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A KrF oscillator system with uniform profiles
OPTICS COMMUNICATIONS
Optics Communications 106 ( 1994) 113- 122 North-Holland
Full length article
A KrF oscillator system with uniform profiles A.V. Deniz ScienceApplicationsInternationalCorporation,McLean, VA22102, USA
and S.P. Obenschain PlasmaPhysicsDivision,US. NavalResearch Laboratory, Washington.DC 20375, USA Received 1 June 1993; revised manuscript received 19 August 1993
A RrF oscillator system that has produced highly uniform M-top focal distributions is described. The oscillator system is part of a large laser system that will utilize the echelon-free induced spatial incoherence technique to obtain uniform illumination of planar targets for fusion research.With this system, focal profiles with small long scale length nonuniformities have been obtained. The nonuniformity was determined by performing a least-squares fit to a series of proftles, and calculating the deviation of each fit from a flat-top profde. With a linear fit, the deviation averaged over the series is f 0.5%, and with a quadratic fit, it is f 1.4%. Details of the oscillator system confguration, focal uniformity measurement techniques, and resulting focal profiles are presented.
1. Introduction One of the requirements for high-gain direct-drive inertial confinement fusion is a highly symmetric implosion of the spherical fuel pellet. Ablation pressure nonuniformities less than a few percent are thought to be required. If an ultraviolet wavelength is used (which couples more efficiently to the target than longer wavelengths), then there is only modest lateral smoothing of the ablation pressure [ 11. Therefore, success with direct drive laser fusion requires the development of techniques for highly uniform illumination of fuel pellets. It is perhaps impossible with today’s technology to have a uniform focal illumination with a nearly diffraction limited beam. Progress has instead been made by methods that employ controlled spatial and temporal incoherence with focal profiles that are smooth when averaged over many temporal coherence times [ 2- 111. These smoothing techniques have been shown to reduce laser plasma instabilities [ 12- 18 1. However, with existing high energy glass lasers, peak-to-valley focal nonuniformities still are typically N 10% with these techniques [9-l 11.
The NIKE RrF laser system [ 19-2 1 ] currently under construction at the Naval Research Laboratory, is being built to have focal profiles that are uniform enough to produce ablation pressures flat to within 2% on planar targets. NIKE will use the echelon-free induced spatial incoherence (EFISI ) technique [ 3 ] to produce 60 time diffraction limited flattop focal profiles with at least two kilojoules in 4 ns on target. The EFISI technique is illustrated in fig. 1. An oscillator with spatially incoherent output illuminates an object aperture, whose image is relayed by the two lenses to the image plane. Light from each point in the object aperture illuminates an aperture at the Fourier plane with the same intensity. The light passes through the object aperture, is amplified at the Fourier plane, and is then focused at the image +fb*L--_(
I=+ sysLml
object aperture
0
;o fO”“N
aperture
+f2-
I imageOf ob,cct
Fig. 1. A simplified schematic of the echelon-free induced spatial incoherence technique.
0030-4018/94/%07.00 0 1994 Elsevier Science B.V. All rights reserved. SSDI 0030-4018 (93)E0422-C
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plane. With the EFISI technique, the image profile would not be strongly affected by the spatial nonuniformity of the amplifier gain because light from each point in the object aperture is amplified by the same amount. Also, because the image is already many times diffraction limited, it will not be strongly affected by the phase aberrations of the laser system. There are several requirements that the oscillator must meet: (i) The light at the object aperture must have sufficient spatial incoherence (divergence) to fill the Fourier aperture. (ii) The oscillator output must have sufficient temporal incoherence to produce a time-averaged smooth profile. (iii) The light at the object and Fourier apertures must produce a uniform flat-top profile at the image plane. (iv) The light from each point in the object aperture must illuminate the Fourier aperture with the same profile. Requirements (i) through (iii) are perhaps straightforward. Requirement (iv) is necessary so that the focal profile be insensitive to gain nonuniformities of an amplifier located at the Fourier plane. These requirements are discussed in the next section. Here we report on a KrF oscillator system with unconventional resonator optics that has come close to meeting the above requirements. This oscillator, with a two stage Pockels cell pulse slicer, produces 4 ns flat-top focal profiles (using an f/ 130 lens) with tilts on the order of 1% and the temporal coherence time of 0.6 ps. When a nonuniform amplifier gain was simulated by blocking half the Fourier aperture, the tilts were still less than 3%. This system approached the focal uniformity goals for the NIKE laser. In this oscillator, the laser medium is imaged back onto itself by the resonator optics. Thus a photon that makes a large angle with the longitudinal axis will still pass through the laser medium, even after many transits (unless the angle is so large that the finite size of the laser chamber windows blocks it). The output of this oscillator therefore has greater angular divergence than the output of an oscillator with conventional stable resonator optics. In addition, the Fourier aperture is illuminated more nearly uniformly by each point in the object aperture, as required by the EFISI technique. 114
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The next section describes the oscillator setup in greater detail and presents its measured characteristics. Section 3 presents details of the pulse slicing system. Section 4 presents details of the imager used to measure the profiles. Section 5 presents the focal profiles and the algorithms used to evaluate them. Section 6 summarizes the results and presents the conclusions.
2. The oscillator system As mentioned in the introduction, the oscillator output must be spatially incoherent so that it can produce a beam that is many times diffraction limited. The focal profile does not then depend strongly on the phase errors encountered during propagation. The focal profiles presented are 60 times diffraction limited, although the NIRE system has been designed to propagate beams which are up to 120 times diffraction limited without vignetting. If both the object and Fourier apertures are circular, then the beam is N,, times diffraction limited with N*=Dd/lzf,
(1)
where D is the Fourier aperture diameter, d is the object aperture diameter, A is the wavelength of the light (248 nm), andfis the focal length of the lens between the object and Fourier apertures. The second requirement is that the oscillator output be temporally incoherent. At any given time, the focal profile is a complicated speckle pattern. After one coherence time, it is a different pattern. The measured profiles are time-integrated, so that if the measurement is made over more coherence times, then both the shot-to-shot variation due to the speckle and the random variation from one coherence zone to the next are decreased. The coherence time must be short enough so that these variations are not too large. The third requirement is that light from each part of the object aperture illuminate the Fourier aperture with the same angular energy distribution. This requirement can be stated more precisely as follows: Let r((x, k) be the time-averaged intensity profile at point R within the Fourier aperture due to light from several coherence zones (which have a size of
-d/60) around point x in the object aperture. The requirement becomes VJ((x, R) = 0 .
