Abstract

Flattened Gaussian beams are characterized by a waist profile that passes in a continuous way from a nearly flat illuminated region to darkness. The steepness of the transition region is controlled by an integer parameter N representing the order of the beam. Being expressible as a sum of N Laguerre–Gauss modes, a flattened Gaussian beam turns out to be very simple to study as far as propagation is concerned. We investigate the main features of the field distribution pertaining to a flattened Gaussian beam throughout the space and present experimental results relating to the laboratory production of this type of beam.

© 1996 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  4. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  31. C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
    [CrossRef]

1994

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

P.-A. Bélanger, Y. Champagne, C. Paré, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

1993

1992

1991

1988

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

1983

Y. Kawamura, Y. Itagaki, K. Toyota, S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44–46 (1983).
[CrossRef]

1976

1975

H. J. Landau, “On Szegö eigenvalue distribution theorem and non-Hermitian kernels,” J. Analyse Math. 28, 335–357 (1975).
[CrossRef]

1972

1961

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions: Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–63 (1961).

Bagini, V.

C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

Barakat, R.

R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), pp. 35–80.
[CrossRef]

Bélanger, P. A.

Bélanger, P.-A.

P.-A. Bélanger, Y. Champagne, C. Paré, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

Bollanti, S.

S. Bollanti, P. Di Lazzaro, F. Flora, G. Giordano, T. Letardi, C. Petrucci, G. Schina, C. E. Zheng, “Compact three-electrodes excimer laser IANUS for a POPA optical system,” in High-Power Gas and Solid State Lasers, D. Schuoecker, T. Letardi, M. Bohrer, H. Weber, eds., Proc. SPIE2206, 144–153 (1994).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed., reprinted (Pergamon, Oxford, 1993).

Bowers, M. S.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed., revised (McGraw-Hill, New York, 1986).

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series. Vol. I, reprinted (Gordon & Breach, New York, 1992).

Bussière, S.

Champagne, Y.

P.-A. Bélanger, Y. Champagne, C. Paré, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

Colombeau, B.

Cox, A. J.

D’Anna, J.

De Silvestri, S.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Dew, S. K.

Di Lazzaro, P.

S. Bollanti, P. Di Lazzaro, F. Flora, G. Giordano, T. Letardi, C. Petrucci, G. Schina, C. E. Zheng, “Compact three-electrodes excimer laser IANUS for a POPA optical system,” in High-Power Gas and Solid State Lasers, D. Schuoecker, T. Letardi, M. Bohrer, H. Weber, eds., Proc. SPIE2206, 144–153 (1994).
[CrossRef]

Dobrowolski, J. A.

Dohnalik, T.

Duplain, G.

Flora, F.

S. Bollanti, P. Di Lazzaro, F. Flora, G. Giordano, T. Letardi, C. Petrucci, G. Schina, C. E. Zheng, “Compact three-electrodes excimer laser IANUS for a POPA optical system,” in High-Power Gas and Solid State Lasers, D. Schuoecker, T. Letardi, M. Bohrer, H. Weber, eds., Proc. SPIE2206, 144–153 (1994).
[CrossRef]

Froehly, C.

Giordano, G.

S. Bollanti, P. Di Lazzaro, F. Flora, G. Giordano, T. Letardi, C. Petrucci, G. Schina, C. E. Zheng, “Compact three-electrodes excimer laser IANUS for a POPA optical system,” in High-Power Gas and Solid State Lasers, D. Schuoecker, T. Letardi, M. Bohrer, H. Weber, eds., Proc. SPIE2206, 144–153 (1994).
[CrossRef]

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Gupta, R.

Hrynevych, M.

Ih, C. S.

Itagaki, Y.

Y. Kawamura, Y. Itagaki, K. Toyota, S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44–46 (1983).
[CrossRef]

Kawamura, Y.

Y. Kawamura, Y. Itagaki, K. Toyota, S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44–46 (1983).
[CrossRef]

Kermene, V.

Klingsporn, P. E.

Lachance, R. L.

Landau, H. J.

H. J. Landau, “On Szegö eigenvalue distribution theorem and non-Hermitian kernels,” J. Analyse Math. 28, 335–357 (1975).
[CrossRef]

Laporta, P.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Lavigne, P.

