Abstract

The dark spot effect downstream from the nonlinear hot image is accounted for in this paper. The conditions for the formation of dark spot are carefully discussed. The explanation is based on analytical analysis, and the results are verified by numerical simulations. The dependence of the location of the dark spot on the nonlinear phase delay seems to suggest a probable method for measuring the nonlinear refractive coefficient of materials.

© 2012 Optical Society of America

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References

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  1. J. T. Hunt, K. R. Manes, and P. A. Renard, “Hot-images from obscurations,” Appl. Opt. 32, 5973–5982 (1993).
    [CrossRef]
  2. C. C. Widmayer, D. Milam, and S. P. deSzoeke, “Nonlinear formation of holographic images of obscurations in laser beams,” Appl. Opt. 36, 9342–9347 (1997).
    [CrossRef]
  3. C. C. Widmayer, M. R. Nickels, and D. Milam, “Nonlinear holographic imaging of phase errors,” Appl. Opt. 37, 4801–4805 (1998).
    [CrossRef]
  4. C. C. Widmayer, L. R. Jones, and D. Milam, “Measurement of the nonlinear coefficient of carbon disulfide using holographic self-focusing,” J. Nonlinear Opt. Phys. Mater. 7, 563–570 (1998).
    [CrossRef]
  5. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).
  6. X. Wang, “On the beam intensity irregularities induced by the 3rd order nonlinear effect in ICF drivers,” Ph.D. thesis (Sichuan University, 2003), pp. 39–40.
  7. D. Li, J. Zhao, T. Peng, and Z. Cai, “Theoretical analysis of the image with a local intensity minimum during hot image formation in high-power laser systems,” Appl. Opt. 48, 6229–6233 (2009).
    [CrossRef]
  8. L. Xie, J. Zhao, and F. Jing, “Second-order hot-image from a scatterer in high-power laser systems,” Appl. Opt. 44, 2553–2557 (2005).
    [CrossRef]
  9. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001), p. 51.

2009 (1)

2005 (1)

1998 (2)

C. C. Widmayer, M. R. Nickels, and D. Milam, “Nonlinear holographic imaging of phase errors,” Appl. Opt. 37, 4801–4805 (1998).
[CrossRef]

C. C. Widmayer, L. R. Jones, and D. Milam, “Measurement of the nonlinear coefficient of carbon disulfide using holographic self-focusing,” J. Nonlinear Opt. Phys. Mater. 7, 563–570 (1998).
[CrossRef]

1997 (1)

1993 (1)

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001), p. 51.

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Cai, Z.

deSzoeke, S. P.

Hunt, J. T.

Jing, F.

Jones, L. R.

C. C. Widmayer, L. R. Jones, and D. Milam, “Measurement of the nonlinear coefficient of carbon disulfide using holographic self-focusing,” J. Nonlinear Opt. Phys. Mater. 7, 563–570 (1998).
[CrossRef]

Li, D.

Manes, K. R.

Milam, D.

Nickels, M. R.

Peng, T.

Renard, P. A.

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Wang, X.

X. Wang, “On the beam intensity irregularities induced by the 3rd order nonlinear effect in ICF drivers,” Ph.D. thesis (Sichuan University, 2003), pp. 39–40.

Widmayer, C. C.

Xie, L.

Zhao, J.

Appl. Opt. (5)

J. Nonlinear Opt. Phys. Mater. (1)

C. C. Widmayer, L. R. Jones, and D. Milam, “Measurement of the nonlinear coefficient of carbon disulfide using holographic self-focusing,” J. Nonlinear Opt. Phys. Mater. 7, 563–570 (1998).
[CrossRef]

JETP Lett. (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Other (2)

X. Wang, “On the beam intensity irregularities induced by the 3rd order nonlinear effect in ICF drivers,” Ph.D. thesis (Sichuan University, 2003), pp. 39–40.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001), p. 51.

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Figures (6)

Fig. 1.
Fig. 1.

Formation of nonlinear hot image and downstream dark spot.

Fig. 2.
Fig. 2.

Hot image and dark spot under different values of B, the defect being pure phase-typed with depth π. (a) Intensity distribution at hot image plane, B=0.516; (b) intensity distribution at location 577 mm distant downstream from hot image, B=0.516; (c) intensity distribution at hot image plane, B=0.967; and (d) intensity distribution at location 1082 mm distant downstream from hot image, B=0.967.

Fig. 3.
Fig. 3.

Hot image and dark spot under different values of B, the defect being pure phase-typed with depth π. (a) Intensity distribution at hot image plane, B=0.516; (b) intensity distribution at location 258 mm distant downstream from hot image, B=0.516; (c) intensity distribution at hot image plane, B=0.967; and (d) intensity distribution at location 485 mm distant downstream from hot image, B=0.967.

Fig. 4.
Fig. 4.

Location of dark spot as a function of the characteristic length of the defect. The nonlinear phase delay B is 1.289 rad. (a) A circular defect and (b) an elliptical defect.

Fig. 5.
Fig. 5.

Location of dark spot as a function of the area of the defect. The nonlinear phase delay B is 1.289 rad.

Fig. 6.
Fig. 6.

Location of dark spot as a function of the nonlinear phase delay.

Equations (11)

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t(x1,y1)=1+αp(x1,y1),
UoU(x2,y2)exp[iδnk0d]=(Ub+Us)exp[ik0γ|U(x2,y2)|22d],
Uoexp(iB)[Ub+(1+iB)Us+iBUb2Us*].
v1=exp(iB)iBUb2Us*=α*Bλz1exp[i(kz1+B)]exp[ik2z1(x22+y22)]P*(x2λz1,y2λz1),
α*Biλ2z1z2exp[i(kz1+kz2+B)]×exp[ik2z1(x22+y22)]P*(x2λz1,y2λz1)exp[ik2z2((x3x2)2+(y3y2)2)]dx2dy2.
iα*Bexp[i(2kz1+B)]exp[ik2z1(x32+y32)]p(x3,y3).
iα*Bexp[i(2kz1+B)]p(x3,y3).
iα*ABexp[i(2kz1+B+kzc)]iλzc=ABλzcα*exp[i(2kz1+B+kzc)],
ABλzcα*=1.
zc=|α|ABλ.
zc=2πBr2λ.

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