Abstract

The analytical propagation equation of a nonparaxial Lorentz–Gauss beam in free space is derived on the basis of vectorial Rayleigh–Sommerfeld integral formulae. As special cases of the general formulae, the far-field expression and the scalar paraxial result are also presented. According to the obtained analytical representation, the nonparaxial propagation properties of a Lorentz–Gauss beam in free space are illustrated and analyzed with numerical examples. This research provides an approach to investigate the propagation of a Lorentz–Gauss beam beyond the paraxial approximation.

© 2008 Optical Society of America

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References

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  1. O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
    [Crossref]
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    [Crossref]
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    [Crossref]
  7. S. R. S. Seshadri, “Virtual source for a Hermite-Gaussian beam,” Opt. Lett. 28, 595-597 (2003).
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  8. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).
  9. Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express 15, 11942-11951 (2007).
    [Crossref] [PubMed]
  10. D. Deng, H. Yu, S. Xu, G. Tian, and Z. Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am. B 25, 83-87 (2008).
    [Crossref]
  11. C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol. 39, 598-604 (2007).
    [Crossref]
  12. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparxial beam,” Opt. Commun. 202, 17-20 (2002).
    [Crossref]
  13. Z. Mei and D. Zhao, “Nonparaxial propagation of controllable dark-hollow beams,” J. Opt. Soc. Am. A 25, 537-542 (2008).
    [Crossref]
  14. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, New York, 1980), p. 1147.
  15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  16. O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).
  17. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and 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]

2008 (2)

2007 (3)

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).

Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express 15, 11942-11951 (2007).
[Crossref] [PubMed]

C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol. 39, 598-604 (2007).
[Crossref]

2006 (1)

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[Crossref]

2003 (1)

2002 (2)

1999 (1)

1992 (1)

1990 (1)

1979 (1)

1975 (1)

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. QE-11, 400-402 (1975).
[Crossref]

Abramowitz, M.

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

Agrawal, G. P.

An, Y.

Chen, C. G.

Ciattoni, A.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparxial beam,” Opt. Commun. 202, 17-20 (2002).
[Crossref]

Crosignani, B.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparxial beam,” Opt. Commun. 202, 17-20 (2002).
[Crossref]

Deng, D.

Dumke, W. P.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. QE-11, 400-402 (1975).
[Crossref]

Durst, F.

Fan, Z.

Ferrera, J.

Gawhary, O. E.

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, New York, 1980), p. 1147.

Heilmann, R. K.

Konkola, P. T.

Liang, C.

Luneburg, R. K.

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

Mei, Z.

Naqwi, A.

Pattanayak, D. N.

Porto, P. D.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparxial beam,” Opt. Commun. 202, 17-20 (2002).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, New York, 1980), p. 1147.

Schattenburg, M. L.

Seshadri, S. R. S.

Severini, S.

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[Crossref]

Stegun, I. A.

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

Tian, G.

Wang, L.

C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol. 39, 598-604 (2007).
[Crossref]

Wünsche, A.

Xu, S.

Yu, H.

Zeng, X.

Zhang, Y.

C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol. 39, 598-604 (2007).
[Crossref]

Zhao, D.

Zheng, C.

C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol. 39, 598-604 (2007).
[Crossref]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. QE-11, 400-402 (1975).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparxial beam,” Opt. Commun. 202, 17-20 (2002).
[Crossref]

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).

Opt. Express (1)

Opt. Laser Technol. (1)

C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol. 39, 598-604 (2007).
[Crossref]

Opt. Lett. (1)

Other (3)

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

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, New York, 1980), p. 1147.

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

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Figures (7)

Fig. 1
Fig. 1

Light intensity distributions of a Lorentz–Gauss beam in different reference planes: w 0 = w 0 x = w 0 y = 0.5 λ . (a) z = 0 . (b) z = 50 λ . (c) z = 1000 λ .

Fig. 2
Fig. 2

Light intensity distributions of a Lorentz–Gauss beam in different reference planes: w 0 = 0.5 λ , w 0 x = 0.2 λ , and w 0 y = 1 λ . (a) z = 0 . (b) z = 50 λ . (c) z = 1000 λ .

Fig. 3
Fig. 3

Light intensity distributions of a Lorentz–Gauss beam in different reference planes: w 0 = 0.5 λ , w 0 x = 0.2 λ , and w 0 y = 5 λ . (a) z = 0 . (b) z = 50 λ . (c) z = 1000 λ .

Fig. 4
Fig. 4

Light intensity distributions of a Lorentz–Gauss beam in different reference planes: w 0 = 0.5 λ , and w 0 x = w 0 y = 5 λ . (a) z = 0 . (b) z = 50 λ . (c) z = 1000 λ .

Fig. 5
Fig. 5

Light intensity distributions of a Lorentz–Gauss beam in different reference planes: w 0 = 5 λ , and w 0 x = w 0 y = 0.5 λ . (a) z = 0 . (b) z = 50 λ . (c) z = 1000 λ .

