Abstract

The general astigmatic transform, or two-dimensional non-separable linear canonical transform of a Hermite–Laguerre–Gaussian beam, is investigated by theoretical means. Some corollaries that apply to Hermite–Gaussian and Laguerre–Gaussian beam propagation are presented and discussed.

© 2010 Optical Society of America

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  5. E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys. Usp. 47, 1177–1203 (2004).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2010

E. G. Abramochkin and V. G. Volostnikov, Modern Theory of Gaussian Beams (Fizmatlit, Moscow, 2010). [in Russian].

E. G. Abramochkin and V. G. Volostnikov, “Generalized Hermite-Laguerre-Gauss beams,” Phys. Wave Phenom. 18, 14–22 (2010).
[CrossRef]

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals,” J. Opt. Soc. Am. A 27, 1288–1302 (2010).
[CrossRef]

2009

2008

2007

2005

2004

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince–Gaussian beams,” Opt. Lett. 29, 144–146 (2004).
[CrossRef]

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, 873–880 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys. Usp. 47, 1177–1203 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157–S161 (2004).
[CrossRef]

2001

E. G. Abramochkin, “Hermite–Laguerre–Gaussian functions,” Vestnik Samara State Univ. 4, 19–41 (2001). [in Russian]

2000

A. Belafhal and L. Dalil-Essakali, “Collins formula and propagation of Bessel-modulated Gaussian light beams through an ABCD optical system,” Opt. Commun. 177, 181–188 (2000).
[CrossRef]

R. M. Potvliege, “Waveletlike basis function approach to the propagation of paraxial beams,” J. Opt. Soc. Am. A 17, 1043–1047 (2000).
[CrossRef]

1999

C. F. R. Caron and R. M. Potvliege, “Bessel-modulated Gaussian beams with quadratic radial dependence,” Opt. Commun. 164, 83–93 (1999).
[CrossRef]

1996

M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical system,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

1993

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

1991

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

1990

Yu. A. Anan’ev, Optical Resonators and Gaussian beams (Nauka, Moscow, 1990). [in Russian].

1989

1987

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

1986

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series; Vol. 2: Special Functions (Gordon and Breach Sci. Publ., 1986).

1970

1966

Abramochkin, E.

E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Hermite-Laguerre-Gauss beams,” Phys. Wave Phenom. 18, 14–22 (2010).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, Modern Theory of Gaussian Beams (Fizmatlit, Moscow, 2010). [in Russian].

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157–S161 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys. Usp. 47, 1177–1203 (2004).
[CrossRef]

E. G. Abramochkin, “Hermite–Laguerre–Gaussian functions,” Vestnik Samara State Univ. 4, 19–41 (2001). [in Russian]

Alieva, T.

Anan’ev, Yu. A.

Yu. A. Anan’ev, Optical Resonators and Gaussian beams (Nauka, Moscow, 1990). [in Russian].

Bandres, M. A.

Bastiaans, M. J.

Belafhal, A.

A. Belafhal and L. Dalil-Essakali, “Collins formula and propagation of Bessel-modulated Gaussian light beams through an ABCD optical system,” Opt. Commun. 177, 181–188 (2000).
[CrossRef]

Brychkov, Yu. A.

Yu. A. Brychkov, Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas (CRC Press, 2008).

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series; Vol. 2: Special Functions (Gordon and Breach Sci. Publ., 1986).

Cai, Y.

Caron, C. F. R.

C. F. R. Caron and R. M. Potvliege, “Bessel-modulated Gaussian beams with quadratic radial dependence,” Opt. Commun. 164, 83–93 (1999).
[CrossRef]

Chen, C.

Christodoulides, D. N.

Collins, S. A.

Dalil-Essakali, L.

A. Belafhal and L. Dalil-Essakali, “Collins formula and propagation of Bessel-modulated Gaussian light beams through an ABCD optical system,” Opt. Commun. 177, 181–188 (2000).
[CrossRef]

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]

Gutiérrez-Vega, J. C.

Hesselink, L.

Karimi, E.

Khonina, S. N.

Koç, A.

Kogelnik, H.

Kotlyar, V. V.

Li, T.

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series; Vol. 2: Special Functions (Gordon and Breach Sci. Publ., 1986).

Marrucci, L.

Miyamoto, Y.

Ohtani, T.

Ozaktas, H. M.

Padovani, C.

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

Piccirillo, B.

Potvliege, R. M.

