Abstract

When a lens is not prepared with an exact quadratic profile but is approximated by such a profile, there is considerable arbitrariness in the choice of the quadratic profile. Only one of these choices is optimal and gives the correct description of the physical optics involved. Laser beam self-focusing is chosen as the example in case, and it is shown that the optimal energy-conserving solution is equivalent to the variational and moments theories of self-focusing while at the same time it is paraxial in nature. Hankel-transformation techniques are used to prove this.

© 2004 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), p. 584.
  2. A. K. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge U. Press, London, 1989), p. 566.
  3. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 253.
  4. D. Subbarao, R. Uma, and H. Singh, “Paraxial theory of self-focusing of cylindrical beams. I. ABCD laws,” Phys. Plasmas 5, 3440–3550 (1998).
    [CrossRef]
  5. J. F. Lam, B. Lippman, and F. Tappert, “Self-trapped laser beams in plasmas,” Phys. Fluids 20, 1176–1179 (1977).
    [CrossRef]
  6. D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
    [CrossRef]
  7. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Kokhlov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, A. T. Arechi and E. D. Shulz Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. II, pp. 1151–1228.
  8. M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, “Self-focusing of laser beams in plasmas and semiconductors,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIII, p. 169.
  9. S. Madhekar, S. Konar, and M. S. Sodha, “Self-tapering of elliptic Gaussian beams in an elliptic core nonlinear fiber,” Opt. Lett. 20, 2192–2194 (1995).
    [CrossRef]
  10. D. Subbarao and M. S. Sodha, “Theory of paraxial self-focusing,” J. Appl. Phys. 50, 4604–4610 (1979).
    [CrossRef]
  11. D. Subbarao, R. Uma, and A. K. Ghatak, “Wave-optical theory of fast self-focusing in a plasma,” Laser Part. Beams 1, 367–377 (1983).
    [CrossRef]
  12. G. Fibich and B. Ilan, “An example of the failure of the aberrationless approximation,” J. Opt. Soc. Am. B 17, 1749–1758 (2000).
    [CrossRef]
  13. G. Fibich and G. C. Papanicolaou, “Self-focusing in the critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).
    [CrossRef]
  14. C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation (Springer, New York, 1999).
  15. V. M. Malkin, “On analytical theory for stationary self-focusing of radiation,” Physica D 64, 251–266 (1993).
    [CrossRef]
  16. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with loss and gain variation,” Appl. Opt. 4, 1562–1567 (1965).
    [CrossRef]
  17. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), pp. 166, 296.
  18. F. F. Chen, Introduction to Plasma Physics and Controlled Thermonuclear Fusion, 2nd ed. (Plenum, New York, 1984), Vol. 1, pp. 305–309, 410.
  19. M. Abramovitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  20. M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
    [CrossRef]
  21. D. Subbarao, H. Singh, R. Uma, and S. Bhaskar, “Computer simulation of laser-beam self-focusing in a plasma,” J. Plasma Phys. 61, 449–467 (1999).
    [CrossRef]

2000

1999

G. Fibich and G. C. Papanicolaou, “Self-focusing in the critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).
[CrossRef]

D. Subbarao, H. Singh, R. Uma, and S. Bhaskar, “Computer simulation of laser-beam self-focusing in a plasma,” J. Plasma Phys. 61, 449–467 (1999).
[CrossRef]

1998

D. Subbarao, R. Uma, and H. Singh, “Paraxial theory of self-focusing of cylindrical beams. I. ABCD laws,” Phys. Plasmas 5, 3440–3550 (1998).
[CrossRef]

1995

1993

V. M. Malkin, “On analytical theory for stationary self-focusing of radiation,” Physica D 64, 251–266 (1993).
[CrossRef]

1991

1983

D. Subbarao, R. Uma, and A. K. Ghatak, “Wave-optical theory of fast self-focusing in a plasma,” Laser Part. Beams 1, 367–377 (1983).
[CrossRef]

1979

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

D. Subbarao and M. S. Sodha, “Theory of paraxial self-focusing,” J. Appl. Phys. 50, 4604–4610 (1979).
[CrossRef]

1977

J. F. Lam, B. Lippman, and F. Tappert, “Self-trapped laser beams in plasmas,” Phys. Fluids 20, 1176–1179 (1977).
[CrossRef]

1965

Anderson, D.

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
[CrossRef]

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

Bhaskar, S.

D. Subbarao, H. Singh, R. Uma, and S. Bhaskar, “Computer simulation of laser-beam self-focusing in a plasma,” J. Plasma Phys. 61, 449–467 (1999).
[CrossRef]

Bonnedal, M.

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

Desaix, M.

Fibich, G.

G. Fibich and B. Ilan, “An example of the failure of the aberrationless approximation,” J. Opt. Soc. Am. B 17, 1749–1758 (2000).
[CrossRef]

G. Fibich and G. C. Papanicolaou, “Self-focusing in the critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).
[CrossRef]

Ghatak, A. K.