(2)
Note that r((x, R) need not also be uniform across the Fourier aperture. If eq. (2) is satisfied, then the focal profile will be flat, even if the light is subjected to nonuniform amplification at the Fourier aperture. Equation (2) can be checked by placing a small pinhole in the object aperture and measuring the time-averaged intensity profile at the Fourier aperture. The resulting Fourier intensity profile should be the same regardless of where the pinhole is placed within the object aperture. Adherence to eq. (2) can also be checked by blocking various regions of the Fourier aperture, and observing whether or not the image maintains its flat-top shape. The latter method was used in this work. Figure 2 shows the optical setup for the oscillator used for the results presented here. The oscillator optics image the laser medium back onto itself. All the oscillator configurations investigated used a 1 cmx 2 cm x 80 cm discharge pumped KrF laser medium. The rear optics consist of a positive lens and a highreflectivity flat mirror with an aperture, and the front optic is a 50% reflectivity flat mirror. The distance between each mirror and the lens is equal to the focal length of the lens ( 1 m). Light from point a inside the laser medium will be imaged, after reflection by both the front and rear optics, at point b. This property of imaging the laser medium back onto itself produces a large angular divergence. Conventional discharge oscillator optics did not satisfy these requirements. A stable resonator setup consisting of a flat front mirror and a rear mirror with a large radius of curvature ( 5 m, 10 m, and co) was tested. While the image of the object aperture was flat, the angular divergence was too small to produce the required 60 times diffraction limited beam. The front of the laser medium (which is approximately 10 cm from the chamber window) was re-
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layed with a telescope to the object aperture. This geometry was found empirically to produce the flattest images. The image was not as flat when the center of the discharge was imaged onto the object aperture, or when the object aperture was N 50 cm from the front oscillator mirror outside the oscillator (no telescope was used in the latter case). The images produced by this setup had flat-top focal profiles, but with residual tilts on the order of 10% in the vertical direction. This was most likely due to irregularities in the laser cell and/or electrodes. The tilts also varied during the oscillator pulse. Finally, there was a shot-to-shot variation of the tilts, presumably caused by irreproducibilities of the discharge. All profile tilts were substantially reduced by adding the telescope system shown in fig. 3. The oscillator output was split into two beams and then recombined. One beam passed through a single telescope, and the image was inverted. The other beam traveled the same distance but passed through two telescopes, and the image was not inverted. Combining the beams at the object aperture resulted in images with a very small tilt ( u 1%) , independent of when the oscillator pulse was sliced. The tilts of the profiles also had a very small shot-to-shot variation. The rear reflector of the oscillator, which is located close to a Fourier plane, was apertured to limit the angular divergence of the output at the object aperture. This affected the concavity of the profile at the image plane (after passing through the Fourier aperture). Decreasing the angular divergence by placing a smaller aperture at the rear reflector tended to make the image concave up, while increasing the divergence had the opposite effect. With no rear reflector aperture, the image was slightly concave down. The rear reflector aperture was made smaller until
I:
0-l
KS OYlllat”r
Laser Medium
Fig. 2. The optical setup for the oscillator.
I I Ob,eCl apemlrr
Fig. 3. T’heoptical setup for the oscillator and telescopes. 115
the concavity was nearly eliminated. For the results in this work, the divergence from the object aperture was limited to N 8 mrad in the vertical direction, and N 14 mrad in the horizontal direction. This divergence is sufficient to overfill the Fourier aperture, which requires only 5 mrad. The object aperture diameter of 3 mm was limited by the transverse dimension of the laser medium ( 1 cm by 2 cm), and the Fourier aperture size was limited by the size of the Pockels cells to 0.5 cm X 0.5 cm, which implies a focal length of 1 m. The required angular divergence of the oscillator is 602/d- 5 mrad, which is less than the measured angular divergence. Two KD*P Pockels cells in series were used to slice a 4 ns pulse out of the 30 ns oscillator output. The setup is shown in fig. 4. A pair of dielectric polarizers polarized the beam horizontally before the object aperture. The beam then passes through the Fourier aperture, the two Pockels cells, and the remaining sets of dielectric polarizers. The energy contrast ratio, defined as the ratio of the transmitted fluences with and without voltage applied to the Pockels cells, was 3000 to 1.
3. The imager The imager is a cooled, slow-scan, two-dimensional charge coupled device (CCD) camera. It is capable of measuring the energy profiles to an ao curacy of better than l%, and it has a spatial resolution of 384 by 576. The camera parameters are shown in table 1. The CCD is coated with a phosphor in order to increase its quantum efficiency at 248 nm to -0.25. The imager has a vacuum window to prevent formation of frost on the CCD. This window has an antireflection coating on each surface with a power reflectivity of 0.25% at 1~248 nm, 0” incidence. This re,ecWd beam
Fig. 4. The pulse slicing system.
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Table 1 Parameters of the CCD camera. Parameter
Value
Format Pixel size Readout noise Charge transfer efftciency Full well capacity Quantum efficiency Dark current Exposure time Digitizer resolution CCD temperature Window thickness Window reflectivity
384 by 576 pixels 23pmby23pm 25 electrons 0.99998 160000 electrons/pixel 0.25 at 248 nm 15 electrons/set/pixel 0.1 s 12 bits -45°C l.Ocm 0.25%, each surface
introduced a negligible error in the measurement of the profile because the coherence length of the light ( 1.9 x 10m2 cm) was much shorter than the window thickness. For temporally coherent light, the noise in a cooled CCD measurement is due mainly to the statistical nature of the photoelectric process (the electron shot noise) and the preamplifier noise. For the measurements presented the signal was large enough that the preamplifier noise was negligible. Because the number of photoelectrons in one pixel for several measurements of identical light levels has a Poisson distribution, the electron shot noise ad for one measurement is
%d=~ *
(3)
The noise in one pixel is independent of the noise in another. In addition to the noise in the CCD measurement, there is a random variation ci associated with the temporal incoherence of light. If a pixel in the detector is smaller than a spatial coherence zone, then ci=Ne (z/T)"~9
(4)
where T is the laser pulse length, t is the coherence time, N, is the average number of photoelectrons in the pixel measured in time T,and Oi is the shot-toshot RMS variation of the signal in electrons. In this case, there is correlation in the measurements of adjacent pixels. If on the other hand there are N, coherence zones in a pixel, then Gi=Ne (~lNcT)"~e
(5)
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In this case, the random variation for adjacent pixels is independent. The bandwidth ( E l/7) of the oscillator has been measured to be 1.6 THz, which gives a coherence time of 0.6 ps. Because the random variation associated with the incoherence is not correlated with the CCD measurement noise, the total random variation a, in electrons of the number of electronics in a pixel is given by 2 6,
++t&. -
(6)
Because a typical CCD exhibits a nonuniform response to light, the camera was calibrated by illuminating it with a uniform light source (fig. 5). The oscillator was used as a light source for a commercial integrating sphere. This has the advantage of calibrating the camera for the same wavelength and approximately the same pulse length used in the measurements. The camera was placed N 1 m from the output aperture of the sphere. At that distance, the variation in light across the CCD surface from a lambertian surface is theoretically 1.5 x 10 -‘. In an actual setup, the variation will probably be greater; the camera and integrating sphere might not be perfectly aligned, there might be spurious reflections, and the light output from the integrating sphere might not be perfectly uniform. In order to test the calibration technique, the camera was calibrated at different positions. The distance between the camera and the integrating sphere was varied from 0.4 m to 1.6 m, the camera was moved f 3 mm perpendicular to a line between it and the sphere, and the camera was rotated about that line. The greatest change in the calibration was a 0.3% spatial tilt across the 8.83 mm width of the sensor. This corresponds to a systematic error of less than 0.1% in the calculation of the tilt of a 2 mm diameter profile.
During the calibration, there will be the random variation in the measurements given by eq. (6). There were several coherence zones in each pixel, so eq. (5) was used to calculate the random variation due to the incoherence of light. The size 6 of a coherence zone at the CCD is approximately r3=Al/Dl,
(7)
where 1 is the wavelength of light and 1 is the distance between the CCD and the integrating sphere output aperture, which has a diameter 4. For A= 248 nm, I= 1 m, and DI=5 cm, 6 is 4.96 urn. Using the pixel size from the table, the value of N, (to be used in eq. (5)) is (23 um/4.96 pm)2=21.5. In order to measure the noise, the output from the integrating sphere was measured 64 times. After correcting for variation in’the total energy falling on the CCD, the noise-to-signal ratio (0,/N,) for a given pixel was calculated. At a signal level of 6.6 x lo4 electrons, the result ranged from 2.5 x 10m3 to 3.0 x 1O-‘, compared to an expected value (from eq. (6)) of 3.8~10~~ (T=30 ns, 7~0.6 ps, and N, = 2 1.5 ) . This indicates that the camera is capable of measuring the profiles with low noise. The calibration was determined by summing 64 measurements (with the background subtracted) of the integrating sphere output (flat fields). The average signal level was 1.5 x 1O5 electrons, or 3825 counts. The calibration factor Ci for pixel i is given by
where NP is the number of pixels on the CCD, and J is the summed flat field signal at pixel i. Because & is a sum of 64 measurements, the relative error of Ci calculated by eq. (6) due to the error in fi: is reduced by a factor of 8 from 2.8~ 10m3to 3.5~ 10M4. The relative energy ei falling on pixel i is given by ei=Cj(si-bi)
Fig. 5. Calibration of the camera. The oscillator illuminates an integrating sphere. A negative lens (not shown) is used to spread out the input light to avoid damaging the integrating sphere. A tube (not shown) is used in order to reduce stray light. Apertures inside the tube are necessary in order to reduce stray reflections.
