The topography measurement of fast moving surfaces is aninteresting subject for research that can contribute to widerange of subjects from microscopy to sample surface mea-surements. In particular, we are interested in the surfacestructure of moving projectiles and the back (free) surfaceof shocked targets. In general, one would expect that suchmeasurements should be obtained by holography [1–5]. How-ever, for moving objects, the movement during the laser pulseneeds to be less than one-quarter of the laser wavelength. Forvelocities of the order of 1 km= sec and a 1=2 μm laser wave-length, one needs a laser pulse shorter than 100ps. Further-more, in dynamic holographic experiments, it is difficult toget an absolute measurement scale, in particular on few-micrometer dimensions as required in our experiments. Sinceholography requires a simultaneous measurement of twoshort laser pulses, its implementation is a difficult task in sucha short time. In this paper, we use a new method recently sug-gested and tested for static objects [6,7]. The method is basedon measuring the backscattered intensity from the tested ob-ject at two different imaging planes. The local intensitychanges between the two close imaging planes, separatedby a known distance, give us the required information to re-trieve the object phase. This noninterferometry approach isbased on solving the Maxwell wave equation in the paraxialapproximation. In this case the time-independent wave equa-tion is
�∇2
H þ 2ik∂
∂z
�uðx; y; zÞ ¼ 0; ð1Þ
where ∇H ¼ ð∂=∂x; ∂=∂yÞ, k ¼ 2π=λ, and λ is the laserwavelength. The wave amplitude is uðx; y; zÞ ¼½Iðx; y; zÞ�1=2eiϕðx;y;zÞ; Iðx; y; zÞ is the intensity and ϕðx; y; zÞis the phase. By substituting uðx; y; zÞ into Eq. (1) and takingthe imaginary part, we get the transport intensity equation(TIE) [8] [Eq. (2)]. The topography of a moving surface isrecovered by solving the TIE [7–10]:
k∂Iðx; y; zÞ
∂z¼ −∇H · ½Iðx; y; zÞ∇Hϕðx; y; zÞ�: ð2Þ
The left side of Eq. (2) is the intensity derivative along the op-tical axis, and it can be approximated by the first-order differ-ence equation in Δz, leading to the following equation:
kI2ðx; y; z2Þ − I1ðx; y; z1Þ
Δz¼ −∇H · ½Iðx; y; z1Þ∇Hϕðx; y; z1Þ�:
ð3Þ
The intensities I1ðx; y; z1Þ and I2ðx; y; z2Þ are measured ontwo different imaging planes separated by the distanceΔz ¼ z2 − z1.
There are several numerical methods [7] for solving Eq. (3).This equation was traditionally solved with standard Poissonsolvers. In this case, a suitable boundary condition for phase ϕis required. However, this phase is not known experimentally.
In this paper, Eq. (3) is solved by expanding the solution ϕinto a complete set of polynomials. In this approach we re-duce the partial differential equation to a set of algebraic equa-tions. The most significant advantage of this method is that theboundary conditions are determined by the intensity and canbe easily controlled experimentally by using a laser source.The expansion of phase ϕ is chosen with the complete setof the Zernike polynomials [11,12]. The Zernike polynomialsZiðr=R; θÞ, (i ¼ 1; 2; 3;…), appear to be the most naturalchoice for removal of the optical aberrations as well as fordealing with our experimental boundary conditions. R is de-fined by Iðr ≥ RÞ ¼ 0 and ϕðx; y; z1Þ≡ ϕðx; yÞ ¼ ϕðr; θÞ
Therefore,
ϕðr; θÞ ¼X∞i¼0
φiZiðr=R; θÞ: ð4Þ
For practical purposes, we change the infinite sum to the first26 Zernike polynomials.
