A high-resolution, high-precision beam-shaping system for high-power-laser systems is demonstrated. A liquid-crystal-on-silicon spatial light modulator is run in closed-loop to shape laser-beam amplitude and wavefront. An unprecedented degree of convergence is demonstrated, and important practical issues are discussed. Wavefront shaping for the applications in OMEGA EP laser is demonstrated, and other interesting examples are presented.
©2010 Optical Society of America
The design and applications of complex spatial filters is an active area of research, especially in Fourier transform holography and pattern recognition. Complex filters can be realized by binary masks , phase-only carrier modulation , parity sequences , or phase-only pseudorandom encoding . Recently, similar principles have been applied to laser-beam shaping. Noticeable among these are the methods based on error-diffusion algorithm [5,6] and carrier filtering . The importance of controlling the laser-beam amplitude and wavefront in high-power lasers cannot be overemphasized. There are important applications such as gain precompensation  and hot-spot suppression . Diffraction-limited focusing is an important subject and numerous schemes have been applied [9–12]. As the beam size increases in higher-power systems, these applications require high-spatial-resolution control of the beam shape and dynamic adaptability to system change . Current state-of-the-art, high-power laser systems require a practical beam-shaping system beyond a proof-of-principle demonstration. As with adaptive optics, a practical control system is driven by closed-loop feedback. In this paper, we discuss an adaptive beam-shaping system based on a liquid crystal modulator that allows extremely fine control of a laser beam both in amplitude and wavefront.
It has been shown that a single spatial light modulator (SLM) together with a spatial filter can simultaneously control laser-beam amplitude and wavefront . This scheme—a phase-only carrier method—makes it possible to have point-by-point control of a laser beam’s amplitude and wavefront. The principle is to carry away unnecessary local energy by high-frequency phase modulation, which is spatially filtered. The initial experimental result showed a relative root-mean-square (rms) error as small as 8.5% between the measured intensity and the goal intensity . This result left room for improvement. Several issues are addressed for closed-loop implementation of this method: (1) the inherent wavefront error of the spatial light modulator, (2) energy fluctuation of laser pulses, (3) spatial registration error between the SLM and the sensor, (4) digitization effect of the SLM command resolution, and (5) wavefront accuracy. After discussing the solutions to these problems in Secs. 2–6, the algorithms will be presented in Sec. 7. In Sec. 8, experimental results will be shown in a continuous-wave (cw) source and pulsed-source test-bed setup. Field shaping (both amplitude and wavefront) is demonstrated with excellent convergence as well as amplitude-only and wavefront-only shaping. In the application section, high-spatial-frequency wavefront shaping is demonstrated for application to the OMEGA EP laser. The term “wavefront shaping” is employed as opposed to “adaptive optics” to emphasize that the objective wavefront is not necessarily a flat wavefront.
2. Static wavefront error correction
Many SLM devices have wavefront defects. The SLM used in our experiment is a liquid-crystal-on-silicon (LCOS) SLM from Hamamatsu (model X10468). It operates in a reflective scheme, and the surface area is 12 mm × 16 mm with 600 × 792 pixels. The pixel size is 20 μm with a 95% areal fill factor. The phase retardation in each pixel is controlled by 8-bit command. The maximum stroke of the SLM is 2 μm per reflection at 1.053 μm, custom modified from its original 1-μm dynamic range. This modification reduces the digitization level per wave from 8 bit to 7 bit. The peak-to-valley of the SLM surface wavefront error is 2.1 waves at 1.053 μm per reflection. Self-correcting the wavefront consumes almost all of its dynamic range. Although phase wrapping can bypass this problem to some degree, the phase discontinuity across the wrap usually results in a noticeable modulation in the laser beam as well as complicating the closed-loop algorithm. A simple solution is to compensate the wavefront error with a static corrector. Magnetorheological finishing (MRF)  is an effective technique used to produce a static phase corrector. Nd:YLF laser rods in our laser system have been successfully polished over a 21-mm-diam circular aperture using this technique. The same technique is directly translatable to the 12-mm × 16-mm SLM aperture. The design and the produced MRF corrector are shown in Fig. 1 . The design map is larger than the SLM area to allow for a smooth extrapolated region. The residual wavefront is shown in Fig. 1(c). Since our requirement is over a 12-mm × 12-mm area, the extra region is masked in blue. The rms error within the use area is 0.06 waves and the peak-to-valley error is 0.36 waves on reflection. The MRF process takes one or two iterations. Since amplitude correction in the carrier method requires half-wave modulation, the available correction dynamic range is ~1 wave peak-to-valley.
