Abstract

Hertz vector diffraction theory is applied to a focused TEM00 Gaussian light field passing through a circular aperture. The resulting theoretical vector field model reproduces plane-wave diffractive behavior for severely clipped beams, expected Gaussian beam behavior for unperturbed focused Gaussian beams as well as unique diffracted-Gaussian behavior between the two regimes. The maximum intensity obtainable and the width of the beam in the focal plane are investigated as a function of the clipping ratio between the aperture radius and the beam width in the aperture plane.

©2009 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
  5. G. D. Gillen, S. Guha, and K. Christandl, “Optical dipole traps for cold atoms using diffracted laser light,” Phys. Rev. A 73, 013409 (2006).
    [Crossref]
  6. S. Guha and G. D. Gillen,“Vector diffraction theory of refraction of light by a spherical surface,” J. Opt. Soc. Am. B 24, 1–8 (2007).
    [Crossref]
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  11. B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. 28, 2440–2442 (2003).
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  12. G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express 16, 3504–3514 (2008).
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    [Crossref]
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  16. G. D. Gillen and S. Guha, “Modeling and propagation of near-field diffraction patterns: a more complete approach,” Am. J. Phys. 72, 1195–1201 (2004).
    [Crossref]
  17. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 2003.)
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    [Crossref]
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  21. D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
    [Crossref]
  22. D. R. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. AP-14, 676–683 (1966).
    [Crossref]
  23. I. Ghebregziabher and B. C. Walker, “Effect of focal geometry on radiation from atomic ionization in an ultra-strong and ultrafast laser field,” Phys. Rev. A 76, 023415 (2007).
    [Crossref]
  24. J. M. P. Coelho, M. A. Abreu, and F. C. Rodrigues, “Modelling the spot shape influence on high-speed transmission lap welding of thermoplastic films,” J. Opt. Lasers Eng. 46, 55–61 (2007).
    [Crossref]
  25. A. Yariv, Quantum Electronics, Third Edition, (John Wiley & Sons, New York, 1989.)

2008 (1)

2007 (3)

S. Guha and G. D. Gillen,“Vector diffraction theory of refraction of light by a spherical surface,” J. Opt. Soc. Am. B 24, 1–8 (2007).
[Crossref]

I. Ghebregziabher and B. C. Walker, “Effect of focal geometry on radiation from atomic ionization in an ultra-strong and ultrafast laser field,” Phys. Rev. A 76, 023415 (2007).
[Crossref]

J. M. P. Coelho, M. A. Abreu, and F. C. Rodrigues, “Modelling the spot shape influence on high-speed transmission lap welding of thermoplastic films,” J. Opt. Lasers Eng. 46, 55–61 (2007).
[Crossref]

2006 (1)

G. D. Gillen, S. Guha, and K. Christandl, “Optical dipole traps for cold atoms using diffracted laser light,” Phys. Rev. A 73, 013409 (2006).
[Crossref]

2005 (4)

2004 (2)

K. Duan and B. Lü, “Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture,” J. Opt. Soc. Am. A 21, 1613–1620 (2004).
[Crossref]

G. D. Gillen and S. Guha, “Modeling and propagation of near-field diffraction patterns: a more complete approach,” Am. J. Phys. 72, 1195–1201 (2004).
[Crossref]

2003 (1)

2002 (1)

2000 (1)

M. S. Yeung, “Limitation of the Kirchhoff boundary conditions for aerial image simulations in 157-nm optical lithography,” IEEE Electron. Dev. Lett. 21, 433–435 (2000).
[Crossref]

1994 (1)

1979 (1)

1972 (1)

1966 (1)

D. R. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. AP-14, 676–683 (1966).
[Crossref]

1964 (1)

D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
[Crossref]

1953 (1)

G. Bekefi, “Diffraction of electromagnetic waves by an aperture in a large screen,” J. Appl. Phys. 24, 1123–1130 (1953).
[Crossref]

1897 (1)

Lord Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Philos. Mag. 43, 259–272 (1897).

1894 (1)

A. Sommerfeld, “Zur mathematischen Theorie der Beugungserscheinungen,” Nachr. Kgl. Wiss Göttingen 4, 338–342 (1894).

1883 (1)

G. R. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. (Leipzig)  18, 663–695 (1883).
[Crossref]

Abreu, M. A.

J. M. P. Coelho, M. A. Abreu, and F. C. Rodrigues, “Modelling the spot shape influence on high-speed transmission lap welding of thermoplastic films,” J. Opt. Lasers Eng. 46, 55–61 (2007).
[Crossref]

Agrawal, G. P.