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,
(9)
where Si and bi are respectively the signal and background at pixel i. The signal levels of the profiles presented in the next section are 6.0 x 1O4electrons, or N 1500 counts, while the background levels (which are due to the camera) are _ 50 counts. This implies that the RMS measurement error due to the electron shot noise and the calibration error is 0.4% for one 117
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pixel. When observing longer scale length focal profile nonuniformities that cover many pixels, one can reduce this error by averaging over more than one pixel.
4. The profiles The object aperture is imaged onto the camera through a lens, the Fourier aperture, the pulse slicing system, a demagnifying telescope, and focusing lens. A schematic of the experimenal arrangement without the demagnifying telescope is shown in fig. 4. The telescope is located before the second lens, which focuses the beam onto the camera after attenuation by two reflections by uncoated surfaces and transmission through a 95% flat reflecl.or. The telescope demagnifies the image of the object aperture from 3 mm to 2 mm diameter. The measured profiles were analyzed with the following algorithm: (i) Determine the centroid (x,, yc) of the profile. (ii) Find the edges of the flat region of the horizontal and vertical cross-sections through the centroid. (iii) For the flat region of each cross-section: (a) Perform a linear least-squares fit, and calculate the variation of the fit from a Sat-top (tilt). (b) Perform a quadratic least-squarer, tit, and calculate the mean-to-peak variation of the fit from a flat-top. (c) Calculate the RMS deviation of the measured crosssection from the fits. The centroid is determined by (10)
where Xi is the x position of pixel i. The calculation of y, is analogous. The edges of the flat region of a cross-section is determined by the following algorithm: (i) Find the pixels 1 and I’ where the measured value of the cross-section is just greater than a specified fraction fp of the maximum value. (ii) The edges of the flat re:gion at t pixels from 1 and r towards the center of the cross-section. For example, if 1is the pixel on the left edge of the profile, and pixel numbers increase from left to right, the the 118
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flat region is between pixels I+ t and r- t, inclusive. For the results presented,f, was 0.25, and t was 5 (the cross-section of the profile is 85 pixels across). The results do not depend strongly on& or t. Three aspects of the profiles were evaluated. First, the 4 ns pulse was taken at different times during the oscillator output to check whether the object aperture illumination uniformity changed in time. It was found that the tilts of the profiles do not vary appreciably during the oscillator output. Second, a series of profiles were taken at the same time during the oscillator output to determine the shot-to-shot variation of the profiles. It was found that the observed shot-to-shot RMS variation of the tilts was consistent with the spatial and temporal incoherence of the light. It was also found that the variation of the linear fit from a flat-top profile (a measure of tilt) averaged over the series was f 0.5%, and of the quadratic fit (a measure of peaking or concavity) was + 1.4%. Finally, the Fourier aperture was partially blocked to simulate the effects of a nonuniform gain in an amplifier. The profiles were found to be insensitive to this partial blocking; blocking half the Fourier aperture changed tilts of the profile by only a few percent. Figure 6 shows a typical profile and its cross-sections. The RMS deviation of the measurement from the linear least-squares fit is 1.0% along the vertical direction and 0.8% along the horizontal. If the protile were flat, the deviation would be the same as the total random variation aY (given by eq. (6 ) ) in each datum. Because a pixel is smaller than a coherence zone, eq. (4) can be used to calculate the variation due to the incoherence of the light. With a coherence time of 0.6 ps, a pulse length of 4 ns, and a signal of 6 x 1O4 electrons, the expected RMS relative variation of each datum is 1.3Oh.This indicates that the observed deviation is probably due to the CCD shot noise error and the variation of the light energy due to the laser incoherence and the finite averaging time. Figure 7 shows the tilts of the cross-sections as a function of time during the oscillator output. Eight images were recorded for seven times spaced 5 ns apart during the oscillator output, for a total of 56 images. Each point on the graphs is the tilt of the horizontal or vertical cross-section of one image. Note that the time variation is negligible. Therefore, the
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Tilt
5L
Along
y
I
20
0 time Tilt
40
(nsec) Along
x
1 40
-5t 20
0 time
-,,,,,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,
Fig. 7. Tilts of the oscillator output.
cros5sections
(nsec) a function of time during the
as
Tilt Along y L 12
0
2
4
0 -2
-1
0
1
2
x (mm)
~
;,
-ynT:,
-4
-2
,
0 tilt
2
4
(%)
Fig. 8. A histogram of the tilts of the linear fits to the profdes with the Fourier aperture completely unblocked.
-2
-1
0
1
2
Y ‘(mm) Fig. 6. A typical profile and its cross-sections: (a) threedimensional view, (b) horizontal cross-section through the centroid, and (c) vertical cross-section through the centroid.
output of the oscillator does not vary appreciably when averaged over 4 ns. Figure 8 is a histogram of the tilts for 64 images.
Note that the average tilt is very nearly zero; it is - 0.75% along the vertical direction, and 0.75% along the horizontal. The RMS variation of the tilt is I%, which is close to the expected value of 0.7% (this is calculated in the appendix). The RMS variation of the tilt of a single beam of fig. 3 is N 5% along the vertical direction and u 1% along the horizontal. The larger variation along the vertical direction for a single beam indicates that the telescopes shown in fig. 3 are effective in reducing the shot-to-shot variation of the tilt. Furthermore, the average tilt of a single beam is 5 to lo%, which indicates that the telescopes 119
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are also effective in reducing the average tilt. Figure 9 is a histogram of the mean-to-peak variation of the quadratic fit from a flat-top profile for the 64 images. The mean averaged over the images is + 1.4% along the vertical direction, and + 0.6% along the horizontal. The variation is larger along the vertical direction because there is more shot-to-shot variation of the concavity of the profile, which is not reduced by the telescopes. Figure 10 shows the tilts of the cross-sections with different parts of the Fourier aperture blocked in or-
Variation 101
Along y I
-
I
:!“I? 0
1
IrG
2 variation
Variation 81
3
-I
Along
4
5. summary
x
I
3
2
1
variation
4
(%)
Fig. 9. A histogram of the mean-to-peak variation of the quadratic tits from a flat-top profile with the Fourier aperture completely unblocked.
Tilt -
Along
y
207i
-g
0
z
3
-20 open fourier
top
right
blocking
Tilt
bottom
left blocked)
(half
Alone
x
-1oF
3 open fourier
right bottom left top blocking (half blocked)
Fig. 10. Variation of the tilts of the cross-section when different parts of the Fourier aperture are blocked.
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der to simulate nonuniform amplifier gain. Sixteen images were recorded with (i) the Fourier aperture completely unblocked, (ii) the top half of the Fourier aperture blocked, (iii) the right half blocked, (iv) the bottom half blocked, and (v) the left half blocked, for a total of 80 images. Note that the average tilt changed by a few percent. Note also that when the top or bottom of the Fourier aperture was blocked, the shot-to-shot variation of the tilt of the vertical cross-section increased, because different parts of the inverted and noninverted beams were blocked. The two beams were not exact inverses of each other, and the the tilt of one did not cancel the tilt of the other. This larger shot-to-shot variation does not occur in the horizontal direction because the angular divergence of the oscillator output in the horizontal direction is larger than the divergence in the vertical direction.
