Abstract

I use the angular spectrum representation to compute exactly the Gaussian beam close to the waist (w0) in the case of a highly nonparaxial field (w0<λ). The computation is done in the vectorial case for a polarized Gaussian beam. In the area of the waist, the contribution of the propagating and evanescent waves is discussed. Moreover, the Gaussian wave is developed in terms of a series, which permits one to get analytical expressions for both propagating and evanescent waves when the observation is close to the waist.

© 2006 Optical Society of America

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  1. M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
    [Crossref]
  2. L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
    [Crossref]
  3. G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
    [Crossref]
  4. C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999).
    [Crossref]
  5. 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 approximation," J. Opt. Soc. Am. A 19, 404-412 (2002).
    [Crossref]
  6. G. P. Agrawal and D. N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximations," J. Opt. Soc. Am. 69, 575-578 (1979).
    [Crossref]
  7. G. Zhou, X. Chu, and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
    [Crossref]
  8. A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17-20 (2002).
    [Crossref]
  9. K. Duan and B. Lü, "Polarization properties of vectorial nonparaxial Gaussian beams in the far field," Opt. Lett. 30, 309-310 (2005).
    [Crossref]
  10. A. Ashkin, "Trapping of atoms by resonance radiation pressure," Phys. Rev. Lett. 40, 729-732 (1978).
    [Crossref]
  11. A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. U.S.A. 94, 4853-4860 (1997).
    [Crossref] [PubMed]
  12. K. Belkebir, P. C Chaumet, and A. Sentenac, "Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A 23, 586-595 (2006).
    [Crossref]
  13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 109.
  14. N. I. Petrov, "Evanescent and propagating fields of a strongly focused beam," J. Opt. Soc. Am. A 20, 2385-2389 (2003).
    [Crossref]
  15. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Corrected and Enlarged Edition (Academic, 1980), p. 951.
  16. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Corrected and Enlarged Edition (Academic, 1980), pp. 1064-1067.
  17. G. P. M. Poppe and C. M. J. Wijers, "More efficient computation of the complex error function," ACM Trans. Math. Softw. 16, 38-46 (1990).
    [Crossref]
  18. M. Xiao. "Evanescent waves do contribute to the far field," J. Mod. Opt. 46, 729-733 (1999).
    [Crossref]
  19. E. Wolf and J. Foley, "Do evanescent waves contribute to the far field?" Opt. Lett. 23, 16-18 (1998).
  20. A. Rahmani and G. W. Bryant, "Contribution of evanescent waves to the far field: the atomic point of view," Opt. Lett. 25, 433-435 (2000).
    [Crossref]
  21. T. Setälä, M. Kaivola, and A. Friberg, "Evanescent and propagating electromagnetic fields in scattering from point-dipole structures: reply to comment," J. Opt. Soc. Am. A 19, 1449-1451 (2002).
    [Crossref]

2006 (1)

2005 (2)

K. Duan and B. Lü, "Polarization properties of vectorial nonparaxial Gaussian beams in the far field," Opt. Lett. 30, 309-310 (2005).
[Crossref]

G. Zhou, X. Chu, and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[Crossref]

2003 (1)

2002 (3)

2000 (1)

1999 (2)

1998 (1)

1997 (1)

A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. U.S.A. 94, 4853-4860 (1997).
[Crossref] [PubMed]

1990 (1)

G. P. M. Poppe and C. M. J. Wijers, "More efficient computation of the complex error function," ACM Trans. Math. Softw. 16, 38-46 (1990).
[Crossref]

1983 (1)

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
[Crossref]

1979 (2)

1978 (1)

A. Ashkin, "Trapping of atoms by resonance radiation pressure," Phys. Rev. Lett. 40, 729-732 (1978).
[Crossref]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
[Crossref]

G. P. Agrawal and D. N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximations," J. Opt. Soc. Am. 69, 575-578 (1979).
[Crossref]

Ashkin, A.

A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. U.S.A. 94, 4853-4860 (1997).
[Crossref] [PubMed]

A. Ashkin, "Trapping of atoms by resonance radiation pressure," Phys. Rev. Lett. 40, 729-732 (1978).
[Crossref]

Belkebir, K.

Bryant, G. W.

Chaumet, P. C

Chen, C. G.

Chu, X.

G. Zhou, X. Chu, and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[Crossref]

Ciattoni, A.

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

Crosignani, B.

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

Davis, L. W.

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[Crossref]

Di Porto, P.

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

Duan, K.

K. Duan and B. Lü, "Polarization properties of vectorial nonparaxial Gaussian beams in the far field," Opt. Lett. 30, 309-310 (2005).
[Crossref]

Ferrera, J.

Foley, J.

Friberg, A.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Corrected and Enlarged Edition (Academic, 1980), p. 951.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Corrected and Enlarged Edition (Academic, 1980), pp. 1064-1067.

Heilmann, R. K.

Kaivola, M.

Konkola, P. T.

Lax, M.

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
[Crossref]

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[Crossref]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[Crossref]

Lü, B.

