Abstract

The topography of moving surfaces is recovered by noninterferometric measurements. The phase reconstruction is derived by measuring the intensities of a backscattered pulsed laser light and solving the transport intensity equation (TIE). The TIE is solved by expanding the phase into a series of Zernike polynomials, leading to a set of appropriate algebraic equations. This technique, which enables us to make a direct connection between experiments and the TIE, has been successfully tested in gas gun experiments. In particular, the topographies of a moving projectile and the free surface of a shocked target were recovered.

© 2011 Optical Society of America

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References

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  1. M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
    [CrossRef]
  2. M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
    [CrossRef]
  3. C. F. McMillan and R. K. Whipkey, “Holographic measurement of ejecta from shocked metal surfaces,” Proc. SPIE 1032, 555–558 (1988).
  4. P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.
  5. Y. J. Kang, S. H. Baik, W. J. Ryu, and K. S. Kimd, “Measurement of shock waves using phase-shifting pulsed holographic interferometer,” Opt. Laser Technol. 35, 323–329 (2003).
    [CrossRef]
  6. S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter, “Fast optical computerized topography,” Phys. Lett. A 374, 2798–2800(2010).
    [CrossRef]
  7. S. V. Pinhasi, R. Alimi, L. Perelmutter, and S. Eliezer, “Fast topography retrieval using different solutions of the transport intensity equation,” J. Opt. Soc. Am. A 27, 2285–2292 (2010).
    [CrossRef]
  8. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  9. S. Vinikman-Pinhasi and E. N. Ribak, “Piezoelectric and piezo-optic effects in porous silicon,” Appl. Phys. Lett. 88, 111905–111905–2 (2006).
    [CrossRef]
  10. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1942 (1995).
    [CrossRef]
  11. V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975) Sec. 9.2.

2010 (2)

2006 (1)

S. Vinikman-Pinhasi and E. N. Ribak, “Piezoelectric and piezo-optic effects in porous silicon,” Appl. Phys. Lett. 88, 111905–111905–2 (2006).
[CrossRef]

2003 (1)

Y. J. Kang, S. H. Baik, W. J. Ryu, and K. S. Kimd, “Measurement of shock waves using phase-shifting pulsed holographic interferometer,” Opt. Laser Technol. 35, 323–329 (2003).
[CrossRef]

1999 (1)

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

1997 (1)

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

1995 (1)

1988 (1)

C. F. McMillan and R. K. Whipkey, “Holographic measurement of ejecta from shocked metal surfaces,” Proc. SPIE 1032, 555–558 (1988).

1983 (1)

1934 (1)

V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Alimi, R.

Arad, B.

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

Baik, S. H.

Y. J. Kang, S. H. Baik, W. J. Ryu, and K. S. Kimd, “Measurement of shock waves using phase-shifting pulsed holographic interferometer,” Opt. Laser Technol. 35, 323–329 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975) Sec. 9.2.

Eliezer, S.

S. V. Pinhasi, R. Alimi, L. Perelmutter, and S. Eliezer, “Fast topography retrieval using different solutions of the transport intensity equation,” J. Opt. Soc. Am. A 27, 2285–2292 (2010).
[CrossRef]

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter, “Fast optical computerized topography,” Phys. Lett. A 374, 2798–2800(2010).
[CrossRef]

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

Goldberg, I. B.

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

Gureyev, T. E.

Henis, Z.

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

Hockaday, M. P.

P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.

Horovitz, Y.

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

Kang, Y. J.

Y. J. Kang, S. H. Baik, W. J. Ryu, and K. S. Kimd, “Measurement of shock waves using phase-shifting pulsed holographic interferometer,” Opt. Laser Technol. 35, 323–329 (2003).
[CrossRef]

Kimd, K. S.

Y. J. Kang, S. H. Baik, W. J. Ryu, and K. S. Kimd, “Measurement of shock waves using phase-shifting pulsed holographic interferometer,” Opt. Laser Technol. 35, 323–329 (2003).
[CrossRef]

King, N. S. P.

P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.

Lee, H.

P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.

Maman, S.

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

McMillan, C. F.

C. F. McMillan and R. K. Whipkey, “Holographic measurement of ejecta from shocked metal surfaces,” Proc. SPIE 1032, 555–558 (1988).

Nugent, K. A.

Obst, A.

P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.

Perelmutter, L.

Pinhasi, S. V.

Platts, D.

P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.

Ribak, E. N.

S. Vinikman-Pinhasi and E. N. Ribak, “Piezoelectric and piezo-optic effects in porous silicon,” Appl. Phys. Lett. 88, 111905–111905–2 (2006).
[CrossRef]

Roberts, A.

Roberts, J. P.

P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.

Ryu, W. J.

Y. J. Kang, S. H. Baik, W. J. Ryu, and K. S. Kimd, “Measurement of shock waves using phase-shifting pulsed holographic interferometer,” Opt. Laser Technol. 35, 323–329 (2003).
[CrossRef]

Scannapieco, A. J.

