Abstract

We generalized the nonparaxial field components of Laguerre–Gaussian and flattened Gaussian beams obtained using the angular spectrum method to include symmetric radial and angular expansions and simplified them using an approximate evaluation of the integral equations for the field components. These field components possess series expressions in orders of a natural expansion parameter, which clarifies the physical interpretation of the series expansion. A connection between Laguerre–Gaussian and flat-top Gaussian profiles is obtained.

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References

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2010

G. Zhou, Opt. Commun. 283, 3383 (2010).
[CrossRef]

2007

S. Yan and B. Yao, 32, 3367 (2007).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007), Eq. 6.631 (1).

2006

2004

M. A. Bandres and J. C. Gutiérrez-Vega, 29, 2213 (2004).

1998

B. Quesnel and P. Mora, Phys. Rev. E 58, 3719 (1998).
[CrossRef]

1990

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

1979

1975

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

1972

1968

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

1966

D. R. Rhodes, IEEE Trans. Antennas Propagat. 14, 676(1966).
[CrossRef]

P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields (Pergamon, 1966).

1950

C. J. Bouwkamp, Rep. Prog. Phys. 17, 32 (1950).

Agrawal, G. P.

Bandres, M. A.

M. A. Bandres and J. C. Gutiérrez-Vega, 29, 2213 (2004).

Bouwkamp, C. J.

C. J. Bouwkamp, Rep. Prog. Phys. 17, 32 (1950).

Carter, W. H.

Cicchitelli, L.

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields (Pergamon, 1966).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007), Eq. 6.631 (1).

Gutiérrez-Vega, J. C.

M. A. Bandres and J. C. Gutiérrez-Vega, 29, 2213 (2004).

Hora, H.

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Mora, P.

B. Quesnel and P. Mora, Phys. Rev. E 58, 3719 (1998).
[CrossRef]

Pattanayak, D. N.

Postle, R.

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

Quesnel, B.

B. Quesnel and P. Mora, Phys. Rev. E 58, 3719 (1998).
[CrossRef]

Rhodes, D. R.

D. R. Rhodes, IEEE Trans. Antennas Propagat. 14, 676(1966).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007), Eq. 6.631 (1).

Sepke, S. M.

Umstadter, D. P.

Yan, S.

S. Yan and B. Yao, 32, 3367 (2007).

Yao, B.

S. Yan and B. Yao, 32, 3367 (2007).

Zhou, G.

G. Zhou, Opt. Commun. 283, 3383 (2010).
[CrossRef]

IEEE Trans. Antennas Propagat.

D. R. Rhodes, IEEE Trans. Antennas Propagat. 14, 676(1966).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

G. Zhou, Opt. Commun. 283, 3383 (2010).
[CrossRef]

Opt. Lett.

Phys. Rev. A

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

Phys. Rev. E

B. Quesnel and P. Mora, Phys. Rev. E 58, 3719 (1998).
[CrossRef]

Rep. Prog. Phys.

C. J. Bouwkamp, Rep. Prog. Phys. 17, 32 (1950).

Other

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007), Eq. 6.631 (1).

P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields (Pergamon, 1966).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

M. A. Bandres and J. C. Gutiérrez-Vega, 29, 2213 (2004).

S. Yan and B. Yao, 32, 3367 (2007).

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Equations (14)

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E x ( x , y , 0 ) = n = 0 N l = n n a n , l ( r 2 w 0 ) n exp [ r 2 2 w 0 2 ] exp [ i l ϕ ] ,
A x ( p , q ) = ( k 2 π ) 2 n , l a n , l ( r 2 w 0 ) n exp [ r 2 2 w 0 2 ] × exp [ i l ϕ ] exp [ i k b · r ] r d r d ϕ ,
A x ( p , q ) = k 2 2 π n , l a n , l ( i ) l 0 ( r 2 w 0 ) n exp [ r 2 2 w 0 2 ] × J l ( k b r ) r d r .
A x ( p , q ) = 1 2 π f 2 n , l a n , l ( i ) l ( b 2 f ) l c ! exp [ b 2 2 f 2 ] × L c ( l ) ( b 2 2 f 2 ) ,
E x ( x , y , z ) = 1 2 π f 2 n , l a n , l ( i ) l c ! 0 1 0 2 π ( b 2 f ) l × exp [ b 2 2 f 2 ] L c ( l ) ( b 2 2 f 2 ) × exp [ i k b · r ] exp [ i k z 1 b 2 ] b d b d θ .
exp [ i k z ( 1 b 2 ) ] = m = 0 1 m ! ( b 2 2 ) m ( k z ) m + 1 h m 1 ( 1 ) ( k z ) ,
E x ( x , y , z ) = m = 0 1 m ! ( k z ) m + 1 h m 1 ( 1 ) ( k z ) × n , l a n , l ( i ) l c ! I m , n , l ( r ) ,
I m , n , l ( r ) = 1 f l + 2 0 1 ( b 2 ) 2 m + l exp [ b 2 2 f 2 ] × L c ( l ) ( b 2 2 f 2 ) J 0 ( b k r ) b d b .
I m , n , l ( r ) = 1 f l + 2 0 ( b 2 ) 2 m + l exp [ b 2 2 f 2 ] × L c ( l ) ( b 2 2 f 2 ) J 0 ( b k r ) b d b 1 f l + 2 1 ( b 2 ) 2 m + l exp [ b 2 2 f 2 ] × L c ( l ) ( b 2 2 f 2 ) J 0 ( b k r ) b d b ,
I m , n , l ( r ) = f 2 m j = 0 c ( 1 ) j c + ! Γ ( m + j + l 2 + 1 ) j ! ( c j ) ! ( l + j ) ! × exp [ r 2 2 w 0 2 ] L m + j + l 2 ( r 2 2 w 0 2 ) .
E x ( x , y , z ) = m = 0 f 2 m ( k z ) m + 1 h m 1 ( 1 ) ( k z ) n = 0 N l = n n a n , l × j = 0 c i l ( 1 ) j C m , n , l , j exp [ r 2 2 w 0 2 ] L m + j + l 2 ( r 2 2 w 0 2 ) ,
C m , n , l , j = c + ! c ! Γ ( m + j + l 2 + 1 ) m ! j ! ( c j ) ! ( l + j ) ! .
I error ( r ) 1 f n + 2 1 ( b 2 ) 2 m + n exp [ b 2 2 f 2 ] b d b
1 2 m + N / 2 f N + 2 1 exp [ u 2 f 2 ] u m + N / 2 d u = exp [ 1 2 f 2 ] j = 0 m + N 2 ( m + N 2 ) ! j ! 1 2 m f 2 ( 2 f 2 ) N 2 j + 1 ,

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