Abstract

On the basis of superposition of beams, a group of virtual sources that generate a cosh-Gaussian wave is identified. A closed-form expression is derived for this cosh-Gaussian wave, which, in the appropriate limit, yields the paraxial approximation for the cosh-Gaussian beam. From this expression, the paraxial approximation and the nonparaxial corrections of all orders for the corresponding paraxial cosh-Gaussian beam are determined.

© 2007 Optical Society of America

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References

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G. P. Agrawal and M. Lax, Phys. Rev. A 27, 1963 (1983).
[CrossRef]

1981

M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

1979

A. L. Cullen and P. K. Yu, Phys. Rev. A 366, 155 (1979).

1977

1976

1971

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

1967

Agrawal, G. P.

G. P. Agrawal and M. Lax, Phys. Rev. A 27, 1963 (1983).
[CrossRef]

Belanger, P.-A.

M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Byron, F. W.

F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics, Vol. 2 (Addison-Wesley, 1969).

Casperson, L. W.

Couture, M.

M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Cullen, A. L.

A. L. Cullen and P. K. Yu, Phys. Rev. A 366, 155 (1979).

Deschamps, G. A.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Felsen, L. B.

Fuller, R. W.

F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics, Vol. 2 (Addison-Wesley, 1969).

Hall, D. G.

Lax, M.

G. P. Agrawal and M. Lax, Phys. Rev. A 27, 1963 (1983).
[CrossRef]

R. Sheppard, C. J.

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Seshadri, S. R.

Sherman, G. C.

Shin, S. Y.

Tover, A. A.

Yu, P. K.

A. L. Cullen and P. K. Yu, Phys. Rev. A 366, 155 (1979).

Electron. Lett.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Rev. A

A. L. Cullen and P. K. Yu, Phys. Rev. A 366, 155 (1979).

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

G. P. Agrawal and M. Lax, Phys. Rev. A 27, 1963 (1983).
[CrossRef]

Other

F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics, Vol. 2 (Addison-Wesley, 1969).

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Equations (29)