K. Duan and B. Lü, "Polarization properties of vectorial nonparaxial Gaussian beams in the far field," Opt. Lett. 30, 309-310 (2005).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 109.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[Crossref]

Pattanayak, D. N.

Petrov, N. I.

Poppe, G. P. M.

G. P. M. Poppe and C. M. J. Wijers, "More efficient computation of the complex error function," ACM Trans. Math. Softw. 16, 38-46 (1990).
[Crossref]

Rahmani, A.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Corrected and Enlarged Edition (Academic, 1980), pp. 1064-1067.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Corrected and Enlarged Edition (Academic, 1980), p. 951.

Saghafi, S.

Schattenburg, M. L.

Sentenac, A.

Setälä, T.

Sheppard, C. J. R.

Wijers, C. M. J.

G. P. M. Poppe and C. M. J. Wijers, "More efficient computation of the complex error function," ACM Trans. Math. Softw. 16, 38-46 (1990).
[Crossref]

Wolf, E.

E. Wolf and J. Foley, "Do evanescent waves contribute to the far field?" Opt. Lett. 23, 16-18 (1998).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 109.

Xiao., M.

M. Xiao. "Evanescent waves do contribute to the far field," J. Mod. Opt. 46, 729-733 (1999).
[Crossref]

Zhao, L.

G. Zhou, X. Chu, and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[Crossref]

Zhou, G.

G. Zhou, X. Chu, and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[Crossref]

ACM Trans. Math. Softw. (1)

G. P. M. Poppe and C. M. J. Wijers, "More efficient computation of the complex error function," ACM Trans. Math. Softw. 16, 38-46 (1990).
[Crossref]

J. Mod. Opt. (1)

M. Xiao. "Evanescent waves do contribute to the far field," J. Mod. Opt. 46, 729-733 (1999).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

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

Opt. Laser Technol. (1)

G. Zhou, X. Chu, and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005).
[Crossref]

Opt. Lett. (3)

Phys. Rev. A (3)

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[Crossref]

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[Crossref]

G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983).
[Crossref]

Phys. Rev. Lett. (1)

A. Ashkin, "Trapping of atoms by resonance radiation pressure," Phys. Rev. Lett. 40, 729-732 (1978).
[Crossref]

Proc. Natl. Acad. Sci. U.S.A. (1)

A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. U.S.A. 94, 4853-4860 (1997).
[Crossref] [PubMed]

Other (3)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 109.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Corrected and Enlarged Edition (Academic, 1980), p. 951.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Corrected and Enlarged Edition (Academic, 1980), pp. 1064-1067.

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Figures (4)

Fig. 1
Fig. 1

Gaussian beam with w 0 = λ 2 in the ( x , z ) plane at y = 0 . The left- and right-side are on the x and z components, respectively. (a) and (e) are the logarithm of the modulus of the component of the electric field. (b) and (f) are the logarithms of the modulus of the propagating part, (c) and (g) are the logarithms of the modulus of the evanescent part, (d) and (h) are the ratios in percent between the modulus of the evanescent part and the modulus of the total electric field.

Fig. 2
Fig. 2

Gaussian beam with w 0 = λ 2 in the ( x , y ) plane at z = λ 4 . Same legend as in Fig. 1.

Fig. 3
Fig. 3

Gaussian beam with w 0 = λ 2 versus x λ at z = λ 4 . With crosses the exact solution using Eqs. (15, 16, 17, 18). Solid curve and dashed curve with the series development with l = 10 and l = 30 , respectively. (a) x component for the evanescent wave. (b) x component for the propagating wave. (c) z component for the evanescent wave. (d) z component for the propagating wave.

Fig. 4
Fig. 4

Gaussian beam with w 0 = λ 2 versus x λ at z = λ 4 in the solid curve and z = 5 λ in the dashed curve and with w 0 = λ at z = λ 4 in the dot-dashed curve and z = 5 λ in the dotted curve. Note that the symbols are the results of the series development with l = 30 . (a) x is the component for the evanescent wave. (b) x is the component for the propagating wave. (c) z is the component for the evanescent wave. (d) z component for the propagating wave.

Tables (1)

Tables Icon

Table 1 Value of l Needed to Compute I x and I z a

Equations (35)