Barakat, R.

Bekefi, G.

G. Bekefi, “Diffraction of electromagnetic waves by an aperture in a large screen,” J. Appl. Phys. 24, 1123–1130 (1953).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 2003.)

Carter, W. H.

Chen, C. G.

Christandl, K.

G. D. Gillen, S. Guha, and K. Christandl, “Optical dipole traps for cold atoms using diffracted laser light,” Phys. Rev. A 73, 013409 (2006).
[Crossref]

Coelho, J. M. P.

J. M. P. Coelho, M. A. Abreu, and F. C. Rodrigues, “Modelling the spot shape influence on high-speed transmission lap welding of thermoplastic films,” J. Opt. Lasers Eng. 46, 55–61 (2007).
[Crossref]

Duan, K.

Ferrera, J.

Ghebregziabher, I.

I. Ghebregziabher and B. C. Walker, “Effect of focal geometry on radiation from atomic ionization in an ultra-strong and ultrafast laser field,” Phys. Rev. A 76, 023415 (2007).
[Crossref]

Gillen, G. D.

S. Guha and G. D. Gillen,“Vector diffraction theory of refraction of light by a spherical surface,” J. Opt. Soc. Am. B 24, 1–8 (2007).
[Crossref]

G. D. Gillen, S. Guha, and K. Christandl, “Optical dipole traps for cold atoms using diffracted laser light,” Phys. Rev. A 73, 013409 (2006).
[Crossref]

S. Guha and G. D. Gillen, “Description of light propagation through a circular aperture using nonparaxial vector diffraction theory,” Opt. Express 13, 1424–1447 (2005).
[Crossref] [PubMed]

G. D. Gillen and S. Guha, “Modeling and propagation of near-field diffraction patterns: a more complete approach,” Am. J. Phys. 72, 1195–1201 (2004).
[Crossref]

Guha, S.

S. Guha and G. D. Gillen,“Vector diffraction theory of refraction of light by a spherical surface,” J. Opt. Soc. Am. B 24, 1–8 (2007).
[Crossref]

G. D. Gillen, S. Guha, and K. Christandl, “Optical dipole traps for cold atoms using diffracted laser light,” Phys. Rev. A 73, 013409 (2006).
[Crossref]

S. Guha and G. D. Gillen, “Description of light propagation through a circular aperture using nonparaxial vector diffraction theory,” Opt. Express 13, 1424–1447 (2005).
[Crossref] [PubMed]

G. D. Gillen and S. Guha, “Modeling and propagation of near-field diffraction patterns: a more complete approach,” Am. J. Phys. 72, 1195–1201 (2004).
[Crossref]

Heilmann, R. K.

Hsu, W.

Kirchhoff, G. R.

G. R. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. (Leipzig)  18, 663–695 (1883).
[Crossref]

Konkola, P. T.

Li, Y.

Lü, B.

Pattanayak, D. N.

Rayleigh, Lord

Lord Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Philos. Mag. 43, 259–272 (1897).

Rhodes, D. R.

D. R. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. AP-14, 676–683 (1966).
[Crossref]

D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
[Crossref]

Rodrigues, F. C.

J. M. P. Coelho, M. A. Abreu, and F. C. Rodrigues, “Modelling the spot shape influence on high-speed transmission lap welding of thermoplastic films,” J. Opt. Lasers Eng. 46, 55–61 (2007).
[Crossref]

Schattenburg, M. L.

Sommerfeld, A.

A. Sommerfeld, “Zur mathematischen Theorie der Beugungserscheinungen,” Nachr. Kgl. Wiss Göttingen 4, 338–342 (1894).

Walker, B. C.

I. Ghebregziabher and B. C. Walker, “Effect of focal geometry on radiation from atomic ionization in an ultra-strong and ultrafast laser field,” Phys. Rev. A 76, 023415 (2007).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 2003.)

Yariv, A.

A. Yariv, Quantum Electronics, Third Edition, (John Wiley & Sons, New York, 1989.)

Yeung, M. S.

M. S. Yeung, “Limitation of the Kirchhoff boundary conditions for aerial image simulations in 157-nm optical lithography,” IEEE Electron. Dev. Lett. 21, 433–435 (2000).
[Crossref]

Zhou, G.