P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.

Sheppard, M.

P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.

Shpitalnik, R.

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

Sorenson, P. S.

P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.

Taylor, A. J.

P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.

Teague, M. R.

Vinikman-Pinhasi, S.

S. Vinikman-Pinhasi and E. N. Ribak, “Piezoelectric and piezo-optic effects in porous silicon,” Appl. Phys. Lett. 88, 111905–111905–2 (2006).
[CrossRef]

Watson, S.

P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.

Werdiger, M.

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

Whipkey, R. K.

C. F. McMillan and R. K. Whipkey, “Holographic measurement of ejecta from shocked metal surfaces,” Proc. SPIE 1032, 555–558 (1988).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975) Sec. 9.2.

Zernike, V. F.

V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Appl. Phys. Lett. (2)

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

S. Vinikman-Pinhasi and E. N. Ribak, “Piezoelectric and piezo-optic effects in porous silicon,” Appl. Phys. Lett. 88, 111905–111905–2 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Laser Part. Beams (1)

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

Opt. Laser Technol. (1)

Y. J. Kang, S. H. Baik, W. J. Ryu, and K. S. Kimd, “Measurement of shock waves using phase-shifting pulsed holographic interferometer,” Opt. Laser Technol. 35, 323–329 (2003).
[CrossRef]

Phys. Lett. A (1)

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter, “Fast optical computerized topography,” Phys. Lett. A 374, 2798–2800(2010).
[CrossRef]

Physica (1)

V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Proc. SPIE (1)

C. F. McMillan and R. K. Whipkey, “Holographic measurement of ejecta from shocked metal surfaces,” Proc. SPIE 1032, 555–558 (1988).

Other (2)

P. S. Sorenson, A. Obst, N. S. P. King, A. J. Scannapieco, H. Lee, M. Sheppard, J. P. Roberts, D. Platts, A. J. Taylor, S. Watson, and M. P. Hockaday, “In-line particle field holography at Pegasus,” Report LAUR-94-4331, Los Alamos National Laboratory, Los Alamos New Mexico, USA, 1994.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975) Sec. 9.2.

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Figures (5)

Fig. 1
Fig. 1

Diagnostics area: optical and recording systems and triggering setup in the diagnostic area of the gas gun device. The backreflected light from the object is imaged on two CCD cameras separated by distance Δ z . The optical components are imaging on the CCD’s two planes near the target. PMT, photomultiplier.

Fig. 2
Fig. 2

Trigger measurements and time scale diagnostic system. Top, shooting command and laser flash lamp trigger; middle, projectile velocity and Q-switch trigger; bottom, planarity of the impactor. For a detailed definition of ( a ) , ( b ) ( e 5 ) see text.

Fig. 3
Fig. 3

Topographies of the tested target (a) before and (b), (c) during the dynamic experiment; (b) topography of the target when the first 26 Zernike polynomials are taken into account; (c) target surface after elimination of the horizontal tilt.

Fig. 4
Fig. 4

(a) In flight topography of a moving impactor. Inner circle, topography of laser-illuminated region of target; second circle, projectile border; white dots, locations of shorting pins ( e 1 - e 5 ) on projectile surface. The shorting times (in parentheses) are relative to the first shorted pin, e 1 . (b) Projectile topography after eliminating tilt.

Fig. 5
Fig. 5

Average error in micrometers at different static experiments of the same sample.

Equations (10)

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( H 2 + 2 i k z ) u ( x , y ; z ) = 0 ,
k I ( x , y ; z ) z = H · [ I ( x , y ; z ) H ϕ ( x , y ; z ) ] .
k I 2 ( x , y ; z 2 ) I 1 ( x , y ; z 1 ) Δ z = H · [ I ( x , y ; z 1 ) H ϕ ( x , y ; z 1 ) ] .
ϕ ( r , θ ) = i = 0 φ i Z i ( r / R , θ ) .
r = ( x 2 + y 2 ) 1 / 2 ; θ = arctg ( y / x ) ; p = ( / r , 1 r / θ ) ,
cos θ 2 , 3 = ( 1 + ( λ φ 2 , 3 2 π R ) 2 ) 1 2 θ 2 , 3 λ φ 2 , 3 2 π R .
i = 1 26 M i j φ i = R 2 F j ,
M i j 0 2 π 0 R I ( r , θ ; z 1 ) p Z i ( r / R , θ ) · p Z j ( r / R , θ ) r d r d θ F j ( R , z ) R 2 0 2 π 0 R k I 2 ( r , θ ; z 2 ) I 1 ( r , θ ; z 1 ) Δ z Z j ( r / R , θ ) r d r d θ .
h ( r , θ ) = λ 2 π ϕ ( r , θ ) .
I ( r , θ ) > 0 r < R I ( r , θ ) = 0 r R .

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