, F. C. Crnz, and J. C. Bergquist, "CW second harmonic generation with elliptical Gaussian beam," Opt. Commun. 123, 207-214 (1996).
[CrossRef]

Bin, Z.

Bychkov, Yu. A.

A. P. Prudnikov, Yu. A. Bychkov, and O. I. Marychev, Integrals and Series: Volume 2: Special Functions (Nauka, 1983).

Cai, Y.

Crnz, F. C.

A. Steinbach, M. Ranner, F. C. Crnz, and J. C. Bergquist, "CW second harmonic generation with elliptical Gaussian beam," Opt. Commun. 123, 207-214 (1996).
[CrossRef]

Friberg, A. T.

Gutierrez-Vega, J.

Jaroszewicz, Z.

Jefimovs, K.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, "Astigmatic Bessel laser beams," J. Mod. Opt. 51, 677-686 (2004).

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, "Astigmatic Bessel laser beams," J. Mod. Opt. 51, 677-686 (2004).

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, "Astigmatic Bessel laser beams," J. Mod. Opt. 51, 677-686 (2004).

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).

Lin, Q.

Marychev, O. I.

A. P. Prudnikov, Yu. A. Bychkov, and O. I. Marychev, Integrals and Series: Volume 2: Special Functions (Nauka, 1983).

Mitreska, Z.

Z. Mitreska, "Diffraction of elliptical Gaussian light beams on rectangular profile grating of transmittance," Pure Appl. Opt. 3, 995-1004 (1994).
[CrossRef]

Paakkonen, P.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, "Astigmatic Bessel laser beams," J. Mod. Opt. 51, 677-686 (2004).

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Bychkov, and O. I. Marychev, Integrals and Series: Volume 2: Special Functions (Nauka, 1983).

Ranner, M.

A. Steinbach, M. Ranner, F. C. Crnz, and J. C. Bergquist, "CW second harmonic generation with elliptical Gaussian beam," Opt. Commun. 123, 207-214 (1996).
[CrossRef]

Schwarz, U. T.

Seshadri, S.

Simonen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, "Astigmatic Bessel laser beams," J. Mod. Opt. 51, 677-686 (2004).

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).

Steinbach, A.

A. Steinbach, M. Ranner, F. C. Crnz, and J. C. Bergquist, "CW second harmonic generation with elliptical Gaussian beam," Opt. Commun. 123, 207-214 (1996).
[CrossRef]

Thaning, A.

Turunen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, "Astigmatic Bessel laser beams," J. Mod. Opt. 51, 677-686 (2004).

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, 5157-5161 (2004).
[CrossRef]

Zhu, L.

Appl. Opt.

J. Mod. Opt.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, P. Paakkonen, and J. Turunen, "Astigmatic Bessel laser beams," J. Mod. Opt. 51, 677-686 (2004).

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, "An analysis of the angular momentum of a light field in terms of angular harmonics," J. Mod. Opt. 48, 1543-1557 (2001).

J. Opt. A, Pure Appl. Opt.

Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A, Pure Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, 5157-5161 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

A. Steinbach, M. Ranner, F. C. Crnz, and J. C. Bergquist, "CW second harmonic generation with elliptical Gaussian beam," Opt. Commun. 123, 207-214 (1996).
[CrossRef]

Opt. Lett.

Pure Appl. Opt.

Z. Mitreska, "Diffraction of elliptical Gaussian light beams on rectangular profile grating of transmittance," Pure Appl. Opt. 3, 995-1004 (1994).
[CrossRef]

Other

A. P. Prudnikov, Yu. A. Bychkov, and O. I. Marychev, Integrals and Series: Volume 2: Special Functions (Nauka, 1983).

V.A.Soifer, ed., Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Propagation of the LG mode ( n , m ) = ( 5 , 2 ) with the elliptic distortion in free space (negative). The image size is 5 × 5 mm , 256 × 256   pixels . The characteristic radius of the Gaussian beam is σ = 0.391 , wavelength is λ = 0.63 μ m , the ellipticity coefficient is 0.66.

Fig. 2
Fig. 2

Propagation of the LG mode ( n , m ) = ( 3 , 1 ) with elliptic distortion in free space. The ellipticity coefficient is 0.66.

