Abstract

Based on the angular spectrum representation and the method of stationary phase, an analytical vectorial structure of an apertured Gaussian beam in the far field has been derived without any approximation that is concise and convenient to calculate. The analytical expressions of the energy flux of the transverse electric (TE) term, the transverse magnetic (TM) term, and the apertured Gaussian beam are also presented in the far field. The energy flux distributions of the TE term, the TM term, and the apertured Gaussian beam are numerically demonstrated in the far-field plane. The influences of the linearly polarized angle, the f-parameter, and the truncation parameter on the energy flux distributions in the far field of the TE term, the TM term, and the apertured Gaussian beam are discussed in detail.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. T. Eyyuboğlu, “Hermite hyperbolic/sinusoidal Gaussian beams in ABCD systems,” Optik (Stuttgart) 118, 289–295 (2007).
    [CrossRef]
  2. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Propagation of partially coherent beams after a source plane ring aperture,” PIER 4, 84–90 (2008).
  3. D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
    [CrossRef]
  4. P. Belland and J. P. Crenn, “Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture,” Appl. Opt. 21, 522–527 (1982).
    [CrossRef] [PubMed]
  5. S. Wang, E. Bernabeu, and J. Alda, “ABCD matrix for weakly apertured Gaussian beams in the far field,” Appl. Opt. 30, 1584–1585 (1991).
    [CrossRef] [PubMed]
  6. G. Toker, A. Brunfeld, and J. Shamir, “Diffraction of apertured Gaussian beams: solution by expansion in Chebyshev polynomials,” Appl. Opt. 32, 4706–4712 (1993).
    [CrossRef] [PubMed]
  7. Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt. 36, 772–778 (1997).
    [CrossRef] [PubMed]
  8. E. M. Drège, N. G. Skinner, and D. M. Byrne, “Analytical far-field divergence angle of a truncated Gaussian beam,” Appl. Opt. 39, 4918–4925 (2000).
    [CrossRef]
  9. B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. 28, 2440–2442 (2003).
    [CrossRef] [PubMed]
  10. K. Duan and B. Lü, “Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture,” Opt. Express 11, 1474–1480 (2003).
    [CrossRef] [PubMed]
  11. H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22, 647–653 (2005).
    [CrossRef]
  12. G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun. 281, 1929–1934 (2008).
    [CrossRef]
  13. R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A 18, 1678–1680 (2001).
    [CrossRef]
  14. G. Zhou, “Analytical vectorial structure of Laguerre–Gaussian beam in the far field,” Opt. Lett. 31, 2616–2618 (2006).
    [CrossRef] [PubMed]
  15. H. Guo, J. Chen, and S. Zhuang, “Vector plane wave spectrum of an arbitrary polarized electromagnetic wave,” Opt. Express 14, 2095–2100 (2006).
    [CrossRef] [PubMed]
  16. D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. 32, 2711–2713 (2007).
    [CrossRef] [PubMed]
  17. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
    [CrossRef]

2008 (2)

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Propagation of partially coherent beams after a source plane ring aperture,” PIER 4, 84–90 (2008).

G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun. 281, 1929–1934 (2008).
[CrossRef]

2007 (2)

H. T. Eyyuboğlu, “Hermite hyperbolic/sinusoidal Gaussian beams in ABCD systems,” Optik (Stuttgart) 118, 289–295 (2007).
[CrossRef]

D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. 32, 2711–2713 (2007).
[CrossRef] [PubMed]

2006 (2)

2005 (1)

2003 (3)

2001 (1)

2000 (1)

1997 (1)

1993 (1)

1991 (1)

1982 (1)

1972 (1)

Alda, J.

Baykal, Y.

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Propagation of partially coherent beams after a source plane ring aperture,” PIER 4, 84–90 (2008).

Belland, P.

Bernabeu, E.

Bosch, S.

Brunfeld, A.

Byrne, D. M.

Cai, Y.

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Propagation of partially coherent beams after a source plane ring aperture,” PIER 4, 84–90 (2008).

Carnicer, A.

Carter, W. H.

Chen, J.

Chu, X.

G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun. 281, 1929–1934 (2008).
[CrossRef]

Crenn, J. P.

Deng, D.

Drège, E. M.

Duan, K.

Eyyuboglu, H. T.

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Propagation of partially coherent beams after a source plane ring aperture,” PIER 4, 84–90 (2008).

H. T. Eyyuboğlu, “Hermite hyperbolic/sinusoidal Gaussian beams in ABCD systems,” Optik (Stuttgart) 118, 289–295 (2007).
[CrossRef]

Guo, H.

Guo, Q.

Jiang, Z.

Liu, Z.

Lu, Q.

Lü, B.

