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Target-plane intensities on the short-pulse beamlines of OMEGA EP, a petawatt-class laser, are characterized on-shot using the focal-spot diagnostic (FSD), an indirect wavefront-based measurement. Phase-retrieval methods are employed using on-shot and offline camera-based far-field measurements to improve the wavefront measurements and yield more-accurate, repeatable focal-spot predictions. Incorporation of these techniques has improved the mean cross-correlation between the FSD predictions and direct far-field fluence measurements in the target chamber from 0.78 to 0.94.
©2012 Optical Society of America
1. Introduction
Intensity distribution at the target plane is a critical parameter in many experiments conducted on high-energy laser systems. However, measurement of the focal-spot intensity on a high-energy shot at the target plane is a challenging task. Direct viewing of a high-energy shot at the target plane with a camera, requiring attenuation of ~1020 W/cm2 intensities, is not feasible. Other target diagnostics, such as x-ray pinhole cameras, have poor resolution and do not provide a true measure of focal-spot intensity because of the complexities of the target interaction. Remote camera measurements can be made with a sampled diagnostic beam, but it is difficult to account for the effect of non-common-path optics, such as the final optics that focus the beam to target and the optics for the diagnostic. On the short-pulse beamlines of the OMEGA EP laser at the University of Rochester’s Laboratory for Laser Energetics [1], a focal-spot diagnostic (FSD) using a remote wavefront-measurement system was developed to make on-shot target-plane focal-spot predictions [2]. As discussed below, this approach has been challenged by limitations in the wavefront sensor and calibration errors, both of which can lead to erroneous wavefront measurements that in turn yield poor focal-spot predictions.
Phase retrieval has proven to be an effective technique for estimating wavefront based on a series of intensity measurements [3–5] and has recently been applied to retrieve near-field phase based on offline target-plane intensity measurements in high-energy laser systems [6, 7]. Often, these intensity measurements are made simultaneously in two different planes, for example, a pupil plane and a far-field plane, or two or more far-field planes at different defocus distances. A variety of algorithms have been used to estimate the phases of the optical fields in these planes based on the rules of propagation between them. These can generally be classified into two categories: iterative transform methods and gradient-search methods [4]. In the former, an initial estimate is made of the phase profile in one plane, and the field is propagated back and forth between the planes with the magnitude of the field being replaced with the measured value and the phase being based on the calculated phase at each iteration [3]. In the latter categories, a merit function is produced and minimized by utilizing its gradient over a discrete set of variables [4, 5]. Gradient-search algorithms can utilize search parameters that are point by point (independently varying the wavefront at discrete points over the planes of interest) or modal (modifying the coefficients of a modal expansion of the wavefront).
In this paper, we demonstrate the application of various phase-retrieval techniques to improve the target-plane focal-spot predictions on OMEGA EP. As a result of this work, the FSD now produces target-plane predictions that reliably match direct measurements of the target-plane focal-spot intensity obtained at low energy (<10 mJ).
2. Overview of the focal-spot measurement
The OMEGA EP short-pulse system utilizes optical parametric chirped-pulse amplification (OPCPA) in combination with Nd:glass amplifiers to generate a high-energy, high-peak-power laser pulse [1]. Figure 1 shows a diagram of the final section of the OMEGA EP short- pulse system, indicating the location of key diagnostics relative to the beam for the on-shot focal-spot measurement. Stretched, amplified pulses propagate into the vacuum grating compression chamber, containing four tiled-grating assemblies (TGA’s) that recompress the stretched pulses to the picosecond level. Each TGA is comprised of three closely spaced grating segments that are interferometrically aligned to produce an effective meter-scale grating [8]. To prevent damage associated with illumination of the edges of the grating tiles, the short-pulse beams in OMEGA EP are apodized to produce the three-segment beam profile shown in Fig. 1. The beam then propagates to a leaky diagnostic mirror that reflects 99.5% of the beam fluence into a main beam, while the remainder is transmitted as a diagnostic beam. The main beam is transported to the target chamber via a series of mirrors and focused by an f/2 (with respect to the beam diagonal) off-axis parabolic (OAP) mirror. The diagnostic beam is provided to a suite of laser diagnostics, the short-pulse–diagnostic package (SPDP).