Fig. 3
Fig. 3

Propagation of the LG mode ( n , m ) = ( 2 , 0 ) with elliptic distortion in free space. The ellipticity coefficient is 0.66.

Fig. 4
Fig. 4

Propagation of the LG mode ( n , m ) = ( 5 , 2 ) with weakly elliptic distortion in free space. The ellipticity coefficient is 0.91.

Fig. 5
Fig. 5

42-order binary DOE to generate the LG modes: (a) DOE phase, (b) arrangement of the modes on the diffraction orders, (c) estimated intensity distribution in the lens focal plane.

Fig. 6
Fig. 6

Experimental intensity distribution (for the converging illuminating beam) for an axial DOE inclination of 45 deg .

Fig. 7
Fig. 7

Experimental intensity distributions (for the converging illuminating beam) for an axial DOE inclination of 20 deg .

Fig. 8
Fig. 8

Experimental intensity distribution for the LG mode ( n , m ) = ( 5 , 4 ) at various DOE axial inclinations: approximately (a) 20 deg , (b) 45 deg , (c) 60 deg , (d) 75 deg .

Fig. 9
Fig. 9

Experimental intensity distribution for an axial DOE inclination of 75 deg .

Equations (64)

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

0 x m 2 exp ( p x ) J m ( b x ) L n m ( c x ) d x = ( b 2 ) m ( p c ) n p m + n + 1 exp ( b 2 4 p ) L n m ( b 2 c 4 p c 4 p 2 ) ,
Ψ m n ( r , φ ) = ( r 2 w 0 ) m exp ( r 2 w 0 2 ) L n m ( 2 r 2 w 0 2 ) exp ( i m φ ) ,
Ψ m n ( ρ , θ , z ) = ( i ) m + 1 k z exp ( i k ρ 2 2 z + i m θ ) 0 ( r 2 w 0 ) m exp ( r 2 w 0 2 + i k r 2 2 z ) L n m ( 2 r 2 w 0 2 ) J m ( k r ρ z ) r d r = w 0 w ( z ) [ ρ 2 w ( z ) ] m exp [ ρ 2 w 2 ( z ) + i k ρ 2 2 R ( z ) + i m θ i ( 2 n + m + 1 ) tan 1 ( z z 0 ) ] L n m ( 2 ρ 2 w 2 ( z ) ) ,
I m n ( ρ , z ) = Ψ m n ( ρ , θ , z ) 2 .
Ψ m n ( 1 ) ( r , φ ) = ( r w 0 ) 2 n + m exp ( r 2 w 0 2 + i m φ ) ,
0 r 2 n + m exp ( p r 2 ) J m ( c r ) r d r
= n ! c m 2 m + 1 p m + n + 1 exp ( c 2 4 p ) L n m ( c 2 4 p ) .
Ψ m n ( 1 ) ( ρ , θ , z ) = ( i ) m + 1 k z exp ( i k ρ 2 2 z + i m θ ) 0 ( r w 0 ) 2 n + m exp ( r 2 w 0 2 + i k r 2 2 z ) J m ( k r ρ z ) r d r = ( i ) m + 1 z 0 n ! z ( 1 i z 0 z ) n m 2 1 exp [ i k ρ 2 2 z + i m θ ] x m 2 exp ( x ) L n m ( x ) ,