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

Letardi, T.

S. Bollanti, P. Di Lazzaro, F. Flora, G. Giordano, T. Letardi, C. Petrucci, G. Schina, C. E. Zheng, “Compact three-electrodes excimer laser IANUS for a POPA optical system,” in High-Power Gas and Solid State Lasers, D. Schuoecker, T. Letardi, M. Bohrer, H. Weber, eds., Proc. SPIE2206, 144–153 (1994).
[CrossRef]

López-Olazagasti, E.

J. Ojeda-Castaneda, G. Saavedra, E. López-Olazagasti, “Super-Gaussian beams of continuous order as GRIN modes,” Opt. Commun. 102, 21–24 (1993).
[CrossRef]

Magni, V.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Malyak, P. H.

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series. Vol. I, reprinted (Gordon & Breach, New York, 1992).

Martínez-Herrero, R.

Mejías, P. M.

Metcalf, H.

Morin, M.

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

Namba, S.

Y. Kawamura, Y. Itagaki, K. Toyota, S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44–46 (1983).
[CrossRef]

Ojeda-Castaneda, J.

J. Ojeda-Castaneda, G. Saavedra, E. López-Olazagasti, “Super-Gaussian beams of continuous order as GRIN modes,” Opt. Commun. 102, 21–24 (1993).
[CrossRef]

Palma, C.

C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

Paré, C.

P.-A. Bélanger, Y. Champagne, C. Paré, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

P. A. Bélanger, R. L. Lachance, C. Paré, “Super-Gaussian output from a CO2laser by using a graded-phase mirror resonator,” Opt. Lett. 17, 739–741 (1991).
[CrossRef]

Parent, A.

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

Parsons, R. R.

Perrone, M. R.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Petrucci, C.

S. Bollanti, P. Di Lazzaro, F. Flora, G. Giordano, T. Letardi, C. Petrucci, G. Schina, C. E. Zheng, “Compact three-electrodes excimer laser IANUS for a POPA optical system,” in High-Power Gas and Solid State Lasers, D. Schuoecker, T. Letardi, M. Bohrer, H. Weber, eds., Proc. SPIE2206, 144–153 (1994).
[CrossRef]

Piegari, A.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions: Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–63 (1961).

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series. Vol. I, reprinted (Gordon & Breach, New York, 1992).

Saavedra, G.

J. Ojeda-Castaneda, G. Saavedra, E. López-Olazagasti, “Super-Gaussian beams of continuous order as GRIN modes,” Opt. Commun. 102, 21–24 (1993).
[CrossRef]

Saviot, A.

Scaglione, S.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Schina, G.

S. Bollanti, P. Di Lazzaro, F. Flora, G. Giordano, T. Letardi, C. Petrucci, G. Schina, C. E. Zheng, “Compact three-electrodes excimer laser IANUS for a POPA optical system,” in High-Power Gas and Solid State Lasers, D. Schuoecker, T. Letardi, M. Bohrer, H. Weber, eds., Proc. SPIE2206, 144–153 (1994).
[CrossRef]

Serna, J.

Sheppard, C. J. R.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Slepian, D.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions: Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–63 (1961).

Svelto, O.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Toyota, K.

Y. Kawamura, Y. Itagaki, K. Toyota, S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44–46 (1983).
[CrossRef]

Vampouile, M.

Verly, P. G.

Waldorf, A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed., reprinted (Pergamon, Oxford, 1993).

Xie, C.

Zheng, C. E.

S. Bollanti, P. Di Lazzaro, F. Flora, G. Giordano, T. Letardi, C. Petrucci, G. Schina, C. E. Zheng, “Compact three-electrodes excimer laser IANUS for a POPA optical system,” in High-Power Gas and Solid State Lasers, D. Schuoecker, T. Letardi, M. Bohrer, H. Weber, eds., Proc. SPIE2206, 144–153 (1994).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions: Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–63 (1961).