R. M. Potvliege, “Waveletlike basis function approach to the propagation of paraxial beams,” J. Opt. Soc. Am. A 17, 1043–1047 (2000).
[CrossRef]

C. F. R. Caron and R. M. Potvliege, “Bessel-modulated Gaussian beams with quadratic radial dependence,” Opt. Commun. 164, 83–93 (1999).
[CrossRef]

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series; Vol. 2: Special Functions (Gordon and Breach Sci. Publ., 1986).

Santamato, E.

Santarsiero, M.

M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical system,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

Siviloglou, G. A.

Skidanov, R. V.

Soifer, V. A.

Takeda, M.

Volostnikov, V.

E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Hermite-Laguerre-Gauss beams,” Phys. Wave Phenom. 18, 14–22 (2010).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, Modern Theory of Gaussian Beams (Fizmatlit, Moscow, 2010). [in Russian].

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157–S161 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys. Usp. 47, 1177–1203 (2004).
[CrossRef]

Wada, A.

Wünsche, A.

Zhou, G.

Zito, G.

Appl. Opt.

J. Opt. A, Pure Appl. Opt.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157–S161 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

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

C. F. R. Caron and R. M. Potvliege, “Bessel-modulated Gaussian beams with quadratic radial dependence,” Opt. Commun. 164, 83–93 (1999).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical system,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

A. Belafhal and L. Dalil-Essakali, “Collins formula and propagation of Bessel-modulated Gaussian light beams through an ABCD optical system,” Opt. Commun. 177, 181–188 (2000).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Usp.

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys. Usp. 47, 1177–1203 (2004).
[CrossRef]

Phys. Wave Phenom.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Hermite-Laguerre-Gauss beams,” Phys. Wave Phenom. 18, 14–22 (2010).
[CrossRef]

Vestnik Samara State Univ.

E. G. Abramochkin, “Hermite–Laguerre–Gaussian functions,” Vestnik Samara State Univ. 4, 19–41 (2001). [in Russian]

Other

E. G. Abramochkin and V. G. Volostnikov, Modern Theory of Gaussian Beams (Fizmatlit, Moscow, 2010). [in Russian].

Yu. A. Anan’ev, Optical Resonators and Gaussian beams (Nauka, Moscow, 1990). [in Russian].

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series; Vol. 2: Special Functions (Gordon and Breach Sci. Publ., 1986).

Yu. A. Brychkov, Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas (CRC Press, 2008).

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Figures (2)

Fig. 1
Fig. 1

Intensity and phase of the beam exp ( i Ψ 0 ) G 4 , 2 ( R ( π 4 ) r 2 | π 4 ( arg σ y arg σ x ) 2 ) for c x = 1 , c y = 3 2 , and various l. The vector r 2 = r 1 | σ x σ y | is chosen instead of r 1 used in the text to compensate for the scaled increase in the beam as l + . The frame number N is related to the detection plane l by the equation L = tan ( π N 16 ) , i.e., N = 0 , 4 , and 8 correspond to the initial beam and its transformations in the Rayleigh plane and in the Fourier plane, correspondingly.

Fig. 2
Fig. 2

Same as Fig. 1, but for c x = 2 2 , c y = 0 . HG beams are obtained for N = 2 and N = 6 .

Tables (1)

Tables Icon

Table 1 HLG Beams with the Parameter Shifts at Angles That Are Multiples of π 2

Equations (81)

