Abstract

On the basis of the vectorial Rayleigh–Sommerfeld formulas and by means of the relation between Hermite and Laguerre polynomials, the analytical expressions for the propagation of the Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams beyond the paraxial approximation are derived, with the corresponding far-field propagation expressions and that for the Gaussian beams being given as special cases of the results. Some detailed comparisons of our results with the expansion series and paraxial expressions are made, which show the advantages of our results over the expansion series. With the results obtained, some typical intensity patterns of nonparaxial HG and LG beams are shown.

© 2005 Optical Society of America

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References

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  1. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  2. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  3. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  4. H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
    [CrossRef]
  5. H. C. Kim, Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
    [CrossRef]
  6. H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
    [CrossRef]
  7. G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
  8. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19, 404–412 (2002).
    [CrossRef]
  9. A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).
    [CrossRef]
  10. Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
    [CrossRef]
  11. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).
  12. K. Duan, B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A 21, 1924–1932 (2004).
    [CrossRef]
  13. X. Zeng, C. Liang, Y. An, “Far-field propagation of an off-axis Gaussian wave,” Appl. Opt. 38, 6253–6256 (1999).
    [CrossRef]
  14. I. Kimel, L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
    [CrossRef]
  15. A. E. Siegman, Lasers (University Science Books, 1986).

2004

2002

1999

H. C. Kim, Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

X. Zeng, C. Liang, Y. An, “Far-field propagation of an off-axis Gaussian wave,” Appl. Opt. 38, 6253–6256 (1999).
[CrossRef]

1998

Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
[CrossRef]

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

1993

I. Kimel, L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

1992

1985

1979

1975

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Agrawal, G. P.

An, Y.

Cao, Q.

Chen, C. G.

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Deng, X.

Duan, K.

Elias, L. R.

I. Kimel, L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

Ferrera, J.

Friberg, A. T.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

Fukumitsu, O.

Heilmann, R. K.

Kim, H. C.

H. C. Kim, Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

Kimel, I.

I. Kimel, L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

Konkola, P. T.

Laabs, H.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lee, Y. H.

H. C. Kim, Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

Liang, C.

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lü, B.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Pattanayak, D. N.

Schattenburg, M. L.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Takenaka, T.

Wünsche, A.

Yokota, M.

Zeng, X.

Appl. Opt.

IEEE J. Quantum Electron.

I. Kimel, L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

H. C. Kim, Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

Phys. Rev. A

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Other

A. E. Siegman, Lasers (University Science Books, 1986).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).

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Figures (4)

Fig. 1
Fig. 1

(a) E m n x H 2 , ψ m n 2 , and ψ ¯ x ( m , n ) 2 of a nonparaxial HG beam; (b) E m n x L 2 , ϕ m n 2 , and ϕ ¯ x ( m , n ) 2 of a nonparaxial LG beam. m = 4 , n = 4 , and the other calculation parameters are f = 0.01 , y = 0 , and z = 20 z 0 .

Fig. 2
Fig. 2

Intensity distributions of a Gaussian beam. The calculation parameters are f = 0.4 , y = 0 , and z = 20 z 0 .

Fig. 3
Fig. 3

(a) E m n x H 2 , ψ m n 2 , and ψ ¯ x ( m , n ) 2 of a nonparaxial HG beam; (b) E m n x L 2 , ϕ m n 2 , and ϕ ¯ x ( m , n ) 2 of a nonparaxial LG beam. m = 4 , n = 4 , and the other calculation parameters are f = 0.01 , y = 0 , and z = 20 z 0 .

Fig. 4
Fig. 4

Transverse-mode patterns for nonparaxial (a) HG and (b) LG beams of different orders.

Equations (32)