D. Subbarao, R. Uma, and A. K. Ghatak, “Wave-optical theory of fast self-focusing in a plasma,” Laser Part. Beams 1, 367–377 (1983).
[CrossRef]

Ilan, B.

Kogelnik, H.

Konar, S.

Lam, J. F.

J. F. Lam, B. Lippman, and F. Tappert, “Self-trapped laser beams in plasmas,” Phys. Fluids 20, 1176–1179 (1977).
[CrossRef]

Lippman, B.

J. F. Lam, B. Lippman, and F. Tappert, “Self-trapped laser beams in plasmas,” Phys. Fluids 20, 1176–1179 (1977).
[CrossRef]

Lisak, M.

Madhekar, S.

Malkin, V. M.

V. M. Malkin, “On analytical theory for stationary self-focusing of radiation,” Physica D 64, 251–266 (1993).
[CrossRef]

Papanicolaou, G. C.

G. Fibich and G. C. Papanicolaou, “Self-focusing in the critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).
[CrossRef]

Singh, H.

D. Subbarao, H. Singh, R. Uma, and S. Bhaskar, “Computer simulation of laser-beam self-focusing in a plasma,” J. Plasma Phys. 61, 449–467 (1999).
[CrossRef]

D. Subbarao, R. Uma, and H. Singh, “Paraxial theory of self-focusing of cylindrical beams. I. ABCD laws,” Phys. Plasmas 5, 3440–3550 (1998).
[CrossRef]

Sodha, M. S.

Subbarao, D.

D. Subbarao, H. Singh, R. Uma, and S. Bhaskar, “Computer simulation of laser-beam self-focusing in a plasma,” J. Plasma Phys. 61, 449–467 (1999).
[CrossRef]

D. Subbarao, R. Uma, and H. Singh, “Paraxial theory of self-focusing of cylindrical beams. I. ABCD laws,” Phys. Plasmas 5, 3440–3550 (1998).
[CrossRef]

D. Subbarao, R. Uma, and A. K. Ghatak, “Wave-optical theory of fast self-focusing in a plasma,” Laser Part. Beams 1, 367–377 (1983).
[CrossRef]

D. Subbarao and M. S. Sodha, “Theory of paraxial self-focusing,” J. Appl. Phys. 50, 4604–4610 (1979).
[CrossRef]

Tappert, F.

J. F. Lam, B. Lippman, and F. Tappert, “Self-trapped laser beams in plasmas,” Phys. Fluids 20, 1176–1179 (1977).
[CrossRef]

Uma, R.

D. Subbarao, H. Singh, R. Uma, and S. Bhaskar, “Computer simulation of laser-beam self-focusing in a plasma,” J. Plasma Phys. 61, 449–467 (1999).
[CrossRef]

D. Subbarao, R. Uma, and H. Singh, “Paraxial theory of self-focusing of cylindrical beams. I. ABCD laws,” Phys. Plasmas 5, 3440–3550 (1998).
[CrossRef]

D. Subbarao, R. Uma, and A. K. Ghatak, “Wave-optical theory of fast self-focusing in a plasma,” Laser Part. Beams 1, 367–377 (1983).
[CrossRef]

Appl. Opt.

J. Appl. Phys.

D. Subbarao and M. S. Sodha, “Theory of paraxial self-focusing,” J. Appl. Phys. 50, 4604–4610 (1979).
[CrossRef]

J. Opt. Soc. Am. B

J. Plasma Phys.

D. Subbarao, H. Singh, R. Uma, and S. Bhaskar, “Computer simulation of laser-beam self-focusing in a plasma,” J. Plasma Phys. 61, 449–467 (1999).
[CrossRef]

Laser Part. Beams

D. Subbarao, R. Uma, and A. K. Ghatak, “Wave-optical theory of fast self-focusing in a plasma,” Laser Part. Beams 1, 367–377 (1983).
[CrossRef]

Opt. Lett.

Phys. Fluids

J. F. Lam, B. Lippman, and F. Tappert, “Self-trapped laser beams in plasmas,” Phys. Fluids 20, 1176–1179 (1977).
[CrossRef]

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

Phys. Plasmas

D. Subbarao, R. Uma, and H. Singh, “Paraxial theory of self-focusing of cylindrical beams. I. ABCD laws,” Phys. Plasmas 5, 3440–3550 (1998).
[CrossRef]

Physica D

V. M. Malkin, “On analytical theory for stationary self-focusing of radiation,” Physica D 64, 251–266 (1993).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.

G. Fibich and G. C. Papanicolaou, “Self-focusing in the critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).
[CrossRef]

Other

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation (Springer, New York, 1999).

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), pp. 166, 296.

F. F. Chen, Introduction to Plasma Physics and Controlled Thermonuclear Fusion, 2nd ed. (Plenum, New York, 1984), Vol. 1, pp. 305–309, 410.

M. Abramovitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), p. 584.

A. K. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge U. Press, London, 1989), p. 566.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 253.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Kokhlov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, A. T. Arechi and E. D. Shulz Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. II, pp. 1151–1228.

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, “Self-focusing of laser beams in plasmas and semiconductors,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIII, p. 169.

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Figures (2)

Fig. 1
Fig. 1

Orbit of the phase point on the grgi plane for slow self-focusing without absorption by use of Eqs. (16) and (19). Closed orbits with different intensities indicate periodic focusing for saturating ponderomotive nonlinearity for ωpr0/2c=3.0.

Fig. 2
Fig. 2

Orbit of the phase point on the grgi plane for slow self-focusing with absorption for saturating ponderomotive nonlinearity. Notice the convergence of the different orbits to the focal point. The calculation is based on Eq. (11) for ωpr0/2c=3.0 and ν/ω=0.4.

Equations (50)

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=0-2r2
2Ez2+2Er2+1rEr+k02E=0,
=(EE*)=L-Φ(EE*),
Φ(EE*)=n=1an(EE*)n,
E(r, z)=0A(α)exp[β(α, z)]J0(αr)αdα,
A(α)=0E(r, 0)J0(αr)rdr.
β(α, z)β0(z)+α2β2(z),
E(r, z)=E0g exp(β0)exp[-r2/r02g],
r2E(r, z)=r20A(α)exp[β(α, z)]J0(αr)αdα=-0d2dα2+1αddαA(α)exp[β(α, z)]×J0(αr)αdα
2z2+k0222α2+1αα+k020-α2
×A(α)exp[β(α, z)]=0,
|A(α)exp[β(α, z)]|,ddαA(α)exp[β(α, z)]0
d2β0dz2+dβ0dz2+k02eff=0,
d2gdz2+2 dβ0dzdgdz+4χr02=0,
β0=-ik00zϕdz,ϕ=eff=ϕr+iϕi.
dgdη=1i(1-¼k022r04g2).
ddη1g=1i¼2k02r04-1g2.
g=-iζζ,
ζ=dζdη,
d2ζdη2=-¼2k02r04ζ.
ζ(Δη)=A sin½k0r020Δη2dη+B cos½k0r020Δη2dη.
g˜=Ag˜0+BCg˜0+D,
ABCD=cos½k0r020Δη2dη-i sin½k0r020Δη2dη-i sin½k0r020Δη2dηcos½k0r020Δη2dη.
Q=σ0 sin½k0r020Δη2dη-iσ1 cos½k0r020Δη2dη=exp-iσ1½k0r020Δη2dη,
σ0=1001,σ1=0110
1g=1f2+i1fdfdη1f2=grgr2+gi2,1fdfdη=-gigr2+gi2.
d2fdη2=-¼2k2r04f+1f3,
β0r-120η 1grdgrdη dη=12 ln grexp(β0r)gr.
E(r, η)=E0 1f expi tan-112df2dηexp(iβ0i)×exp-r2r02f2exp-ir2r021fdfdη.
|E(r, η)|2=E02f2 exp-2r2r02f2.
gr=1f21f4+1fdfdη2,
gi=-1fdfdη1f4+1fdfdη2,
|E(r, z)|2=E02gr2+gi2 exp(2β0r)exp-r2r022grgr2+gi2,
S=c8πϕr0EE*(EE*)2πrdr.
EE*=E02q2 exp(-2r2/r02p),
q=|g|exp(-β0r),p=[Re(g-1)]-1,
S=cE02r02p8πq2ϕrL-0an (E02/q2)nn+1.
0=L-n=1an (2n+1)(n+1)2E02q2n,
2=-2r02pn=1an n(n+1)2E02q2n.
exp-(2n+t)r2r02pH.T.12r02p2n+t exp-α2r02p4(2n+t)
12r02p2n+t exp-α2r02p4t1+2nα2r024(2n+t)t
(H.T)-1 exp-r2tr02p t2n+t4n+t2n+t-r2r02p2nt2n+t.
S=cE02r02p8πq2ϕrL-0an 4nt+t2-t2n(2n+t)2E02q2n.
Φ(EE*)=ωp2/ω2(1-iν2/ω2) Φ¯(EE*)=ωp2/ω2(1-iν2/ω2)[1-exp(-EE*)]=ωp2/ω2(1-iν2/ω2)n=1(-1)nn!(EE*)n
an=ωp2/ω2(1-iν2/ω2)a¯n,a¯n=(-1)nn!.
S¯(x)=n=1a¯n(n+1)2xn=1x0x 1x0xΦ¯(x)dxdx,
S¯2(x)=n=1a¯nn(n+1)2xn=xdS¯dx.
S¯(x)=E(x)x-1,S¯2(x)=(1-e-x)x-E(x)x,
d2Bdz2+k02effB=0.
B(z)=B+ exp-ik00z(eff)1/2dz+B- expik00z(eff)1/2dz,

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