Ψ ̂ m n ( r ) = ( r 2 w 0 ) m exp ( r 2 w 0 2 ) L n m ( 2 r 2 w 0 2 ) .
F γ ( ρ , θ , z ) = ( i ) m + 1 k z exp [ i k ρ 2 2 z + i m tan 1 ( ρ sin θ ρ cos θ z sin γ ) ] 0 Ψ ̂ m n ( r ) exp ( i k r 2 2 z ) J m ( k r z z 2 sin 2 γ + ρ 2 2 ρ z cos θ sin γ ) r d r .
{ ξ = ρ cos θ z sin γ , η = ρ sin θ , }
ξ 2 + η 2 = z 2 sin 2 γ + ρ 2 2 ρ z cos θ sin γ .
I γ ( ξ 2 + η 2 , z ) = F γ ( ρ , θ , z ) 2 .
F ( ξ , η , z ) = i k 2 π Ψ m n ( x , y ) exp ( i k R ) R d x d y ,
F ( ρ , θ , z ) = i k 2 π z 0 0 2 π Ψ m n ( r , φ ) exp [ i k r 2 + ρ 2 + z 2 2 r ρ cos ( θ φ ) ] r d r d φ ,
F γ ( ρ , θ , z ) = i k 2 π z 0 0 2 π Ψ m n ( r , φ ) exp [ i k r cos φ sin γ + i k r 2 + ρ 2 + z 2 2 r ρ cos ( θ φ ) ] r d r d φ .
z 2 + r 2 + ρ 2 2 r ρ cos ( θ φ ) z + r 2 + ρ 2 2 r ρ cos ( θ φ ) 2 z r 4 + ρ 4 + 2 r 2 ρ 2 ( 4 r 3 ρ + 4 r ρ 3 ) cos ( θ φ ) + 4 r 2 ρ 2 cos 2 ( θ φ ) 8 z 3 + .
z 2 + r 2 + ρ 2 2 r ρ cos ( θ ρ ) ( z + ρ 2 2 z ρ 4 8 z 3 ) + ( r 2 2 z r 2 ρ 2 2 z 3 ) r ρ z cos ( θ φ ) r 2 ρ 2 4 z 3 cos 2 ( θ φ ) .
F γ ( ρ , θ , z ) = i k 2 π z exp [ i k ( z + ρ 2 2 z ρ 4 8 z 3 ) ] 0 Ψ ̂ m n ( r ) exp [ i k r 2 2 z ( 1 ρ 2 z 2 ) ] × { 0 2 π exp [ i m φ + i k r cos φ sin γ i k r ρ z cos ( θ φ ) i k r 2 ρ 2 4 z 3 cos 2 ( θ φ ) ] d φ } r d r .
I 0 = exp ( i m θ ) 0 2 π exp [ i m ψ i k r ρ 0 z cos ( ψ ν ) i k r 2 ρ 2 4 z 3 cos 2 ψ ] d ψ ,
ψ = φ θ ,
ρ 0 2 = ( ρ z sin γ cos θ ) 2 + ( z sin γ sin θ ) 2 ,
ν = tan 1 ( z sin γ sin θ ρ z sin γ cos θ ) .
P = k r ρ 0 z , Q = k r 2 ρ 2 4 z 3 .
I 0 = exp ( i m θ ) 0 2 π exp [ i m ψ i P cos ( ψ ν ) i Q cos 2 ψ ] d ψ = exp ( i m θ ) p = i p J p ( Q ) 0 2 π exp [ i 2 p ψ + i m ψ i P cos ( ψ ν ) ] d ψ = 2 π ( i ) m exp [ i m ( θ + ν ) ] p = ( i ) p J p ( Q ) J m + 2 p ( P ) exp ( i 2 p ν ) .
F γ ( ρ , θ , z ) = ( i ) m + 1 k z exp [ i m ( θ + ν ) + i k ( z + ρ 2 2 z ρ 4 8 z 3 ) ] × p = ( i ) p exp ( i 2 p ν ) 0 Ψ ̂ m n ( r ) exp [ i k r 2 2 z ( 1 ρ 2 z 2 ) ] J p ( k r 2 ρ 2 4 z 3 ) J m + 2 p ( k r ρ 0 z ) r d r .
