Abstract

Our work is in the context of the French “laser mégajoule” project, about fusion by inertial confinement. The project leads to the problem of characterizing the inner surface, of the approximately spherical target, by optical shadowgraphy techniques. Our work is entirely based on the basic idea that optical shadowgraphy produces “caustics” of systems of optical rays, which contain a great deal of 3D information about the surface to be characterized. We develop a method of 3D reconstruction based upon this idea plus a “small perturbations” technique. Although computations are made in the special “spherical” case, the method is in fact general and may be extended to several other situations.

© 2007 Optical Society of America

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References

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  1. J. D. Lindl, Inertial Confinement Fusion (Springer, 1998).
  2. S. W. Haan, M. C. Herrmann, P. A. Amendt, D. A. Callahan, T. R. Dittrich, M. J. Edwards, O. S. Jones, M. M. Marinak, D. H. Munro, S. M. Pollaine, J. D. Salmonson, B. K. Spears, and L. J. Suter, "Update on specifications for NIF ignition targets, and their rollup into an error budget," Fusion Sci. Technol. 49, 553-557 (2006).
  3. J. K. Hoffer and L. R. Foreman, "Radioactively induced sublimation in solid tritium," Phys. Rev. Lett. 60, 1310-1313 (1988).
    [CrossRef] [PubMed]
  4. M. Martin, C. Gauvin, A. Choux, P. Baclet, and G. Pascal, "The cryogenic target for ignition on the LMJ: useful tools to achieve nominal temperature and roughness conditions of the DT solid layer," Fusion Sci. Technol. 49, 600-607 (2006).
  5. F. Gillot, A. Choux, L. Jeannot, G. Pascal, and P. Baclet, "Characterization of the DT layer of ICF targets by optical techniques," Fusion Sci. Technol. 49, 626-634 (2006).
  6. V. I. Arnold, A. Varchenko, and S. Goussein-Zadé, Singularités des Applications Différentiables. I: Classification des Points Critiques, des Caustiques et des Fronts d'Onde (Editions Mir, 1986).
    [PubMed]
  7. V. Zakalyukin, "Applications of flag contact singularities," New Developments in Singularity Theory Nato Series (Kluwer, 2001), pp. 41-70.
  8. D. H. Edgell, W. Seka, R. S. Craxton, L. M. Elasky, D. R. Harding, R. L. Keck, L. D. Lund, and M. D. Wittman, "Characterization of cryogenic direct-drive ICF targets during studies and just prior to shot time," Fusion Sci. Technol. 49, 616-625 (2006).
  9. A. I. Nikitenko and S. M. Tolokonnikov, "Optimal 'tomography' of 2-layered targets: 3D parameters reconstruction from shadow images," Fusion Sci. Technol. 51, 705-716 (2007).
  10. D. Marr and E. Hildreth, "Theory of edge detection," Proc. R. Soc. London , Ser. B 207, 187-217 (1980).
    [CrossRef]
  11. A. Choux, "Commande optimale d'un système de conformation cryogénique d'une couche solide d'isotopes de l'hydrogène dans un microballon par chauffage infra rouge," Ph.D. dissertation (University of Burgundy, 2006).
  12. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Chelsea, 1955).
  13. H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics (Cambridge U. Press, 1996).
    [CrossRef]

2007

A. I. Nikitenko and S. M. Tolokonnikov, "Optimal 'tomography' of 2-layered targets: 3D parameters reconstruction from shadow images," Fusion Sci. Technol. 51, 705-716 (2007).

2006

S. W. Haan, M. C. Herrmann, P. A. Amendt, D. A. Callahan, T. R. Dittrich, M. J. Edwards, O. S. Jones, M. M. Marinak, D. H. Munro, S. M. Pollaine, J. D. Salmonson, B. K. Spears, and L. J. Suter, "Update on specifications for NIF ignition targets, and their rollup into an error budget," Fusion Sci. Technol. 49, 553-557 (2006).

M. Martin, C. Gauvin, A. Choux, P. Baclet, and G. Pascal, "The cryogenic target for ignition on the LMJ: useful tools to achieve nominal temperature and roughness conditions of the DT solid layer," Fusion Sci. Technol. 49, 600-607 (2006).

