Spatial filter pinhole development for the National ... - OSA Publishing

Spatial filter pinhole development for the National Ignition Facility James E. Murray, David Milam, Charles D. Boley, Kent G. Estabrook, and John A. Caird

We present results from a major experimental effort to understand the behavior of spatial filter pinholes and to identify and demonstrate a pinhole that will meet the requirements of the National Ignition Facility 共NIF兲. We find that pinhole performance depends significantly on geometry and material. Cone pinholes are found to stay open longer and to cause less backreflection than pinholes of more conventional geometry. We show that a ⫾150-␮rad stainless-steel cone pinhole will pass a full-energy NIF ignition pulse with required margins for misalignment and for smoothing by spectral dispersion. On the basis of a model fitted to experimental results, a ⫾125-␮rad stainless-steel cone pinhole is also projected to meet these requirements. © 2000 Optical Society of America OCIS codes: 120.3180, 320.4240, 330.6110, 350.5400.

1. Introduction

Spatial filters have been used for more than 20 years to control the nonlinear growth of spatial noise during propagation of high-power laser beams.1,2 A spatial filter consists of a pair of lenses that focus and recollimate a beam, with a pinhole at the common focus. The pinhole acts as a low-pass filter for the spatial noise on the input beam, allowing only the lower spatial frequencies to be transmitted. An important problem associated with the use of spatial filters arises from the interaction of the beam with the material of the pinhole. Usually, a pinhole that is sufficiently small to filter unwanted spatial frequencies is also small enough that the beam intensity striking the edge of the pinhole creates a plasma. If this plasma propagates into the central part of the beam, it can distort or block the transmitted pulse, causing pinhole closure. It can also lead to backreflections, owing to the nonlinear interaction of the main laser pulse with plasma waves. Several early investigations reported experimental observations of pinhole closure3–5 and backreflections,3,4 including the observation that misalignment3,5 increases the severity of these effects. Evidence was also reported4 that

The authors are with the Lawrence Livermore National Laboratory, University of California, Livermore, California 94551. The e-mail address for J. E. Murray is [email protected]. Received 16 August 1999; revised manuscript received 10 January 2000. 0003-6935兾00兾091405-16$15.00兾0 © 2000 Optical Society of America

refraction rather than absorption is the dominant mechanism that causes the onset of pinhole closure, and the closure speed was estimated as 2 ⫻ 107 to 5 ⫻ 107 cm兾s. Recently a cone-shaped pinhole was introduced.6 This is a new geometry designed to stay open longer than hole-in-plate pinholes. Initial experimental results showed improved performance, with a closure speed of ⬃107 cm兾s. Evidence was also reported that pinhole materials with high atomic number Z perform better than low-Z materials. Another recent investigation of pinhole closure,7 similar to but less extensive than the one reported here, also observed an advantage for high-Z pinhole materials. However, it reported no dependence of pinhole closure on the geometry of the pinhole. A recent study8 suggested that cone pinholes of clear or absorbing glass would produce appreciably less plasma than metallic pinholes. It presented threshold data for plasma generation in support of this suggestion. The closure speeds reported in the literature suggest that pinhole closure could be a major problem for the National Ignition Facility 共NIF兲, which will use pulses with durations as long as 20 ns. 共The NIF is a 192beam Nd:glass laser system designed for inertial confinement fusion experiments. It is part of the Department of Energy’s program to maintain the nation’s nuclear weapons stockpile by computerized simulation rather than by testing.兲 Pinholes for the NIF will have acceptance angles between ⫾100 and ⫾200 ␮rad, with angles of ⫾150 ␮rad or less being strongly preferred. 共The acceptance angle is the pinhole ra20 March 2000 兾 Vol. 39, No. 9 兾 APPLIED OPTICS

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Fig. 2. The three types of pinhole tested on the Beamlet laser. Fig. 1. Schematic diagram of the pinhole region in a spatial filter.

dius divided by the focal length of the spatial filter lens.兲 The corresponding pinhole radii range from 1.2 to 6 mm. For the closure speeds reported in the literature, closure times would vary from 2.5 to 60 ns. The former would not permit operation of the laser, whereas the latter would allow for a large safety margin. In addition, there are other considerations. Closure times are influenced not only by the angular aperture of the pinhole but also by beam alignment, by the quality of the laser beam, and by the design and material of the pinhole. Inasmuch as the interplay of these parameters was not described previously, existing results are insufficient for the design of pinholes for the NIF. The primary goal of the research presented in this paper is to demonstrate the smallest pinhole that will pass a 20-ns NIF ignition pulse. Further, the pinhole is required to tolerate a misalignment of ⫾10% of the pinhole radius and a ⫾7.5-␮rad increase in beam divergence that is required for smoothing by spectral dispersion 共SSD兲.9 In this paper we present general information about closure and the geometry of pinholes and describe the three components of our development effort: 共1兲 measurement of closure times for proposed pinholes,10 with high-energy NIF-like pulses in the Beamlet laser facility11; 共2兲 off-line experiments12 to measure the general characteristics of the plasmas that close pinholes; and 共3兲 development of a model that summarizes results of these experiments. During the course of this study we demonstrated that a cone-shaped, stainless-steel 共SS兲 pinhole with an angular acceptance of ⫾150 ␮rad will transmit the NIF ignition pulse, with the required margin for misalignment and additional beam divergence. The phenomenological model, noted above, permits extrapolation of our limited data and renders these results applicable to laser systems other than the NIF. This model predicts that a ⫾125-␮rad SS cone pinhole would also satisfy the NIF requirement. 2. Description of Pinhole Closure

The term pinhole closure is somewhat of a misnomer. What happens is illustrated in Fig. 1, which is a schematic of the pinhole region of a spatial filter along with the focal distribution of a beam that is directed into the page. The rim of the pinhole is 1406

APPLIED OPTICS 兾 Vol. 39, No. 9 兾 20 March 2000

indicated by the outer circle. The focal distribution of the laser beam is represented by a central hatched region that contains the low-divergence light and by a diamond-shaped skirt that is blocked by the pinhole. The diamondlike shape corresponds to the diffractive lobes of the rectangular NIF beam. Intensities in the skirt are expected to be substantial, ranging from a few gigawatts to a few terawatts per square centimeter. When the skirt strikes the pinhole material, the ablated plasma expands into the pinhole and interacts with the transmitted light. The term closure suggests that part of the light is blocked by an opaque plasma, requiring an electron density approaching the critical value of approximately 1021 cm⫺3 at 1053 nm. Instead, refraction in a less dense plasma causes redirection of light in the focal distribution. This in turn causes redistribution of light in the near field of the transmitted beam and, therefore, increased intensity modulation. This general description of pinhole closure is supported by both calculations and our experimental findings. Simple estimates indicate that significant intensity modulation can result from passage of the beam through a plasma of typical size 共several millimeters兲 with an electron density as low as 1018 cm⫺3. Our experimental findings show that an increase in near-field intensity modulation is the most sensitive indicator of the onset of pinhole closure. 3. Pinhole Geometry

We tested the three types of pinhole, shown schematically at the top of Fig. 2. The washer type is a circular hole in a flat plate, with the plate oriented at approximately normal incidence to the beam. The four-leaf type is divided into four azimuthal segments, with the segments separated along the beam line to reduce plasma convergence at the axis of the pinhole. Our version used straight segments, as shown in the left photograph in Fig. 2. The pinhole could be oriented so as to present a diamond-shaped aperture that matched the diamond shape of the focal distribution, or it could be rotated 45° to present a square. The third type is the cone pinhole.6 The right-hand photograph in Fig. 2 shows three cone pinholes constructed from SS, Au, and Ta. Because the operation of the cone type is not so obvious as the other two, and because it performed better than the other two, we proceed to describe it in some detail.

