Abstract

On the basis of angular spectrum representation and the stationary-phase method, a far-field expression for nonparaxial Gaussian beams diffracted at a circular aperture is derived, which permits us to study the far-field nonparaxial properties of apertured Gaussian beams both analytically and numerically. It is shown that for the apertured case, the f-parameter and the truncation parameter affect the beam’s far-field properties. The f-parameter plays the more important role in determining the beam nonparaxiality than does the truncation parameter, whereas the truncation parameter additionally influences the beam diffraction. A comparison with the paraxial case is made. For the unapertured case our results reduce to the previous ones.

© 2003 Optical Society of America

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References

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  1. M. Lax, W. H. Louisell, and W. B. McKnight, �??From Maxwell to paraxial wave optic,�?? Phys. Rev. A 11, 1365- 1370 (1975).
    [CrossRef]
  2. T. Takenaka, M. Yokota, and O. Fukumitsu, �??Propagation of light beams beyond the paraxial approximation,�?? J. Opt. Soc. Am. A 2, 826-829 (1985).
    [CrossRef]
  3. H. Laabs, �??Propagation of Hermite�??Gaussian beams beyond the paraxial approximation,�?? Opt. Commun. 147, 1-4 (1998).
    [CrossRef]
  4. 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]
  5. Q. Cao and X. Deng, �??Corrections to the paraxial approximation of an arbitrary free-propagation beam,�?? J. Opt. Soc. Am. A 15, 765-774 (1998).
    [CrossRef]
  6. W. H. Carter, �??Electromagnetic field of a Gaussian beam with an elliptical cross section,�?? J. Opt. Soc. Am, 62, 1195-1201 (1972).
    [CrossRef]
  7. G. P. Agrawal and D. N. Pattanayak, �??Gaussain 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, and M. L. Schattenburg, �??Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,�?? J. Opt. Soc. Am. 19, 404-412 (2002).
    [CrossRef]
  9. S. R. S Seshadri, �??Virtual source for the Bessel-Gaussian beam,�?? Opt. Lett. 27, 988-1000 (2002).
  10. L. Mandel and E. Wolf, Optical Coherence and quantum Optics, (Cambridge U. Press, Cambridge, UK, 1995).
  11. X. Zeng, C. Liang, and Y. An, �??Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,�?? Appl. Opt. 36, 2042-2047 (1997).
    [CrossRef] [PubMed]
  12. S. Nemoto, �??Nonparaxial Gaussian beams,�?? Appl. Opt. 29, 1940-1946 (1990).
    [CrossRef] [PubMed]

Appl. Opt. (2)

J. Opt. Soc. Am (2)

W. H. Carter, �??Electromagnetic field of a Gaussian beam with an elliptical cross section,�?? J. Opt. Soc. Am, 62, 1195-1201 (1972).
[CrossRef]

G. P. Agrawal and D. N. Pattanayak, �??Gaussain beam propagation beyond the paraxial approximation,�?? J. Opt. Soc. Am. 69, 575-578 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

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

Opt. Lett. (1)

S. R. S Seshadri, �??Virtual source for the Bessel-Gaussian beam,�?? Opt. Lett. 27, 988-1000 (2002).

Phys. Rev. A (1)

M. Lax, W. H. Louisell, and W. B. McKnight, �??From Maxwell to paraxial wave optic,�?? Phys. Rev. A 11, 1365- 1370 (1975).
[CrossRef]

Other (1)

L. Mandel and E. Wolf, Optical Coherence and quantum Optics, (Cambridge U. Press, Cambridge, UK, 1995).

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Figures (3)

Fig. 1.
Fig. 1.

Schematic illustration of a circular aperture located at the plane z=0 in the Cartesian coordinate system.

Fig. 2.
Fig. 2.

Irradiance distributions |E(x,0,10zR )|2 of a Gaussian beam at the plane z=10 zR , (a)f=0.32, δ=0.5; (b)f=0.32, δ=1; (c)f=0.32, δ=2.2; (d)f=0.08, δ=0.5; (e)f=0.08, δ=1; (f)f=0.08, δ=2.2. Solid curve, nonparaxial result; dashed curve, paraxial result.

