Abstract

A uniform asymptotic theory of the free-space paraxial propagation of coherent flattened Gaussian beams is proposed in the limit of nonsmall Fresnel numbers. The pivotal role played by the error function in the mathematical description of the related wavefield is stressed.

© 2013 Optical Society of America

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References

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  1. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  2. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]
  3. R. Borghi, “Elegant Laguerre–Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18, 1627–1633 (2001).
    [CrossRef]
  4. M. Santarsiero and R. Borghi, “On the correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
    [CrossRef]
  5. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002).
    [CrossRef]
  6. This can be easily verified by using common scientific databases such as, for instance, http://scholar.google.com .
  7. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  8. J. Stamnes, “Uniform asymptotic theory of diffraction by apertures,” J. Opt. Soc. Am. 73, 96–109 (1983).
    [CrossRef]
  9. K. Schwarzschild, “Die Beugungsfigure im Fernrohr weit ausserhalb des Focus,” Sitzb. München Akad. Wiss. Math.-Phys. Kl. 28, 271–294 (1898).
    [CrossRef]
  10. K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497 (1998).
    [CrossRef]
  11. NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/ . DLMF Update, Version 1.0.6, May6, 2013.
  12. E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3, 465–469 (1986).
    [CrossRef]
  13. G. Szegö, Orthogonal Polynomials (American Mathematical Society, 1939).
  14. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon and Breach, 1986), Vol. I.
  15. N. M. Temme, “Uniform asymptotic expansions of integrals: a selection of problems,” J. Comput. Appl. Math. 65, 395–417 (1995).
    [CrossRef]
  16. R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, 1973).
  17. M. Eichhorn, G. Stöppler, M. Schellhorn, K. T. Zawilski, and P. G. Schunemann, “Gaussian-versus flat-top-pumping of a mid-IR ZGP RISTRA OPO,” Appl. Phys. B 108, 109–115 (2012).
    [CrossRef]
  18. H. Malik and A. Malik, “Strong and collimated terahertz radiation by super-Gaussian lasers,” Europhys. Lett. 100, 45001 (2012).
    [CrossRef]
  19. M. Gong, Y. Qiu, Q. Liu, L. Huang, P. Yan, and H. Zhang, “Beam quality improvement by amplitude gain control in power amplifier system,” Laser Phys. Lett. 9, 838 (2012).
    [CrossRef]
  20. H. Ma, Z. Liu, X. Xu, and J. Chen, “Simultaneous adaptive control of dual deformable mirrors for full-field beam shaping with the improved stochastic parallel gradient descent algorithm,” Opt. Lett. 38, 326–328 (2013).
    [CrossRef]
  21. E. Mironov, A. Voitovich, and O. Palashov, “Apodizing diaphragm based on the Faraday effect,” Opt. Commun. 295, 170–175 (2013).
    [CrossRef]
  22. Y. A. Brichkov, Handbook of Special Functions (CRC Press, 2008).

2013 (2)

2012 (3)

M. Eichhorn, G. Stöppler, M. Schellhorn, K. T. Zawilski, and P. G. Schunemann, “Gaussian-versus flat-top-pumping of a mid-IR ZGP RISTRA OPO,” Appl. Phys. B 108, 109–115 (2012).
[CrossRef]

H. Malik and A. Malik, “Strong and collimated terahertz radiation by super-Gaussian lasers,” Europhys. Lett. 100, 45001 (2012).
[CrossRef]

M. Gong, Y. Qiu, Q. Liu, L. Huang, P. Yan, and H. Zhang, “Beam quality improvement by amplitude gain control in power amplifier system,” Laser Phys. Lett. 9, 838 (2012).
[CrossRef]

2002 (1)

2001 (1)

1999 (1)

1998 (1)

K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497 (1998).
[CrossRef]

1996 (1)

1995 (1)

N. M. Temme, “Uniform asymptotic expansions of integrals: a selection of problems,” J. Comput. Appl. Math. 65, 395–417 (1995).
[CrossRef]

1994 (1)

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

1986 (1)

1983 (1)

1898 (1)

K. Schwarzschild, “Die Beugungsfigure im Fernrohr weit ausserhalb des Focus,” Sitzb. München Akad. Wiss. Math.-Phys. Kl. 28, 271–294 (1898).
[CrossRef]

Ambrosini, D.

Bagini, V.

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Brichkov, Y. A.

Y. A. Brichkov, Handbook of Special Functions (CRC Press, 2008).

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon and Breach, 1986), Vol. I.

Chen, J.

Dingle, R. B.

R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, 1973).

Eichhorn, M.