3. Energy fluctuation
In iterative laser-beam shaping, the energy of the input laser beam must stay constant because the diagnostic at the output cannot distinguish the effect of beam shaping from the effect of input-energy fluctuation. This condition is not generally satisfied. Regenerative laser amplifiers, for example, have shot-to-shot fluctuations because of their sensitivity to the injection. The effect of energy fluctuation at the output beam diagnostic can be nullified by normalizing the output laser beam with an independent energy measurement. A different approach is used in this work. Two small areas, called “anchors,” are selected within the region of interest and the iteration proceeds in two steps: In the first step, the program does not control the beam over one of the anchors. It is then guaranteed that any change in the integrated energy inside the first anchor comes from energy fluctuation only, not from beam shaping. The rest of the beam is rescaled according to the measured energy change in the first anchor. When the extra-anchor region converges within the criterion (<2% relative rms, see Sec. 8), the program proceeds to the second step in which the energy of the beam is rescaled using the second anchor area. In the second step, most of the beam-shaping action occurs over the first anchor area. There are subtle differences between the first step and the second step. In the second step, the beam can be shaped over the anchor area, whereas this is not the case in the first step. This difference in the second step erases the slight discontinuity over the boundary of the anchors.
4. Spatial registration and interpolation
In an ordinary adaptive optic loop, the influence functions of each deformable mirror (DM) actuator are characterized beforehand. As the number of actuators increases [as in our SLM (600 × 800)], precharacterization becomes impractical. The phase retardation of an SLM is quite predictable with respect to its command voltage. This suggests a control mode where the wavefront map is directly imposed on the SLM. It is called “direct” method in contrast to “indirect” method, which uses unprocessed raw slopes . Small errors associated with any nonlinearity in the SLM response are removed by an iterative correction scheme. In contrast to the precharacterization approach where the spatial registration between the SLM pixels and the wavefront sensor is automatically resolved, the spatial registration has to be separately characterized in a direct zonal control mode. With a reasonably good imaging system assumed, the characterization of linear transformations such as relative rotation, translation, and magnification factors, should allow the zonal method to work.
Table 1 shows differences in the approach of the wavefront closed-loop control in published works. The indirect method is more efficient in that the wavefront reconstruction step can be omitted, whereas the influence matrix approach is not suitable for devices with a large number of control points.
The effects of spatial misregistration are shown in Fig. 2 . A staircase beam-shaping process is simulated with different spatial registration errors. In the case of rotation error [Fig. 2(d)], ripples appear in the radial direction around the beam edge. Similarly small errors in the estimation of magnification factors or translation cause ripples in the beam. The simulations suggest that the spatial registration should be less than half-pixel of the measurement system. Numerical optimization of the linear transformation parameters has successfully prevented ripple artifacts. A grid pattern imposed on the SLM is compared with the measured pattern, and the difference is minimized by optimizing the parameters.
5. Digitization effect
The wave retardation on the SLM is controlled with an 8-bit signal. With a 2-wave dynamic range, the control provides only 64 steps over the half-wave range, which covers 0% to 100% transmission. As the steps are controlled with balance to preserve the average value of the modulation (Sec. 7), the actual available steps are reduced to 32. In terms of wavefront control, 0.008 waves/step (2 waves/256 steps) is the maximum resolution. Some digitization effects are unavoidable in either intensity or wavefront shaping. A measured example in flat wavefront shaping shows this effect in Fig. 3 . A similar effect is observed in intensity shaping. These problems can be handled with dithering, which adds artificial noise in the command map before it is digitized . The information lost through digitization is encoded in the local mean value of noise. The purpose of dithering is cosmetic in most cases. The amount of dithering should not be excessive in order to minimize high-spatial frequency error. Adding Gaussian random noise with a standard deviation of 0.002 waves was found to be sufficient to mask the digitization effect [Fig. 3(b)]. The random noise spreads the focal spot size by 1%. The discontinuity in phase may turn into amplitude modulation on propagation. Assuming 8-bit digitization, a numerical simulation suggests that the relative change in intensity is estimated to be 5% on free-space propagation equivalent to the Fresnel number 1000, compared with continuous phase control.
6. Wavefront accuracy
Most adaptive optic systems are designed to produce a flat wavefront. This mode of operation does not require the reconstructed wavefront to be accurate because the wavefront converges to zero. In more general cases, the wavefront may be controlled to an arbitrary shape. There is an important application that precompensates laser-beam wavefront into a non-flat wavefront. Precompensation becomes more important as the aperture size and laser power increase where there are few options available for implementing a large-aperture, high-resolution deformable mirror. Since the iteration process uses a wavefront map instead of an influence matrix of slopes (Sec. 4), it is important to provide an accurately reconstructed wavefront. In the case of wavefront sensors based on slopes or phase-differences measurement, it has been shown that the frequency response of wavefront reconstruction has larger errors at higher spatial frequencies [17,18]. Since the band-limited reconstructor  is known to have unity frequency response regardless of the spatial frequency, this specific reconstructor was used for high-resolution wavefront shaping. Figure 4 illustrates the reconstruction-dependent frequency-response effects.