Mao, H.

H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22, 647–653 (2005).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

Martínez-Herrero, R.

Mejías, P. M.

Shamir, J.

Skinner, N. G.

Toker, G.

Wang, S.

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

S. Wang, E. Bernabeu, and J. Alda, “ABCD matrix for weakly apertured Gaussian beams in the far field,” Appl. Opt. 30, 1584–1585 (1991).
[CrossRef] [PubMed]

Zhang, W.

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

Zhao, D.

H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22, 647–653 (2005).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

Zheng, J.

G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun. 281, 1929–1934 (2008).
[CrossRef]

Zhou, G.

G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun. 281, 1929–1934 (2008).
[CrossRef]

G. Zhou, “Analytical vectorial structure of Laguerre–Gaussian beam in the far field,” Opt. Lett. 31, 2616–2618 (2006).
[CrossRef] [PubMed]

Zhuang, S.

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun. 281, 1929–1934 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Optik (Stuttgart) (1)

H. T. Eyyuboğlu, “Hermite hyperbolic/sinusoidal Gaussian beams in ABCD systems,” Optik (Stuttgart) 118, 289–295 (2007).
[CrossRef]

PIER (1)

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Propagation of partially coherent beams after a source plane ring aperture,” PIER 4, 84–90 (2008).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Contour graphs of energy flux distributions of the TE term, the TM term, and the apertured Gaussian beam with α = 0 in the far-field plane z = 10 z r . (a),(d),(g) The TE term. (b),(e),(h) The TM term. (c),(f),(i) The apertured Gaussian beam. (a)–(c) f = 0.05 and δ = 0.8 . (d)–(f) f = 0.5 and δ = 0.8 . (g)–(i) f = 0.5 and δ = 1.2 .

Fig. 2
Fig. 2

Contour graphs of energy flux distributions of the TE term, the TM term, and the unapertured Gaussian beam with α = 0 in the far-field plane z = 10 z r . (a),(d) The TE term. (b),(e) The TM term. (c),(f) The unapertured Gaussian beam. (a)–(c) f = 0.05 and δ = . (d)–(f) f = 0.5 and δ = .

Fig. 3
Fig. 3

Contour graphs of energy flux distributions of the TE term, the TM term, and the apertured Gaussian beam with α = 45 ° in the far-field plane z = 10 z r . (a),(d) The TE term. (b),(e) The TM term. (c),(f) The apertured Gaussian beam. (a)–(c) f = 0.05 and δ = 0.8 . (d)–(f) f = 0.5 and δ = 0.8 .