Fig. 1 Diagram of final optics in the short-pulse OMEGA EP system showing focal-spot diagnostics. WFS: wavefront sensor; CCD: charge-coupled device
The FSD utilizes a high-resolution Shack–Hartmann wavefront sensor (HASO 128, Imagine Optic SA [9]) installed in the SPDP, which is denoted FSD WFS in Fig. 1. The FSD wavefront sensor provides a measurement of the (temporally and spectrally averaged) near-field amplitude and phase of the sample beam that can be numerically propagated to calculate the focal-spot–fluence distribution [10]. Of more interest is the wavefront of the main beam within the target chamber, which is obtained by adding a transfer wavefront ΔW that corrects for the non-common-path optics between the sensor and the target plane. A calibration process to measure ΔW in OMEGA EP is described in detail in Ref. 2.
In close proximity to the FSD wavefront sensor is the far-field camera—a 16-bit, cooled charge-coupled–device (CCD) camera that records the focal-spot–fluence distribution of the sample beam. Both the FSD and this far-field CCD acquire data on every shot. In addition, for some low-energy shots, a focal-spot microscope (FSM) has been inserted into the target chamber to directly image the focal spot at the target plane onto another 16-bit CCD camera [2]. These two cameras provide the direct focal-spot intensity measurements for the phase-retrieval algorithms discussed in the next section. The image data from both of these CCD’s are also used to evaluate the quality of the phase-retrieval results.
In principle, a wavefront sensor can also be provided in the target chamber, simultaneously with the FSM, to directly measure the wavefront at the target plane at low energy. This is being considered as a future upgrade that could simplify the calibration process and improve accuracy, though the engineering effort required to deploy such an instrument in a vacuum environment is substantial.
To evaluate a focal-spot measurement, the far-field distribution calculated from the FSD near-field measurement is compared to the directly measured far field from the appropriate CCD camera (either the far-field CCD or the FSM). The agreement is quantified by the following cross-correlation:
where FFSD and FCCD are the far-field–fluence distributions from the FSD and the camera, respectively, and Δx and Δy are spatial offset parameters for the 2-D cross-correlation function. The cross-correlation is normalized such that a perfect agreement results in a value of unity. In practice, a value C > 0.9 has been found to normally indicate an acceptable measurement.Initial focal-spot–intensity predictions from the FSD wavefront sensor were often quite poor, with cross-correlations typically far below 0.9. Further investigation revealed a number of causes of poor focal-spot prediction, which have been mitigated by applying phase retrieval from the on-shot, far-field CCD images and FSM images. This will be discussed in the next section.
3. Phase-retrieval applications
3.1 Differential piston error
Initially, the FSD predictions provided good estimates of the encircled energy, but the morphology of the focal spot did not agree well even with the nearby far-field CCD. One cause was the segmented nature of the OMEGA EP beam profile. The FSD wavefront sensor, being a Shack–Hartmann type, fundamentally measures the local wavefront slope. Spatial integration algorithms used to reconstruct the wavefront [11] possess an inherent uncertainty in the average phase (piston term) in each discrete segment of the beam. The resulting differential piston uncertainty significantly impacts the predicted focal spot. Fortunately, the relative average phases in the outer beam segments can be retrieved from the far-field CCD image, thereby removing the differential piston uncertainty.
To retrieve the differential piston phases in the two outer segments, one searches the resulting 2-D phase space to minimize the error between the FSD prediction and the CCD measurement. The algorithm proceeds from the near-field intensity and wavefront measured by the FSD wavefront sensor. The piston phases are added to the outer tiles of the wavefronts and the predicted far-field fluence distribution is then calculated. This is compared to the CCD-measured far-field fluence by generating the root-sum-square (rss) error,
where the fluences are co-registered and appropriately normalized. Note that this merit function has periodic boundary conditions in the piston phases, leading to an inconsequential 2π uncertainty. A grid search followed by a quasi-Newton minimization search [12] has proven to be a reliable means of finding the optimum solution for all cases encountered.An experiment was conducted wherein a narrowband laser was propagated through the beamline and the compressor to the SPDP. A mask was used to obscure varying numbers of beam segments, and six simultaneous focal-spot measurements were acquired by the far-field CCD and the FSD wavefront sensor with each configuration. Focal-spot predictions from the FSD wavefront sensor were generated with and without retrieving the differential piston phases and cross-correlated with the far-field CCD image. The results of this experiment are summarized in Fig. 2 , which shows the average and range of cross-correlations measured while illuminating one-, two-, and three-beam segments. For each configuration, the red squares and error bars were generated with the outer-segment piston phases retrieved, while the blue diamonds were generated without performing that step (optimizing wavefront continuity across beam segments). The error bars represent the minimum and maximum values over the six measurements. Note that when the average relative phase in the outer segments was not retrieved, the cross-correlation and variability became progressively worse as more beam segments were introduced. Conversely, the corrected measurements remained at a high value (>0.97 for all measurements) for all of the shots.