1562 J. Opt. Soc. Am. A / Vol. 28, No. 8 / August 2011 Pinhasi et al.
the coefficients φ2 and φ3 are related accordingly to the tiltangles θ2 and θ3 relative to the z ¼ 0 plane. The tilt angles are
cos θ2;3 ¼�1þ
�λφ2;3
2πR
�2�
−12
⇒ θ2;3 ≈λφ2;3
2πR : ð5Þ
Substituting Eq. (4) into Eq. (3), multiplying by Zernikepolynomials, and integrating over rdrdθ leads to the followingalgebraic relation:
X26i¼1
Mijφi ¼ R2Fj; ð6Þ
where
Mij≡
Z2π
0
ZR
0Iðr;θ;z1Þ∇pZiðr=R;θÞ ·∇pZjðr=R;θÞrdrdθ
FjðR;zÞ≡R−2
Z2π
0
ZR
0kI2ðr;θ;z2Þ− I1ðr;θ;z1Þ
ΔzZjðr=R;θÞrdrdθ:
ð7ÞEquation (6) is a system of algebraic equations where the so-lution consists of the Zernike coefficients φi. With the knowl-edge of φi, the phase ϕðr; θÞ can be reconstructed accordingto Eq. (4). The surface topography hðr; θÞ is calculated fromthe relation
hðr; θÞ ¼ λ2π ϕðr; θÞ: ð8Þ
The intensities I1ðx; y; z1Þ and I2ðx; y; z2Þ, separated by aknown distanceΔz, are measured experimentally. The experi-
ments were carried out using a laser light source where theilluminated region has a spot of radius R and is well defined:
Iðr; θÞ > 0 r < RIðr; θÞ ¼ 0 r ≥ R
: ð9Þ
In the above boundary conditions, the solution for phase ϕis unique up to an arbitrary constant. For topography calcula-tion, our interest is in height differences only, and thereforethe topography is uniquely defined.
The experimental system is described in Fig. 1. The impactexperiments were performed by accelerating an aluminum im-pactor in a gas gun. The aluminum (Al-1100) projectile impac-tor (25mm diameter and 1mm thickness) with velocity of theorder of 1km=s is accelerated into the aluminum target(20mm diameter, 2mm thickness). The topography of the tar-get is measured before and after the shock wave experiment.A doubled frequency Nd:YAG (0:532 μm) laser source withpulse duration of 7 ns was used to illuminate the target.Two imaging planes separated by a measured distance Δzare imagined on the two CCD cameras (1220 × 1220 pixels).The imaging optics are adjusted to obtain two similar magni-fications on the two CCDs. The distance between the twolenses in the imaging system is equal to the sum of their focallengths, resulting the same optical magnification on both ima-ging planes.
In the dynamic experiments, triggers are required to oper-ate the laser flash lamp and to open the laser Q-switch accord-ing to the motion of the projectile. The time scale of theexperiment is adjusted in such a way that our 7ns laser pulseirradiates the target (or the projectile) when we want to mea-sure its surface topography. The zero time is defined when theimpactor (projectile) starts moving. The time systemmeasure-ments of Fig. 2 show also the timing of two optical pins thatmeasure the impactor velocity and the timing of five-pindevices that are electrically shortened when the target (orthe projectile) surface hits these pins. The shortened pins
Fig. 1. (Color online) 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 Δz. The optical components are imaging onthe CCD’s two planes near the target. PMT, photomultiplier.
Pinhasi et al. Vol. 28, No. 8 / August 2011 / J. Opt. Soc. Am. A 1563
serve as an indication of the planarity, or rather the lack ofplanarity, of the shocked target or the flying projectile. Forcompleteness, Fig. 2 also shows the time of a magnetic sensorthat confirms the motion of our projectile.
In order to synchronize the laser pulse and the arrival timeof the shock wave to the free surface, we use a three-leveltriggering system as shown in Fig. 2. The experiment startswith the shooting command in the computer (signal a). Afterthe impactor starts moving, a magnetic sensor gives us a sig-nal (b) as the confirmation of the impactor motion. At the endof the gun barrel, an optical pin laser diode (c) gives the triggerto operate the laser flash lamp. The flash lamp laser requires atrigger of 180 μs before opening the Q-switch in order to getmaximum laser efficiency. The second optical pin (d) isplaced 50mm away from the first optical pin, for projectilevelocity measurement. At the impact plane, between the pro-jectile and the target, there is a set of five shorting pins (ei;i ¼ 1…5). The first shorting pin triggers the laser Q-switchwhile the others give an indication about the planarity ofthe impactor.