Ψ m n ( x , y ; α ) = ( 2 x 2 + 2 α 2 y 2 w 0 2 ) m 2 exp ( x 2 + α 2 y 2 w 0 2 ) L n m ( 2 x 2 + 2 α 2 y 2 w 0 2 ) exp [ i m tan 1 ( α y x ) ] .
x = α r cos φ ,
y = r sin φ , 0 α 1 ,
Ψ m n ( r , φ ; α ) = ( α r 2 w 0 ) m exp ( α 2 r 2 w 0 2 ) L n m ( 2 α 2 r 2 w 0 2 ) exp ( i m φ ) .
F α ( ρ , θ , z ) = i k 2 π z exp [ i k ρ 2 2 z ( cos 2 θ + α 2 sin 2 θ ) ] × 0 0 2 π Ψ ̂ m n ( α r ) exp [ i m φ + i k r 2 2 z ( α 2 cos 2 φ + sin 2 φ ) i k α r ρ z cos ( θ φ ) ] r d r d φ .
ξ = ρ cos θ ,
η = α ρ sin θ .
F α ( ρ , θ , z ) = i k 2 π z exp [ i k ρ 2 2 z ( cos 2 θ + α 2 sin 2 θ ) ] 0 Ψ ̂ m n ( α r ) exp [ i k r 2 4 z ( 1 + α 2 ) ] × { 0 2 π exp [ i m φ i k r 2 4 z ( 1 α 2 ) cos 2 φ i k α r ρ z cos ( θ φ ) ] d φ } r d r .
I ̂ 0 = 0 2 π exp [ i m φ i k r 2 4 z ( 1 α 2 ) cos 2 φ i k α r ρ z cos ( θ φ ) ] d φ .
A = k α r ρ z , B = k r 2 ( 1 α 2 ) 4 z .
I ̂ 0 = p = ( i ) p J p ( B ) 0 2 π exp [ i 2 p φ + i m φ i A cos ( φ θ ) ] d φ = 2 π p = ( i ) p + m J p ( B ) J m + 2 p ( A ) exp [ i ( m + 2 p ) θ ] .
F α ( ρ , θ , z ) = ( i ) m + 1 k z exp [ i k ρ 2 2 z ( cos 2 θ + α 2 sin 2 θ ) ] × p = ( i ) p exp [ i ( 2 p + m ) θ ] 0 Ψ ̂ m n ( α r ) exp [ i k r 2 4 z ( 1 + α 2 ) ] J p [ k r 2 ( 1 α 2 ) 4 z ] J m + 2 p ( k α r ρ z ) r d r .
F α ( ρ , θ , z ) ( i ) m + 1 z exp ( i m θ ) 0 Ψ ̂ m n ( α r ) J m ( k α r ρ z ) r d r = ( i ) m ( 1 ) n exp ( i m θ ) ( w 0 α 2 σ ) ( ρ 2 σ ) m exp ( ρ 2 σ 2 ) L n m ( 2 ρ 2 σ 2 ) ,
F α ( ρ = 0 , θ , z ) 0 Ψ ̂ m n ( α r ) exp [ i k r 2 4 z ( 1 + α 2 ) ] J m 2 [ k r 2 4 z ( 1 α 2 ) ] r d r ,
J p ( x ) ( x 2 ) p Γ ( p + 1 ) , x 0 ,
F α ( ρ = 0 , z ) Γ 1 ( m 2 + 1 ) [ k ( 1 α 2 ) 8 z ] m 2 ( α 2 w 0 ) m 0 r 2 m exp [ α 2 r 2 w 0 2 + i k r 2 ( 1 + α 2 ) 4 z ] L n m ( 2 α 2 r 2 w 0 2 ) r d r .
0 x m exp ( p x ) L n m ( c x ) d x = Γ ( m + n + 1 ) ( p c ) n n ! p m + n + 1
0 r 2 m exp [ α 2 r 2 w 0 2 + i k r 2 ( 1 + α 2 ) 4 z ] L n m ( 2 α 2 r 2 w 0 2 ) r d r = Γ ( m + n + 1 ) ( 1 ) n exp [ i ( 2 n + m + 1 ) η ] 2 n ! [ α 4 w 0 4 + k 2 ( 1 + α 2 ) 2 16 z 2 ] m + 1 2 ,
F α 1 ( ρ , θ , z ) S ( ρ , θ ) p = ( i ) p ( ϵ 2 ) p Γ 1 ( p + 1 ) exp ( i 2 p θ ) δ ( p ) × 0 ( 2 α r w 0 ) m r 2 p exp [ α 2 r 2 w 0 2 + i k r 2 ( 1 + α 2 ) 4 z ] L n m ( 2 α 2 r 2 w 0 2 ) J m + 2 p ( k α r ρ z ) r d r ,