The main results of  are summarized here for the discussions that follow. The phase-only carrier method modifies the input field as
A closed-loop implementation of the carrier method will be presented for amplitude and wavefront shaping. The main flow of the amplitude-shaping algorithm is shown in Fig. 5(a) . The measured fluence at nth step, Fn, is renormalized to according to the anchoring technique introduced in Sec. 3. is interpolated into the reference space using the linear transformation, resulting in . The reference space is set to the spatial light modulator space. In the next step, is compared with the previous fluence map, , and the objective map in the reference space, . The nth envelope command cn is created based on these comparisons. If the incremental difference is within the measurement noise or digitization limit of the SLM, the algorithm stops. Otherwise, it repeats with the new modulation depth
The wavefront shaping, as sketched in Fig. 5(b), starts from wavefront measurement Wn. Similar to the energy renormalization step in fluence shaping, the piston of the wavefront is adjusted to a fixed value (i.e., the mid-point of the two extremes of the SLM dynamic range) at a fixed point in space because the piston value is indeterminate in the reconstructed wavefront. The fixed point is selected such that the required wavefront retardation in the SLM will fall within its control capability. The next step is to convert the piston-adjusted wavefront into the interpolated wavefront in the SLM space The zonal reconstruction wavefront map occasionally has a noisy edge that causes the algorithm to diverge near the edge. The edge noise is smoothed at this step. Alternatively, the wavefront map can be stabilized using modal reconstruction. The next step is to modify the command map bias as follows:
Dithering noise is added to the command map at each step either in the fluence or in the wavefront-shaping algorithm. Since the phase bias f and the modulation depth c can be controlled simultaneously, the two loops can be joined. The current system runs at 1 Hz and converges in ten iterations using a gain of 0.3. Twenty iterations are needed when using the two-step anchoring technique.
Both cw-source and pulsed-source test-bed setups have been used to demonstrate the beam shaping described in this paper. The wavelength of both sources is 1.053 μm. The cw-source setup is shown in Fig. 6 . The cw-source is expanded and collimated through divergence off the single-mode fiber tip and a singlet lens. Since the SLM works only at horizontal polarization, a thin-film polarizer is inserted. The beam is truncated with a square apodizer and sent to the SLM. The static phase corrector is placed directly in front of the SLM. The reflected beam is redirected by a beam splitter and imaged to the Shack–Hartmann wavefront sensor (Imagine Optic). The sensor has a 133 × 133 lenslet array with a 114-μm pitch. The wavefront sensor provides an intensity map of the integrated intensity per lenslet. The SLM plane is imaged to the wavefront sensor by a spatial filter. The carrier period is set at 80 μm, corresponding to 4 pixels. The initial beam profiles are shown in Fig. 7 . Four examples of amplitude shaping are shown in Fig. 8 . The degree of convergence is quantified using the relative rms metricFigs. 8(e) and 8(f)], where the change is 0.05 waves rms. The change in measured wavefront in this case appears to be caused by the centroid shift in the Shack–Hartmann sensor, associated with a rapid intensity slope around the edge, rather than the effect of the beam-shaping system. Large slab amplifiers have less gain toward the edges due to the bleaching of gain by amplified spontaneous emission. Gain precompensation is needed to compensate for this effect .
Wavefront shaping is demonstrated in Fig. 9 . The wavefront can be controlled to pixel-scale precision of the wavefront sensor. Figures 9(a) and 9(b) show the shaping of a small Gaussian wavefront spot. The lineout of the experimental map agrees well with the design at each point. Shown in (c) and (d) of Fig. 9 is another example of shaping a defocus term. The discrepancy between the objective and the achieved wavefront is less than 0.003 waves rms. The ultimate spatial-frequency bandwidth of wavefront control is limited by the wavefront-sampling resolution, assuming the SLM has a much higher spatial resolution. The number of points that can be pixelwise controlled is 17689 (133 × 133) in the current setup, an order of magnitude higher than the currently available commercial deformable mirrors (kilo-DM, 1000 actuator MEMS DM, Boston micromachines). Since the wavefront-sensing resolution can be increased with a higher-resolution Shack–Hartmann or with different wavefront-sensing techniques, this approach is a promising technique for high-resolution adaptive optics. The amplitude change attributable to wavefront shaping is characterized using relative rms with the two fluences in Eq. (7) defined differently. F 1 and F 2 are now beam fluences after and before shaping, respectively. The change in beam fluence after wavefront shaping is less than 4%.