(%)
-’
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A laser oscillator system has been developed with a 4 ns pulse output, 3000 to 1 energy contrast ratio, and flat focal profile. Its angular divergence is large enough to produce 60 times diffraction-limited images. The coherence time was 0.6 ps, and the pulse duration was 30 ns. The focal images of 4 ns slices of the beam had the desired flat cross-sections (tilts of N 1%). The shape of the focal profile should not depend strongly on the gain profiles of the laser amplifiers if they are placed at the Fourier aperture of the EFISI system. This was tested by partially blocking the Fourier aperture and observing that the shape of the focal profile was not strongly affected (the tilts were typically on the order of a few percent). The top, bottom, left, and right halves of the Fourier aperture were blocked. Partially blocking the Fourier aperture simulates an extremely nonuniform amplifier gain; typical nonuniformities of actual amplifiers used in the NIKE laser are only approximately 20%. The oscillator system described here comes close to fulfilling the goals for the NIKE system. The existing system is now being used to test the ability of the NIKE system to maintain uniform focal profdes after several stages of amplification. In future work, we will attempt obtain more nearly uniform beams
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which are less affected by partial blocking of the Fourier aperture. We will also attempt to produce more than 60 times diffraction limited beams, so that phase aberrations in the laser system will have even less effect on the focal profile.
Acknowledgements
We would like to thank our colleagues for useful discussions and assistance: Mark S. Pronko, Robert H. Lehmberg, Thomas Lehecka, Carl J. Pawley, Warren Webster, Julius Goldhar, and Andrew J. Schmitt. The data was archived on disk using a modified version of National Center for Supercomputer Activities’ Hierarchical Data Format. This work was supported by the United States Department of Energy.
Appendix
This appendix presents a calculation of the variance of the slope of the linear least-squares tit. The variance is determined by the variance of the data used to calculate the slope. The random variation of each datum in the measured profile is due to both the CCD measurement noise and the random variation associated with the temporal incoherence of the light. The CCD measurement noise is due mainly to the electron short noise, and is given by eq. ( 3 ) . The random variation due to the incoherence is given by eq. ( 4) or ( 5 ) . The total random variation a, (given by eq. (6) ) causes the calculated slope m to have a variance al. In order to calculate a,, let ya be a set of Nr random variables, and yai be the value of sample i of ycl. Let ‘x, be a set of Nr numbers. The y, are the measurements at position x,; Nf is the number of pixels across the profile. Define
(lla)
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(lid) where N is the number of profiles measured. Note that the X~ can be chosen so that /3r is zero. Note also that in this case, eq. (1 Id) can be approximated by an integral. Let L be the diameter of the profile, and let Ax=L/Nfi Then 82= $&:A-?
(12)
L/2 Z-
N r L
r
J -L/2
=NfL2/12.
X2dX
(13)
(14)
The variance will be calculated by first using the standard formula for the slope of the linear leastsquares tit to a set of data. The variance will depend on &. However, a$ will be negligible if the pixels cr and /3 are not close together, because the size of a coherence zone is approximately the same size as a pixel. Using a rough approximation, the ratio a,/~, is calculated. The slope m of the linear least-squares fit to the data pairs (x,, ya) is a random variable given by (after setting j?r= 0): (15)
Let a; be the variance of m. Because x, is a number (rather than a random variable),
(16) For the results presented, there are 60 coherence zones and 85 pixels across a profile; each coherence zone spans approximately one pixel. In addition, ci- 750 electrons (from eq. (4), with T=4 ns, r=O.6 ps, and N,=6.0x104 electrons), while ad-250 electrons (from eq. ( 3 ) ). The total variation at one pixel is therefore (eq. (6 ) ) 790 electrons, which is a relative variation of 1.3 x 10m2. Because the variation due to the measurement ( occa) is independent for each pixel, while the variation due to the incoherence of light (ci) is somewhat correlated, the rather arbitrary assumption can be made: 121
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References ifa=j3, [l] S.E. Bodner, J. Fusion Energy 1 (1981) 221.
= ko:, , if la-jIl=l, =o,
otherwise .
(17)
Equation ( 16 ) then becomes - L : (x:+kx~_,x,+kx,+,x,) oL- /!I: a=1
&a,
(18)
where x, = 0 for (Y= 0 and cy= Nf+ 1 so that there is no contribution from the norrexistent points 0 and Nr+ 1. Because the signal level is nearly the same for all points on the profile, the variances of each of the ya will all be approximately equal: (19)
(21) Approximating /3*by N,L*/ 12, this becomes
(22) For a 2 mm diameter image with Nf=85 and a,/aY=0.27/mm. Wilth 0,/N,= 1.3x lo-*, a,/N,=3Sx IO-‘/mm, or a 0.7% shot-to-shot RMS variation of the tilt of the p:rofile. With k=0.75, u,,Juy=0.30/mm, and with k=0.25, am/a,= 0.23/mm. This indicates that the value of k does not strongly affect the RMS variation of the tilts. kE0.5,
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[2] R.H. Lehmberg and S.P. Obenschain, Optics Comm. 46 (1983) 27. [ 31 R.H. Lehmberg and J. Goldhar, Fusion Tech. 11 (1987) 532. [4] Ximing Deng, Xiangchun Liang, Zezun Chen, Wenyan Yu and Renyong Ma, Appl. Optics 25 (1986) 377. [ 5 ] A.J. Schmitt and J.H. Gardner, J. Appl. Phys. 60 ( 1986) 6. [ 61 S. Skupsky, R.W. Short, T. Kessler, R.S. Craxton, S. Letzring and J.M. Soures, J. Appl. Phys. 66 (1989) 3456. [7] D.G. Colombant and A.J. Schmitt, J. Appl. Phys. 67 ( 1990) 2303. [8] Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka and C. Yamam&a, Phys. Rev. Lett. 53 ( 1984) 1057. [9] D. Veron, G. ThieIl and C. Gouedard, Optics Comm. 97 (1993) 259. [ 1OlD.M. Penington, M.A. Henesian, H.T. Powell, C.E. Thompson and T.L. Weiland, Conf. on Lasers and electrooptics, 1993, Technical Digest Series, Vol. 11 (1993). [ 111 H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Am&i, T. Jitsuno and S. Nakei, J. Appl. Phys. 73 (1993) 2122. [ 121 S.P. Obenschain, J. Grun, M.J. Herbst, K.J. Keamey, C.K. Manka, E.A. McLean, A.N. Mostovych, J.A. Stamper, R.R. Whitlock, S.E. Bodner, J.H. Gardner and R.H. Lehmberg, Phys. Rev. Lett. 56 (1986) 2807. [ 131 J. Grun, M.H. Emery, C.K. Manka, T.N. Lee, E.A. Mclean, A. Mostovych, J. Stamper, S. Bodner, S.P. Obenschain and B.H. Ripen, Phys. Rev. Lett. 58 (1987) 2672. [ 141 A.N. Mostovych, S.P. Obenschain, J.H. Gardner, J. Grun, K.J. Keamey, C.K. Manka, E.A. McLean and C.J. Pawley, Phys. Rev. Lett. 59 (1987) 1193, [ 151 S.P. Obenschain, C.J. Pawley, A.N. Mostovych, J.A. Stamper, J.H. Gardner, A.J. Schmitt and S.E. Bcdner, Phys. Rev. Lett. 62 (1989) 768. 1610. Willi, T. Afshar-rad and S. Coe, Phys. Fluids B 2 ( 1990) 1318. 171 T.A. Peyser, C.K. Mat&a, S.P. Obenschain and K.J. Keamey, Phys. Fluids B 3 ( 199 1) 1479. 181 D.K. Bradley, J.A. Delettrez and C.P. Verdon, Phys. Rev. Lett. 68 (1992) 2774. 191 C.J. Pawley et al., Proc. Intern. Conf. on Lasers ‘90, eds. D.G. Harris and J. Herbelin ( 199 1) p. 491. [20] M.S. Pronko et al., Proc. Intern. Conf. on Lasers ‘91, eds. F.J. Duartes and D.G. Harris (1992) p. 691. [21] S.E. Bodner et al., Fourteenth Intern. Conf. on Plasma physics and controlled nuclear fusion research, IAEA-CN56/B-2-1(C), (1992).