Am. J. Phys. (1)

G. D. Gillen and S. Guha, “Modeling and propagation of near-field diffraction patterns: a more complete approach,” Am. J. Phys. 72, 1195–1201 (2004).
[Crossref]

Ann. Phys. (1)

G. R. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. (Leipzig)  18, 663–695 (1883).
[Crossref]

IEEE Electron. Dev. Lett. (1)

M. S. Yeung, “Limitation of the Kirchhoff boundary conditions for aerial image simulations in 157-nm optical lithography,” IEEE Electron. Dev. Lett. 21, 433–435 (2000).
[Crossref]

IEEE Trans. Antennas Propag. (1)

D. R. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. AP-14, 676–683 (1966).
[Crossref]

J. Appl. Phys. (1)

G. Bekefi, “Diffraction of electromagnetic waves by an aperture in a large screen,” J. Appl. Phys. 24, 1123–1130 (1953).
[Crossref]

J. Opt. Lasers Eng. (1)

J. M. P. Coelho, M. A. Abreu, and F. C. Rodrigues, “Modelling the spot shape influence on high-speed transmission lap welding of thermoplastic films,” J. Opt. Lasers Eng. 46, 55–61 (2007).
[Crossref]

J. Opt. Soc. Am. (2)

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

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

Nachr. Kgl. Wiss Göttingen (1)

A. Sommerfeld, “Zur mathematischen Theorie der Beugungserscheinungen,” Nachr. Kgl. Wiss Göttingen 4, 338–342 (1894).

Opt. Express (2)

Opt. Lett. (2)

B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. 28, 2440–2442 (2003).
[Crossref] [PubMed]

K. Duan and B. Lü, “Polarization properties of vectorial nonparaxial Gaussian beams in the far field,” Opt. Lett. 2005, 308–310 (2005).
[Crossref]

Philos. Mag. (1)

Lord Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Philos. Mag. 43, 259–272 (1897).

Phys. Rev. A (2)

G. D. Gillen, S. Guha, and K. Christandl, “Optical dipole traps for cold atoms using diffracted laser light,” Phys. Rev. A 73, 013409 (2006).
[Crossref]

I. Ghebregziabher and B. C. Walker, “Effect of focal geometry on radiation from atomic ionization in an ultra-strong and ultrafast laser field,” Phys. Rev. A 76, 023415 (2007).
[Crossref]

Proc. IEEE (1)

D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
[Crossref]

Other (2)

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 2003.)

A. Yariv, Quantum Electronics, Third Edition, (John Wiley & Sons, New York, 1989.)

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Figures (7)

Fig. 1.
Fig. 1. Theoretical setup for most calculations, where ωa is the e -1 width of the electric field of the incident Gaussian field in the aperture plane, ωo is the minimum beam waist, and zG is the on-axis location of the Gaussian focal plane.
Fig. 2.
Fig. 2. Calculated Gaussian behavior for a/ωa = 4 using GHVDT. Figures (a) and (b) are the normalized intensity versus x in the aperture plane and versus z, respectively, for zG = 0. Figures (c) and (d) are the normalized intensity versus x in the focal plane and versus z, respectively, for zG = 0.01 m. Calculated intensity distributions using a purely Gaussian beam propagation model, Yariv, are included for comparison purposes.
Fig. 3.
Fig. 3. Calculated plane-wave diffraction behavior using GHVDT. Calculations are of the normalized intensity versus on-axis distance from the aperture for aperture to wavelength ratios, a/λ, of (a) 5, (b) 5.5, and (c) 10. The incident peak intensity in the aperture plane is 10-4, normalized to the theoretical peak intensity in the focal plane at z = 0.01 m. Calculated intensity distributions using a purely plane wave vector diffraction theory, HVDT, are included for comparison purposes.
Fig. 4.
Fig. 4. Calculation of the z-component of the Poynting vector versus both x and y. Calculations are in the aperture plane for an incident Gaussian beam with the beam waist, ωo , in the aperture plane and an aperture radius of a = 5λ = 0.78ωo .
Fig. 5.
Fig. 5. Calculations of the normalized on-axis intensity for the regime between pure diffractive and pure Gaussian behavior for zG = 0.01 m and (a) a = 100λ = 0.16ωa , (b) a = 200λ= 0.31ωa , (c) a = 318λ= 0.5ωa , and (d) a = 636λ=ωa . All intensities are normalized to the peak unperturbed focal spot intensity. Calculated intensity distributions using a purely plane wave vector diffraction theory, HVDT, are included to illustrate the transition of the GHVDT model from the diffraction regime to the focused diffracted-Gaussian regime.
Fig. 6.
Fig. 6. Calculation of the maximum normalized intensity beyond the aperture as a function of the clipping ratio, a/ωa , where zG = 0.01 m.
Fig. 7.
Fig. 7. Calculation of the e -2 width of the beam intensity profile in the focal plane as a function of the clipping ratio, a/ωa , where zG = 0.01 m.