IEEE J. Quantum Electron.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

J. Analyse Math.

H. J. Landau, “On Szegö eigenvalue distribution theorem and non-Hermitian kernels,” J. Analyse Math. 28, 335–357 (1975).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

P.-A. Bélanger, Y. Champagne, C. Paré, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

J. Ojeda-Castaneda, G. Saavedra, E. López-Olazagasti, “Super-Gaussian beams of continuous order as GRIN modes,” Opt. Commun. 102, 21–24 (1993).
[CrossRef]

Y. Kawamura, Y. Itagaki, K. Toyota, S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44–46 (1983).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

Other

S. Bollanti, P. Di Lazzaro, F. Flora, G. Giordano, T. Letardi, C. Petrucci, G. Schina, C. E. Zheng, “Compact three-electrodes excimer laser IANUS for a POPA optical system,” in High-Power Gas and Solid State Lasers, D. Schuoecker, T. Letardi, M. Bohrer, H. Weber, eds., Proc. SPIE2206, 144–153 (1994).
[CrossRef]

As usual, circ(r) is defined as 1 if r≤ 1 and 0 elsewhere, rbeing the radial coordinate in R2.

M. Born, E. Wolf, Principles of Optics, 6th ed., reprinted (Pergamon, Oxford, 1993).

R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), pp. 35–80.
[CrossRef]

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed., revised (McGraw-Hill, New York, 1986).

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series. Vol. I, reprinted (Gordon & Breach, New York, 1992).

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Function fN versus ξ [see Eq. (1)], for different values of N.

Fig. 2
Fig. 2

Function UN versus r [see Eq. (2)], for different values of N.

Fig. 3
Fig. 3

Modula of the expansion coefficients c n ( N ) for a FGB of order N [see Eq. (10)] versus n, for different values of N.

Fig. 4
Fig. 4

Three-dimensional plots of the intensity of propagated FGB as a function of r and NF, for (a) N = 4, (b) N = 16, and (c) N = 49. The other parameters are w0 = 1 mm and λ = 0.5 μm.

Fig. 5
Fig. 5

FGB on-axis intensity as a function of z [see Eq. (20)], for different values of N (solid curves). The dotted curve represents the on-axis intensity associated with a circular hole of radius w0 [see Eq. (16)]. The other parameters are w0 = 1 mm and λ = 0.5 μm.

Fig. 6
Fig. 6

Two-dimensional Fourier transform of function UN(r) [see Eq. (26)] normalized to the value w02, for different values of N. The dotted curve represents a two-dimensional Fourier transform of circ(r/w0) [see Eq. (28)].

Fig. 7
Fig. 7

Behavior of the M2 quality factor for a FGB as a function of N. Dotted curve, exact values [Eq. (41)]; solid curve, approximation (45).

Fig. 8
Fig. 8

Experimental values (circles) of intensity profile for a FGB on its waist plane, with N = 14 and w0 = 0.53 mm. The solid curve indicates the theoretical curve |UN(r)|2.

Fig. 9
Fig. 9

Experimental near-field propagated intensity profile of the FGB of Fig. 8 (circles) at (a) z = 182 mm, (b) z = 273 mm, and (c) z = 546 mm. Solid curves indicate the theoretical behavior as predicted by Eq. (13).

Fig. 10
Fig. 10

Experimental transverse distributions of the intensity of the FGB of Fig. 8 at (a) z = 0, (b) z = 182 mm, (c) z = 273 mm, and (d) 546 mm.

Fig. 11
Fig. 11

Experimental behavior of on-axis intensity of the FGB of Fig. 8 (circles), together with the theoretical behavior (solid curve) as predicted by Eq. (20).

Fig. 12
Fig. 12

Experimental far-field intensity profile of the FGB of Fig. 8 (circles), together with the theoretical behavior (solid curve) as predicted by Eq. (26).

Fig. 13
Fig. 13

Experimental transverse distribution of the far-field intensity of the FGB of Fig. 8.

