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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    [Crossref]
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    [Crossref]
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    [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]
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    [Crossref]
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    [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).
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2002 (2)

1998 (1)

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]

1997 (1)

1992 (1)

1990 (1)

1985 (1)

1979 (1)

1975 (1)

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

1972 (1)

Agrawal, G. P.

An, Y.

Cao, Q.

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]

Carter, W. H.

Chen, C. G.

Deng, X.

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]

Ferrera, J.

Fukumitsu, O.

Heilmann, R. K.

Konkola, P. T.

Laabs, H.

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, and W. B. McKnight, “From Maxwell to paraxial wave optic,” Phys. Rev. A 11, 1365– 1370 (1975).
[Crossref]

Liang, C.

Louisell, W. H.

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

Mandel, L.

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

McKnight, W. B.

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

Nemoto, S.

Pattanayak, D. N.

Schattenburg, M. L.

Seshadri, S. R. S

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

Takenaka, T.

Wolf, E.

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

Wünsche, A.

Yokota, M.

Zeng, X.

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

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

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 (2)

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

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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