Fig. 6
Fig. 6

Light intensity distributions of a Lorentz–Gauss beam in different reference planes: w 0 = 5 λ , w 0 x = 0.2 λ , and w 0 y = 0.5 λ . (a) z = 0 . (b) z = 50 λ . (c) z = 1000 λ .

Fig. 7
Fig. 7

Light intensity distributions of a Lorentz–Gauss beam in different reference planes: w 0 = 5 λ , w 0 x = 0.2 λ , and w 0 y = 5 λ . (a) z = 0 . (b) z = 50 λ . (c) z = 1000 λ .

Equations (40)

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( E x ( x 0 , y 0 , 0 ) E y ( x 0 , y 0 , 0 ) ) = ( E 0 w 0 x w 0 y [ 1 + ( x 0 w 0 x ) 2 ] [ 1 + ( y 0 w 0 y ) 2 ] 0 exp [ k ( x 0 2 + y 0 2 ) 2 z r ] ) ,
E x ( r ) = 1 2 π E x ( x 0 , y 0 , 0 ) G ( r , ρ 0 ) z d x 0 d y 0 ,
E y ( r ) = 1 2 π E y ( x 0 , y 0 , 0 ) G ( r , ρ 0 ) z d x 0 d y 0 ,
E z ( r ) = 1 2 π ( E x ( x 0 , y 0 , 0 ) G ( r , ρ 0 ) x + E y ( x 0 , y 0 , 0 ) G ( r , ρ 0 ) y ) d x 0 d y 0 ,
with G ( r , ρ 0 ) = exp ( i k r ρ 0 ) r ρ 0 ,
r ρ 0 = ( r 2 + ρ 0 2 2 x x 0 2 y y 0 ) 1 2 r + ρ 0 2 2 x x 0 2 y y 0 2 r ,
G ( r , ρ 0 ) = 1 r exp [ i k ( r + ρ 0 2 2 x x 0 2 y y 0 2 r ) ] .
E x ( r ) = i E 0 z λ r 2 w 0 x w 0 y exp ( i k r ) 1 [ 1 + ( x 0 w 0 x ) 2 ] [ 1 + ( y 0 w 0 y ) 2 ] exp { i k 2 r [ ( 1 + i r z r ) ρ 0 2 2 x x 0 2 y y 0 ] } d x 0 d y 0 ,
E y ( r ) = 0 ,
E z ( r ) = i E 0 λ r 2 w 0 x w 0 y exp ( i k r ) x x 0 [ 1 + ( x 0 w 0 x ) 2 ] [ 1 + ( y 0 w 0 y ) 2 ] exp { i k 2 r [ ( 1 + i r z r ) ρ 0 2 2 x x 0 2 y y 0 ] } d x 0 d y 0 .
T 1 ( j , z ) = 1 1 + ( j 0 w 0 j ) 2 exp ( i k 2 r ( A j 0 2 2 j j 0 ) ) d j 0 = w 0 j 2 exp ( i k j 2 2 A r ) 1 w 0 j 2 + j 0 2 exp [ i k A 2 r ( j A j 0 ) 2 ] d j 0 ,
f 1 ( τ ) f 2 ( τ ) = f 1 ( η ) f 2 ( τ η ) d η ,
T 1 ( j , z ) = w 0 j 2 exp ( i k j 2 2 A r ) [ f 1 ( j A ) f 2 ( j A ) ] ,
f 1 ( τ ) = 1 w 0 j 2 + τ 2 ,
f 2 ( τ ) = exp ( i k A τ 2 2 r ) .
f 1 ( τ ) f 2 ( τ ) = f 1 ( ξ ) f 2 ( ξ ) exp ( i ξ τ ) d ξ ,
T 1 ( j , z ) = w 0 j i π r 2 k A exp ( i k j 2 2 A r ) exp ( w 0 j ξ ) exp ( i r ξ 2 2 k A ) exp ( i ξ j A ) d ξ .
T 1 ( j , z ) = w 0 j i π r 2 k A exp ( i k j 2 2 A r ) { 0 exp [ i r ξ 2 2 k A ( w 0 j + i j A ) ξ ] d ξ + 0 exp [ i r ξ 2 2 k A ( w 0 j i j A ) ξ ] d ξ } .
T 1 ( j , z ) = π w 0 j 2 [ V j + ( j , z ) + V j ( j , z ) ] exp ( i k j 2 2 A r ) ,
V j ± ( j , z ) = exp [ k A 2 i r ( w 0 j ± i j A ) 2 ] { 1 erf [ k A 2 i r ( w 0 j ± i j A ) ] } ,
erf ( x ) = 2 π 0 x exp ( s 2 ) d s
E x ( r ) = i π 2 E 0 z 4 λ r 2 exp ( ρ 2 w 2 ( r ) ) [ V x + ( x , z ) + V x ( x , z ) ] [ V y + ( y , z ) + V y ( y , z ) ] exp [ i ( k r z r ρ 2 r w 2 ( r ) ) ] ,
T 2 ( x , z ) = x 0 1 + ( x 0 w 0 x ) 2 exp ( i k 2 r ( A x 0 2 2 x x 0 ) ) d x 0 = w 0 x 2 exp ( i k x 2 2 A r ) [ f 3 ( x A ) f 2 ( x A ) ] ,
f 3 ( τ ) = τ w 0 x 2 + τ 2 .
