Abstract

The exact expressions of the electromagnetic field pertinent to Gaussian and flattened Gaussian linearly polarized boundary distributions have been derived in closed-form terms for any point lying on the axis. The obtained results allow the fields to be predicted for an arbitrary transverse beam size. Numerical results showing the differences between the exact results and those obtained within the paraxial framework are also presented.

© 2002 Optical Society of America

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References

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    [CrossRef]
  5. H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
    [CrossRef]
  6. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
    [CrossRef]
  7. J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton, U. Press, Princeton, N.J., 1995).
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    [CrossRef]
  11. A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B 17, 809–819 (2000).
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  14. H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
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  15. H.-C. Kim, Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
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  16. H. Ostenberg, L. W. Smith, “Closed solutions of Rayleigh diffraction integral for axial points,” J. Opt. Soc. Am. 51, 1050–1054 (1961).
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  17. J. C. Heurtley, “Scalar Rayleigh–Sommerfeld and Kirchhoff diffraction integrals: a comparison of exact evaluations for axial points,” J. Opt. Soc. Am. 63, 1003–1008 (1973).
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  18. P. Varga, P. Torok, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
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  19. C. J. R. Sheppard, “High-aperture beams,” J. Opt. Soc. Am. A 18, 1579–1587 (2001).
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  20. H. Heyman, L. Felsen, “Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A 18, 1588–1611 (2001).
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  21. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).
  22. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  23. I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).
  24. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  25. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
    [CrossRef]
  26. M. Santarsiero, R. Borghi, “Correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
    [CrossRef]
  27. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation features of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]

2001 (3)

2000 (2)

1999 (3)

M. Santarsiero, R. Borghi, “Correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
[CrossRef]

H.-C. Kim, Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

1998 (3)

P. Varga, P. Torok, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

S. R. Seshadri, “Electromagnetic Gaussian beams,” J. Opt. Soc. Am. A 15, 2712–2792 (1998).
[CrossRef]

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

1997 (1)

1996 (3)

1994 (2)

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

1988 (1)

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

1985 (1)

1975 (1)

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

1973 (1)

1972 (1)

1961 (1)

Ambrosini, D.

Bagini, V.

Borghi, R.

Bosch, S.

Carnicer, A.

Carter, W. H.

Ciattoni, A.

Crosignani, B.

De Silvestri, S.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Di Porto, P.

Enoch, S.

Felsen, L.

Friberg, A. T.

Fukumitsu, O.

Gori, F.

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Gralak, B.

Hall, D. G.

Heurtley, J. C.

Heyman, H.

Joannopoulos, J.

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton, U. Press, Princeton, N.J., 1995).

Kettunen, V.

Kim, H.-C.

H.-C. Kim, Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

Kuittinen, M.

Laabs, H.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

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

Laporta, P.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Lax, M.

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

Lee, Y. H.

H.-C. Kim, Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

Louisell, W. H.

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

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).

Magni, V.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Marti´nez-Herrero, R.

Maystre, D.

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

McKnight, W. B.

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

Meade, R.

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton, U. Press, Princeton, N.J., 1995).

Meji´as, P. M.

Ostenberg, H.

Pacileo, A. M.

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Santarsiero, M.

Savchencko, A. Yu.

Schirripa Spagnolo, G.

Seshadri, S. R.

Sheppard, C. J. R.

Smith, L. W.

Svelto, O.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Takenaka, T.

Tayeb, G.

Torok, P.

P. Varga, P. Torok, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

Turunen, J.

Vahimaa, P.

Varga, P.

P. Varga, P. Torok, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

Winn, J.

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton, U. Press, Princeton, N.J., 1995).

Yariv, A.

Yokota, M.

Zel’dovich, B. Ya.

IEEE J. Quantum Electron. (2)

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (9)

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

Opt. Commun. (4)

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

H.-C. Kim, Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

P. Varga, P. Torok, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

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

Pure Appl. Opt. (1)

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

Other (4)

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton, U. Press, Princeton, N.J., 1995).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

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Figures (7)

Fig. 1
Fig. 1

Behavior of the relative error for the modulus of the on-axis optical field in Eq. (14) as a function of z/λ, for different values of the ratio w0/λ.

Fig. 2
Fig. 2

Behavior of the exact normalized on-axis field amplitude (solid curve) as a function of z/λ, for a Gaussian beam with w0=λ/2, together with the paraxial prediction (dotted curve).

Fig. 3
Fig. 3

Behavior of the function fN(ξ) versus ξ for some values of the order N.

Fig. 4
Fig. 4

Behavior of the relative error Eq. (14) as a function of z/λ, for a FG beam having w0=λ and for different values of the order N.

Fig. 5
Fig. 5

Behavior of the relative error [Eq. (14)] as a function of z/λ, for a FG of order N=10 and for different values of w0=λ.

Fig. 6
Fig. 6

Normalized on-axis field amplitude (solid curve) as a function of z/λ, for a FG beam of order N=10 having w0=λ/2, together with the paraxial prediction (dotted curve).

Fig. 7
Fig. 7

Same as in Fig. 6 but for w0=3λ/2.

Equations (25)

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E(r, z)=-12πE(r, 0)zexp(ikR)Rd2r,
Ez(r, z)=12πEx(r, 0)xexp(ikR)R+Ey(r, 0)yexp(ikR)Rd2r.
E(r, 0)=e^xexp-|r|2w02,
Ex(0, z)=-12πexp-|r|2w02zexp(ik|r|2+z2)|r|2+z2d2r,
Ez(0, z)=-12πexp-|r|2w02exp(ik|r|2+z2)|r|2+z2×ik-1|r|2+z2xd2r.
Ex(0, z)=-120exp-ρ2w02zexp(ikρ2+z2)ρ2+z2d(ρ2),
Ez(0, z)=-12π02πcos θdθ0exp-ρ2w02exp(ikρ2+z2)ρ2+z2×ik-1ρ2+z2ρ2dρ,
Ex(0, z)=-z z20exp-ρ2w02exp(ikρ2+z2)ρ2+z2d(ρ2)=-zF1w02, z2,
F(p, η)=η0exp(-pξ) exp(ikξ+η)ξ+ηdξ.
F(p, η)=ηexp(pη)ηexp(-pt) exp(ikt)tdt.
2ηexp(-pξ2+ikξ)dξ
=2 exp-k24pηexp-pξ-ik2p2dξ=πpexp-k24p1-erfpη1-ik2pη,
ηexp(-pt) exp(ikt)tdt
=πpexp-k24p1-erfpη1-ik2pη,
F(p, η)=ηπpexp-k24p×exp(pη)1-erfpη1-ik2pη=πpexp-k24p×exp(pη)1-erfpη1-ik2pη-exp(ikη)η.
1-erf(u)exp(-u2)πu,
F(p, η)-exp(ikη)η11+i[(2pη)/k],
zw0-iπw0λ1
Ex(0, x)exp(ikz) 11+i(2z/kw02).
r(z)=|Exnpar(0, z)-Expar(0, z)||Exnpar(0, z)|,
E(N)(r, 0)=e^xfN|r|(N+1)1/2w0,
fN(ξ)=exp(-ξ2)m=0N1m! ξ2m,
E(N)(r, 0)=e^x[LNexp(-pr2)]p=(N+1)/w02,
LN=m=0N(-p)mm!mpm.
Ex(N)(0, z)=-z[LNF(p, z2)]p=(N+1)/w02.

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