Equations (45)

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

E = k 2 + ( ) ,
H = ik ε o μ o × .
x ( r ) = 1 2 π ( ix ( r o ) z o ) z o = 0 e ikρ ρ d x o d y o .
ρ = r r o .
E ix ( r o ) = E 0 exp [ i ( i ln ( 1 + z 0 q 0 ) + k r 0 2 2 ( q 0 + z 0 ) + k z 0 ) ] ,
E ix ( r o ) = E o ( 1 + z o q o ) exp [ ik r o 2 2 ( q o + z o ) ik z o ] ,
q o = i k 2 w 0 2 ,
r o 2 = x o 2 + y o 2 ,
ix ( r o ) = E o k 2 1 ( 1 + z o z G q o ) exp [ ik r o 2 2 ( q o + z o z G ) ik ( z o z G ) ] .
( ix z o ) z o = 0 = E o k 2 A ( x o , y o )
A ( x o , y o ) = 1 1 z G q o ( ik r o 2 2 ( q o z G ) 2 1 q 0 z G ik ) exp ( ik r o 2 2 ( q 0 z G ) ) .
E x = k 2 x + 2 x x 2 ,
E y = 2 x y x ,
E z = 2 x z x ,
H x = 0 ,
H y = ik ε o μ o x z ,
H z = ik ε o μ o x y .
x 1 x ω o , y 1 y ω o , z 1 z z n ,
x 01 x 0 ω o , y 01 y 0 ω o , z G 1 z G z n , and q 1 q o z n .
E x ( r 1 ) = E o 2 π p 1 A 1 f 1 [ ( 1 + s 1 ) ( 1 + 3 s 1 ) ( x 1 x 01 ) 2 ρ 1 2 ] d x 01 d y 01 ,
E y ( r 1 ) = E o 2 π p 1 A 1 f 1 ( 1 + 3 s 1 ) ( x 1 x 01 ) ( y 1 y 01 ) ρ 1 2 d x 01 d y 01 ,
E z ( r 1 ) = E o z 1 2 π A 1 f 1 [ ( 1 + 3 s 1 ) ( x 1 x 01 ) ρ 1 2 ] d x 01 d y 01 ,
H x ( r 1 ) = 0 ,
H y ( r 1 ) = i p 1 z 1 H 0 2 π A 1 f 1 s 1 d x 01 d y 01 ,
H z ( r 1 ) = i H 0 2 π A 1 f 1 s 1 ( y 1 y 01 ) d x 01 d y 01 ,
A 1 = 1 1 + 2 i z G 1 ( i r 01 2 2 ( q 1 z G 1 ) 2 1 ( q 1 z G 1 ) i p 1 2 ) exp ( i r 01 2 2 ( q 1 z G 1 ) ) ,
f 1 = e i p 1 ρ 1 ρ 1 ,
s 1 = 1 i p 1 ρ 1 ( 1 + 1 i p 1 ρ 1 ) ,
ρ 1 2 = ( x 1 x 01 ) 2 + ( y 1 y 01 ) 2 + p 1 2 z 1 2 ,
r 01 2 = x 01 2 + y 01 2 .
ω a 2 = ω o 2 ( 1 + z G 2 z R 2 ) .
S z S o = 0.49 + 0.50 erf [ 0.266 ( a ω a ) 2 + 2.25 a ω a 2.14 ] .
log ( ω ω o ) = log ( a ω a ) + log ( 1.33 ) .
ρ 1 p 1 z 1 + ( x 1 x 01 ) 2 + ( y 1 y 01 ) 2 2 p 1 z 1 .
E z ( r 1 ) = E 0 4 ( 1 + 3 s 11 ) p 1 3 z 1 2 d 1 2 ( κ 1 + κ 2 + κ 3 ) e i p 1 2 z 1 e c 1 ( x 1 2 + y 1 2 ) ,
κ 1 = a x 1 d 1 4 ( b 1 2 a 1 )
κ 2 = a b 1 x 1 d 1 4
κ 3 = b x 1 d 1 2 [ a a 1 2 d 1 4 ( x 1 2 + y 1 2 ) + b ]
s 11 = 1 i p 1 2 z 1 ( 1 + 1 i p 1 2 z 1 )
a = i 2 ( 1 + 2 i z G 1 ) ( q 1 z G 1 ) 2
b = 1 1 + 2 i z G 1 ( 1 q 1 z G 1 + i p 1 2 )
a 1 = i 2 z 1
b 1 = i 2 ( q 1 z G 1 )
c 1 = a 1 b 1 a 1 + b 1
d 1 = ( a 1 + b 1 ) .

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