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

( x 2 + y 2 + 2 i k l ) F = 0 .
| F ( r , l ) | l = 0 = F 0 ( r ) L 2 ( R 2 ) ,
F ( r , l ) = F R l [ F 0 ( ρ ) ] ( r ) = k 2 π i l R 2 exp ( i k 2 l | r ρ | 2 ) F 0 ( ρ ) d ρ ,
H n , m ( r ) = exp ( r 2 ) H n ( 2 x ) H m ( 2 y ) ,
L n , ± m ( r ) = exp ( r 2 ) r m e ± i m φ L n m ( 2 r 2 ) ,
F R l [ F 0 ( ρ w 0 ) ] ( r ) = 1 | σ | exp ( 2 i l r 2 k w 0 4 | σ | 2 i γ arg σ ) F 0 ( r w 0 | σ | ) .
G n , m ( r | 0 ) = ( i ) m H n , m ( r ) ,
G n , m ( r | π 4 ) = ( 1 ) min 2 max min ! L min , n m ( r ) .
G n , m ( x , y | α ) = ( i ) m exp ( r 2 ) ( 8 cos 2 α ) ( n + m ) 2 × k 0 ( 1 ) k 8 k k ! [ x 2 + y 2 ] k ( x α ) n ( y α ) m ,
G n , m ( x β , y β | α ) = ( cos 2 α cos 2 β ) ( n + m ) 2 exp ( 2 i x y sin 2 ( β α ) cos 2 α cos 2 β ) × k = 0 min ( 2 sin 2 ( β α ) cos 2 α ) k k ! ( m k ) ( n k ) × G n k , m k ( x α , y α | β ) ,
x θ = x cos θ + i y sin θ cos 2 θ , y θ = y cos θ + i x sin θ cos 2 θ
F [ exp ( i ψ ( ρ , α ) ) H n , m ( ρ ) ] ( r ) = 1 2 2 exp ( i ψ ( r , α ) 8 π i 4 ( n + m ) ) × G n , m ( R ( α ) r 2 2 | α ) ,
R ( θ ) = ( cos θ sin θ sin θ cos θ )
F [ f ( ρ ) ] ( r ) = 1 2 π R 2 exp [ i ( r ρ ) ] f ( ρ ) d ρ
F ( r ) = 1 2 π i det B R 2 exp ( i 2 [ ( ρ B 1 A ρ ) + ( r D B 1 r ) 2 ( r B 1 ρ ) ] ) f ( ρ ) d ρ .
F ( r ) = F [ exp ( i a ρ 2 + i b ψ ( ρ , β ) ) f ( ρ ) ] ( r ) .
F [ exp ( i a ρ 2 + i b ψ ( ρ , β ) ) G n , m ( ρ | α ) ] ( r ) = ( i ) n + m 2 | σ + σ | exp ( i Ψ ( r ) 4 | σ + σ | 2 + i ( n m ) ϕ + i ( n + m + 1 ) ω + ) G n , m ( R ( γ ) S R ( β ) r | θ ) .
Ψ ( r ) = a ( 1 + a 2 b 2 ) r 2 + b ( 1 + b 2 a 2 ) ψ ( r , β ) ,
S = 1 2 ( 1 | σ + | 0 0 1 | σ | ) ,
sin 2 θ = sin 2 α cos 2 ω + cos 2 α sin 2 β sin 2 ω ,
exp ( 2 i γ ) cos 2 θ = cos 2 α cos 2 β + i ( sin 2 α sin 2 ω cos 2 α sin 2 β cos 2 ω ) ,
exp ( i ϕ ± i θ ) = exp ( i ω ) ( cos γ sin γ ) ( cos α cos β + i sin α sin β ) exp ( i ω ) ( sin γ ± cos γ ) × ( cos α sin β i sin α cos β ) .
G n , m ( R ( α ) r | π 4 ) = e i ( n m ) α G n , m ( r | π 4 ) .
F [ exp ( i a ρ 2 ) G n , m ( ρ | α ) ] ( r ) = ( i ) n + m 2 1 + a 2 exp ( i a r 2 4 ( 1 + a 2 ) + i ( n + m + 1 ) arctan a ) × G n , m ( r 2 1 + a 2 | α ) .
F [ exp ( i b ψ ( ρ , β ) ) G n , m ( ρ | α ) ] ( r ) = ( i ) n + m 2 1 + b 2 exp ( i b ψ ( r , β ) 4 ( 1 + b 2 ) i ( n m ) ϕ ) × G n , m ( R ( γ β ) r 2 1 + b 2 | θ ) ,
F [ exp ( i b ψ ( ρ , π 4 ) ) G n , m ( ρ | α ) ] ( r ) = ( i ) n + m 2 1 + b 2 exp ( i b ψ ( r , π 4 ) 4 ( 1 + b 2 ) ) × G n , m ( r 2 1 + b 2 | α + arctan b ) .
F [ exp ( i ψ ( ρ , 0 ) ) G n , m ( ρ | α ) ] ( r ) = ( i ) n + m 2 2 exp ( i ψ ( r , 0 ) 8 + i ( n m ) π 4 ) × G n , m ( R ( α ) r 2 2 | 0 ) .