F. Gillot, A. Choux, L. Jeannot, G. Pascal, and P. Baclet, "Characterization of the DT layer of ICF targets by optical techniques," Fusion Sci. Technol. 49, 626-634 (2006).

D. H. Edgell, W. Seka, R. S. Craxton, L. M. Elasky, D. R. Harding, R. L. Keck, L. D. Lund, and M. D. Wittman, "Characterization of cryogenic direct-drive ICF targets during studies and just prior to shot time," Fusion Sci. Technol. 49, 616-625 (2006).

1988

J. K. Hoffer and L. R. Foreman, "Radioactively induced sublimation in solid tritium," Phys. Rev. Lett. 60, 1310-1313 (1988).
[CrossRef] [PubMed]

1980

D. Marr and E. Hildreth, "Theory of edge detection," Proc. R. Soc. London , Ser. B 207, 187-217 (1980).
[CrossRef]

Fusion Sci. Technol.

M. Martin, C. Gauvin, A. Choux, P. Baclet, and G. Pascal, "The cryogenic target for ignition on the LMJ: useful tools to achieve nominal temperature and roughness conditions of the DT solid layer," Fusion Sci. Technol. 49, 600-607 (2006).

F. Gillot, A. Choux, L. Jeannot, G. Pascal, and P. Baclet, "Characterization of the DT layer of ICF targets by optical techniques," Fusion Sci. Technol. 49, 626-634 (2006).

D. H. Edgell, W. Seka, R. S. Craxton, L. M. Elasky, D. R. Harding, R. L. Keck, L. D. Lund, and M. D. Wittman, "Characterization of cryogenic direct-drive ICF targets during studies and just prior to shot time," Fusion Sci. Technol. 49, 616-625 (2006).

A. I. Nikitenko and S. M. Tolokonnikov, "Optimal 'tomography' of 2-layered targets: 3D parameters reconstruction from shadow images," Fusion Sci. Technol. 51, 705-716 (2007).

S. W. Haan, M. C. Herrmann, P. A. Amendt, D. A. Callahan, T. R. Dittrich, M. J. Edwards, O. S. Jones, M. M. Marinak, D. H. Munro, S. M. Pollaine, J. D. Salmonson, B. K. Spears, and L. J. Suter, "Update on specifications for NIF ignition targets, and their rollup into an error budget," Fusion Sci. Technol. 49, 553-557 (2006).

Phys. Rev. Lett.

J. K. Hoffer and L. R. Foreman, "Radioactively induced sublimation in solid tritium," Phys. Rev. Lett. 60, 1310-1313 (1988).
[CrossRef] [PubMed]

Proc. R. Soc. London

D. Marr and E. Hildreth, "Theory of edge detection," Proc. R. Soc. London , Ser. B 207, 187-217 (1980).
[CrossRef]

Other

A. Choux, "Commande optimale d'un système de conformation cryogénique d'une couche solide d'isotopes de l'hydrogène dans un microballon par chauffage infra rouge," Ph.D. dissertation (University of Burgundy, 2006).

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Chelsea, 1955).

H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics (Cambridge U. Press, 1996).
[CrossRef]

V. I. Arnold, A. Varchenko, and S. Goussein-Zadé, Singularités des Applications Différentiables. I: Classification des Points Critiques, des Caustiques et des Fronts d'Onde (Editions Mir, 1986).
[PubMed]

V. Zakalyukin, "Applications of flag contact singularities," New Developments in Singularity Theory Nato Series (Kluwer, 2001), pp. 41-70.

J. D. Lindl, Inertial Confinement Fusion (Springer, 1998).

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Figures (12)

Fig. 1
Fig. 1

Building with the 10 m diameter shooting sphere.

Fig. 2
Fig. 2

Laser-megajoule project target.

Fig. 3
Fig. 3

Reconstructed deformation.

Fig. 4
Fig. 4

Infrared heating of the target.

Fig. 5
Fig. 5

(Color online) Microshell inside the cylindrical cavity.

Fig. 6
Fig. 6

(Color online) Visible first bright ring.

Fig. 7
Fig. 7

(Color online) Some optical paths.