Fig. 4. Single beam line of the NIF.

Fig. 3. Cross section of a cone pinhole.

The interior surface of a cone pinhole is a cylindrically symmetric section of a cone with the large end toward the beam, as shown in Fig. 3. The beam enters from the left, and the right end is positioned at the focal plane of the spatial filter lens. The basic design philosophy of the cone pinholes is the following: 共1兲 Maximize the reflection or refraction of incoming light to minimize absorption and plasma generation on the interior surface of the cone. 共2兲 Enlarge the input diameter of the cone as much as possible to reduce plasma generation at the input end. Given an input beam with a specific f-number, F# 共defined as the ratio of the focal length of the spatial filter lens to the width of the square input beam兲, and a desired pinhole diameter, dout, these items constrain the remaining pinhole parameters as follows: 共1兲 To maximize the reflection of incoming light 共actually refraction from the generated plasma兲, we must minimize the cone half-angle ␣. However, it is also necessary that the rejected light from the pinhole not add to the near-field intensity distribution of the transmitted beam. Thus it must miss the clear aperture of the lens at the output of the spatial filter. This leads to a minimum value for ␣: ␣ ⬎ ␣min ⫽ tan⫺1关共2F#兲⫺1兴.

(1)

共2兲 Enlargement of the input diameter, for a given ␣, leads to longer cones. The maximum pinhole length L is obtained from the need to prevent light reflected from one side of the cone from hitting the far side of the cone, where it would add to the existing intensity at the output of the pinhole. The dotted lines in Fig. 3 show marginal rays from opposite sides of the beam, which reflect from the leading edge of the pinhole. These rays must pass through the output hole without hitting the interior surface, leading to the following maximum value for L: L ⬍ Lmax ⫽ dout兾共␣ ⫹ ␣min兲.

(2)

␣ ⫽ 1.3␣min ⫽ 25 mrad,

(4)

L ⫽ 0.7Lmax ⫽ 29 mm.

(5)

Larger pinholes used the same ␣ and had L increase linearly with dout. 4. Variation of Closure Time with F# of the Spatial Filter

Each beam line of the NIF contains six pinholes that must transmit high-energy pulses. An arm consists of a four-pass cavity and a booster section, as shown in Fig. 4. A seed pulse is injected into the transport spatial filter 共TSF兲 and makes one pass through the booster amplifier on its way into the cavity. It then makes four passes through the cavity amplifier, the plasma-electrode Pockels cell 共PEPC兲, and the cavity spatial filter 共CSF兲 and is then switched back into the booster section for a final pass through the booster amplifier and the TSF. The angles of the mirrors that form the cavity are separated slightly so that each pass uses a different pinhole. Although the pulse energy at the TSF is substantially larger than at the CSF, the pass-four CSF pinhole will be the most difficult to keep open. The final pinhole in the TSF must transmit the total pulse energy, which may be as large as 20 kJ per beam. To reach that output energy will require 14.8 kJ at the pass-four pinhole in the CSF. However, the F# of the TSF 共80兲 greatly exceeds that of the CSF 共31兲. Using scaling arguments, we proceed to show that the F# difference is more important than the energy difference. We can determine the scaling of closure with F# by considering two spatial filters with the same types of pinhole and with the same cutoff angles. We assume that beams of equal quality are passed with zero misalignment through each of the filters. Then the closure time ␶ scales directly as some characteristic distance x, which depends on the pinhole diameter and beam size, and inversely as the propagation speed c of the phase disturbance: ␶1 x1 c2 ⫽ . ␶2 x2 c1

The input diameter follows from ␣ and L: din ⫽ dout ⫹ 2␣L.

the limits of these constraints. The following parameters were used for the ⫾100-␮rad cone pinholes:

(3)

The design of all the cone pinholes for the Beamlet was somewhat conservative, in that it did not press

(6)

Because x is a characteristic distance traveled by the phase disturbance before it interacts significantly with the beam, it will be significantly shorter than the pinhole radius. Because, however, dimensions 20 March 2000 兾 Vol. 39, No. 9 兾 APPLIED OPTICS

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in the far field scale directly with f-number, x will scale in the same manner as the pinhole diameter d, under the assumptions of equal beam quality and cutoff angles. Thus we may take x1兾x2 ⫽ d1兾d2. Speed c can be related to the f-number by means of results from off-line tests11 described in Section 6 below. The relevant result here is that the propagation speed of a constant phase contour is approximately proportional to the intensity I that hits the pinhole edge. This holds for intensities below ⬃300 GW兾cm2 for SS and below ⬃70 GW兾cm2 for Ta. Inasmuch as the skirt intensity in the leading edge of the NIF ignition pulse is comparable to or less than either of these values, the speed ratio becomes

冉冊

2

c2 I2 P2 A1 P2 F1# ⫽ ⫽ ⫽ , c1 I1 P1 A2 P1 F2#

(7)

where P is the power through the spatial filter and A is the beam area in the pinhole plane. The pinhole diameter can also be related to F# and the acceptance angle of the spatial filter, ␦, through d1 ␦1 F1#D1 ⫽ , d2 ␦2 F2#D2

(8)

where D is the beam diameter at the spatial filter lens. As we assume that the acceptance angles are equal, we have, finally,

冉冊

3

␶1 D1 P2 F1# ⫽ . ␶2 D2 P1 F2#

(9)

Thus the closure time scales directly as the beam diameter, inversely as the power into the spatial filter, and directly as the cube of the f-number. For the NIF the f-numbers of the CSF and the TSF are 31 and 80, respectively, and the beams are of equal sizes. In addition, the power into the TSF exceeds that into the fourth pass of the CSF by at most the unsaturated gain of the booster amplifier, which is ⬃3. 共At the end of a NIF ignition pulse, the booster amplifier gain will be saturated to only a few percent.兲 Thus the pass-four pinhole of the CSF is expected to close six or more times faster than the TSF pinhole. Note that the assumption of equal beam quality for the CSF and the TSF is not strictly true. Between these two filters there are two SF lenses and six amplifier disks, all of which add phase errors. Their influence on the total beam quality, however, is relatively small. Another consideration is ⌬B 共the intensity-dependent phase retardation兲, which can be significantly higher at the TSF than at the CSF. However, the leading edge of an ignition pulse, which is the first concern for closure, is sufficiently less intense than the peak at either filter that the contribution to ⌬B is negligible. 5. Full-Scale Pinhole Closure Experiments on the Beamlet Laser

To simulate NIF operating conditions as closely as possible, we conducted full-scale pinhole closure ex1408


Fig. 5. Beamlet pulse shapes used to simulate NIF ignition pulses at the TSF 共dashed curve兲 and the CSF 共solid curve兲.