F [ exp ( i a ρ 2 + i 1 + a 2 ψ ( ρ , β ) ) G n , m ( ρ | π 4 ) ] ( r ) = ( i ) n + m 2 2 1 + a 2 4 exp [ i ψ ( r , β ) 8 1 + a 2 + i ( n m ) ( β + π 4 ) + i 2 ( n + m + 1 ) arctan a ] × G n , m [ R ( π 4 ) S R ( β ) r | 0 ] ,
S = 1 2 ( 1 a 1 0 0 1 + a 1 ) , a 1 = a 1 + a 2 .
F [ exp ( ± i ( ξ c η ) 2 + ( η c ξ ) 2 1 c 2 ) G n , m ( ρ | α ) ] ( r ) = ( i ) n + m 2 2 c 1 exp ( i r 2 8 c 1 ± i ( n + m + 1 ) π 4 ) × G n , m ( r 1 | α arctan c ) ,
F [ exp ( ± 2 i ( ξ c η ) ( η c ξ ) 1 c 2 ) G n , m ( ρ | α ) ] ( r ) = ( i ) n + m 2 2 c 1 exp ( i x y 4 c 1 i ( n + m + 1 ) arctan c ) × G n , m ( r 1 | α ± π 4 ) ,
c 1 = 1 + c 2 1 c 2 , r 1 = ( x + c y , y + c x ) 2 2 1 + c 2 .
a = ± 1 + c 2 1 c 2 , b = 2 c 1 c 2 , β = π 4 ,
ω + = ± π 4 , ω = arctan c
a = 2 c 1 c 2 , b = ± 1 + c 2 1 c 2 , β = π 4 ,
ω + = arctan c , ω = ± π 4
F [ exp ( i sin 2 β ψ ( ρ , β ) ) G n , m ( ρ | π 8 ) ] ( r ) = ( i ) n + m 2 1 + sin 2 2 β exp ( i sin 2 β ψ ( r , β ) 4 ( 1 + sin 2 2 β ) + i ( n m ) ϕ ) G n , m ( R ( 2 β ) r 2 1 + sin 2 2 β | π 8 ) ,
LGA n , m ( r ) = exp ( i c x x 2 + i c y y 2 ) G n , m ( r | π 4 ) .
F R l [ LGA n , m ( ρ w 0 ) ] ( r ) = k w 0 2 i l exp ( i k r 2 2 l ) F [ exp ( i k w 0 2 ρ 2 2 l + i c x ξ 2 + i c y η 2 ) G n , m ( ρ | π 4 ) ] ( k w 0 r l ) .
a = k w 0 2 2 l + c x + c y 2 , b = c x c y 2 , β = 0 , α = π 4 .
L = 2 l k w 0 2 , σ x = 1 + c x L + i L , σ y = 1 + c y L + i L ,
F R l [ LGA n , m ( ρ w 0 ) ] ( r ) = 1 | σ x σ y | exp [ i Ψ ( r ) + i Ψ 0 ] × G n , m [ R ( π 4 ) r 1 | π 4 ω ] ,
Ψ ( r ) = c x + ( 1 + c x 2 ) L w 0 2 | σ x | 2 x 2 + c y + ( 1 + c y 2 ) L w 0 2 | σ y | 2 y 2 ,
Ψ 0 = ( n m ) π 4 ( n + m + 1 ) arg σ x + arg σ y 2 ,
r 1 = ( x w 0 | σ x | , y w 0 | σ y | ) .
( 1 + c x c y ) L 2 + ( c x + c y ) L + 1 = 0 .
R exp ( 2 x 2 2 x ( b + c ) ) H n ( x + b ) H m ( x + c ) d x = π ( 1 ) m 2 ( n + m + 1 ) 2 exp ( ( b + c ) 2 2 ) H n + m ( b c 2 ) ,
R exp ( x 2 + 2 a x ) H n ( x + b ) H m ( x + c ) d x = π exp ( a 2 ) 2 max min ! ( a + a 1 ) | n m | × L min | n m | [ 2 ( a + b ) ( a + c ) ] ,
F n , m ( r ) = F [ exp ( i a ρ 2 + i b ψ ( ρ , β ) ) G n , m ( ρ | α ) ] ( r ) .
R exp ( i x ξ c ξ 2 ) d ξ = π c exp ( x 2 4 c ) = π | c | exp ( x 2 4 c i 2 arg c ) ,
F [ exp ( i a ρ 2 + i b ψ ( ρ , β ) ρ 2 ) ] ( r ) = 1 2 σ + * σ * exp ( ( 1 i a ) r 2 + i b ψ ( r , β ) 4 σ + * σ * ) = 1 2 | σ + σ | exp ( A r 2 + B ψ ( r , β ) 4 | σ + σ | 2 + i arg σ + + arg σ 2 ) ,
σ ± = 1 + i ( a ± b ) ,
A = ( 1 + a 2 + b 2 ) + i a ( 1 + a 2 b 2 ) ,
B = 2 a b + i b ( 1 + b 2 a 2 ) .
n , m = 0 G n , m ( r | α ) s n t m n ! m ! = exp ( r 2 ψ ( s , t , α ) + 2 2 ( x S i y T ) ) ,