δ ( p ) = { 1 , ρ 0 ( 1 ) p , p < 0 } ,
S ( ρ , θ ) = ( 1 ) m + 1 k z exp [ i m θ + i k ρ 2 2 z ( cos 2 θ + α 2 sin 2 θ ) ] ,
0 x m + p 2 exp ( c x ) L n m ( c x ) J m + p ( b x ) d x = ( p c ) n p m + n + 1 ( b 2 ) m exp ( b 2 4 p ) L n m ( b 2 c 4 p c p 2 ) ,
F α 1 ( ρ , θ , z ) S ( ρ , θ ) 0 ( 2 α r w 0 ) m L n m ( 2 α 2 r 2 w 0 2 ) exp [ α 2 r 2 w 0 2 + i k r 2 ( 1 + α 2 ) 4 z ] { J m ( k α r ρ z ) i ϵ r 2 2 e i 2 θ J m + 2 ( k α r ρ z ) + O ( ϵ 2 ) } r d r ,
p = 0 t 2 p J m + 2 p ( x ) ( 2 p ) ! = 1 2 x m 2 J m ( x 2 2 t x ) { ( x 2 t ) m 2 + ( x + 2 t ) m 2 } .
p = 0 t 2 p J m + 2 p ( x ) ( 2 p ) ! J m ( x ) + t 2 2 J m + 2 ( x ) + O ( t 4 ) .
F α 1 ( ρ , θ , z ) S ( ρ , θ ) 0 ( 2 α r w 0 ) m L n m ( 2 α 2 r 2 w 0 2 ) exp [ α 2 r 2 w 0 2 + i k r 2 ( 1 + α 2 ) 4 z ] × J m [ r ( k α ρ z ) 2 ± 2 i ϵ e i θ ( k α ρ z ) ] r d r ,
S ( ρ , θ ) = S ( ρ , θ ) 1 2 ( k α ρ z ) m 2 [ ( k α ρ z 2 i ϵ e i θ ) m 2 + ( k α ρ z + 2 i ϵ e i θ ) m 2 ] .
F α 1 ( ρ , θ , z ) S ( ρ , θ ) 0 ( 2 α r w ) m L n m ( 2 α 2 r 2 w 2 ) exp [ α 2 r 2 w 2 + i k r 2 ( 1 + α 2 ) 4 z ] J m ( k α ρ r z ) r d r ,
ρ = ± 1 α z k ( 1 α 2 ) exp [ i ( θ π 4 ) ] .
ρ 0 = 1 α z ( 1 α 2 ) k .
F α 1 ( ρ , θ , z ) 2 F 0 ( ρ ) i k ( 1 α 2 ) 8 z e 2 i θ F 1 ( ρ ) 2 ,
F 0 ( ρ ) = k z 0 ( 2 α r w 0 ) m L n m ( 2 α 2 r 2 w 0 2 ) exp [ α 2 r 2 w 0 2 + i k r 2 ( 1 + α 2 ) 4 z ] J m ( k α ρ r z ) r d r ,
F 1 ( ρ ) = k z 0 ( 2 α r w 0 ) m r 2 L n m ( 2 α 2 r 2 w 0 2 ) exp [ α 2 r 2 w 0 2 + i k r 2 ( 1 + α 2 ) 4 z ] J m + 2 ( k α ρ r z ) r d r .
Ψ m n ( x ) = ( 2 x 2 w 0 2 ) m 2 exp ( x 2 w 0 2 ) L n m ( 2 x 2 w 0 2 ) .
0 t ( m 1 2 ) exp ( c t 2 ) L n m ( c t ) exp ( p t ) d t = 2 2 π n ! exp ( p 2 2 c i π m 2 ) 2 n + m 2 c m + 1 2 H n + m ( i p 2 c ) H n ( i p 2 c ) , c > 0 ,
i k 2 π f ψ m n ( x ) exp [ i k f ( x ξ + y η ) ] d x d y = ( i ) m + 1 δ ( η ) n ! w 0 2 π 2 n + m + 1 2 exp [ ( ξ σ ) 2 ] H n + m ( ξ σ ) H n ( ξ σ ) ,
n HG = n LG ,
m HG = n LG + m LG .

Metrics