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E x ( r ) = 1 2 π E x ( r 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0 ,
E y ( r ) = 1 2 π E y ( r 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0 ,
E z ( r ) = 1 2 π [ E x ( r 0 , 0 ) G ( r , r 0 ) x + E y ( r 0 , 0 ) G ( r , r 0 ) y ] d x 0 d y 0 ,
G ( r , r 0 ) = exp ( i k r r 0 ) r r 0 ,
r r 0 r + x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r ,
E x ( r ) = i z λ r exp ( i k r ) r E x ( r 0 , 0 ) exp ( i k x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r ) d x 0 d y 0 ,
E y ( r ) = i z λ r exp ( i k r ) r E y ( r 0 , 0 ) exp ( i k x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r ) d x 0 d y 0 ,
E z ( r ) = i λ r exp ( i k r ) r [ E x ( r 0 , 0 ) ( x x 0 ) + E y ( r 0 , 0 ) ( y y 0 ) ] exp ( i k x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r ) d x 0 d y 0 .
E m n x H ( r 0 , 0 ) = H m ( 2 x 0 w 0 ) H n ( 2 y 0 w 0 ) exp [ ( x 0 2 + y 0 2 w 0 2 ) ] ,
E m n y H ( r 0 , 0 ) = 0 ,
exp [ ( x a ) 2 b ] H n ( 2 x ) d x = π b ( 1 2 b ) n 2 H n ( 2 a 1 2 b ) ,
E m n x H ( x , y , z ) = i z B 2 k f 2 r 2 exp ( i k r ) P x m P y n ,
E m n y H ( x , y , z ) = 0 ,
E m n z H ( x , y , z ) = i B 2 k f 2 r 2 exp ( i k r ) P y n ( x P x m m 2 k f P x m 1 1 2 2 k f P x m + 1 ) ,
P α u = ( 1 2 B ) u 2 H u ( 2 A α 1 2 B ) exp ( A 2 α 2 B ) , α = x , y .
r r 0 r x x 0 + y y 0 r .
E x f m n ( x , y , z ) = ( 1 ) ( m + n ) 2 i z 2 k f 2 r 2 exp ( i k r ) H m ( x 2 r f ) H n ( y 2 r f ) exp ( x 2 + y 2 4 r 2 f 2 ) ,
E y f m n ( x , y , z ) = 0 ,
E z f m n ( x , y , z ) = ( 1 ) ( m + n ) 2 i z 2 k f 2 r 2 exp ( i k r ) exp ( x 2 + y 2 4 r 2 f 2 ) H n ( y 2 r f ) [ x H m ( x 2 r f ) + m i 2 r f H m 1 ( x 2 r f ) i 2 2 k f H m + 1 ( x r f ) ] .
E x ( x , y , z ) = 1 ( 1 + 2 i k f 2 r ) z r exp ( i k r ) exp [ i k 2 ( 1 + 2 i k f 2 r ) r ( x 2 + y 2 ) ] ,
E z ( x , y , z ) = 2 i k f 2 r ( 1 + 2 i k f 2 r ) 2 x r exp ( i k r ) exp [ i k 2 ( 1 + 2 i k f 2 r ) r ( x 2 + y 2 ) ] .
E x ( x , y , z ) = 1 2 i k f 2 z r 2 exp ( i k r ) exp ( 1 4 f 2 x 2 + y 2 r 2 ) ,
E z ( x , y , z ) = 1 2 i k f 2 x r 2 exp ( i k r ) exp ( 1 4 f 2 x 2 + y 2 r 2 ) .
E m n L = ( 2 ρ 0 ) m L n m ( 2 ρ 0 2 ) exp ( ρ 0 2 ) exp ( i m θ ) ,
( 2 ρ 0 ) m L n m ( 2 ρ 0 2 ) exp ( i m θ ) = ( 1 ) n 2 2 n + m n ! s = 0 n t = 0 m i t ( n s ) ( m t ) H M ( 2 ξ 0 ) H N ( 2 η 0 ) ,
E m n L = ( 1 ) n 2 2 n + m n ! s = 0 n t = 0 m i t ( n s ) ( m t ) H M ( 2 ξ 0 ) H N ( 2 η 0 ) exp ( ρ 0 2 ) .
E m n x L = ( 1 ) n 2 2 n + m n ! s = 0 n t = 0 m i t ( n s ) ( m t ) E M N x H ( x , y , z ) ,
E m n y L = 0 ,
E m n z L = ( 1 ) n 2 2 n + m n ! s = 0 n t = 0 m i t ( n s ) ( m t ) E M N z H ( x , y , z ) .
E m n x f L = ( 1 ) n 2 2 n + m n ! s = 0 n t = 0 m i t ( n s ) ( m t ) E x f M N ( x , y , z ) ,
E m n y L = 0 ,
E m n z L = ( 1 ) n 2 2 n + m n ! s = 0 n t = 0 m i t ( n s ) ( m t ) E z f M N ( x , y , z ) .

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