Equations (61)

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f N ( ξ ) = exp ( - ξ 2 ) n = 0 N ξ 2 n n ! .
U N ( r ) = A 0 f N ( N + 1 r w 0 ) = A 0 exp [ - ( N + 1 ) r 2 w 0 2 ] n = 0 N 1 n ! ( N + 1 r w 0 ) 2 n ,
f N ( ξ ) = - 2 ξ 2 N + 1 N ! exp ( - ξ 2 ) ,
f N ( ξ ) = - 2 ξ 2 N N ! ( 2 N + 1 - 2 ξ 2 ) exp ( - ξ 2 ) ,
U N max = A 0 2 N + 1 w 0 N ! ( N + 1 2 ) N + 1 / 2 × exp [ - ( N + 1 2 ) ] .
U N max A 0 w 0 2 N π .
V γ ( r ) = A 0 exp [ - ( r / w 0 ) γ ] ,
V N max A 0 w 0 e γ ;
U N ( r ) = A 0 n = 0 N c n ( N ) L n [ 2 ( N + 1 ) r 2 w 0 2 ] exp [ - ( N + 1 ) r 2 w 0 2 ] ,
c n ( N ) = ( - 1 ) n m = n N 1 2 m ( m n ) .
c n + 1 ( N ) = - c n ( N ) + ( - 1 ) n 2 N ( N + 1 n + 1 ) ,             n = 0 , , N - 1 , c 0 ( N ) = 2 - 1 / 2 N ,
c N - n ( N ) = ( - 1 ) N - n ( 2 - c n ( N ) ) ,
U N ( r ; z ) = A 0 w N ( 0 ) w N ( z ) exp { i [ k z - Φ N ( z ) ] } × exp { [ i k 2 R N ( z ) - 1 w N 2 ( z ) ] r 2 } × n = 0 N c n ( N ) L n [ 2 r 2 w N 2 ( z ) ] exp [ - 2 i n Φ N ( z ) ] ,
w N ( z ) = w N ( 0 ) 1 + [ λ z π w N 2 ( 0 ) ] 2 ,
R N ( z ) = z { 1 + [ π w N 2 ( 0 ) λ z ] 2 } ,
Φ N ( z ) = arctan [ λ z π w N 2 ( 0 ) ] ;
w N ( 0 ) = w 0 N + 1 .
N F = w 0 2 / λ z .
U N ( 0 ; z ) = A 0 w N ( 0 ) w N ( z ) exp { i [ k z - Φ N ( z ) ] } × n = 0 N c n ( N ) exp [ - 2 i n Φ N ( z ) ] = A 0 w N ( 0 ) w N ( z ) exp { i [ k z - Φ N ( z ) ] } × n = 0 N ( - 1 ) n m = n N 1 2 m ( m n ) × exp [ - 2 i n Φ N ( z ) ] ,
U N ( 0 ; z ) = A 0 w N ( 0 ) w N ( z ) exp { i [ k z - Φ N ( z ) ] } × m = 0 N 1 2 m n = 0 m ( m n ) × { - exp [ - 2 i Φ N ( z ) ] } n = A 0 w N ( 0 ) w N ( z ) exp { i [ k z - Φ N ( z ) ] } × m = 0 N { 1 - exp [ - 2 i Φ N ( z ) ] 2 } m .
U N ( 0 ; z ) = A 0 w N ( 0 ) w N ( z ) exp { i [ k z - Φ N ( z ) ] } × 1 - sin N + 1 Φ N ( z ) exp { - i ( N + 1 ) [ Φ N ( z ) - π / 2 ] } cos Φ N ( z ) exp [ - i Φ N ( z ) ] .
I N ( 0 ; z ) = A 0 2 w N 2 ( 0 ) w N 2 ( z ) 1 + sin 2 N + 2 Φ N ( z ) - 2 sin N + 1 Φ N ( z ) cos { ( N + 1 ) [ Φ N ( z ) - π / 2 ] } cos 2 Φ N ( z ) .
I ( 0 ; z ) = 4 A 0 2 sin 2 ( k w 0 2 / 4 z ) ,
w N = A w N ( 0 ) 1 + [ F ( A , B ) ] 2 .
R N = A B 1 + [ F ( A , B ) ] - 2 1 + B C { 1 + [ F ( A , B ) ] - 2 } ,
Φ N = arctan [ F ( A , B ) ] ,