M. Eichhorn, G. Stöppler, M. Schellhorn, K. T. Zawilski, and P. G. Schunemann, “Gaussian-versus flat-top-pumping of a mid-IR ZGP RISTRA OPO,” Appl. Phys. B 108, 109–115 (2012).
[CrossRef]

Gong, M.

M. Gong, Y. Qiu, Q. Liu, L. Huang, P. Yan, and H. Zhang, “Beam quality improvement by amplitude gain control in power amplifier system,” Laser Phys. Lett. 9, 838 (2012).
[CrossRef]

Gori, F.

Huang, L.

M. Gong, Y. Qiu, Q. Liu, L. Huang, P. Yan, and H. Zhang, “Beam quality improvement by amplitude gain control in power amplifier system,” Laser Phys. Lett. 9, 838 (2012).
[CrossRef]

Li, Y.

Liu, Q.

M. Gong, Y. Qiu, Q. Liu, L. Huang, P. Yan, and H. Zhang, “Beam quality improvement by amplitude gain control in power amplifier system,” Laser Phys. Lett. 9, 838 (2012).
[CrossRef]

Liu, Z.

Ma, H.

Malik, A.

H. Malik and A. Malik, “Strong and collimated terahertz radiation by super-Gaussian lasers,” Europhys. Lett. 100, 45001 (2012).
[CrossRef]

Malik, H.

H. Malik and A. Malik, “Strong and collimated terahertz radiation by super-Gaussian lasers,” Europhys. Lett. 100, 45001 (2012).
[CrossRef]

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon and Breach, 1986), Vol. I.

Mielenz, K. D.

K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497 (1998).
[CrossRef]

Mironov, E.

E. Mironov, A. Voitovich, and O. Palashov, “Apodizing diaphragm based on the Faraday effect,” Opt. Commun. 295, 170–175 (2013).
[CrossRef]

Pacileo, A. M.

Palashov, O.

E. Mironov, A. Voitovich, and O. Palashov, “Apodizing diaphragm based on the Faraday effect,” Opt. Commun. 295, 170–175 (2013).
[CrossRef]

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon and Breach, 1986), Vol. I.

Qiu, Y.

M. Gong, Y. Qiu, Q. Liu, L. Huang, P. Yan, and H. Zhang, “Beam quality improvement by amplitude gain control in power amplifier system,” Laser Phys. Lett. 9, 838 (2012).
[CrossRef]

Santarsiero, M.

Schellhorn, M.

M. Eichhorn, G. Stöppler, M. Schellhorn, K. T. Zawilski, and P. G. Schunemann, “Gaussian-versus flat-top-pumping of a mid-IR ZGP RISTRA OPO,” Appl. Phys. B 108, 109–115 (2012).
[CrossRef]

Schirripa Spagnolo, G.

Schunemann, P. G.

M. Eichhorn, G. Stöppler, M. Schellhorn, K. T. Zawilski, and P. G. Schunemann, “Gaussian-versus flat-top-pumping of a mid-IR ZGP RISTRA OPO,” Appl. Phys. B 108, 109–115 (2012).
[CrossRef]

Schwarzschild, K.

K. Schwarzschild, “Die Beugungsfigure im Fernrohr weit ausserhalb des Focus,” Sitzb. München Akad. Wiss. Math.-Phys. Kl. 28, 271–294 (1898).
[CrossRef]

Stamnes, J.

Stöppler, G.

M. Eichhorn, G. Stöppler, M. Schellhorn, K. T. Zawilski, and P. G. Schunemann, “Gaussian-versus flat-top-pumping of a mid-IR ZGP RISTRA OPO,” Appl. Phys. B 108, 109–115 (2012).
[CrossRef]

Szegö, G.

G. Szegö, Orthogonal Polynomials (American Mathematical Society, 1939).

Temme, N. M.

N. M. Temme, “Uniform asymptotic expansions of integrals: a selection of problems,” J. Comput. Appl. Math. 65, 395–417 (1995).
[CrossRef]

Voitovich, A.

E. Mironov, A. Voitovich, and O. Palashov, “Apodizing diaphragm based on the Faraday effect,” Opt. Commun. 295, 170–175 (2013).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Xu, X.

Yan, P.

M. Gong, Y. Qiu, Q. Liu, L. Huang, P. Yan, and H. Zhang, “Beam quality improvement by amplitude gain control in power amplifier system,” Laser Phys. Lett. 9, 838 (2012).
[CrossRef]

Zauderer, E.

Zawilski, K. T.