periments on the Beamlet laser.11 The Beamlet, a prototype for one of the 192 beams of the NIF, also contained a four-pass cavity and a booster section. It was built to demonstrate the practicality of the new design features and specifications for the NIF. Because the two lasers have similar beam apertures and the same number of amplifier slabs in both the cavity and the booster amplifiers, we assumed that the focal distribution for the NIF would be similar to that of the Beamlet. The major difference relevant to pinhole closure was that the two spatial filters of the Beamlet had f-numbers of 26, as opposed to the cavity 共F# ⫽ 31兲 and transport 共F# ⫽ 80兲 spatial filters of the NIF. In fact, this difference was advantageous in that the entire output energy of the Beamlet was available for the study of pinhole closure on the TSF, which was a reasonable match to the limiting, passfour pinhole in the CSF of the NIF. Each spatial filter on the Beamlet had lenses with a focal length of 900 cm. The NIF will use a variety of pulse powers and temporal shapes, but we assumed at the outset that the inertial confinement ignition pulse would be the most difficult for pinhole closure. The dashed curve in Fig. 5 shows the desired 1053-nm pulse shape at the output of an arm of the NIF. This pulse must have an energy of ⬃20 kJ to deliver 1.8 MJ on target at 351 nm from all beam lines. The corresponding power in the leading-edge foot will be ⬃360 GW, and the power in the intense 3-ns segment will be ⬃3600 GW, giving a contrast of 10:1. Gain saturation in the booster amplifier causes the pulse shapes to be different in the cavity and transport filters. The solid curve in Fig. 5 shows the shape at the pass-four CSF pinhole needed to get the 10:1, 20-kJ pulse at the output. Its peak is approximately the same, 3500 GW, but the leading edge of its foot is only 170 GW, giving a contrast of 21:1. A major concern is that the leading edge of the foot will generate a plasma that has a full 20 ns to expand into the pinhole and interfere with the passage of the pulse. Therefore, a necessary condition is that the pinhole must stay open during the passage of a 20-ns square pulse that represents only the foot of the shaped pulse. This foot-only pulse is shown as the hatched region in Fig. 5. For the 21:1 shape, it contains 22% of the total pulse energy. The majority of our experiments on the Beamlet used 20-ns square

Fig. 6. Near-field images at beam powers below and above threshold for pinhole closure.

pulses; a smaller number used either 10:1 or 21:1 NIF ignition pulses. A.

Diagnostics

In addition to routine diagnostics for output energy, pulse shape, and near-field and far-field distributions, several additional diagnostics were employed. Two time-resolved near-field diagnostics were added: a streak camera with its slit exposed to a horizontal strip of the near field and a framing camera, called the gated optical imager, or GOI, whose 150-ps gate was timed to the last nanosecond of the pulse. A schlieren far-field diagnostic extended the range of the usual far-field camera from ⫾66 to ⫾250 ␮rad. Finally, an interferometer was added to the TSF to monitor the change in phase in the pinhole during the passage of the main pulse. A second streak camera on the interferometer gave time resolution to the phase measurement, and a second GOI on the interferometer was used to watch for closure on the pinhole axis or from directions other than that monitored by the streak camera. 1. Near-Field Diagnostics The most sensitive diagnostics for determining pinhole closure were the time-resolved near-field diagnostics. Pinhole closure near threshold affects only the modulation of the near-field distribution transmitted through the spatial filter. However, it affects the tail of the pulse first, because the plasma requires time to propagate into the beam. As a result, timeaveraged diagnostics are relatively insensitive to closure threshold, because the unperturbed distribution from the bulk of the pulse dominates the perturbed portion at the end. A gated image of the near-field intensity gave the most sensitive indication of whether a pinhole closed on a specific shot. A GOI was delayed to look within the last nanosecond of the main pulse. Examples of GOI images for above and below closure are shown in Fig. 6. To determine when a pinhole closed during a pulse, we monitored the beam modulation as a function of time with a streak camera looking at a horizontal section of the transmitted near field. Figure 7 shows an example of the processed near-field streak data for a case with obvious pinhole closure. We characterize beam modulation with a calculated parameter

Fig. 7. Typical near-field streak-camera results showing definition of closure.

called contrast, defined as the standard deviation of the fluctuations in intensity, normalized to the average. The plot shows the contrast, which was calculated along the spatial dimension of the streak image, as a function of time for a 20-ns square pulse. 共The heavier curve is a smoothed version of the contrast.兲 The contrast starts at just under 0.1 and decreases slightly for the first ⬃12 ns. This initial decrease is typical and is attributed to saturation in the amplifiers. At ⬃12 ns a sharp increase in contrast begins, which is typical of pinhole closure. Our definition of closure time was the time at which the contrast increases by 20% above the initial background. In this case, closure occurred at ⬃13 ns. The estimated accuracy of this method of determining closure time was ⫾1 ns. 2. Schlieren Far-Field Diagnostic The schlieren far-field diagnostic provided a means of measuring misalignment of the beam in the pinhole and the beam intensity on the edge of the pinhole. This intensity was used in the analysis of our results. It can also be used to relate our results to pinhole closure in other laser systems. The schlieren diagnostic used a ⫾66-␮rad hole in a high-reflectivity mirror as the schlieren block, with the schlieren image obtained from the reflected beam. The diagnostic measured the incident and reflected energies to permit calibration of the far-field image on each shot. The left-hand image in Fig. 8 shows schlieren data with a ⫾150-␮rad pinhole in the TSF. The central shadow results from the ⫾66-␮rad schlieren block. The bright horizontal and vertical bands are the diffractive lobes of the square beam. The outer circular shadow is the shadow of the 150␮rad pinhole.

Fig. 8. Schlieren image of a ⫾150-␮rad pinhole with and without guidelines superimposed to determine alignment. 20 March 2000 兾 Vol. 39, No. 9 兾 APPLIED OPTICS

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Fig. 9. Measured far-field intensity through the horizontal diffraction lobes of the Beamlet laser 共solid curve兲 and a smoothed fit 共dashed curve兲. The intensity is normalized to a beam power of 1 GW.

The right-hand image in Fig. 8 shows a determination of beam misalignment for this shot. A circular aperture was centered on the shadow of the pinhole edge to define the center of the pinhole. The dashed vertical and horizontal lines were centered on the vertical and horizontal diffraction bands to define the center of the beam. 共These lines were rotated slightly to match a small rotation of the CCD camera that recorded the image.兲 The offset between the two centers indicates the misalignment. Note that the relative position of the central schlieren block is irrelevant to misalignment. For the data in Fig. 8, the center of the beam is 12 ␮rad to the right and 7 ␮rad down relative to the center of the pinhole, yielding a radial misalignment of 14 ␮rad. We found the repeatability of this determination to be ⫾3 ␮rad, based on several measurements of each of the data points. Use of the schlieren images to determine the intensity on the rim of the pinhole was more complicated. One needs a record of the far field of the beam that was incident upon the pinhole in the TSF. A calibrated schlieren image for a shot gives the far-field distribution of light that was transmitted by the pinhole on that shot, but the true far field is likely to be somewhat different. Rays that passed near the rim of the pinhole might have been deflected out of the beam by refraction in the plasma at the pinhole. We believe that loss of edge rays explains why the outer edges of schlieren images like that in Fig. 8 are rounded and not a crisp image of the edge of the pinhole. The imperfection of schlieren images was corrected by use of a standard profile that was obtained from schlieren images recorded without a pinhole in the TSF. A lineout through the horizontally oriented diffractive lobes of such a focal distribution is shown in Fig. 9. It is the average of two low-energy shots that were taken with ⫾200-␮rad pinholes in the CSF and with no pinhole in the TSF. It was normalized to provide intensities in gigawatts per square centimeter for a total beam power through the pinhole of 1410