Fig. 3.
Fig. 3.

(a) Variation of the far-field divergence angle θ versus the parameter 1/f for different values of δ=0.5, 1, and 3; (b) variation of the far-field divergence angle θ versus truncation parameter for different values of f=0.08, 0.20, and 0.32. Solid curve, nonparaxial result; dashed curve, paraxial result.

Equations (28)

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E 0 ( x , y , 0 ) = exp ( x 2 + y 2 w 0 2 ) ,
E ( x , y , 0 ) = t ( x , y ) exp ( x 2 + y 2 w 0 2 ) ,
t ( x , y ) = { 1 c x 2 + y 2 a 2 0 otherwise .
E ( x , y , z ) = A ( p , q ) exp [ ik ( px + qy + mz ) ] d p d q ,
l A ( p , q ) = ( k 2 π ) 2 E 0 ( x , y , 0 ) t ( x , y ) exp [ i k ( px + qy ) ] d x d y
= ( k 2 π ) 2 x 2 + y 2 a 2 E 0 ( x , y , 0 ) exp [ i k ( px + qy ) ] d x d y
= ( k 2 π ) 2 0 a 0 2 π exp ( ρ 2 w 0 2 ) exp [ i k ρ ( p cos θ + q sin θ ) ]
= k 2 2 π 0 a exp ( ρ 2 w 0 2 ) ρ J 0 [ k ρ p 2 + q 2 ] ,
m = { ( 1 p 2 q 2 ) 1 2 p 2 + q 2 1 i ( p 2 + q 2 1 ) 1 2 p 2 + q 2 > 1 ,
E ( x , y , z ) = p 2 + q 2 1 A ( p , q ) exp [ ik ( px + qy + mz ) ] d p d q .
E ( x , y , z ) = k 2 2 π 0 a exp ( ρ 2 w 0 2 ) ρ u ( ρ ; x , y , z ) d ρ ,
u ( ρ ; x , y , z ) = p 2 + q 2 1 J 0 ( k ρ p 2 + q 2 ) exp [ ik ( px + qy + mz ) ] d p d q .
u ( ρ ; x , y , z ) = i 2 π k z r 2 exp ( ikr ) J 0 ( k ρ x 2 + y 2 r ) .
J 0 ( v ) = s = 0 ( 1 ) s v 2 s 2 2 s ( s ! ) 2 ,
E ( x , y , z ) = i kz r 2 e ikr 0 a exp ( ρ 2 w 0 2 ) ρ J 0 ( k ρ x 2 + y 2 r )
= i z 2 k r 2 e ikr m = 0 ( 1 ) m 2 2 m f 2 m + 2 ( m ! ) 2 ( x 2 + y 2 r 2 ) m [ Γ ( 1 + m , δ 2 ) m ! ] ,
f = 1 k w 0 ( f -parameter ) ,
δ = a w 0 ( truncation paramerer ) ,
r = ( x 2 + y 2 + z 2 ) 1 2 ,
r z + x 2 y 2 2 z .
E p ( x , y , z ) = i 2 k exp ( ikz ) z exp [ ik 2 z ( x 2 + y 2 ) ]
× m = 0 ( 1 ) m 2 2 m f 2 m + 2 ( m ! ) 2 h m [ Γ ( 1 + m , δ 2 ) m ! ] ,
h = x 2 + y 2 z 2 .
E p ( x , y , z ) = i λz exp ( ikz ) exp [ i k 2 z ( x 2 + y 2 ) ]
× x 0 2 + y 0 2 a 2 E ( x 0 , y 0 , 0 ) exp { ik 2 z ( x x 0 + y y 0 ) } d x 0 d y 0 .
E ( x , y , z ) = i k w 0 2 2 z r 2 exp ( ikr ) exp ( k 2 w 0 2 4 x 2 + y 2 r 2 ) .
E np [ w ( z ) , 0 , z ] 2 = 1 2 E np ( 0 , 0 , z ) 2 .
θ = lim z arctan w ( z ) z .

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