F ( r , s , t ) = n , m = 0 F n , m ( r ) s n t m n ! m ! = exp ( ψ ( s , t , α ) ) × F [ exp ( i a ρ 2 + i b ψ ( ρ , β ) ρ 2 ) ] ( r 1 ) = 1 2 σ + * σ * exp ( ψ ( s , t , α ) ( 1 i a ) r 1 2 + i b ψ ( r 1 , β ) 4 σ + * σ * ) ,
F n , m ( r ) = | s n t m F ( r , s , t ) | s = t = 0 = F 0 , 0 ( r ) × [ some polynomial on x , y of order n + m ] ,
F ̃ n , m ( r ) = F 0 , 0 ( r ) z 1 n z 2 m G n , m P ( V r | θ ) = 1 2 σ + * σ * exp ( i Im [ A r 2 + B ψ ( r , β ) ] 4 | σ + σ | 2 ) × z 1 n z 2 m G n , m ( V r | θ )
V = ( v 11 v 12 v 21 v 22 ) GL 2 ( R ) .
G n , m ( r | θ ) = exp ( r 2 ) G n , m P ( r | θ ) .
F ̃ ( r , s , t ) = n , m = 0 F ̃ n , m ( r ) s n t m n ! m ! = F 0 , 0 ( r ) n , m = 0 G n , m P ( V r | θ ) ( z 1 s ) n ( z 2 t ) m n ! m ! = F 0 , 0 ( r ) exp ( ψ ( z 1 s , z 2 t , θ ) + 2 2 [ v x ( z 1 s cos θ + z 2 t sin θ ) i v y ( z 2 t cos θ z 1 s sin θ ) ] ) ,
Re [ A r 2 + B ψ ( r , β ) ] 4 | σ + σ | 2 = Re [ A ( X 2 + Y 2 ) + B ( X 2 Y 2 ) ] 4 | σ + σ | 2 = X 2 4 | σ + | 2 + Y 2 4 | σ | 2 = | V r | 2 V = 1 2 R ( γ ) ( 1 | σ + | 0 0 1 | σ | ) R ( β ) ,
z 1 cos θ = i exp ( i ω + ) [ exp ( i ω ) A cos γ exp ( i ω ) B * sin γ ] ,
z 1 sin θ = exp ( i ω + ) [ exp ( i ω ) A sin γ + exp ( i ω ) B * cos γ ] ,
z 2 cos θ = i exp ( i ω + ) [ exp ( i ω ) A * cos γ exp ( i ω ) B sin γ ] ,
z 2 sin θ = exp ( i ω + ) [ exp ( i ω ) A * sin γ + exp ( i ω ) B cos γ ] ,
A = cos α cos β + i sin α sin β ,
B = cos α sin β + i sin α cos β ,
ω ± = arg σ + ± arg σ 2 { 2 ω + = arg ( 1 + b 2 a 2 + 2 i a ) , 2 ω = arg ( 1 + a 2 b 2 + 2 i b ) . }
| z 1 cos θ | 2 + | z 1 sin θ | 2 = | z 2 cos θ | 2 + | z 2 sin θ | 2 = 1 ,
z 1 cos θ × z 2 cos θ + z 1 sin θ × z 2 sin θ = exp ( 2 i ω + ) ,
z 1 = i exp ( i ω + + i ϕ ) ,
z 2 = i exp ( i ω + i ϕ ) .
{ exp ( i ϕ ) [ cos θ cos γ i sin θ sin γ ] = A exp ( i ω ) , exp ( i ϕ ) [ cos θ sin γ + i sin θ cos γ ] = B * exp ( i ω ) . }
cos 2 θ cos 2 γ = cos 2 α cos 2 β ,
cos 2 θ sin 2 γ + i sin 2 θ = exp ( 2 i ω ) [ cos 2 α sin 2 β + i sin 2 α ] .
sin 2 θ = sin 2 α cos 2 ω + cos 2 α sin 2 β sin 2 ω ,
exp ( 2 i γ ) cos 2 θ = cos 2 α cos 2 β + i [ sin 2 α sin 2 ω cos 2 α sin 2 β cos 2 ω ] .
0 = sin 2 γ × cos 2 θ cos 2 γ cos 2 γ × cos 2 θ sin 2 γ = cos 2 α cos 2 β sin 2 γ [ sin 2 α sin 2 ω cos 2 α sin 2 β cos 2 ω ] cos 2 γ .
exp ( i ϕ ± i θ ) = A exp ( i ω ) ( cos γ sin γ ) B * exp ( i ω ) ( sin γ ± cos γ ) .
F n , m ( r ) = ( i ) n + m 2 | σ + σ | exp ( i ( n m ) ϕ + i ( n + m + 1 ) ω + ) × exp ( i Im [ A r 2 + B ψ ( r , β ) ] 4 | σ + σ | 2 ) G n , m ( V r | θ ) ,

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