1 GW. The data from approximately 70 –200 ␮rad came from the schlieren images. The central part of the distribution came from a normal 共not schlieren兲 far-field diagnostic. Between these regions, we interpolated with a smooth line. The rapid falloff starting at ⬃170 ␮rad is due to the clipping of the distribution by the ⫾200-␮rad pinholes in the CSF. There was some asymmetry in the focal distribution, with the horizontal lobe having a higher intensity at angles greater than ⬃100 ␮rad. The dashed curve in Fig. 9 is a smoothly varying fit to the horizontal data. It was adopted as the standard shape of the far-field distribution. In pinhole closure experiments, the intensity at a given time on the pinhole edge was read from a version of this standard curve that was scaled in amplitude to account for the beam power at that time in the pulse. We tested the general validity of this procedure by comparing normalized lineouts through several schlieren images with one another and with this standard curve. We found that the wings of the far field were quite reproducible to beam powers as high as 5 TW. The normalized, on-axis intensity decreased with the number of shots during a day, but the distribution at angles greater than 100 ␮rad remained constant. At angles near the radius of the pinhole under test, intensities that were read from the standard shape were larger than those in the rounded edges of calibrated schlieren images. Although the use of a standard shape eliminated most of the questions regarding far-field intensity, this standard curve ignores the fact that schlieren images were not smooth distributions. Figure 8 clearly shows that the far field consisted of a distinct speckle field, whose peak-to-average ratio was typically 3:1 and occasionally as much as 6:1. The size of the smallest speckle is in reasonable agreement with the diffraction-limited angle for the full-aperture beam, 2␭兾D ⬇ 6 ␮rad, where D ⫽ 34 cm is the edge dimension of the square Beamlet beam. The rapid spatial variation of the speckle field causes difficulty in assigning an intensity to each point in the far field. However, we have two pieces of evidence that closure is determined by the spatial average of the far-field intensity rather than by the peaks and valleys of the speckle. First, the speckle field in the vicinity of the pinhole edge, i.e., at ⫾100 ␮rad or more, varied significantly from shot to shot as a result of beam misalignments, which were as large as 20 ␮rad. If closure depended on the peak intensity of the speckle, the closure times for a given pinhole should have changed by at least the 3:1 peak-to-average ratio of the speckle field, because the closure speed increases approximately linearly with intensity. Instead, our results were quite stable when fluctuations in beam power were taken into account, as is shown in Subsection 5.B. Second, if we assume that at least one of the speckles was incident upon the edge, regardless of the alignment, and that speckle determined closure, then the introduction of smoothing by spectral dispersion should have produced a substantial increase in closure times. Instead, the introduction of ⫾7.5 ␮rad of SSD,

tered along its path. The phase change is given by ⫺⌬␾, with ⌬␾ ⫽ ⫺

Fig. 10. Schematic diagram of the interferometer installed on the Beamlet laser and typical streak-camera output data.

which averaged each speckle over an area approximately five times larger in a time equal to the inverse of the applied bandwidth, or ⬃12 ps, had no effect on pinhole closure.9 We conclude that the peaks of the speckle field do not significantly affect closure and that closure is determined by the smooth spatial average of the far-field intensity distribution. This result also makes sense from a physical point of view. Plasma expanding from the peak intensity of a speckle on the interior surface of a cone would spread both laterally and axially, rather than only in the axial direction as for a locally uniform intensity. Therefore, the electron density from a speckle would decrease more rapidly with distance than the density from a uniform intensity. As will be seen in Subsection 5.C below, the plasma generated at the pinhole edge must propagate quite far into the pinhole, of the order of 20 –30 ␮rad for the smallest pinholes tested 共⫾100 ␮rad兲. This is several times the size of the approximately 6-␮rad speckle. Consequently, the electron density from a speckle would tend to homogenize with the uniform background plasma before it propagated sufficiently into the beam to cause closure. In summary, we used the smoothed standard curve in Fig. 9 to assign the intensity loading of the pinholes. The curve shows that the far-field intensity on a ⫾100␮rad pinhole on the Beamlet is 0.38 GW兾cm2 for 1.0 GW through the spatial filter, with perfect alignment and no spreading attributable to SSD. Therefore, under the same conditions, the 170-GW leading edge of the 21:1-contrast ignition pulse produced an intensity of 65 GW兾cm2 on the Beamlet pinhole. 3. Interferometry A Mach–Zehnder interferometer provided a measurement of the phase change in the pinhole during the passage of pulses. Figure 10 is a schematic of the interferometer installed on the Beamlet TSF. Its design was based on the interferometer used in offline measurements discussed in Section 6 below. The probe beam used in the interferometer was counterpropagating, with a wavelength of 532 nm, to help to protect the detectors from scattered main-beam light at 1053 nm. Each ray of the probe beam accumulated a phase governed by the plasma encoun-

2␲ ␭

兰冑

共 1 ⫺ n e兾n c ⫺ 1兲dl,

where ne is the local electron density and nc is the critical electron density 共⬃4 ⫻ 1021 cm⫺3兲 at a wavelength of 532 nm. We work with the positive quantity ⌬␾ for convenience, although the phase change itself is negative. The probe propagated at a small angle with respect to the main beam, 2.25 mrad in the vertical direction. The angle of the probe introduced some distortion of the interferogram in the vertical direction, but the primary data recorder was a streak camera with its slit oriented horizontally at the center of the pinhole, minimizing the effect of this distortion. The streak camera showed phase changes as a function of time along a horizontal strip of the probe beam. The streaked interferogram in Fig. 10 shows typical pinhole closure data. The pulse length was 3.5 ns. A wedge introduced with interferometer alignment increased the background phase linearly 共from left to right in the figure兲, resulting in straight fringes with uniform spacings until the time of pulse arrival at the pinhole. When ⌬␾ became significant with respect to 2␲, ⌬␾ noticeably reduced the phase at both edges of the interferogram. This increased the phase gradient on the left side 共decreased fringe spacing兲 and decreased it on the right 共increased fringe spacing兲. Note that the streaked interferogram shows the phase change in the pinhole after the passage of the 3.5-ns pulse as well as during the pulse. The duration of the streak is ⬃40 ns. We had hoped to be able to use the interferometer to determine closure times for shots that did not close during the 20-ns laser pulse. Unfortunately, the error bars that resulted from extracting phase magnitude and time from the interferograms were so large that most of the interferometer data were not useful. We did, however, confirm the basic exponential character of the plasma expansion away from the pinhole edge, which was also observed with modeling12 and off-line measurements 共Section 6兲, and we did show that a shot simulating a NIF ignition shot was quite far from closure 共Subsection 5.C兲. A second GOI, which was of the same type as the one on the near-field camera, was also installed on the interferometer to give a two-dimensional view of the interferometer fringes at the end of the pulse. Results from this diagnostic and the far-field and schlieren far-field diagnostics are discussed in the next subsections. B.

Results with 20-ns Square Pulses

Most of our pinhole closure tests were conducted with 20-ns square pulses to simulate the leading foot of the shaped NIF ignition pulse. To simplify comparison with shots of different energies and closure times, we found it convenient to define the closure energy Ec as the energy required for closing 20 March 2000 兾 Vol. 39, No. 9 兾 APPLIED OPTICS

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Fig. 12. Closure energies for ⫾100-␮rad pinholes and square temporal pulses. Fig. 11. Inverse of closure time versus average pulse power for the ⫾100-␮rad SS cone pinhole, showing a constant value of Ec ⫽ 2.76 kJ for all nine shots.

a pinhole for any square pulse. Taking this as proportional to the product of the pulse power and the closure time for a square temporal pulse, we see from Section 4 that, for any two pinholes of the same geometry and material,