Fig. 8
Fig. 8

(Color online) Definitions of ρ, μ, and R u ( ρ ) .

Fig. 9
Fig. 9

(Color online) Radial profile.

Fig. 10
Fig. 10

(Color online) Concentration of rays around the caustic.

Fig. 11
Fig. 11

(Color online) Superposition of the detected caustic.

Fig. 12
Fig. 12

(Color online) Comparison between cuts of original and reconstructed surfaces.

Equations (24)

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R u * ( ρ ) = h 1 ( ρ ) + u h 2 ( ρ ) ,
h 1 ( ρ ) = ρ cos   2 Ψ ,
h 2 ( ρ ) = 1 f h 1 ( ρ ) + tan   2 Ψ ,
Ψ = arcsin   ρ r e x t arcsin   n e x t ρ n μ b r e x t + arcsin   n e x t ρ n μ b r i n t arcsin   n e x t ρ n DT r i n t + arcsin   n e x t ρ n DT r DT ,
S :   ( ρ , u ) ( R u * ( ρ ) , u ) .
R u * ( ρ ) ρ = 0 .
R 0 * ρ = 0 .
R c * = h 1 ( ρ * ) .
R ( ρ , θ ) = R * ( ρ ) + a 1 ( ρ ) ε 1 ( ρ , θ ) + a 2 ( ρ ) ε 2 ( ρ , θ ) + O 2 ,
α ( ρ , θ ) = θ + a 3 ( ρ ) ε 3 ( ρ , θ ) + O 2 ,
R ρ ( ρ , θ ) = 0 .
θ ( ρ , α ) = α a 3 ( ρ ) ε 3 ( ρ , α ) + O 2 .
R ( ρ , α ) = R * ( ρ ) + a 1 ( ρ ) ε 1 ( ρ , α ) + a 2 ( ρ ) ε 2 ( ρ , α ) + O 2 ,
R c ( α ) = R * ( ρ * ) + a 1 ( ρ * ) ε 1 ( ρ * , α ) + a 2 ( ρ * ) ε 2 ( ρ * , α ) + O 2 ,
0 = R * ρ ( ρ * + δ ρ ) + a 1 ρ ( ρ * + δ ρ ) ε 1 ( ρ * + δ ρ , α ) + a 1 ( ρ * + δ ρ ) ε 1 ρ ( ρ * + δ ρ , α ) + a 2 ρ ( ρ * + δ ρ ) × ε 2 ( ρ * + δ ρ , α ) + a 2 ( ρ * + δ ρ ) ε 2 ρ ( ρ * + δ ρ , α ) + O 2 .
0 = R * ρ ( ρ * ) + δ ρ 2 R * ρ 2 ( ρ * ) + a 1 ρ ( ρ * ) ε 1 ( ρ * , α ) + a 1 ( ρ * ) ε 1 ρ ( ρ * , α ) + a 2 ρ ( ρ * ) ε 2 ( ρ * , α ) + a 2 ( ρ * ) ε 2 ρ ( ρ * , α ) + δ ρ O 1 + O 2 .
R c ( α ) = R * ( ρ * + O 1 ) + a 1 ( ρ * + O 1 ) ε 1 ( ρ * + O 1 , α ) + a 2 ( ρ * + O 1 ) ε 2 ( ρ * + O 1 , α ) + O 2 ,
R c ( α ) = R * ( ρ * ) + R * ρ ( ρ * ) O 1 + a 1 ( ρ * ) ε 1 ( ρ * , α ) + a 2 ( ρ * ) ε 2 ( ρ * , α ) + O 2 .
ε 3 ( ρ * , θ ) = 1 R cos 2 ( φ * ) ε 1 ( φ * , θ ) θ + O 2 .
θ ( ρ * , α ) = α + O 1 ,
ε ( θ , φ ) = i = 1 i = k λ i e i ( θ , φ ) ,
Θ :   ε ( θ , φ ) Δ R c ( θ ) = R c ( θ ) R * ,
R c ( α ) R * ( ρ * ) + a 1 ( ρ * ) ε 1 ( ρ * , α ) + a 2 ( ρ * ) ε 2 ( ρ * , α ) ,
surface   S × tangent plane     caustic,

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