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E p ( ρ , 0 ) = exp [ ρ 2 w 0 2 ] cosh ( Ω 0 x ) cosh ( Ω 0 y ) ,
E p ( ρ , 0 ) = exp ( a 2 2 w 0 2 ) 4 { exp [ ( ( x a 2 w 0 ) 2 + ( y a 2 w 0 ) 2 ) ] + exp [ ( ( x + a 2 w 0 ) 2 + ( y + a 2 w 0 ) 2 ) ] + exp [ ( ( x a 2 w 0 ) 2 + ( y + a 2 w 0 ) 2 ) ] + exp [ ( ( x + a 2 w 0 ) 2 + ( y a 2 w 0 ) 2 ) ] } ,
( 2 x 2 + 2 y 2 + 2 z 2 + k 2 ) E ( ρ , z ) = S ex δ ( z z ex ) [ δ ( x b ) δ ( y b ) + δ ( x + b ) δ ( y + b ) + δ ( x b ) δ ( y + b ) + δ ( x + b ) δ ( y b ) ] ,
E ( ρ , z ) = 1 4 π 2 E ( q , z ) ¯ exp ( i ρ q ) d q ,
E ( q , z ) ¯ = E ( ρ , z ) exp ( i ρ q ) d ρ ,
E ( ρ , z ) = 1 4 π 2 exp ( i ρ q ) i S ex 2 ξ exp [ i ξ ( z z ex ) ] { exp [ i ( u + v ) b ] + exp [ i ( u + v ) b ] + exp [ i ( u v ) b ] + exp [ i ( u v ) b ] } d q
E ( ρ , z ) = 1 4 π 2 exp ( i ρ q ) i S ex 2 k exp [ i k ( z z ex ) ] exp [ i q 2 2 k ( z z ex ) ] { exp [ i ( u + v ) b ] + exp [ i ( u + v ) b ] + exp [ i ( u v ) b ] + exp [ i ( u v ) b ] } d q .
E ( ρ , z ) = S ex exp [ i k ( z z ex ) ] 4 π ( z z ex ) { exp [ i k ( x b ) 2 + ( y b ) 2 2 ( z z ex ) ] + exp [ i k ( x + b ) 2 + ( y + b ) 2 2 ( z z ex ) ] + exp [ i k ( x b ) 2 + ( y + b ) 2 2 ( z z ex ) ] + exp [ i k ( x + b ) 2 + ( y b ) 2 2 ( z z ex ) ] } .
b = a 2 ,
z ex = i k w 0 2 2 = i z 0 ,
S ex = i π z 0 exp ( a 2 2 w 0 2 k z 0 ) .
E p ( ρ , z ) = i z 0 z i z 0 exp ( a 2 2 w 0 2 + i k ( z + a 2 4 ( z i z 0 ) ) ) exp ( i k + ρ 2 2 ( z i z 0 ) ) cosh ( i k a x 2 ( z i z 0 ) ) × cosh ( i k a y 2 ( z i z 0 ) ) .
E ( ρ , z ) = z 0 4 π exp ( i ρ q ) exp ( a 2 2 w 0 2 k z 0 ) 2 ξ exp [ i ξ ( z i z 0 ) ] { exp [ i ( u + v ) a 2 ] + exp [ i ( u + v ) a 2 ] + exp [ i ( u v ) a 2 ] + exp [ i ( u v ) a 2 ] } d q .
i 2 π exp [ i k ( k 2 u 2 v 2 ) 1 2 z ] ( k 2 u 2 v 2 ) 1 2 exp [ i ( x u + v y ) ] d u d v = exp [ i k ( x 2 + y 2 + z 2 ) 1 2 ] ( x 2 + y 2 + v 2 ) 1 2 ,
exp [ i ( u x 0 + v y 0 ) ] F ( u , v ) exp [ i ( u x + v y ) ] d u d v = f ( x x 0 , y y 0 ) ,
E ( ρ , z ) = i z 0 4 exp ( a 2 2 w 0 2 k z 0 ) [ exp ( i k r 1 ) r 1 + exp ( i k r 2 ) r 2 + exp ( i k r 3 ) r 3 + exp ( i k r 4 ) r 4 ] ,
r 1 = [ ( x a 2 ) 2 + ( y a 2 ) 2 + ( z i z 0 ) 2 ] 1 2 ,
r 2 = [ ( x + a 2 ) 2 + ( y + a 2 ) 2 + ( z i z 0 ) 2 ] 1 2 ,
r 3 = [ ( x + a 2 ) 2 + ( y a 2 ) 2 + ( z i z 0 ) 2 ] 1 2 ,
r 4 = [ ( x a 2 ) 2 + ( y + a 2 ) 2 + ( z i z 0 ) 2 ] 1 2 ,
E ( ρ , z ) = E ( 0 ) ( ρ , z ) + n = 1 f 2 n E ( 2 n ) ( ρ , z ) ,
E ( 0 ) ( ρ , z ) = j = 1 4 E j ( 0 ) ( ρ , z ) ,
E ( 2 n ) ( ρ , z ) = j = 1 4 E j ( 2 n ) ( ρ , z ) ,
E j ( 0 ) ( ρ , z ) = i 4 exp ( a 2 2 w 0 2 + i k z ) Q exp ( i Q ρ j 2 ) ,
E j ( 2 n ) ( ρ , z ) = ( 1 ) n ( Q ρ j ) 2 n L n n ( i Q ρ j 2 ) E j ( 0 ) ( ρ , z ) ,
Q = ( z z 0 i ) 1 , ρ 1 2 = ( x a 2 ) 2 + ( y a 2 ) 2 w 0 2 ,
ρ 2 2 = ( x + a 2 ) 2 + ( y + a 2 ) 2 w 0 2 ,
ρ 3 2 = ( x + a 2 ) 2 + ( y a 2 ) 2 w 0 2 ,
ρ 4 2 = ( x a 2 ) 2 + ( y + a 2 ) 2 w 0 2 , j = 1 , 2 , 3 , 4 ,

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