冉冊

3

Ec1 D1 F1# ⫽ , Ec2 D2 F2#

(11)

which is independent of beam power. Further, for a given f-number and beam diameter, Ec is a constant. This is valid as long as the intensity on the pinhole edge is less than ⬃300 GW兾cm2 for SS and ⬃70 GW兾 cm2 for Ta. Beyond these points, the initial linear increase in closure velocity with intensity begins to fall off. As an example, Fig. 11 shows 1兾␶ plotted against average pulse power for nine shots on a ⫾100-␮rad SS cone pinhole with a range of pulse energies. Each shot used square pulses, so each should have given the same value of Ec by the above relationship. The least-squares fit through the data points 共and the origin兲 shows that each of the shots is consistent with the single value of Ec ⫽ 2.76 kJ. The alignment of the beam on the pinhole is an independent parameter in these data points, and it is somewhat surprising that the points fit this relationship well with no accounting for misalignment. We believe that the reason is that most of the data points have comparable misalignments. Misalignments for all the shots shown in Fig. 11 fall within the range of 1– 6 ␮rad. With the estimated measurement accuracy of ⫾3 ␮rad, there is little distinction among these values. We constructed the pinhole types shown in Fig. 2 with several materials and tested them with 20-ns square pulses. Figure 12 shows Ec plotted as a function of atomic mass for these measurements. All pinholes were nominally ⫾100 ␮rad, although the SS and Ta cones turned out to be 103 and 107 ␮rad after manufacturing. The four-leaf pinholes were rotated to the diamond orientation to match the diamondlike shape of the far field. The graph shows that cone 1412


pinholes perform better than other types in all cases tested. It also shows that improved performance results from higher atomic mass materials. This dependence on atomic mass was also observed with modeling13 and in the off-line measurements 共discussed in Section 6兲. The Au cone is an exception, in that it did not perform as well as the Ta cone. We believe that this result was due to an inadequate surface finish on the Au cone, as explained below. Note that the improvement in performance of the cone over the diamond-oriented four-leaf type is even larger than indicated by the data. The quoted size of the four-leaf pinhole was measured along the side of the square, whereas the quoted size of the cone pinhole was its diameter. Consequently, the ratio of solid angles subtended by four-leaf and cone pinholes of the same quoted size differed by 4兾␲ ⫽ 1.27, and the highest-intensity parts of the far field, which fall into the corners of the four-leaf pinhole, were substantially farther from the axis for the four-leaf pinhole than for the cone pinhole. One practical difficulty with the cone pinholes is that the interior surface must be quite smooth for the pinhole to work properly. The initial SS cones were good 共surface roughness average of 4 – 8 ␮in.兲, but we had some difficulty fabricating the Ta and Au cones. The first attempt with Ta resulted in a surface roughness average of ⬃30 ␮in. and gave the disappointing closure energy of 2.4 kJ. A subsequent try was much better 共⬃8 ␮in.兲 and gave the closure energy of 4.8 kJ shown in Fig. 12. We attribute the performance difference to the improvement in surface finish. The initial Au cone had a surface roughness of 8 –15 ␮in., with a few regions as large as 32 ␮in. It gave a disappointing closure energy of 3.7 kJ, shown in Fig. 12. Subsequent Au cones had much better surface finishes, in the 2– 4-␮in. range, but they were not completed in time to be tested. We found it necessary to take several cleanup shots on the best Ta cone pinhole before it produced the final closure energy of 4.8 kJ. The ⫾100-␮rad SS cone did not seem to require clean-up shots, although the record is obscured by the fact that we tested it first and started out with low pulse energies. Unfortunately, we took only five shots on the Ta cone pinhole, and we tested only a single ⫾100 SS cone, so

we do not know how general is this need for clean-up shots. If this turns out to be a general requirement, it would be a substantial operational disadvantage for Ta cone pinholes compared with SS cones. Recently a procedure for fabricating cone pinholes from clear or absorbing glass was described.8 The idea is to take advantage of a high plasma-generation threshold for glass at high angles of incidence to make cone pinholes with high closure thresholds. This could also solve the surface roughness problem by producing smooth, fire-polished surfaces. Kurnit et al.8 reported a high plasma-generation threshold for NG4 glass13 at the near-grazing angles of incidence at which cone pinholes function. Reported thresholds are higher than the intensity in the foot of the ignition pulse, which would imply a delay in plasma generation until the peak of the pulse and correspondingly longer closure times. However, our measurements, which are discussed in Section 6, indicate that, once they were above the plasmageneration threshold, glass pinholes would not perform so well as metal pinholes, which would limit their usefulness to below the plasma-generation regime. For this reason as well as time constraints, we did not attempt to fabricate or test cone pinholes made from glass. C.

Results with Shaped Pulses

To relate these 20-ns square-pulse results to those for the shaped pulses required for the NIF, we tested a few cone pinholes with both square and shaped pulses. We found that the shaped pulse with 21:1 contrast, indicated by the solid curve in Fig. 5, closed a ⫾100-␮rad SS cone pinhole for 8.3 kJ, three times the pulse energy of the foot-only square pulse 共shown by the hatched region in Fig. 5兲. To demonstrate a pinhole that would pass the required pulse energy and shape, we tested a ⫾150␮rad SS cone with a 12.9-kJ, 21:1 contrast ignition pulse and a 15.5-kJ, 10:1 contrast ignition pulse. The 15.5-kJ test included ⫾7.5 ␮rad of SSD. We found no indication of closure from any of the diagnostics for either pulse. Inasmuch as the intensity in the foot of the 10:1 contrast pulse is larger by a factor of 2 than that for the 21:1 pulse, it is more difficult to keep the 10:1 shape open with the same total pulse energy. This demonstrates that the ⫾150-␮rad SS cone would pass a 15.5-kJ, 21:1 contrast pulse as well. We used the Mach-Zehnder interferometer to estimate how close the ⫾150-␮rad SS cone pinhole was to closure at the end of the 10:1 contrast pulse by comparing its phase shift with that for a smaller pinhole at closure. Figure 13 shows the streaked interferometer output and the phase shift at the end of the pulse. The solid curve in the graph shows ⌬␾ as a function of position in the pinhole at the end of the pulse, and the dashed curve shows ⌬␾ measured at the time of closure for a ⫾100-␮rad SS cone pinhole. Comparison of the two curves shows that the ⫾150␮rad pinhole was quite far from closing, because the plasma would have had to propagate 50 –75 ␮rad

Fig. 13. Streaked interferogram showing the phase shift in a ⫾150-␮rad SS cone pinhole during a 15.5-kJ, 10:1 ignition pulse with ⫾7.5 ␮rad of SSD; 共at the right兲 phase shift at the end of that pulse compared with the phase shift at closure in a ⫾100-␮rad SS cone pinhole.

further to match the phase distribution for the smaller pinhole at closure. The second GOI showed a two-dimensional view of the interferometer fringes at the end of the pulse. We used it to watch for closure on the pinhole axis and from directions other than that monitored by the streak camera 共horizontal兲. Figure 14 shows the interferogram for a ⫾100-␮rad Ta cone, ⬃4 ns after closure. The elliptical boundary of the fringes corresponds to the clear aperture of the pinhole as seen from the angle of the probe beam. It shows compression of fringes along the horizontal on one side and expansion on the other, as observed with the streaked interferograms. In the vertical direction, the fringe motion is more complicated because of the additional effects of the angle of the probe, but it shows no unusual or anomalous behavior. In particular, it shows no evidence of localized jetting7 into the center of the pinhole. Nor does it show on-axis closure owing to radial convergence of plasma from the conical walls, as seen for washer pinholes in the off-line tests described below. D.