Fig. 2 The effect of retrieving the differential piston phase on FSD prediction. The cross-correlation between the FSD and far-field (FF) CCD focal spots for a monochromatic beam is plotted for a varying number of beam segments. The mean and range of six measurements for each configuration are displayed.
3.2 Systematic wavefront errors
A second error that can lead to erroneous focal-spot predictions is the difference between the wavefront profile at the exit pupil of the CCD focusing lens and the wavefront measured at the FSD sensor. This diagnostic wavefront error is static (assuming no system configuration changes) and has been inferred via phase retrieval from simultaneous FSD and far-field CCD measurements over multiple laser shots.
The classical iterative-transform phase-retrieval methods (the Gerchberg–Saxton algorithm [3] and various refinements such as the input–output algorithm of Fienup [4]) were attempted on this problem but were generally unsuccessful, producing unphysical discontinuities in the wavefront. However, a modal phase-retrieval method, based on the gradient-search algorithm reported by Fienup [5], has proven to be successful and robust. A block diagram of the algorithm is shown in Fig. 3 . The inputs to the algorithm are measured near-field intensity and wavefront from the FSD wavefront sensor, and measured focal-spot intensity from the far-field CCD camera. These quantities are measured simultaneously over a number of shots with varying wavefront to improve the accuracy of the final result. The algorithm produces a single low-order SPDP wavefront correction (the diagnostic wavefront error), expressed as a 2-D Legendre polynomial, fourth-order being sufficient for our purposes. By necessity, the algorithm also solves for the differential piston phases in the left and right beam segments for each measurement.
Fig. 3 Block diagram of the modal phase-retrieval algorithm used for measuring the diagnostic wavefront error.
The algorithm proceeds from the near-field intensities and wavefronts measured by the FSD wavefront sensor. The current iteration of the diagnostic wavefront contribution is added to all the wavefronts, and the piston phases are added to the outer tiles of the appropriate wavefronts. The far-field fluence distributions are then calculated from the near-field intensity and corrected wavefront. These are compared to the measured values to generate the rss errors, which are summed over all shots to produce an overall merit function.
A nonlinear optimization algorithm is employed to modify the optimization parameters for the subsequent iterations. In most cases, a quasi-Newton minimization, which is a true gradient-search algorithm, is used [12]. Alternatively, to attempt to find a global solution, a simulated annealing algorithm [13] has initially been employed to identify the approximate solution, followed by a gradient-search algorithm to obtain a more-accurate estimate. The algorithm continues until the merit function is minimized.
Similarly, there are systematic errors in the measurement of the transfer wavefront ΔW. The two-step calibration process utilizes sources on the diagnostic table and in the target chamber [2]. Wavefront errors in these sources and in optics inserted especially for the measurement can lead to static errors in the transfer wavefront.
A static correction to these errors was obtained using the modal phase-retrieval process described above with data from a number of low-energy shots in which the FSM was inserted, providing far-field fluence data at the target plane. The diagnostic wavefront error and differential piston corrections were applied to the FSD measurements on each shot using the far-field CCD data as discussed previously. The transfer wavefront measured during FSD calibration was then applied to obtain an initial estimate for the wavefront converging to the target plane. The static correction was then retrieved using the modal phase-retrieval process described above. As before, a fourth-order Legendre polynomial has been used for the transfer wavefront correction, which is now applied at the end of the calibration process.
3.3 Estimation of chromatic aberrations
A complicating factor in the development of phase-retrieval techniques for OMEGA EP is the presence of significant chromatic aberration. The full spectrum produced by the OPCPA front end is approximately 8-nm wide, and this full spectrum is present in the low-energy far-field measurements presented in this paper. On a high-energy shot, the gain spectrum of the Nd:glass amplifiers in the OMEGA EP beamline reduce the spectral width to ~3.3 nm FWHM. Despite these relatively narrow spectral widths, chromatic aberrations, especially residual angular dispersion in the stretcher and compressor, have been observed to significantly reduce the spatial bandwidth of the focal spot, smoothing the far-field features and reducing the peak fluence. To illustrate this effect, Fig. 4 displays far-field CCD images with two different laser sources being propagated through the beamline. The focal spot displayed in Fig. 4(a) utilized the narrowband laser discussed in Sec. 3.1 and is characterized by a high-contrast speckle pattern.