In the case of shaping the wavefront and amplitude simultaneously, the convergence is comparable to the amplitude-only or wavefront-only shaping results. An example of shaping a flat amplitude and wavefront is shown in Fig. 10 . Figures 10(a) and 10(b) and Figs. 10(c) and 10(d) are fluence and wavefront maps of the same beam. The degree of fluence convergence is 0.8% in relative rms, and the wavefront agreement between the objective and the measurement is 0.002 waves in rms, excluding the tilt terms. The residual tilt terms come from thermal fluctuations.
The setup for a pulsed-laser source is shown in Fig. 11 . The expanded and collimated beam from a Nd:glass regenerative amplifier is truncated by a square apodizer. The apodizer plane is imaged to the SLM through the two Keplerian telescopes. The reflected beam off the SLM is redirected by the polarizer through the Faraday isolation scheme and imaged to the wavefront sensor. The pinhole in the second telescope filters high-order harmonics for beam shaping. The initial beam fluence and wavefront profiles are shown in Fig. 12 . The beam-shaping loop was run to produce a flat amplitude and wavefront beam either with or without the static phase corrector; Fig. 12 corresponds to the case without the corrector and the results are shown in Fig. 13 . The degree of fluence convergence is 7% and the rms difference between the goal and measured wavefront is 0.013 waves. The digitization effect is visible in the fluence shaping [Fig. 13(a)]. The residual wavefront error shown in Fig. 13(c) is a snapshot of the wavefront fluctuations coming from the source laser; it cannot be improved further.
The short-pulse beamlines in the OMEGA EP Laser System  at the Laboratory for Laser Energetics employ two identical deformable mirrors in each beamline, one in the multipass amplifier cavity and the other in the pulse-compression chamber. Both deformable mirrors have 39 actuators patterned hexagonally over a 38- × 38-cm area. The fitting error arising from limited actuator density of the large-area deformable mirror results in a high-order residual wavefront (0.15 waves rms) in the corrected beam, as shown in Fig. 14(a) . Because of the high-spatial-frequency content of the residual wavefront, the radius encircling 80% of the focal spot’s energy is 7× larger than that of the diffraction-limited spot [Figs. 14(b) and 14(c)]. In principle, the high-resolution beam-shaping system described in this paper can be used to precorrect for the high-spatial-frequency wavefront error that is uncorrected by the large-area deformable mirror. This concept was tested in the pulsed-beam test-bed by correcting the wavefront error shown in Fig. 14(a). The residual wavefront, shown in Fig. 14(d), has a residual rms error of 0.015 waves, or about 10× correction.
One of the appropriate locations for this beam-shaping system is in the front-end laser source [optical parametric chirped-pulse amplification (OPCPA)], where the beam size is ~1 cm and the energy density is less than 200 mJ/cm2 at 1.053-μm wavelength. The SLM’s damage threshold has been measured on samples from Hamamatsu with 2.4-ns square pulses at 5 Hz—the same pulse width and repetition rate of the OMEGA EP OPCPA pulses. The preliminary damage-test results are encouraging. No damage was observed at 300 mJ/cm2 for 14 h at a site, the damage threshold of which was 1 J/cm2. A separate radius-of-curvature measurement at 100, 150, and 200 mJ/cm2 running at 5 Hz indicates no change in the liquid crystal birefringence due to power increase within error bars. All these measurements are over a small portion of the area (~0.5-mm spot size) under electronically inactive conditions; they must be interpreted with caution. The other expected challenges for implementing the system into OMEGA EP are phase-to-amplitude conversion effect at an intermediate plane as well as imperfect imaging conditions such as axial astigmatism  and high-order image distortion. The former might impose constraints on the maximum slope in the wavefront shaping and the latter could result in decrease in the spatial resolvability.
Closed-loop operation of a beam-shaping system has been successfully demonstrated using the phase-only carrier method. The key elements of this system are a high-resolution spatial light modulator, a static phase corrector, and a high-resolution wavefront sensor as well as a spatial-filtered image relay. The issues of SLM’s wavefront defect, energy fluctuation, spatial misregistration, digitization, and wavefront-measurement accuracy were discussed and solutions were presented. The algorithms for achieving amplitude and wavefront shaping were laid out based on these considerations. Many examples proved that the spatial resolvability and precision of the system are excellent. This technique is promising for overcoming many challenges in high-power lasers, such as gain precompensation, hot-spot suppression, damage-spot shadowing, and high-order wavefront correction. We also expect that the current system will find many other interesting applications in low-energy lasers.
The authors thank Hamamatsu Photonics for their generous donation of damage test samples and N. Mukozaka for technical discussions. S.-W. Bahk gratefully acknowledges much help from I. A. Begishev, R. G. Roides, and C. Dorrer at the Laboratory for Laser Energetics. 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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