Optics Communications 106 ( 1994) 113- 122 North-Holland
Full length article
A KrF oscillator system with uniform profiles A.V. Deniz ScienceApplicationsInternationalCorporation,McLean, VA22102, USA
and S.P. Obenschain PlasmaPhysicsDivision,US. NavalResearch Laboratory, Washington.DC 20375, USA Received 1 June 1993; revised manuscript received 19 August 1993
A RrF oscillator system that has produced highly uniform M-top focal distributions is described. The oscillator system is part of a large laser system that will utilize the echelon-free induced spatial incoherence technique to obtain uniform illumination of planar targets for fusion research.With this system, focal profiles with small long scale length nonuniformities have been obtained. The nonuniformity was determined by performing a least-squares fit to a series of proftles, and calculating the deviation of each fit from a flat-top profde. With a linear fit, the deviation averaged over the series is f 0.5%, and with a quadratic fit, it is f 1.4%. Details of the oscillator system confguration, focal uniformity measurement techniques, and resulting focal profiles are presented.
1. Introduction One of the requirements for high-gain direct-drive inertial confinement fusion is a highly symmetric implosion of the spherical fuel pellet. Ablation pressure nonuniformities less than a few percent are thought to be required. If an ultraviolet wavelength is used (which couples more efficiently to the target than longer wavelengths), then there is only modest lateral smoothing of the ablation pressure [ 11. Therefore, success with direct drive laser fusion requires the development of techniques for highly uniform illumination of fuel pellets. It is perhaps impossible with today’s technology to have a uniform focal illumination with a nearly diffraction limited beam. Progress has instead been made by methods that employ controlled spatial and temporal incoherence with focal profiles that are smooth when averaged over many temporal coherence times [ 2- 111. These smoothing techniques have been shown to reduce laser plasma instabilities [ 12- 18 1. However, with existing high energy glass lasers, peak-to-valley focal nonuniformities still are typically N 10% with these techniques [9-l 11.
The NIKE RrF laser system [ 19-2 1 ] currently under construction at the Naval Research Laboratory, is being built to have focal profiles that are uniform enough to produce ablation pressures flat to within 2% on planar targets. NIKE will use the echelon-free induced spatial incoherence (EFISI ) technique [ 3 ] to produce 60 time diffraction limited flattop focal profiles with at least two kilojoules in 4 ns on target. The EFISI technique is illustrated in fig. 1. An oscillator with spatially incoherent output illuminates an object aperture, whose image is relayed by the two lenses to the image plane. Light from each point in the object aperture illuminates an aperture at the Fourier plane with the same intensity. The light passes through the object aperture, is amplified at the Fourier plane, and is then focused at the image +fb*L--_(
I=+ sysLml
object aperture
0
;o fO”“N
aperture
+f2-
I imageOf ob,cct
Fig. 1. A simplified schematic of the echelon-free induced spatial incoherence technique.
0030-4018/94/%07.00 0 1994 Elsevier Science B.V. All rights reserved. SSDI 0030-4018 (93)E0422-C
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plane. With the EFISI technique, the image profile would not be strongly affected by the spatial nonuniformity of the amplifier gain because light from each point in the object aperture is amplified by the same amount. Also, because the image is already many times diffraction limited, it will not be strongly affected by the phase aberrations of the laser system. There are several requirements that the oscillator must meet: (i) The light at the object aperture must have sufficient spatial incoherence (divergence) to fill the Fourier aperture. (ii) The oscillator output must have sufficient temporal incoherence to produce a time-averaged smooth profile. (iii) The light at the object and Fourier apertures must produce a uniform flat-top profile at the image plane. (iv) The light from each point in the object aperture must illuminate the Fourier aperture with the same profile. Requirements (i) through (iii) are perhaps straightforward. Requirement (iv) is necessary so that the focal profile be insensitive to gain nonuniformities of an amplifier located at the Fourier plane. These requirements are discussed in the next section. Here we report on a KrF oscillator system with unconventional resonator optics that has come close to meeting the above requirements. This oscillator, with a two stage Pockels cell pulse slicer, produces 4 ns flat-top focal profiles (using an f/ 130 lens) with tilts on the order of 1% and the temporal coherence time of 0.6 ps. When a nonuniform amplifier gain was simulated by blocking half the Fourier aperture, the tilts were still less than 3%. This system approached the focal uniformity goals for the NIKE laser. In this oscillator, the laser medium is imaged back onto itself by the resonator optics. Thus a photon that makes a large angle with the longitudinal axis will still pass through the laser medium, even after many transits (unless the angle is so large that the finite size of the laser chamber windows blocks it). The output of this oscillator therefore has greater angular divergence than the output of an oscillator with conventional stable resonator optics. In addition, the Fourier aperture is illuminated more nearly uniformly by each point in the object aperture, as required by the EFISI technique. 114
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The next section describes the oscillator setup in greater detail and presents its measured characteristics. Section 3 presents details of the pulse slicing system. Section 4 presents details of the imager used to measure the profiles. Section 5 presents the focal profiles and the algorithms used to evaluate them. Section 6 summarizes the results and presents the conclusions.
2. The oscillator system As mentioned in the introduction, the oscillator output must be spatially incoherent so that it can produce a beam that is many times diffraction limited. The focal profile does not then depend strongly on the phase errors encountered during propagation. The focal profiles presented are 60 times diffraction limited, although the NIRE system has been designed to propagate beams which are up to 120 times diffraction limited without vignetting. If both the object and Fourier apertures are circular, then the beam is N,, times diffraction limited with N*=Dd/lzf,
(1)
where D is the Fourier aperture diameter, d is the object aperture diameter, A is the wavelength of the light (248 nm), andfis the focal length of the lens between the object and Fourier apertures. The second requirement is that the oscillator output be temporally incoherent. At any given time, the focal profile is a complicated speckle pattern. After one coherence time, it is a different pattern. The measured profiles are time-integrated, so that if the measurement is made over more coherence times, then both the shot-to-shot variation due to the speckle and the random variation from one coherence zone to the next are decreased. The coherence time must be short enough so that these variations are not too large. The third requirement is that light from each part of the object aperture illuminate the Fourier aperture with the same angular energy distribution. This requirement can be stated more precisely as follows: Let r((x, k) be the time-averaged intensity profile at point R within the Fourier aperture due to light from several coherence zones (which have a size of
-d/60) around point x in the object aperture. The requirement becomes VJ((x, R) = 0 .
(2)
Note that r((x, R) need not also be uniform across the Fourier aperture. If eq. (2) is satisfied, then the focal profile will be flat, even if the light is subjected to nonuniform amplification at the Fourier aperture. Equation (2) can be checked by placing a small pinhole in the object aperture and measuring the time-averaged intensity profile at the Fourier aperture. The resulting Fourier intensity profile should be the same regardless of where the pinhole is placed within the object aperture. Adherence to eq. (2) can also be checked by blocking various regions of the Fourier aperture, and observing whether or not the image maintains its flat-top shape. The latter method was used in this work. Figure 2 shows the optical setup for the oscillator used for the results presented here. The oscillator optics image the laser medium back onto itself. All the oscillator configurations investigated used a 1 cmx 2 cm x 80 cm discharge pumped KrF laser medium. The rear optics consist of a positive lens and a highreflectivity flat mirror with an aperture, and the front optic is a 50% reflectivity flat mirror. The distance between each mirror and the lens is equal to the focal length of the lens ( 1 m). Light from point a inside the laser medium will be imaged, after reflection by both the front and rear optics, at point b. This property of imaging the laser medium back onto itself produces a large angular divergence. Conventional discharge oscillator optics did not satisfy these requirements. A stable resonator setup consisting of a flat front mirror and a rear mirror with a large radius of curvature ( 5 m, 10 m, and co) was tested. While the image of the object aperture was flat, the angular divergence was too small to produce the required 60 times diffraction limited beam. The front of the laser medium (which is approximately 10 cm from the chamber window) was re-
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layed with a telescope to the object aperture. This geometry was found empirically to produce the flattest images. The image was not as flat when the center of the discharge was imaged onto the object aperture, or when the object aperture was N 50 cm from the front oscillator mirror outside the oscillator (no telescope was used in the latter case). The images produced by this setup had flat-top focal profiles, but with residual tilts on the order of 10% in the vertical direction. This was most likely due to irregularities in the laser cell and/or electrodes. The tilts also varied during the oscillator pulse. Finally, there was a shot-to-shot variation of the tilts, presumably caused by irreproducibilities of the discharge. All profile tilts were substantially reduced by adding the telescope system shown in fig. 3. The oscillator output was split into two beams and then recombined. One beam passed through a single telescope, and the image was inverted. The other beam traveled the same distance but passed through two telescopes, and the image was not inverted. Combining the beams at the object aperture resulted in images with a very small tilt ( u 1%) , independent of when the oscillator pulse was sliced. The tilts of the profiles also had a very small shot-to-shot variation. The rear reflector of the oscillator, which is located close to a Fourier plane, was apertured to limit the angular divergence of the output at the object aperture. This affected the concavity of the profile at the image plane (after passing through the Fourier aperture). Decreasing the angular divergence by placing a smaller aperture at the rear reflector tended to make the image concave up, while increasing the divergence had the opposite effect. With no rear reflector aperture, the image was slightly concave down. The rear reflector aperture was made smaller until
I:
0-l
KS OYlllat”r
Laser Medium
Fig. 2. The optical setup for the oscillator.