M. Eichhorn, G. Stöppler, M. Schellhorn, K. T. Zawilski, and P. G. Schunemann, “Gaussian-versus flat-top-pumping of a mid-IR ZGP RISTRA OPO,” Appl. Phys. B 108, 109–115 (2012).
[CrossRef]

Zhang, H.

M. Gong, Y. Qiu, Q. Liu, L. Huang, P. Yan, and H. Zhang, “Beam quality improvement by amplitude gain control in power amplifier system,” Laser Phys. Lett. 9, 838 (2012).
[CrossRef]

Appl. Phys. B (1)

M. Eichhorn, G. Stöppler, M. Schellhorn, K. T. Zawilski, and P. G. Schunemann, “Gaussian-versus flat-top-pumping of a mid-IR ZGP RISTRA OPO,” Appl. Phys. B 108, 109–115 (2012).
[CrossRef]

Europhys. Lett. (1)

H. Malik and A. Malik, “Strong and collimated terahertz radiation by super-Gaussian lasers,” Europhys. Lett. 100, 45001 (2012).
[CrossRef]

J. Comput. Appl. Math. (1)

N. M. Temme, “Uniform asymptotic expansions of integrals: a selection of problems,” J. Comput. Appl. Math. 65, 395–417 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Res. Natl. Inst. Stand. Technol. (1)

K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497 (1998).
[CrossRef]

Laser Phys. Lett. (1)

M. Gong, Y. Qiu, Q. Liu, L. Huang, P. Yan, and H. Zhang, “Beam quality improvement by amplitude gain control in power amplifier system,” Laser Phys. Lett. 9, 838 (2012).
[CrossRef]

Opt. Commun. (2)

E. Mironov, A. Voitovich, and O. Palashov, “Apodizing diaphragm based on the Faraday effect,” Opt. Commun. 295, 170–175 (2013).
[CrossRef]

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

Opt. Lett. (2)

Sitzb. München Akad. Wiss. Math.-Phys. Kl. (1)

K. Schwarzschild, “Die Beugungsfigure im Fernrohr weit ausserhalb des Focus,” Sitzb. München Akad. Wiss. Math.-Phys. Kl. 28, 271–294 (1898).
[CrossRef]

Other (7)

R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, 1973).

NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/ . DLMF Update, Version 1.0.6, May6, 2013.

G. Szegö, Orthogonal Polynomials (American Mathematical Society, 1939).

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon and Breach, 1986), Vol. I.

Y. A. Brichkov, Handbook of Special Functions (CRC Press, 2008).

This can be easily verified by using common scientific databases such as, for instance, http://scholar.google.com .

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

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Figures (7)

Fig. 1.
Fig. 1.

FG profiles FGN(ξ) evaluated, for different values of N, through Eq. (1) (solid curves) and through the asymptotic estimate in Eq. (25) (dashed curves).

Fig. 2.
Fig. 2.

(a) Behavior of the modulus of the 2nth-order ξ-derivative in Eq. (27) (open circles), together with the asymptotic estimate in Eq. (29) (solid curve). N=30. (b) The same plot as in (a) but on a vertical logarithmic scale.

Fig. 3.
Fig. 3.

Behavior of the amplitude of a FG beam of order N=30 propagated at the Fresnel number NF=10. Open circles: exact values provided by Eq. (5). Solid curve: erfc-based asymptotic estimate by keeping only the leading term in the asymptotics expansion in Eq. (22). The dotted curve represents the (normalized) initial FG profile. The phase distribution is wrapped.

Fig. 4.
Fig. 4.

Same as in Fig. 3 but for NF=5.

Fig. 5.
Fig. 5.

Same as in Fig. 3 but for NF=2.

Fig. 6.
Fig. 6.

Same as in Fig. 5 but now including the term b0 in the asympotic expansion of Eq. (22).

Fig. 7.
Fig. 7.

(a) Amplitude and (b) phase distributions obtained by keeping only the leading erfc-based term (dotted curve), by including the term b0 (dashed curve), and by adding also a1 and a2 (solid curve). NF=1.