T 2 ( x , z ) = i π w 0 x 2 2 [ V x + ( x , z ) V x ( x , z ) ] exp ( i k x 2 2 A r ) .
E z ( r ) = i π 2 E 0 4 λ r 2 exp ( ρ 2 w 2 ( r ) ) [ V y + ( y , z ) + V y ( y , z ) ] [ ( x i w 0 x ) V x + ( x , z ) + ( x + i w 0 x ) V x ( x , z ) ] exp [ i ( k r z r ρ 2 r w 2 ( r ) ) ] .
E x ( r ) = i z λ r 2 exp ( i k r i k ρ 2 2 r ) E x ( x 0 , y 0 , 0 ) exp { i k 2 r [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } d x 0 d y 0 .
E x ( r ) = i π 2 E 0 z 4 λ r 2 exp ( i k r i k ρ 2 2 r ) [ V x + ( x , z ) + V x ( x , z ) ] [ V y + ( y , z ) + V y ( y , z ) ] exp ( ρ 2 w 0 2 ( 1 + i r z r ) ) .
E z ( r ) = i π 2 E 0 x 4 λ r 2 exp ( i k r i k ρ 2 2 r ) [ V x + ( x , z ) + V x ( x , z ) ] [ V y + ( y , z ) + V y ( y , z ) ] exp ( ρ 2 w 0 2 ( 1 + i r z r ) ) i λ r 2 exp ( i k r i k ρ 2 2 r ) V 0 ( x 0 , y 0 ) exp ( x 0 2 + y 0 2 w 0 2 ) exp { i k 2 r [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } d x 0 d y 0 ,
V 0 ( x 0 , y 0 ) = E 0 x 0 w 0 x w 0 y [ 1 + ( x 0 w 0 x ) 2 ] [ 1 + ( y 0 w 0 y ) 2 ] .
A ( p , q ) = E 0 x 0 exp [ i 2 π ( p x 0 + q y 0 ) ] w 0 x w 0 y [ 1 + ( x 0 w 0 x ) 2 ] [ 1 + ( y 0 w 0 y ) 2 ] d x 0 d y 0 = { i π 2 E 0 w 0 x exp ( 2 π p w 0 x ) exp ( 2 π q w 0 y ) , p > 0 i π 2 E 0 w 0 x exp ( 2 π p w 0 x ) exp ( 2 π q w 0 y ) , p < 0 } .
V ( x , y , z ) = A ( p , q ) exp [ i k z i π λ ( p 2 + q 2 ) z ] d p d q = i π E 0 w 0 x 2 [ P x + ( x , z ) P x ( x , z ) ] [ P y + ( y , z ) + P y ( y , z ) ] ,
P j ± ( j , z ) = exp ( k 2 i z ( w 0 j ± i j ) 2 ) [ 1 erf ( k 2 i z ( w 0 j ± i j ) ) ] .
E z ( r ) = i π 2 E 0 4 λ r 2 exp ( i k r i k ρ 2 2 r ) [ V y + ( y , z ) + V y ( y , z ) ] [ ( x i w 0 x ) V x + ( x , z ) + ( x + i w 0 x ) V x ( x , z ) ] exp ( ρ 2 w 0 2 ( 1 + i r z r ) ) .
G ( r , ρ 0 ) = 1 r exp [ i k ( r x x 0 + y y 0 r ) ] .
E x ( r ) = i π 2 E 0 z 4 λ r 2 exp ( f 2 f x 2 + f 2 f y 2 ) { exp ( x f x r ) [ 1 erf ( f f x x 2 f r ) ] + exp ( x f x r ) [ 1 erf ( f f x + x 2 f r ) ] } { exp ( y f y r ) [ 1 erf ( f f y y 2 f r ) ] + exp ( y f y r ) [ 1 erf ( f f y + y 2 f r ) ] } exp ( i k r ) ,
E z ( r ) = i π 2 E 0 4 λ r 2 exp ( f 2 f x 2 + f 2 f y 2 ) { ( x i w 0 x ) exp ( x f x r ) [ 1 erf ( f f x + x 2 f r ) ] + ( x + i w 0 x ) exp ( x f x r ) [ 1 erf ( f f x x 2 f r ) ] } { exp ( y f y r ) [ 1 erf ( f f y y 2 f r ) ] + exp ( y f y r ) [ 1 erf ( f f y + y 2 f r ) ] } exp ( i k r ) ,
where f = 1 k w 0 , f x = 1 k w 0 x , f y = 1 k w 0 y .
r = z + ρ 2 2 z = z .
E x ( r ) = π 2 E 0 4 i λ z exp [ i k z x 2 + y 2 w 0 2 ( 1 + i z z r ) ] [ P x + ( x 1 + i z z r , z 1 + i z z r ) + P x ( x 1 + i z z r , z 1 + i z z r ) ] × [ P y + ( y 1 + i z z r , z 1 + i z z r ) + P y ( y 1 + i z z r , z 1 + i z z r ) ] .

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