Results with Short Pulses

We took a few pinhole closure shots at high peak powers, 3–5 TW, and found two significant differences. First, we saw clear evidence for saturation of the linear relationship between phase-contour velocity and intensity with SS cone pinholes. Three shots with a 3.8-ns square pulse 共2.1, 2.7, and 3.1 TW兲 through a ⫾100-␮rad SS cone pinhole gave no indi-

Fig. 14. Two-dimensional interferometer fringes several nanoseconds after closure in a ⫾100-␮rad Ta cone pinhole. 20 March 2000 兾 Vol. 39, No. 9 兾 APPLIED OPTICS

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Fig. 15. Measured backreflected energies for three pinhole types.

observe such backreflection with a weak secondary pulse that was injected by mistake at 24 ns after a 2-TW, 200-ps main pulse. An image of the backreflected energy showed that it originated in the center of the TSF pinhole, which was a ⫾100-␮rad C washer. The estimated backreflection fraction was inconsequential, ⬃10⫺5, but the result confirms that backreflections from the pinhole plasma are possible. We conclude that a major misalignment to a pinhole can create a risk of significant backreflected energy. 6. Off-Line Measurements

cation of closure, although the distribution of phase change in the pinhole at the end of the pulse indicated that the highest-power shot was close to closure. Assuming that the pinhole closed at the end of the pulse, we infer a closure energy of 11.6 kJ at 3.1 TW, which is four times larger than the 2.76 kJ measured at powers less than 190 GW 共see Fig. 11兲. For the far-field distribution in Fig. 9 the corresponding power densities at the pinholes for no misalignment were 1.2 TW兾cm2 and 72 GW兾cm2, respectively. If phase-contour velocity were linear with intensity over this intensity range, the closure energies would have been the same. Second, we measured significant backreflections from pinholes during these high-peak-power experiments. Figure 15 shows backreflected energy measured at the output of the Beamlet front end. The data are plotted as a function of power through the TSF. The points were obtained with 200-ps pulses and three types of ⫾100 ␮rad pinhole. For carbonwasher pinholes, the backreflection increased exponentially with power, and both the C washer and the Ta four-leaf pinholes produced large backreflections for pulses with peak powers above ⬃2 TW. The detector saturated for backreflected energies above ⬃100 mJ, as indicated by the shaded area. Hence data points in that area represent larger backreflections than indicated. In fact, the backreflections from the two C-washer data points at 4.0 and 4.4 TW must have produced backreflected energies approximately 100 times those indicated, because they damaged injection optics. Imaging of the backreflected energy in the plane of the pinhole showed conclusively that the backreflected energy came from the faces of the pinholes rather than from the plasma expanding into the aperture of the pinhole. This was true for the cone pinholes as well as for the washer and four-leaf types. These measurements also showed that SS cone pinholes give inconsequential backreflections over the same range of powers into the TSF, as shown in Fig. 15. This constitutes another important advantage of the cone pinholes compared with the other pinhole geometries. None of our Beamlet laser shots produced a measurable backreflection from the pinhole plasma during the passage of the laser pulse. This result holds despite the fact that in some cases closure occurred as early as 7 ns into a 20-ns pulse. We did, however, 1414


Relationships among plasma expansion velocity, irradiation intensity, and pinhole material were studied with experiments on the Optical Sciences laser at the Lawrence Livermore National Laboratory.11 This laser provided rectangular-waveform pulses of 1-, 5-, or 20-ns duration and energies ranging to ⬃40 J in a 30-mm-diameter beam that was formed by four stages of image relaying. During most of the experiments, half of the beam was intercepted by a vertically oriented knife-edge sample in vacuum. The range of irradiation intensity, up to 500 GW兾cm2, was selected to include the expected loading of the pass-four pinhole in the NIF cavity filter. To reach 500 GW兾cm2 on the Optical Sciences laser we used beams with diameters of either 1.4 or 2.0 mm that were generated by imaging the output plane of the laser onto the sample with suitable demagnification. The expansion of the plasma along the horizontal axis was monitored by time-resolved two-beam interferometry. This diagnostic was the prototype for the one used in the Beamlet experiments discussed above in Section 5. The interferometer was nearly the same as that on the Beamlet 共Fig. 10兲. Both the probe and the reference arms of the interferometer contained a pair of lenses that relayed the plane of the knife-edge sample to the mixing mirror of the interferometer. A single lens was used to image the mixing mirror onto the slit of a streak camera. During experiments with the drive beam at normal incidence on the knife-edge, the probe beam was a 40-mJ, 40-ns, 1053-nm pulse that propagated forward but off axis by 5°–7° relative to the drive beam. During tests of plasma velocity at a large angle of incidence, an on-axis counterpropagating 527-nm, 40-ns probe pulse was used. Interferograms that were recorded for C and Ta knife-edges at normal incidence are shown in Fig. 16. The main pulse had an intensity of 130 GW兾cm2 and a duration of 5 ns. The image of the knife-edge is at the left in each interferogram. The horizontal axis measures distance from the knife-edge, and time proceeds from top to bottom. The bright, 5-ns-duration pulse at the left of each image is light that scattered from the drive pulse into the acceptance angle of the streak camera in the probe channel. The onset of this scattering, the onset of a phase disturbance, or both, mark the arrival of the 5-ns drive pulse. The phase disturbance was caused by passage of the probe beam through a plasma as described above. With passage of time, the plasma expanded away

Fig. 16. Streaked interferograms for C and Ta at 130 GW兾cm2. Time runs from top to bottom.

from the knife-edge, causing an increase in the distance at which phase shift of a particular magnitude occurred. Measurement of the speed of propagation of ⌬␾ was a primary goal of the experiments. We elected to use the speed of propagation of a 2␲ 共onewave兲 shift, c1, measured during the irradiation interval of the drive pulse to characterize the expansion on a given shot. The principal results of our experiments can be summarized in a few figures. We found that c1 varied with the knife-edge material and with the intensity of the drive beam. The material dependence is apparent in Fig. 16. For C, the phase disturbance moved rapidly across the 1.5-mm-wide zone of observation. Near the C knife-edge, the gradient in phase was sufficiently large during the 5-ns drive pulse to scatter the probe beam completely out of the 20-mrad collection angle of the interferometer 共or the beam was simply reflected, if ne ⬎ nc兲, as evidenced by the dark, fringe-free shadow adjacent to the knife-edge. However, the shadow at the edge disappeared ⬃5-ns after the end of the drive pulse, and ⌬␾ was substantially down from its maximum by the end of the streak trace. For the Ta plasma, ⌬␾ obviously moved more slowly, as the phase disturbance was negligible at distances greater than 0.5 mm. However, the plasma did not recover so quickly, as evidenced by the persistence beyond the end of the drive pulse of the dark, fringe-free shadow near the Ta edge. Figure 17 shows positions of the first 2␲ of phase shift as a function of time for several materials irradiated at ⬃130 GW兾cm2. The data vary linearly with time during the 5-ns drive pulse for all the materials. Values of c1 were taken to be the slopes that were measured in this time interval. In light-

Fig. 17. Trajectories of 2␲-phase shift for five materials.