Equations (43)

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

FGN(ξ)=exp(ξ2)m=0N1m!ξ2m,
1=exp(ξ2)exp(ξ2),
UN(r;0)=FGN(rw0N+1),
FGN(ξ)=m=0N(1)m(N+1m+1)Lm(ξ2)exp(ξ2),
UN(r;z)=exp(ikz)1+iN+1πNFexp[N+11+iN+1πNF(rw0)2]×GN[11+iN+1πNF,N+11+iN+1πNF(rw0)2],
GN(t,s)=n=0N(1)n(N+1n+1)tnLn(s).
s=r2w02(N+1)t.
Ln(s)=exp(s)n!0dξexp(ξ)ξnJ0(2sξ),
n=0N(1)nn!(N+1n+1)xn=LN(1)(x),
exp(s)GN(t,s)=0dξexp(ξ)J0(2sξ)LN(1)(ξt),
exp(ξt2)N+1LN(1)(ξt)2J1[2(N+1)ξt]2(N+1)ξt,
exp(s)GN(t,s)2t0dηexp(pη2)J0(2rw0ηt)J1(2ηt),
p=1N+1(1t2).
UN(r;z)exp(ikz)J[2(N+1)1+2iN+1πNF,rw0],
J(u,ρ)=1exp(uρ2)udξexp(ξ)I0(2ρξu),
I0(x)exp(x)2πx,|x|,
J(u,ρ)1uπρ1ξdξexp[u(ξρ)2].
J(u,ρ)1uπ1dξexp[u(ξρ)2]uπρ1dξ(ξρ)exp[u(ξρ)2].
uπ1dξexp[u(ξρ)2]=12erfc[u(1ρ)],
uπρ1dξ(ξρ)exp[u(ξρ)2]=12uπρexp[u(1ρ)2]1+ρ+14uπρ1dξξexp[u(ξρ)2](ξ+ρ)2,
14uπρ1dξξexp[u(ξρ)2](ξ+ρ)2132uρ2erfc[u(1ρ)]12uπρexp[u(1ρ)2]1+ρ116uρ2ρ(4ρ+3ρ+1)(1+ρ)2,
J(u,ρ)=112erfc[u(1ρ)]m=0am(ρ)um12πexp[u(1ρ)2]1+ρm=0bm(ρ)um+1/2,|u|1,
a0=1,a1=116ρ2,a2=5512ρ4,,b0=1ρ,b1=116ρ24ρ+3ρ+1(1+ρ)2,,
UN(r;0)J[2(N+1)rw0],
FGN(ξ)112erfc[2(N+1ξ)],
[d2ndξ2nFGN(ξN+1)]ξ=0=0,n=1,2,,N.
[d2ndξ2nFGN(ξN+1)]ξ=01π(2N+1)nexp[2(N+1)]H2n1(2N+1),
[d2ndξ2nFGN(ξN+1)]ξ=0(n1)!2π(8N+1)n×exp[(N+1)]sin[2N+14n1],
|[d2ndξ2nFGN(ξN+1)]ξ=0|(n1)!2π(8N+1)nexp[(N+1)].
1dξf(ξ)exp[u(ξρ)2],
f(ρ)1dξexp[u(ξρ)2]+1dξ[f(ξ)f(ρ)]exp[u(ξρ)2].
1dξf(ξ)exp[u(ξρ)2]=f(ρ)πu12erfc[u(1ρ)]12u1f(ξ)f(ρ)ξρd{exp[u(ξρ)2]}=f(ρ)πu12erfc[u(1ρ)]+12uf(1)f(ρ)1ρexp[u(1ρ)2]+12u1ddξ[f(ξ)f(ρ)ξρ]exp[u(ξρ)2]dξ,
f(ξ)=1ξ(ξ+ρ)2,
14uπρ1dξξexp[u(ξρ)2](ξ+ρ)2132uρ2erfc[u(1ρ)]+exp[u(1ρ)2]8uuπρ4ρρρ2ρ14ρρ(1+ρ)2(1ρ),
4ρρρ2ρ14ρρ(1+ρ)2(1ρ)=14ρρ(1+ρ)34ρρρ2ρ11ρ=4ρ+3ρ+14ρρ(1+ρ)3,
14uπρ1dξξexp[u(ξρ)2](ξ+ρ)2=132uρ2erfc[u(1ρ)]12uπρexp[u(1ρ)2]1+ρ116uρ2ρ(4ρ+3ρ+1)(1+ρ)2+132πu2ρ31dξξ+4ρξ(ξ+ρ)4exp[u(ξρ)2],
[d2ndξ2nFGN(ξN+1)]ξ=012(N+1)n{d2ndζ2nerfc[2(N+1ζ)]}ζ=0=12(N+1)n[d2ndζ2nerf(2ζ)]ζ=N+1,
exp(x22)H2n1(x)(1)n4n1(2n)!n!sin(x4n1),
exp(x22)H2n1(x)(1)n4n122nn!πnsin(x4n1),
exp(x22)H2n1(x)(4)n2π(n1)!sin(x4n1),
logΓ(n)nlogN+18(N+1)0,n1,
logΓ(n)nlognn,n1,
nlogn+nnlog(N+1)(N+1)0,n1,

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