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

[ E x ( ρ 0 , 0 ) E y ( ρ 0 , 0 ) ] = exp ( ρ 0 2 w 0 2 ) ( cos   α sin   α ) ,
[ E x ( ρ 0 , 0 ) E y ( ρ 0 , 0 ) ] = exp ( ρ 0 2 w 0 2 ) ( cos   α sin   α ) circ ( ζ ) ,
circ ( ζ ) = { 1 , 0 ζ < 1 0 , ζ 1. }
E ( ρ , z ) = A ( p , q ) exp [ i k ( p x + q y + γ z ) ] d p d q ,
A ( p , q ) = A x ( p , q ) e x + A y ( p , q ) e y γ 1 [ p A x ( p , q ) + q A y ( p , q ) ] e z ,
[ A x ( p , q ) A y ( p , q ) ] = k λ ( cos   α sin   α ) 0 R exp ( ρ 0 2 w 0 2 ) J 0 ( k ρ 0 b ) ρ 0 d ρ 0 ,
E ( ρ , z ) = E TE ( ρ , z ) + E TM ( ρ , z ) ,
E TE ( ρ , z ) = b 2 [ q A x ( p , q ) p A y ( p , q ) ] ( q e x p e y ) exp [ i k ( p x + q y + γ z ) ] d p d q ,
E TM ( ρ , z ) = γ 1 b 2 [ p A x ( p , q ) + q A y ( p , q ) ] ( p γ e x + q γ e y b 2 e z ) exp [ i k ( p x + q y + γ z ) ] d p d q .
H ( ρ , z ) = H TE ( ρ , z ) + H TM ( ρ , z ) ,
H TE ( ρ , z ) = η b 2 [ q A x ( p , q ) p A y ( p , q ) ] ( p γ e x + q γ e y b 2 e z ) exp [ i k ( p x + q y + γ z ) ] d p d q ,
H TM ( ρ , z ) = η γ 1 b 2 [ p A x ( p , q ) + q A y ( p , q ) ] ( q e x p e y ) exp [ i k ( p x + q y + γ z ) ] d p d q ,
E TE ( ρ , z ) = k λ 0 R exp ( ρ 0 2 w 0 2 ) T ( ρ 0 , ρ , z ) ρ 0 d ρ 0 ,
T ( ρ 0 , ρ , z ) = b 2 [ q   cos   α p   sin   α ] J 0 ( k ρ 0 b ) ( q e x p e y ) exp [ i k ( p x + q y + γ z ) ] d p d q .
T ( ρ 0 , ρ , z ) = i λ z   sin ( θ α ) ρ r 2 J 0 ( k ρ ρ 0 r ) exp ( i k r ) ( y e x x e y ) ,
J 0 ( k ρ ρ 0 r ) = n = 0 ( 1 ) n ( k ρ ρ 0 ) 2 n 2 2 n ( n ! ) 2 r 2 n .
E TE ( ρ , z ) = i z   sin ( θ α ) exp ( i k r ) ( y e x x e y ) n = 0 ( 1 ) n ρ 2 n 1 [ Γ ( 1 + n , δ 2 ) n ! ] 2 2 n + 1 k ( n ! ) 2 f 2 n + 2 r 2 n + 2 ,
f = 1 / k w 0 ,     δ = R / w 0 .
Γ ( 1 + n , x ) = x exp ( t ) t n d t .
E TM ( ρ , z ) = i   cos ( θ α ) exp ( i k r ) ( x z e x + y z e y ρ 2 e z ) n = 0 ( 1 ) n ρ 2 n 1 [ Γ ( 1 + n , δ 2 ) n ! ] 2 2 n + 1 k ( n ! ) 2 f 2 n + 2 r 2 n + 2 .
H TE ( ρ , z ) = i η z   sin ( θ α ) exp ( i k r ) ( x z e x + y z e y ρ 2 e z ) n = 0 ( 1 ) n ρ 2 n 1 [ Γ ( 1 + n , δ 2 ) n ! ] 2 2 n + 1 k ( n ! ) 2 f 2 n + 2 r 2 n + 3 ,
H TM ( ρ , z ) = i η   cos ( θ α ) exp ( i k r ) ( y e x x e y ) n = 0 ( 1 ) n ρ 2 n 1 [ Γ ( 1 + n , δ 2 ) n ! ] 2 2 n + 1 k ( n ! ) 2 f 2 n + 2 r 2 n + 1 .
S z TE = 1 2 Re [ E TE ( ρ , z ) × H TE ( ρ , z ) ] z = η z 3 sin 2 ( θ α ) 2 k 2 r 5 { n = 0 ( 1 ) n ρ 2 n [ Γ ( 1 + n , δ 2 ) n ! ] 2 2 n + 1 ( n ! ) 2 f 2 n + 2 r 2 n } 2 ,
S z TM = η z cos 2 ( θ α ) 2 k 2 r 3 { n = 0 ( 1 ) n ρ 2 n [ Γ ( 1 + n , δ 2 ) n ! ] 2 2 n + 1 ( n ! ) 2 f 2 n + 2 r 2 n } 2 ,
S z = S z TE + S z TM = η z [ z 2 + ρ 2 cos 2 ( θ α ) ] 2 k 2 r 5 { n = 0 ( 1 ) n ρ 2 n [ Γ ( 1 + n , δ 2 ) n ! ] 2 2 n + 1 ( n ! ) 2 f 2 n + 2 r 2 n } 2 .
E TE ( ρ , z ) = z r z   sin ( θ α ) i ρ r 2 exp ( i k r ) ( y e x x e y ) n = 0 1 n ! ( ρ 2 4 f 2 r 2 ) n = z r z   sin ( θ α ) i ρ r 2 exp ( ρ 2 4 f 2 r 2 + i k r ) ( y e x x e y ) ,
E TM ( ρ , z ) = i z r ρ r 2 cos ( θ α ) exp ( ρ 2 4 f 2 r 2 + i k r ) ( x z e x + y z e y ρ 2 e z ) ,
H TE ( ρ , z ) = i η z r z ρ r 3 sin ( θ α ) exp ( ρ 2 4 f 2 r 2 + i k r ) ( x z e x + y z e y ρ 2 e z ) ,
H TM ( ρ , z ) = i η z r ρ r cos ( θ α ) exp ( ρ 2 4 f 2 r 2 + i k r ) ( y e x x e y ) .
S z = η z r 2 z [ z 2 + ρ 2 cos 2 ( θ α ) ] r 5 exp ( ρ 2 2 f 2 r 2 ) ,
S z TE = η z r 2 z 3 r 5 sin 2 ( θ α ) exp ( ρ 2 2 f 2 r 2 ) ,
S z TM = η z r 2 z r 3 cos 2 ( θ α ) exp ( ρ 2 2 f 2 r 2 ) .

Metrics