Fig. 18. Variation of c1 with atomic mass at 130 GW兾cm2.

element plasmas, c1 remained constant for more than 5 ns after the end of the drive pulse, after which it decreased rapidly to zero and changed sign, indicating a reduction in plasma density or extent along the beam path. The value of c1 for the heavy-element plasmas decreased slightly immediately after the drive pulse but remained positive to the end of the observation period. Figure 18 shows measured values of c1 for the data in Fig. 17. For the metals, these values of c1 were found to vary with atomic mass M as c1 ⫽ 共3.7 ⫻ 107 cm兾s兲兾M0.49, where the exponent was obtained by least-squares curve fitting. The variation of c1 with irradiation intensity is illustrated for one material, SS, in Fig. 19. The measured velocities in steel were found to vary linearly with intensity, within experimental uncertainty. 共In Section 7 below, we employ a refined fit, based on these points and simulations at higher intensities, in which the slope decreases slightly with intensity.兲 The vertical error bars are due largely to an uncertainty of ⬃0.5 rad in fringe position. We also analyzed the streaked interferograms to determine the spatial variation of phase shift at a particular instant. All the interferograms recorded during these experiments indicated that the phase shift decreased exponentially with distance perpendicular to the knife-edge; two examples are shown in Figs. 20 and 21. Figure 20 shows measurements at

Fig. 19. Variation of c1 with intensity for SS. 20 March 2000 兾 Vol. 39, No. 9 兾 APPLIED OPTICS

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Fig. 22. Left, streak image showing an on-axis blockage for a washer pinhole at 18 GW兾cm2; right, similar image showing no on-axis blockage for a four-leaf pinhole at 30 GW兾cm2. Fig. 20. Phase shift versus distance for SS at near-normal incidence and a peak intensity of 500 GW兾cm2.

near-normal incidence on a SS edge, made with the 1053-nm, off-angle probe beam. It shows the phase shift at 1-ns intervals after the beginning of a 5-ns, 500-GW兾cm2 drive pulse. Figure 21 shows measurements made with a 5-ns, p-polarized, 250-GW兾 cm2 pulse 共measured in the plane normal to the beam兲 at an incidence angle of 85°. It shows phase shifts for NG3 absorbing glass,13 SS, and Ta at 2 ns beyond the end of the 5-ns pulse, measured with the 527-nm, on-axis probe. The variation of phase shift with distance for both data sets was exponential to within experimental uncertainty and is consistent with simulations for metals.14 Figure 21 also shows that 250 GW兾cm2 is above the plasma-generation threshold for NG3 glass at an incidence angle of 85°, which is consistent with the results of Kurnit et al.8 Because the measured phase shifts for NG3 glass are faster than those for SS, which in turn are faster than those for Ta, these results also imply that NG3 would not be so good a material for cone pinholes as either SS or Ta at 250 GW兾cm2 and an incidence angle of 85°. The final topic of study during Optical Sciences laser experiments was measurement of the phase shifts during closure of washer and four-leaf pinholes. Experience with the Nova laser at the Lawrence Livermore National Laboratory indicates that

Fig. 21. Phase shift versus distance at 7 ns for 85° angle of incidence and intensity of 250 GW兾cm2. 1416


multiple-leaf pinholes stay open during passage of pulses that close washer pinholes. A suspected difference was that, for a washer pinhole, plasma from every point on the perimeter of a planar pinhole could converge in the center of the hole, whereas this could not occur in a four-leaf pinhole with sufficient separation of the blades. We used the interferometer to observe the phase shift in a circular washer pinhole with a diameter of 0.75 mm and in a rectangular four-leaf pinhole with a transverse dimension of ⬃0.9 mm. Two of the interferograms are shown in Fig. 22. For each of these pinholes there was a significant phase shift near the edge of the pinhole. However, the most striking feature is that the on-axis region of the planar pinhole became effectively opaque while there was still transmittance in the region between the center and the edge. Such on-axis accumulation was not observed with the four-leaf design. 7. Effects of Misalignment on Closure

Beam misalignment is an important issue in pinhole performance. We used the schlieren diagnostic to measure misalignment on each shot, as described in Section 5. Figures 23–26 show closure data for ⫾100-␮rad pinholes with a 20-ns square pulse, plotted as a function of misalignment. Because of the scarcity of data, the points are grouped for a range of beam powers. Figure 23 is for the square- and diamond-oriented four-leaf Ta pinholes, and Fig. 24 is for diamond SS pinholes. Figures 25 and 26 pertain to the SS and Ta cone pinholes, respectively. We used the horizontal component of the measured misalignment only, because we found experimental evidence that the horizontal slit of the streak camera might have missed the onset of closure along the vertical direction. For the diamond-oriented pinholes, the data indicate a decrease in closure time with misalignment. In the remaining cases, unfortunately, the data tend to congregate near a single point 共four points for the square Ta and seven points for the SS cone at intermediate powers兲 or are scattered in power. To interpret the limited closure data and to permit extrapolation to systems other than the Beamlet, we have developed a phenomenological model, the results of which are included in Figs. 23–26. A schematic view of the focal plane, as envisaged in

Fig. 23. Data and model predictions for a ⫾100-␮rad, four-leaf, square-oriented Ta pinhole and diamond-oriented pinhole.

the model, is shown in Fig. 27. We consider a round beam within a round pinhole, with the center of the beam displaced a distance rmis ⫽ f␣ from the center of the pinhole, where ␣ is the misalignment angle and f is the focal length of the spatial filter lens. The radius rb of the beam is arbitrarily chosen to enclose 99% of the azimuthally averaged energy; this is ⬃45 ␮rad. To apply the model to the square- and diamond-oriented four-leaf pinholes,

Fig. 24. Data and model predictions for a ⫾100-␮rad, four-leaf, diamond-oriented SS pinhole.

we chose an effective pinhole radius. For the square orientation, this is half of the side of the square, whereas for the diamond orientation it is the average distance of a blade from the pinhole center. We suppose that the pinhole closes when the plasma ablated from the nearest edge, traveling at a fixed speed, penetrates to a fraction ␤ of the beam radius. The choice of this speed is somewhat ambiguous. It might seem reasonable to choose the 1-wave speed, but the model works more satisfactorily with a faster speed, corresponding to a smaller wave index. In practice, we chose the 0.1wave speed. The wave speeds are inferred from the off-line measurements of phase change versus position, with time as parameter, like that of Fig. 20. Both experiment and simulations14 indicate that the speed of the nth wave varies logarithmically with n, according to cn ⫽ c1 ⫺ c0 ln n, with c0 a positive speed smaller than the 1-wave speed c1. Figure 28 shows power-series fits to values of c1 and c0 obtained from the off-line data for SS and Ta as well as from simulations at higher intensity.14 Details of the fits are given in Table 1.

Fig. 25. Data and model predictions for a ⫾100-␮rad, SS cone pinhole. 20 March 2000 兾 Vol. 39, No. 9 兾 APPLIED OPTICS

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Table 1. Coefficients for the Fitted Speeds c0 and c1 for SS and Taa

Metal SS Ta

Fitted Speed c0 c1 c0 c1

a1 202 453 121 353

a2 ⫺0.177 ⫺0.288 ⫺0.453 ⫺0.142

a3

a4 ⫺5

6.95 ⫻ 10 8.59 ⫻ 10⫺5 6.87 ⫻ 10⫺4 2.01 ⫻ 10⫺3

⫺9.52 ⫻ 10⫺9 ⫺9.04 ⫻ 10⫺9 – –

a See Section 7. The speeds are given by c ⫽ 100 共a1I ⫹ a2I2 ⫹ a3I3 ⫹ a4I4兲 cm兾s, with I in gigawatts per square centimeter.

Fig. 26. Data and model predictions for a ⫾100-␮rad, Ta cone pinhole.

From these assumptions we can write the closure time as ␶⫽

rpin ⫺ rmis ⫺ ␤rb , ␥cn关I共rpin ⫺ rmis兲兴

with rpin the pinhole radius. In the denominator, note that cn depends on the intensity, which in turn depends on the pinhole radius and misalignment. The far-field intensity distribution was taken directly from the Beamlet experiments 共Fig. 9兲. We have introduced a second parameter, ␥, that is used to scale the off-line measurements of c0 and c1, which were taken at normal incidence, to the angle of incidence of our cone pinholes. Thus the adjustable pa-

Fig. 27. Schematic view of the pinhole focal plane with parameters defined for the model.