I I Ob,eCl apemlrr
Fig. 3. T’heoptical setup for the oscillator and telescopes. 115
the concavity was nearly eliminated. For the results in this work, the divergence from the object aperture was limited to N 8 mrad in the vertical direction, and N 14 mrad in the horizontal direction. This divergence is sufficient to overfill the Fourier aperture, which requires only 5 mrad. The object aperture diameter of 3 mm was limited by the transverse dimension of the laser medium ( 1 cm by 2 cm), and the Fourier aperture size was limited by the size of the Pockels cells to 0.5 cm X 0.5 cm, which implies a focal length of 1 m. The required angular divergence of the oscillator is 602/d- 5 mrad, which is less than the measured angular divergence. Two KD*P Pockels cells in series were used to slice a 4 ns pulse out of the 30 ns oscillator output. The setup is shown in fig. 4. A pair of dielectric polarizers polarized the beam horizontally before the object aperture. The beam then passes through the Fourier aperture, the two Pockels cells, and the remaining sets of dielectric polarizers. The energy contrast ratio, defined as the ratio of the transmitted fluences with and without voltage applied to the Pockels cells, was 3000 to 1.
3. The imager The imager is a cooled, slow-scan, two-dimensional charge coupled device (CCD) camera. It is capable of measuring the energy profiles to an ao curacy of better than l%, and it has a spatial resolution of 384 by 576. The camera parameters are shown in table 1. The CCD is coated with a phosphor in order to increase its quantum efficiency at 248 nm to -0.25. The imager has a vacuum window to prevent formation of frost on the CCD. This window has an antireflection coating on each surface with a power reflectivity of 0.25% at 1~248 nm, 0” incidence. This re,ecWd beam
Fig. 4. The pulse slicing system.
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Table 1 Parameters of the CCD camera. Parameter
Value
Format Pixel size Readout noise Charge transfer efftciency Full well capacity Quantum efficiency Dark current Exposure time Digitizer resolution CCD temperature Window thickness Window reflectivity
384 by 576 pixels 23pmby23pm 25 electrons 0.99998 160000 electrons/pixel 0.25 at 248 nm 15 electrons/set/pixel 0.1 s 12 bits -45°C l.Ocm 0.25%, each surface
introduced a negligible error in the measurement of the profile because the coherence length of the light ( 1.9 x 10m2 cm) was much shorter than the window thickness. For temporally coherent light, the noise in a cooled CCD measurement is due mainly to the statistical nature of the photoelectric process (the electron shot noise) and the preamplifier noise. For the measurements presented the signal was large enough that the preamplifier noise was negligible. Because the number of photoelectrons in one pixel for several measurements of identical light levels has a Poisson distribution, the electron shot noise ad for one measurement is
%d=~ *
(3)
The noise in one pixel is independent of the noise in another. In addition to the noise in the CCD measurement, there is a random variation ci associated with the temporal incoherence of light. If a pixel in the detector is smaller than a spatial coherence zone, then ci=Ne (z/T)"~9
(4)
where T is the laser pulse length, t is the coherence time, N, is the average number of photoelectrons in the pixel measured in time T,and Oi is the shot-toshot RMS variation of the signal in electrons. In this case, there is correlation in the measurements of adjacent pixels. If on the other hand there are N, coherence zones in a pixel, then Gi=Ne (~lNcT)"~e
(5)
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In this case, the random variation for adjacent pixels is independent. The bandwidth ( E l/7) of the oscillator has been measured to be 1.6 THz, which gives a coherence time of 0.6 ps. Because the random variation associated with the incoherence is not correlated with the CCD measurement noise, the total random variation a, in electrons of the number of electronics in a pixel is given by 2 6,
++t&. -
(6)
Because a typical CCD exhibits a nonuniform response to light, the camera was calibrated by illuminating it with a uniform light source (fig. 5). The oscillator was used as a light source for a commercial integrating sphere. This has the advantage of calibrating the camera for the same wavelength and approximately the same pulse length used in the measurements. The camera was placed N 1 m from the output aperture of the sphere. At that distance, the variation in light across the CCD surface from a lambertian surface is theoretically 1.5 x 10 -‘. In an actual setup, the variation will probably be greater; the camera and integrating sphere might not be perfectly aligned, there might be spurious reflections, and the light output from the integrating sphere might not be perfectly uniform. In order to test the calibration technique, the camera was calibrated at different positions. The distance between the camera and the integrating sphere was varied from 0.4 m to 1.6 m, the camera was moved f 3 mm perpendicular to a line between it and the sphere, and the camera was rotated about that line. The greatest change in the calibration was a 0.3% spatial tilt across the 8.83 mm width of the sensor. This corresponds to a systematic error of less than 0.1% in the calculation of the tilt of a 2 mm diameter profile.
During the calibration, there will be the random variation in the measurements given by eq. (6). There were several coherence zones in each pixel, so eq. (5) was used to calculate the random variation due to the incoherence of light. The size 6 of a coherence zone at the CCD is approximately r3=Al/Dl,
(7)
where 1 is the wavelength of light and 1 is the distance between the CCD and the integrating sphere output aperture, which has a diameter 4. For A= 248 nm, I= 1 m, and DI=5 cm, 6 is 4.96 urn. Using the pixel size from the table, the value of N, (to be used in eq. (5)) is (23 um/4.96 pm)2=21.5. In order to measure the noise, the output from the integrating sphere was measured 64 times. After correcting for variation in’the total energy falling on the CCD, the noise-to-signal ratio (0,/N,) for a given pixel was calculated. At a signal level of 6.6 x lo4 electrons, the result ranged from 2.5 x 10m3 to 3.0 x 1O-‘, compared to an expected value (from eq. (6)) of 3.8~10~~ (T=30 ns, 7~0.6 ps, and N, = 2 1.5 ) . This indicates that the camera is capable of measuring the profiles with low noise. The calibration was determined by summing 64 measurements (with the background subtracted) of the integrating sphere output (flat fields). The average signal level was 1.5 x 1O5 electrons, or 3825 counts. The calibration factor Ci for pixel i is given by
where NP is the number of pixels on the CCD, and J is the summed flat field signal at pixel i. Because & is a sum of 64 measurements, the relative error of Ci calculated by eq. (6) due to the error in fi: is reduced by a factor of 8 from 2.8~ 10m3to 3.5~ 10M4. The relative energy ei falling on pixel i is given by ei=Cj(si-bi)
Fig. 5. Calibration of the camera. The oscillator illuminates an integrating sphere. A negative lens (not shown) is used to spread out the input light to avoid damaging the integrating sphere. A tube (not shown) is used in order to reduce stray light. Apertures inside the tube are necessary in order to reduce stray reflections.