The recovered topographies of the target before and duringthe dynamic experiment are shown in Fig. 3 for an Al–Al colli-sion with a 800m=s impactor velocity. In Fig. 3(a) we can seethe free surface topography at static conditions. Figure 3(b)shows the topography of the free surface about 100 ns aftershock wave arrival. The dynamic topography includes tiltand deformation of the free surface. In the method we presentin this paper, the tilt can be eliminated. Instead of taking all 26Zernike polynomials in the summation, according to Eq. (4),we can eliminate from the summation the polynomials wewant to subtract in order to get the deformations. In Fig. 3(c)we can see the surface topography during the dynamic experi-ment after elimination of the horizontal tilt. The compressionpressure in our dynamic experiments is about 80kbar. Thedynamic yield stress for Al with velocity of 1 km=s is around8 kbar. Therefore it is conceivable that this collision hasreached the plastic region of the target.
We also measured the topography of a 1mm aluminum pro-jectile (Al-1100) during its flight (not in the same experimentwith the free surface measurement). Figure 4(a) shows thetopography of an Al impactor during flight at a velocity of800m=s. We can see that there is an approximately 8mrad tiltin the horizontal direction and also a 3mrad tilt in the verticaldirection. Figure 4(b) shows the surface topography of theprojectile after “removing” the tilt. This topography has thesame structure as the static measurement.
The experimental space resolution depends on the CCD’spixel size and on the imaging system’s magnification. For ouroptical system, where the CCD’s pixel size is 4:4 μm and themagnification is around 3×, the resolution in the x–y plane is∼� 25 μm in agreement with the measurement of the USAF1951 target. In the z direction, the resolution mainly dependson the paraxial approximation, the optical system quality, and
Fig. 2. (Color online) Trigger measurements and time scale diagnos-tic system. Top, shooting command and laser flash lamp trigger; mid-dle, projectile velocity and Q-switch trigger; bottom, planarity of theimpactor. For a detailed definition of ðaÞ; ðbÞ……ðe5Þ see text.
Fig. 3. Topographies of the tested target (a) before and (b), (c) dur-ing the dynamic experiment; (b) topography of the target when thefirst 26 Zernike polynomials are taken into account; (c) target surfaceafter elimination of the horizontal tilt.
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the lenses’ focal depth. The errors in the topography wereevaluated experimentally in the following way: we measurethe topography of the same sample 20 times (in staticconditions) and calculate the average variance of the results(relative to the first measurement). As we can see in Fig. 5,those statistical reconstruction errors are up to �0:5 μm. Thereconstruction accuracy was calculated using a well-definedphase plate [7]. The inaccuracy of the surface measurementis �0:5 μm. In the dynamic experiments, we need to add theerrors due to the sample motion during the laser pulse. Weevaluated those errors as �2 μm. The total error of the topo-graphy reconstruction in our dynamic experiments is �3 μm.
In summary, a new noninterferometric diagnostic methodto measure the topography of fast moving surfaces is pres-ented. This method is successfully tested for moving objectsas well as for moving shocked targets with velocities of about1 km=s. This method can also be implemented for higher ve-locities, using shorter laser pulse duration or alternativelyusing fast frame cameras recording instead of our CCDs.
ACKNOWLEDGMENTSThis work was supported by the joint Council for HigherEducation–Israel Atomic Energy Commission (VATAT–IAEC)common lab foundation. We are grateful to Dr. R. Alimi forhelp with numerical advice.
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Fig. 5. (Color online) Average error in micrometers at different staticexperiments of the same sample.
Fig. 4. (a) In flight topography of a moving impactor. Inner circle,topography of laser-illuminated region of target; second circle, projec-tile border; white dots, locations of shorting pins (e1-…e5) on projec-tile surface. The shorting times (in parentheses) are relative to the firstshorted pin, e1. (b) Projectile topography after eliminating tilt.
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