Fig. 28. Measured c0 and c1 for SS and Ta, and fits to the data used in the calculations. 1418


rameters are ␤ and ␥. By construction, ␶ decreases with misalignment, because the distance decreases and the speed increases. The closure time drops to zero for a misalignment of rpin ⫺ ␤rb, as is certainly a simplification. For the four-leaf pinhole data, we found the optimum value of ␤ to be 1.9. In this case we had ␥ ⫽ 1, as these pinhole faces were approximately normal to the beam. For the cone data, we used the same value of ␤ and found the optimum values of ␥ to be 0.16 and 0.32 for SS and Ta, respectively. As shown in Figs. 23–26, the model matches the data and gives a closure time that decreases most rapidly for small misalignments. For the Ta diamond pinhole at 165 GW兾cm2, for example, the closure time decreases from 17 to 10 ns as the misalignment increases from 0 to 10 ␮rad; as the misalignment increases by another 10 ␮rad, ␶ decreases by only 5 ns. Similarly, for the SS cone at ⬃200 GW, ␶ decreases from 18 to 7 ns as the misalignment increases to 10 ␮rad; it then decreases to zero before the misalignment reaches 20 ␮rad. We believed that the agreement between model and data was adequate to allow us to try predicting closure for conditions not too far from existing data. We used the model to predict performance of a SS cone pinhole for the 14.8-kJ, 21:1 ignition pulse. To account for the difference in closure times between a square pulse, for which most of the data were taken, and the 21:1 ignition pulse, we scaled to an equivalent square-pulse energy. The scaling factor was 1兾3, the measured ratio of the closure energies for the two pulse shapes for a ⫾100-␮rad SS cone. Figure 29 shows model predictions for a SS cone pinhole and the square-pulse equivalent of the ignition pulse. The graph shows closure time versus misalignment with pinhole size as the parameter. It indicates a very strong dependence on both pinhole size and misalignment. Although the sensitivity to misalignment is not visible in the data, notably in the cluster of data points from 161 to 198 GW in Fig. 25, we believe that the sensitivity is real and that the error bars associated with the misalignment measurement hide the dependence in the data. The model calculations shown in Fig. 29 predict that a ⫾120-␮rad SS cone would remain open for 25 ns for an ignition pulse with 10-␮rad misalignment. Addition of 5 ␮rad in pinhole size for the ⫾7.5 ␮rad of

Fig. 29. Model predictions for SS cone pinholes and 21:1 contrast ignition pulses. The parameters are misalignments in microradians.

added beam divergence for SSD, which is more than required as indicated by measurements,9 implies that a ⫾125-␮rad SS cone pinhole would meet all the NIF requirements. A similar calculation for the Ta cone pinhole, with the same scaling factor used to account for the difference between square and 21:1 ignition pulses, predicts that Ta would not be significantly better than SS. Part of the explanation of this result, which seems to be in conflict with our measurements and modeling of material dependence, comes from the ⫾3␮rad error bar on the misalignment measurements and the sensitivity of the model to misalignment. The 6-␮rad span of this error bar 共in Figs. 23–26兲 corresponds to a factor-of-1.8 difference in closure time for the SS cone at 250 GW, which is more than the measured improvement from SS to Ta. In addition, the Ta scaling factor that accounts for the pulseshape differences may not be the same as for SS. We have more confidence in the SS model and its prediction for an ignition pulse, because the model is based on nine data points as opposed to two for the Ta model and because the pulse-shape scaling factor was measured. 8. Conclusions

The principal results from the three components of our pinhole development program are as follows: 1. Full-scale pinhole closure measurements on the Beamlet facility showed that a. A ⫾150-␮rad SS cone pinhole will easily pass a full-energy NIF ignition pulse through any of the spatial filters in a NIF beam line, with margin for ⫾10% misalignment and ⫾7.5␮rad of divergence for SSD. b. The cone pinhole geometry produces much less backreflected energy from the faces of the pinhole than other pinhole geometries. c. Pinhole closure depends strongly on pinhole material and geometry, with the heavier metals and the cone geometry performing better than others. 2. Off-line measurements of the general characteristics of pinhole closure plasmas showed that

a. The phase shift decreases exponentially from the edges of a pinhole. b. The phase contour speed depends linearly on intensity below 300 GW兾cm2 for SS and 70 GW兾cm2 for Ta, at normal incidence. c. The phase-contour speed is inversely proportional to the square root of the atomic mass of the pinhole material for metals, at normal incidence. 3. A simple model that includes the effects of pinhole diameter, beam power, and alignment agrees reasonably well with the Beamlet data. It predicts that a. Closure depends strongly on pinhole diameter and misalignment. b. On NIF, a ⫾125-␮rad SS cone will pass the ignition pulse, meeting all requirements. We are grateful for the expert technical assistance of N. D. Nielsen, W. D. Sell, and the entire Beamlet operations staff. In addition, we have had helpful discussions with B. B. Afeyan, M. D. Feit, R. Kirkwood, and A. M. Rubenchik. We also acknowledge the assistance of R. Kirkwood in setting up the two GOI’s and Y. A. Zakharenkov in the reduction of interferometric data. This research was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract W-7405ENG-48. References and Notes 1. W. W. Simmons, J. S. Guch, F. Rainer, and J. E. Murray, “A high energy spatial filter for removal of small scale beam instabilities in high power solid state lasers,” internal rep. UCRL-76873 共Lawrence Livermore National Laboratory, University of California, Livermore, Calif., 1975兲. 2. J. T. Hunt, J. A. Glaze, W. W. Simmons, and P. A. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. 17, 2053–2057 共1978兲. 3. J. S. Pearlman and J. P. Anthes, “Closure of pinholes under intense laser radiation,” Appl. Opt. 16, 2328 –2331 共1977兲. 4. J. M. Auerbach, N. C. Holmes, J. T. Hunt, and G. J. Linford, “Closure phenomena in pinholes irradiated by Nd laser pulses,” Appl. Opt. 18, 2495–2499 共1979兲. 5. S. A. Dimakov, S. I. Zavgorodneva, L. V. Koval’chuk, and A. Yu. Rodionov, “Investigation of the threshold of formation of a plasma screening radiation in a spatial filter,” Sov. J. Quantum Electron. 19, 803– 805 共1989兲. 6. P. M. Celliers, K. G. Estabrook, R. J. Wallace, J. E. Murray, L. B. Da Silva, B. J. MacGowan, B. M. Van Wonterghem, and K. R. Manes, “Spatial filter pinhole for high-energy pulsed lasers,” Appl. Opt. 37, 2371–2378 共1998兲. 7. R. G. Bikmatov, C. D. Boley, I. N. Burdonsky, V. M. Chernyak, A. V. Fedorov, A. Y. Goltsov, V. N. Kondrashov, S. N. Koptyaev, N. G. Kovalsky, V. N. Kuznetsov, D. Milam, J. E. Murray, M. I. Pergament, V. M. Petryakov, R. V. Smirnov, V. I. Sokolov, and E. V. Zhuzhukalo, “Pinhole closure in spatial filters of large scale ICF laser systems,” in Third International Conference on Solid State Lasers for Application to Inertial Confinement Fusion, W. H. Lowdermilk, ed., Proc. SPIE 3492, 510 –523 共1999兲. 20 March 2000 兾 Vol. 39, No. 9 兾 APPLIED OPTICS

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