Fig. 4 Effect of chromatic aberrations on focusing is displayed in the far-field CCD images plotted on a logarithmic scale, using (a) a narrowband laser source and (b) a broadband (8-nm spectral width) OPCPA source.
In contrast, the OPCPA source (8-nm spectral width) was used to produce the focal spot in Fig. 4(b). This broadband focal spot appears “blurred” due to chromatic aberration, i.e., it has lost the fine-scale speckle structure observable in Fig. 4(a). Note that the reduction in peak fluence from the monochromatic case can be a factor of 2 for the full 8-nm OPCPA spectrum, although the effect is considerably less pronounced for gain-narrowed high-energy shots, as expected.
Because the FSD wavefront sensor is not spectrally sensitive, it cannot provide information on the chromatic aberration and instead produces a spectrally averaged wavefront measurement. However, the far-field CCD image has been used to estimate the degree of blurring produced by chromatic effects. By comparing Fourier transforms of the measured far-field CCD image and the far-field intensity predicted by the FSD wavefront sensor (using a monochromatic propagation model), the reduction of the spatial bandwidth due to chromatic aberration is estimated. Figure 5 illustrates the process. An initial estimate for the focal spot of the diagnostic beam, shown in Fig. 5(a) is obtained by monochromatically propagating the near-field intensity and phase (as measured by the FSD wavefront sensor) into the far field. The various phase-retrieval methods described previously are employed. The chromatically aberrated diagnostic beam focal spot is also directly measured by the far-field CCD, as shown in Fig. 5(b). Next, Fourier transforms are calculated for both of the focal spots in Figs. 5(a) and 5(b), and envelope functions that characterize the dropoff with spatial frequency are generated by smoothing the magnitudes of the Fourier transforms. A chromatic transfer function H(η,γ) is then defined,
where Ibb is the broadband fluence measured by the far-field CCD and Imono is the monochromatic fluence calculated from the FSD wavefront sensor. Note that it is important to smooth the Fourier transforms in Eq. (3) in order to remove local zeros in the denominator.Fig. 5 Estimation of chromatic effects on focal-spot structure. (a) Initial estimate of diagnostic-beam focal spot based on monochromatic propagation from near-field intensity and wavefront. (b) Measured diagnostic-beam focal spot from far-field CCD. (c) Chromatic transfer function obtained by dividing the envelopes of the Fourier transforms of the focal spot in (a) by that in (b) and averaging over a population of shots. (d) Resulting estimate of focal spot when the chromatic transfer function in (c) is applied to the calculated focal spot in (a).
Figure 5(c) displays the chromatic transfer function obtained by averaging the results of calculating data from five shots. An impulse response, generated by inverse Fourier transform of H, is convolved with the FSD far-field prediction in Fig. 5(a) to produce a focal spot that estimates the effect of chromatic aberration. This final focal-spot estimate is displayed in Fig. 5(d). Comparison of Figs. 5(b) and 5(d) shows that the blurring produced by the chromatic aberration is reproduced well, and, significantly, the peak fluence predicted in Fig. 5(d) is much closer to the actual measured peak fluence than the monochromatic estimate in Fig. 5(a). Note that this chromatic blurring step, using a typical chromatic transfer function, is now performed throughout the phase-retrieval process in order to improve convergence of the phase-retrieval algorithms.
This approach provides only an estimate of the effects of chromatic aberration on the focal spot and is, in fact, only appropriate if residual angular dispersion is the dominant mechanism. Other sources of chromatic wavefront error will affect the focal spot in a more-complex way than can be modeled with an intensity impulse response. For example, spatial chirp of the beam on the second and third gratings in the compressor caused by diffraction by the first grating produces a complex chromatic term. The surface errors in those gratings result in a wavefront error that is offset horizontally in the pupil as a function of wavelength. Since such complex polychromatic errors are not addressed with this method, this will be an area of further development.
4. Results
To evaluate the performance of the phase-retrieval techniques described in the previous section, they have been applied to a number of on-shot focal-spot measurements and compared to available direct focal-spot measurements. First, the accuracy of the sample-beam focal-spot measurement is considered by comparing the FSD result with the far-field CCD.