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,
(9)
where Si and bi are respectively the signal and background at pixel i. The signal levels of the profiles presented in the next section are 6.0 x 1O4electrons, or N 1500 counts, while the background levels (which are due to the camera) are _ 50 counts. This implies that the RMS measurement error due to the electron shot noise and the calibration error is 0.4% for one 117
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pixel. When observing longer scale length focal profile nonuniformities that cover many pixels, one can reduce this error by averaging over more than one pixel.
4. The profiles The object aperture is imaged onto the camera through a lens, the Fourier aperture, the pulse slicing system, a demagnifying telescope, and focusing lens. A schematic of the experimenal arrangement without the demagnifying telescope is shown in fig. 4. The telescope is located before the second lens, which focuses the beam onto the camera after attenuation by two reflections by uncoated surfaces and transmission through a 95% flat reflecl.or. The telescope demagnifies the image of the object aperture from 3 mm to 2 mm diameter. The measured profiles were analyzed with the following algorithm: (i) Determine the centroid (x,, yc) of the profile. (ii) Find the edges of the flat region of the horizontal and vertical cross-sections through the centroid. (iii) For the flat region of each cross-section: (a) Perform a linear least-squares fit, and calculate the variation of the fit from a Sat-top (tilt). (b) Perform a quadratic least-squarer, tit, and calculate the mean-to-peak variation of the fit from a flat-top. (c) Calculate the RMS deviation of the measured crosssection from the fits. The centroid is determined by (10)
where Xi is the x position of pixel i. The calculation of y, is analogous. The edges of the flat region of a cross-section is determined by the following algorithm: (i) Find the pixels 1 and I’ where the measured value of the cross-section is just greater than a specified fraction fp of the maximum value. (ii) The edges of the flat re:gion at t pixels from 1 and r towards the center of the cross-section. For example, if 1is the pixel on the left edge of the profile, and pixel numbers increase from left to right, the the 118
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flat region is between pixels I+ t and r- t, inclusive. For the results presented,f, was 0.25, and t was 5 (the cross-section of the profile is 85 pixels across). The results do not depend strongly on& or t. Three aspects of the profiles were evaluated. First, the 4 ns pulse was taken at different times during the oscillator output to check whether the object aperture illumination uniformity changed in time. It was found that the tilts of the profiles do not vary appreciably during the oscillator output. Second, a series of profiles were taken at the same time during the oscillator output to determine the shot-to-shot variation of the profiles. It was found that the observed shot-to-shot RMS variation of the tilts was consistent with the spatial and temporal incoherence of the light. It was also found that the variation of the linear fit from a flat-top profile (a measure of tilt) averaged over the series was f 0.5%, and of the quadratic fit (a measure of peaking or concavity) was + 1.4%. Finally, the Fourier aperture was partially blocked to simulate the effects of a nonuniform gain in an amplifier. The profiles were found to be insensitive to this partial blocking; blocking half the Fourier aperture changed tilts of the profile by only a few percent. Figure 6 shows a typical profile and its cross-sections. The RMS deviation of the measurement from the linear least-squares fit is 1.0% along the vertical direction and 0.8% along the horizontal. If the protile were flat, the deviation would be the same as the total random variation aY (given by eq. (6 ) ) in each datum. Because a pixel is smaller than a coherence zone, eq. (4) can be used to calculate the variation due to the incoherence of the light. With a coherence time of 0.6 ps, a pulse length of 4 ns, and a signal of 6 x 1O4 electrons, the expected RMS relative variation of each datum is 1.3Oh.This indicates that the observed deviation is probably due to the CCD shot noise error and the variation of the light energy due to the laser incoherence and the finite averaging time. Figure 7 shows the tilts of the cross-sections as a function of time during the oscillator output. Eight images were recorded for seven times spaced 5 ns apart during the oscillator output, for a total of 56 images. Each point on the graphs is the tilt of the horizontal or vertical cross-section of one image. Note that the time variation is negligible. Therefore, the
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Tilt
5L
Along
y
I
20
0 time Tilt
40
(nsec) Along
x
1 40
-5t 20
0 time
-,,,,,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,
Fig. 7. Tilts of the oscillator output.
cros5sections
(nsec) a function of time during the
as
Tilt Along y L 12
0
2
4
0 -2
-1
0
1
2
x (mm)
~
;,
-ynT:,
-4
-2
,
0 tilt
2
4
(%)
Fig. 8. A histogram of the tilts of the linear fits to the profdes with the Fourier aperture completely unblocked.
-2
-1
0
1
2
Y ‘(mm) Fig. 6. A typical profile and its cross-sections: (a) threedimensional view, (b) horizontal cross-section through the centroid, and (c) vertical cross-section through the centroid.
output of the oscillator does not vary appreciably when averaged over 4 ns. Figure 8 is a histogram of the tilts for 64 images.
Note that the average tilt is very nearly zero; it is - 0.75% along the vertical direction, and 0.75% along the horizontal. The RMS variation of the tilt is I%, which is close to the expected value of 0.7% (this is calculated in the appendix). The RMS variation of the tilt of a single beam of fig. 3 is N 5% along the vertical direction and u 1% along the horizontal. The larger variation along the vertical direction for a single beam indicates that the telescopes shown in fig. 3 are effective in reducing the shot-to-shot variation of the tilt. Furthermore, the average tilt of a single beam is 5 to lo%, which indicates that the telescopes 119
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are also effective in reducing the average tilt. Figure 9 is a histogram of the mean-to-peak variation of the quadratic fit from a flat-top profile for the 64 images. The mean averaged over the images is + 1.4% along the vertical direction, and + 0.6% along the horizontal. The variation is larger along the vertical direction because there is more shot-to-shot variation of the concavity of the profile, which is not reduced by the telescopes. Figure 10 shows the tilts of the cross-sections with different parts of the Fourier aperture blocked in or-
Variation 101
Along y I
-
I
:!“I? 0
1
IrG
2 variation
Variation 81
3
-I
Along
4
5. summary
x
I
3
2
1
variation
4
(%)
Fig. 9. A histogram of the mean-to-peak variation of the quadratic tits from a flat-top profile with the Fourier aperture completely unblocked.
Tilt -
Along
y
207i
-g
0
z
3
-20 open fourier
top
right
blocking
Tilt
bottom
left blocked)
(half
Alone
x
-1oF
3 open fourier
right bottom left top blocking (half blocked)
Fig. 10. Variation of the tilts of the cross-section when different parts of the Fourier aperture are blocked.
120
der to simulate nonuniform amplifier gain. Sixteen images were recorded with (i) the Fourier aperture completely unblocked, (ii) the top half of the Fourier aperture blocked, (iii) the right half blocked, (iv) the bottom half blocked, and (v) the left half blocked, for a total of 80 images. Note that the average tilt changed by a few percent. Note also that when the top or bottom of the Fourier aperture was blocked, the shot-to-shot variation of the tilt of the vertical cross-section increased, because different parts of the inverted and noninverted beams were blocked. The two beams were not exact inverses of each other, and the the tilt of one did not cancel the tilt of the other. This larger shot-to-shot variation does not occur in the horizontal direction because the angular divergence of the oscillator output in the horizontal direction is larger than the divergence in the vertical direction.