U ˜ N ( ρ ) = 2 π 0 U N ( r ) J 0 ( 2 π ρ r ) r d r = 2 π n = 0 N α n n ! 0 r 2 n + 1 J 0 ( 2 π ρ r ) × exp ( - α r 2 ) d r ,
α = N + 1 w 0 2 ,
U ˜ N ( ρ ) = π α exp ( - π 2 α ρ 2 ) n = 0 N L n ( π 2 α ρ 2 ) ,
U ˜ N ( ρ ) = π α exp ( - π 2 α ρ 2 ) L n ( 1 ) ( π 2 α ρ 2 ) = π w 0 2 N + 1 L n ( 1 ) ( π 2 w 0 2 N + 1 ρ 2 ) exp ( - π 2 w 0 2 N + 1 ρ 2 ) ,
lim n [ 1 n β L n ( β ) ( x n ) ] = x - β / 2 J β ( 2 x ) ,
lim N U ˜ N ( ρ ) = J 1 ( 2 π w 0 ρ ) ρ w 0 ,
M 2 = 2 π σ r σ ρ ,
σ r 2 = 0 U ( r ) 2 r 3 d r 0 U ( r ) 2 r d r ,
σ ρ 2 = 0 U ˜ ( ρ ) 2 ρ 3 d ρ 0 U ˜ ( ρ ) 2 ρ d ρ .
0 U N ( r ) 2 r 3 d r = A 0 2 w 0 4 8 ( N + 1 ) 2 R N ,
0 U ˜ N ( ρ ) 2 ρ 3 d ρ = A 0 2 S N ,
0 U N ( r ) 2 r d r = 0 U ˜ N ( ρ ) 2 ρ d ρ = A 0 2 w 0 2 4 ( N + 1 ) T N ,
σ r 2 = w 0 2 2 ( N + 1 ) R N T N ,
σ ρ 2 = 4 ( N + 1 ) w 0 2 S N T N ;
M 2 = 2 π 2 R N S N T N .
S N = 1 4 π 2 ( 2 N + 1 ) ! ( N ! ) 2 2 2 N + 1 ,
R N = n = 0 N m = 0 N n + m + 1 2 n + m ( n + m n ) ,
T N = n = 0 N [ m = n N 1 2 m ( m n ) ] 2 .
M 2 = 2 - N N ! T N ( 2 N + 1 ) ! R N .
R N 8 N 2 w 0 4 0 w 0 r 3 d r = 2 N 2 ,
T N 4 N w 0 2 0 w 0 r d r = 2 N ,
M 2 ( 2 N + 1 ) ! 2 N + 1 / 2 N ! .
M 2 ( N / π ) 1 / 4 .
M 2 γ / 2.
c n + 1 ( N ) = ( - 1 ) n + 1 m = n + 1 N 1 2 m ( m n + 1 ) .
c n + 1 ( N ) = ( - 1 ) n + 1 2 k = n N - 1 1 2 k ( k + 1 n + 1 ) ,
( k + 1 n + 1 ) = ( k n ) + ( k n + 1 ) ,
c n + 1 ( N ) = ( - 1 ) n + 1 2 [ k = n N - 1 1 2 k ( k n ) + k = n + 1 N - 1 1 2 k ( k n + 1 ) ] .
c n + 1 ( N ) = - 1 2 { ( - 1 ) n k = n N 1 2 k ( k n ) - ( - 1 ) n + 1 k = n + 1 N 1 2 k ( k n + 1 ) - ( - 1 ) n 2 N [ ( N n ) + ( N n + 1 ) ] } = - 1 2 [ c n ( N ) - c n + 1 ( N ) - ( - 1 ) n 2 N ( N + 1 n + 1 ) ] ,
c 0 ( N ) = 2 - 1 2 N ,
c N ( N ) = ( - 1 ) N 2 N ,
| c 0 ( N ) | + | c N ( N ) | = 2.
| c n - 1 ( N ) | + | c N - n + 1 ( N ) | = 2 ,
| c n ( N ) | + | c N - n ( N ) | = 2.
| c n ( N ) | + | c N - n ( N ) | = ( - 1 ) n c n ( N ) + ( - 1 ) N - n c N - n ( N ) = ( - 1 ) n [ - c n - 1 ( N ) + ( - 1 ) n - 1 2 N ( N + 1 n ) ] + ( - 1 ) N - n [ - c N - n + 1 ( N ) + ( - 1 ) N - n 2 N ( N + 1 N - n + 1 ) ] = | c n - 1 ( N ) | + | c N - n + 1 ( N ) | + 1 2 N [ ( N + 1 N + 1 - n ) - ( N + 1 n ) ] = | c n - 1 ( N ) | + | c N - n + 1 ( N ) | ,

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