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E x ( r ) = + + A x ( k x , k y ) exp [ i ( k x x + k y y + k z z ) ] d k x d k y ,
E y ( r ) = + + A y ( k x , k y ) exp [ i ( k x x + k y y + k z z ) ] d k x d k y ,
E z ( r ) = + + [ k x k z A x ( k x , k y ) + k y k z A y ( k x , k y ) ] exp [ i ( k x x + k y y + k z z ) ] d k x d k y ,
k z = ( k 0 2 k x 2 k y 2 ) for propagating waves ,
k z = i ( k x 2 + k y 2 k 0 2 ) for evanescent waves .
E x ( x , y , 0 ) = E 0 x exp ( ρ 2 2 w 0 2 ) ,
E y ( x , y , 0 ) = E 0 y exp ( ρ 2 2 w 0 2 ) ,
A l ( k x , k y ) = E 0 l w 0 2 2 π exp ( k 2 w 0 2 2 ) ,
E l ( r ) = E 0 l I x ( r ) ,
I x ( r ) = 0 + w 0 2 f ( k ) exp ( i k z z ) J 0 ( k ρ ) k d k ,
E z ( r ) = i ( sin θ E 0 x + cos θ E 0 y ) I z ( r ) ,
I z ( r ) = 0 + k 2 w 0 2 k z f ( k ) exp ( i k z z ) J 1 ( k ρ ) d k ,
I x ( r ) = ( 0 k 0 0 + i ) w 0 2 f ( k ) exp ( i k z z ) J 0 ( k ρ ) k z d k z ,
I z ( r ) = ( 0 k 0 0 + i ) w 0 2 f ( k ) exp ( i k z z ) J 1 ( k ρ ) k d k z .
I x , pro ( r ) = 0 k 0 w 0 2 exp ( w 0 2 ( k 0 2 k z 2 ) 2 ) exp ( i k z z ) J 0 ( ρ k 0 2 k z 2 ) k z d k z ,
I x , eva ( r ) = 0 w 0 2 exp ( w 0 2 ( k 0 2 + α 2 ) 2 ) exp ( α z ) J 0 ( ρ k 0 2 + α 2 ) α d α ,
I z , pro ( r ) = 0 k 0 w 0 2 exp ( w 0 2 ( k 0 2 k z 2 ) 2 ) exp ( i k z z ) J 1 ( ρ k 0 2 k z 2 ) k 0 2 k z 2 d k z ,
I z , eva ( r ) = i 0 w 0 2 exp ( w 0 2 ( k 0 2 + α 2 ) 2 ) exp ( α z ) J 1 ( ρ k 0 2 + α 2 ) k 0 2 + α 2 d α ,
I x ( r ) = w 0 2 f ( k 0 ) l = 0 C l m = 0 l ( 1 ) m k 0 2 m m ! ( l m ) ! ( 0 k 0 0 + i ) exp ( k z 2 w 0 2 2 ) exp ( i k z z ) k z 2 m + 1 d k z ,
I z ( r ) = w 0 2 f ( k 0 ) ρ k 0 2 2 l = 0 C l m = 0 l + 1 ( 1 ) m k 0 2 m m ! ( l + l m ) ! ( 0 k 0 0 + i ) exp ( k z 2 w 0 2 2 ) exp ( i k z z ) k z 2 m d k z ,
I m , eva = 0 + i exp ( k z 2 w 0 2 2 ) exp ( i k z z ) k z m d k z = i m + 1 m ! w 0 m + 1 D m + 1 ( z w 0 ) ,
D 1 ( z w 0 ) = π 2 w ( i z 2 w 0 ) ,
D 2 ( z w 0 ) = 1 z w 0 D 1 ( z w 0 ) ,
D m + 1 ( z w 0 ) = 1 m [ D m 1 ( z w 0 ) z w 0 D m ( z w 0 ) ] ,
I m , pro = 0 k 0 exp ( k z 2 w 0 2 2 ) exp ( i k z z ) k z m d k z = exp ( w 0 2 k 0 2 2 ) exp ( i k 0 z ) k 0 m 1 w 0 2 i z w 0 2 I m 1 , pro m 1 w 0 2 I m 2 , pro ,
I 1 , pro = 1 w 0 2 [ exp ( w 0 2 k 0 2 2 ) exp ( i k 0 z ) 1 i z I 0 , pro ] ,
I 0 , pro = i w 0 π 2 [ w ( i z 2 w 0 ) exp ( w 0 2 k 0 2 2 ) × exp ( i k 0 z ) w ( i z w 0 2 + w 0 k 0 2 ) ] .
I x , eva ( r ) = f ( k 0 ) [ 1 z w 0 π 2 w ( i z 2 w 0 ) ] ,
I x , pro ( r ) = exp ( i z k 0 ) [ 1 z w 0 π 2 w ( i z w 0 2 + w 0 k 0 2 ) ] I x , eva ( r ) ,
I z , eva ( r ) = i ρ 2 w 0 2 f ( k 0 ) [ z π 2 ( w 0 4 k 0 2 + z 2 + w 0 2 w 0 ) w ( i z 2 w 0 ) ] ,
I z , pro ( r ) = ρ 2 exp ( i z k 0 ) [ k 0 + i z w 0 2 i π 2 ( w 0 4 k 0 2 + z 2 + w 0 2 w 0 3 ) w ( i z w 0 2 + w 0 k 0 2 ) ] I z , eva ( r ) .
I x , eva ( r ) = w 0 2 z 2 f ( k 0 ) + O ( 1 z 4 ) ,
I x , pro ( r ) = i w 0 2 k 0 z exp ( i z k 0 ) + O ( 1 z 2 ) ,
I z , eva ( r ) = ρ i k 0 2 w 0 2 2 z f ( k 0 ) + O ( 1 z 3 ) ,
I z , pro ( r ) = ρ i k 0 2 w 0 2 2 z f ( k 0 ) ρ w 0 2 k 0 z 2 exp ( i z k 0 ) + O ( 1 z 3 ) .

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