(%)
-’
0
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A laser oscillator system has been developed with a 4 ns pulse output, 3000 to 1 energy contrast ratio, and flat focal profile. Its angular divergence is large enough to produce 60 times diffraction-limited images. The coherence time was 0.6 ps, and the pulse duration was 30 ns. The focal images of 4 ns slices of the beam had the desired flat cross-sections (tilts of N 1%). The shape of the focal profile should not depend strongly on the gain profiles of the laser amplifiers if they are placed at the Fourier aperture of the EFISI system. This was tested by partially blocking the Fourier aperture and observing that the shape of the focal profile was not strongly affected (the tilts were typically on the order of a few percent). The top, bottom, left, and right halves of the Fourier aperture were blocked. Partially blocking the Fourier aperture simulates an extremely nonuniform amplifier gain; typical nonuniformities of actual amplifiers used in the NIKE laser are only approximately 20%. The oscillator system described here comes close to fulfilling the goals for the NIKE system. The existing system is now being used to test the ability of the NIKE system to maintain uniform focal profdes after several stages of amplification. In future work, we will attempt obtain more nearly uniform beams
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which are less affected by partial blocking of the Fourier aperture. We will also attempt to produce more than 60 times diffraction limited beams, so that phase aberrations in the laser system will have even less effect on the focal profile.
Acknowledgements
We would like to thank our colleagues for useful discussions and assistance: Mark S. Pronko, Robert H. Lehmberg, Thomas Lehecka, Carl J. Pawley, Warren Webster, Julius Goldhar, and Andrew J. Schmitt. The data was archived on disk using a modified version of National Center for Supercomputer Activities’ Hierarchical Data Format. This work was supported by the United States Department of Energy.
Appendix
This appendix presents a calculation of the variance of the slope of the linear least-squares tit. The variance is determined by the variance of the data used to calculate the slope. The random variation of each datum in the measured profile is due to both the CCD measurement noise and the random variation associated with the temporal incoherence of the light. The CCD measurement noise is due mainly to the electron short noise, and is given by eq. ( 3 ) . The random variation due to the incoherence is given by eq. ( 4) or ( 5 ) . The total random variation a, (given by eq. (6) ) causes the calculated slope m to have a variance al. In order to calculate a,, let ya be a set of Nr random variables, and yai be the value of sample i of ycl. Let ‘x, be a set of Nr numbers. The y, are the measurements at position x,; Nf is the number of pixels across the profile. Define
(lla)
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(lid) where N is the number of profiles measured. Note that the X~ can be chosen so that /3r is zero. Note also that in this case, eq. (1 Id) can be approximated by an integral. Let L be the diameter of the profile, and let Ax=L/Nfi Then 82= $&:A-?
(12)
L/2 Z-
N r L
r
J -L/2
=NfL2/12.
X2dX
(13)
(14)
The variance will be calculated by first using the standard formula for the slope of the linear leastsquares tit to a set of data. The variance will depend on &. However, a$ will be negligible if the pixels cr and /3 are not close together, because the size of a coherence zone is approximately the same size as a pixel. Using a rough approximation, the ratio a,/~, is calculated. The slope m of the linear least-squares fit to the data pairs (x,, ya) is a random variable given by (after setting j?r= 0): (15)
Let a; be the variance of m. Because x, is a number (rather than a random variable),
(16) For the results presented, there are 60 coherence zones and 85 pixels across a profile; each coherence zone spans approximately one pixel. In addition, ci- 750 electrons (from eq. (4), with T=4 ns, r=O.6 ps, and N,=6.0x104 electrons), while ad-250 electrons (from eq. ( 3 ) ). The total variation at one pixel is therefore (eq. (6 ) ) 790 electrons, which is a relative variation of 1.3 x 10m2. Because the variation due to the measurement ( occa) is independent for each pixel, while the variation due to the incoherence of light (ci) is somewhat correlated, the rather arbitrary assumption can be made: 121
Volume 106, number 1,2,3
a* -u* a/3-
aor,
OPTICS COMMUNICATIONS
References ifa=j3, [l] S.E. Bodner, J. Fusion Energy 1 (1981) 221.
= ko:, , if la-jIl=l, =o,
otherwise .
(17)
Equation ( 16 ) then becomes - L : (x:+kx~_,x,+kx,+,x,) oL- /!I: a=1
&a,
(18)
where x, = 0 for (Y= 0 and cy= Nf+ 1 so that there is no contribution from the norrexistent points 0 and Nr+ 1. Because the signal level is nearly the same for all points on the profile, the variances of each of the ya will all be approximately equal: (19)
(21) Approximating /3*by N,L*/ 12, this becomes
(22) For a 2 mm diameter image with Nf=85 and a,/aY=0.27/mm. Wilth 0,/N,= 1.3x lo-*, a,/N,=3Sx IO-‘/mm, or a 0.7% shot-to-shot RMS variation of the tilt of the p:rofile. With k=0.75, u,,Juy=0.30/mm, and with k=0.25, am/a,= 0.23/mm. This indicates that the value of k does not strongly affect the RMS variation of the tilts. kE0.5,
122
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[2] R.H. Lehmberg and S.P. Obenschain, Optics Comm. 46 (1983) 27. [ 31 R.H. Lehmberg and J. Goldhar, Fusion Tech. 11 (1987) 532. [4] Ximing Deng, Xiangchun Liang, Zezun Chen, Wenyan Yu and Renyong Ma, Appl. Optics 25 (1986) 377. [ 5 ] A.J. Schmitt and J.H. Gardner, J. Appl. Phys. 60 ( 1986) 6. [ 61 S. Skupsky, R.W. Short, T. Kessler, R.S. Craxton, S. Letzring and J.M. Soures, J. Appl. Phys. 66 (1989) 3456. [7] D.G. Colombant and A.J. Schmitt, J. Appl. Phys. 67 ( 1990) 2303. [8] Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka and C. Yamam&a, Phys. Rev. Lett. 53 ( 1984) 1057. [9] D. Veron, G. ThieIl and C. Gouedard, Optics Comm. 97 (1993) 259. [ 1OlD.M. Penington, M.A. Henesian, H.T. Powell, C.E. Thompson and T.L. Weiland, Conf. on Lasers and electrooptics, 1993, Technical Digest Series, Vol. 11 (1993). [ 111 H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Am&i, T. Jitsuno and S. Nakei, J. Appl. Phys. 73 (1993) 2122. [ 121 S.P. Obenschain, J. Grun, M.J. Herbst, K.J. Keamey, C.K. Manka, E.A. McLean, A.N. Mostovych, J.A. Stamper, R.R. Whitlock, S.E. Bodner, J.H. Gardner and R.H. Lehmberg, Phys. Rev. Lett. 56 (1986) 2807. [ 131 J. Grun, M.H. Emery, C.K. Manka, T.N. Lee, E.A. Mclean, A. Mostovych, J. Stamper, S. Bodner, S.P. Obenschain and B.H. Ripen, Phys. Rev. Lett. 58 (1987) 2672. [ 141 A.N. Mostovych, S.P. Obenschain, J.H. Gardner, J. Grun, K.J. Keamey, C.K. Manka, E.A. McLean and C.J. Pawley, Phys. Rev. Lett. 59 (1987) 1193, [ 151 S.P. Obenschain, C.J. Pawley, A.N. Mostovych, J.A. Stamper, J.H. Gardner, A.J. Schmitt and S.E. Bcdner, Phys. Rev. Lett. 62 (1989) 768. 1610. Willi, T. Afshar-rad and S. Coe, Phys. Fluids B 2 ( 1990) 1318. 171 T.A. Peyser, C.K. Mat&a, S.P. Obenschain and K.J. Keamey, Phys. Fluids B 3 ( 199 1) 1479. 181 D.K. Bradley, J.A. Delettrez and C.P. Verdon, Phys. Rev. Lett. 68 (1992) 2774. 191 C.J. Pawley et al., Proc. Intern. Conf. on Lasers ‘90, eds. D.G. Harris and J. Herbelin ( 199 1) p. 491. [20] M.S. Pronko et al., Proc. Intern. Conf. on Lasers ‘91, eds. F.J. Duartes and D.G. Harris (1992) p. 691. [21] S.E. Bodner et al., Fourteenth Intern. Conf. on Plasma physics and controlled nuclear fusion research, IAEA-CN56/B-2-1(C), (1992).