One representative focal-spot prediction is shown in Fig. 6 . The direct focal-spot measurement from the far-field CCD camera is indicated in Fig. 6(a). The far-field fluence calculated using the uncorrected near-field intensity and phase from the FSD wavefront sensor is plotted in Fig. 6(b) and bears little resemblance to the direct measurement. By applying the various phase-retrieval techniques, however, one can obtain the improved prediction in Fig. 6(c). The corrected FSD prediction has an improved qualitative correspondence with the direct far-field CCD measurement than the uncorrected case. The quantitative agreement is also much better, with the cross-correlation improving from 0.71 to 0.95. The prediction of the peak intensity has also improved to within 10% accuracy, compared to the initial estimate that over-predicted the peak intensity by >2 × .
Fig. 6 An example of improved focal-spot prediction of the sample beam. (a) The direct focal spot measured by the far-field CCD, (b) the initial FSD prediction (cross-correlation = 0.71), and (c) the improved FSD prediction (cross-correlation = 0.95).
The improved focal-spot measurement of the sample beam has also proven to be very reliable and stable over a large number of shots. To illustrate this, the cross-correlation was evaluated over a population of 175 shots, which spanned approximately 18 months. Figure 7(a) is a histogram showing the frequency of cross-correlation values between the FSD and the far-field CCD. The filled bars correspond to the corrected measurements, i.e., those for which all phase-retrieval corrections have been applied. The white bars give the cross-correlation values without these corrections. There is a clear improvement in performance after applying the phase-retrieved corrections, with the mean cross-correlation increasing from 0.83 to 0.96. The consistency of the measurement was also much improved, with the standard deviation of the cross-correlation reducing from 0.044 to 0.010.
Fig. 7 Histograms illustrating the effect of phase-retrieval improvements on a large population of measurements. (a) Sample-beam focal-spot accuracy showing cross-correlation between the FSD prediction and the far-field CCD measurement. (b) Main-beam focal-spot accuracy showing cross-correlation between the FSD prediction and the focal spot microscope (FSM) measurement.
Figure 7(b) shows the more-stringent test of the FSD measurement accuracy: its ability to accurately measure the focal spot remotely in the target chamber. This is a histogram of the cross-correlation between the FSD focal-spot prediction and the FSM for all the available low-energy shots. The measurement accuracy is slightly worse in the target chamber, likely due to the variability in the transfer wavefront correction from campaign to campaign. The measured cross-correlation, however, reliably exceeds 0.9 with >95% probability.
Unfortunately, confirming an accurate target-plane focal-spot prediction at high energy is currently not possible since a high-energy shot cannot be sufficiently attenuated to the level required for the FSM. We must therefore rely on the low-energy results to assess theperformance of the focal-spot measurement in the target plane. To maximize the confidence in the measurement at high energy, we account for all changes to the system configuration for these shots, e.g., attenuators inserted before the SPDP. The wavefront errors contributed by all configuration changes are measured individually off-line. On high-energy shots, the wavefront contributed by each inserted aberrator is removed from the measurement in order to correct for the potential measurement error. Nonlinear effects can also influence the target-plane focal spot at high energy in ways that are undetectable by the FSD. These are minimized by removing transmissive optics in the high-energy path to target and managing the intensities in the diagnostic path to maintain B-integral at a low level.
5. Conclusions
Phase retrieval has been a useful technique for obtaining consistently high quality remote predictions of the on-shot target-plane–fluence distribution. Phase retrieval is now used to evaluate the FSD data on every shot to provide the differential piston phase in the segmented OMEGA EP beam. A modal phase-retrieval technique based on a gradient-search algorithm was also used to retrieve correction wavefronts for the FSD that produce focal spots in consistent agreement with direct camera measurements. Further, an estimate of the effects of chromatic aberration on the focal-spot–fluence distribution is obtained from the sample-beam focal-spot image using a Fourier technique.
Analysis of a large population of on-shot measurements has proven the focal-spot measurement to be reliable. Cross-correlation with direct focal-spot–fluence measurements using the far-field CCD and the focal spot microscope consistently exceeded 0.9, although no direct measurement at the target plane at high energy is currently possible.
Acknowledgments
The authors thank Prof. James Fienup for insightful discussions on phase-retrieval algorithms. This work was supported by the U.S. Department of Energy Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-FC52-08NA28302, the University of Rochester, and the New York State Energy Research and Development Authority. The support of DOE does not constitute an endorsement by DOE of the views expressed in this article.
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