An optical surface of variable concave parabolic shape and a clear aperture of 30 mm was created using two rings to deform a flat 50.8 mm diameter mirror. The deformable mirror assembly was modeled using finite element analysis software as well as analytical solutions. Measured parabolic surface deformation showed good agreement with those models. Mirror performance was quantitatively studied using an interferometer and focal lengths from hundreds of meters down to the meter scale have been achieved. In this publication, the deformable mirror has been applied to compensate on shot thermal lensing in 16 mm diameter and 25 mm diameter Nd:Phosphate glass rod amplifiers by using only a single actuator. The possibility to rapidly change focal lengths across two to three orders of magnitude has applications for remote sensing, such as laser induced breakdown spectroscopy, LIDAR, and control of laser filament formation.
©2006 Optical Society of America
Thermal lensing plays an important role in solid state laser amplification. Due to the limited conversion efficiency of pump power to laser light, a considerable thermal load is deposited in the amplifier material. For rod amplifiers with homogeneous radial pump symmetry along the laser rod, a parabolic temperature profile is formed which leads to thermal lensing. The resulting radial change of index of refraction Δn(r) as a function of radius r inside the laser rod is :
where Q is the generated heat per unit volume, K is the thermal conductivity, and dn/dT is the change of refractive index n with temperature T. The parabolic variation of refractive index change leads to a thermal lensing effect and introduces a lens into the optical system. The rod amplifiers discussed in this paper have roughly the same flashlamp discharge time of 100–200 µs FWHM. Assuming homogeneous radial pump symmetry along the laser rod and a positive value for dn/dt and a constant flashlamp shot rate less than the amplifier cool-down time, a thermal equilibrium will be reached and a positive thermal lens will be formed. This is the most common form in which rod amplifiers are used today and several methods have been developed to deal with this issue [2, 3, 4, 5, 6, 7].
When using large aperture amplifier rods (e.g. larger than 12 mm diameter) and/or gain materials with low thermal conductivity (e.g. Nd:Phosphate glass) with the same pump symmetry as mentioned above the time between laser shots usually exceeds the cool-down time of the gain medium in order to avoid positive thermal lensing effects. When firing a single shot through a rod amplifier with no prior thermal gradient, an instantaneous negative thermal lens is observed during shot if the dn/dt of the laser material is positive. Since this effect only occurs during shot it is very difficult to compensate. In some cases, adaptive optics, using multiple actuator deformable mirrors, are used to compensate such beam distortions. Some facilities adjust the lens separation of successive relay telescopes in order to compensate the additional beam divergence . Since the beam distortions are mainly due to spherical wavefront aberration (which will be shown in the next section), we use our previously developed single actuator deformable mirrors [7, 9, 10] to create a variable focal length concave parabola that can compensate the on-shot thermal lens.
2. Description of the laser system
The 1054 nm laser system used here consists of five subsystems:
- a three stage OPCPA front end at 10 Hz that provides a 2.5 ns FWHM stretched pulse with a chirp of 4 nm/ns at a beam size of 4 mm FWHM and an energy of 30 mJ.
- a 16 mm diameter and a 25 mm diameter Nd:Phosphate glass rod amplifier, both used in double pass configuration. The 1-5 J laser energy output from this stage serves as a seed for successive amplification in the main amplifiers.
- a set of ten Beamlet  type large aperture Nd:Phosphate glass slab amplifiers (41 cm × 41 cm clear aperture) in double pass configuration. This system has recently been activated and energies up to 225 J at a beam diameter of 12 cm have been demonstrated before temporal compression.
- a double pass vacuum grating compressor in off-Littrow configuration consisting of 14 cm × 21 cm and 21 cm × 41 cm gratings with a line spacing of 1480 l/mm. Pulse compressions down to 600 fs with an efficiency of 65 % can be achieved.
- a 1.5 m diameter target chamber for High Energy Density Physics experiments at laser powers up to 100 TW.
2.1. Rod amplifier section
Figure 1 depicts an overview of the rod amplifier section. The 4 mm diameter super-Gaussian output of the last OPCPA stage is used as the object plane for subsequent relay imaging.
A 1:1 vacuum relay creates the first image plane just after a protective Faraday isolator. A separate 1 W cw laser at 1054 nm can be injected at the first polarizer of this Faraday isolator to serve as a co-propagating alignment beam. The beam is then expanded to 10 mm diameter using a spatial filter before amplification in a double passed 16 mm diameter × 235 mm Nd:Phosphate glass rod amplifier. After traversing another Faraday isolator and a two times magnifying vacuum relay telescope, the 20 mm FWHM diameter beam is double passed in a 25 mm diameter × 235 mm Nd:Phosphate glass amplifier. Subsequently, another Faraday isolator is used for back-reflection protection before the beam hits a periscope and is injected into the main amplifier section. A 0.5 % leak from the bottom periscope mirror is used as a probe beam for a Shack-Hartman wavefront sensor. An f=75 cm and f=25 cm lens telescope reduces the beam size to 6.5 mm and images the last relay plane onto the surface of the lenslet array. The lenslet array is backed by a Pulnix TM-9701 progressive scan CCD camera.
3. Characterization of thermal lens
3.1. Shape of thermal lens
In order to study the shape of the thermal lens and its temporal evolution during and after shooting the rod amplifiers, the cw alignment laser was used as a probe laser. The wavefront sensor was calibrated to the cw laser to measure the impact of thermal lens on the wavefront.
Figures 2(a) and 3(a) show the on shot thermal lens when firing the 16 mm and 25 mm rod amplifier. One can clearly see the spherical aberration in the wavefront. Figures 2(b) and 3(b) depict the residual wavefront distortion after mathematical subtraction of spherical aberration. It is clear from the contour plots that good wavefront quality can be achieved if the spherical aberration can be fully removed.
3.2. Thermal lens evolution
In order to characterize the temporal thermal lens evolution, ten measurements per second were taken (and averaged) every second before and after the amplifier shot for a total period of about 4 minutes. To minimize the impact of noise, a 3-D spherical fit was performed and used as the measure of spherical wavefront aberration. Figure 4 shows the peak-to-valley (PV) wavefront deviation for the 16 mm and 25 mm diameter amplifier for a period of 4 minutes.
In addition, Fig. 5 shows an animated GIF of the temporal wavefront evolution when the 25 mm diameter rod amplifier is fired. One can see that a strong instantaneous thermal lens is formed during shot which persists for 5 s and 21 s for the 16 mm and 25 mm diameter rod respectively.
This is due to nonuniform absorption of flashlamp radiation during shot which leads to a larger temperature at the surface of the rod as compared to its center and hence leads to a negative thermal lens. After 20 s/40 s the heat from the flashlamp radiation and amplifier housing has fully propagated into the rod while water cooling has begun to remove heat from the edge region. This leads to a reversed temperature gradient and a positive thermal lens. In the absence of an external heat source, this lens decays with a 1/e fall time of 27 s and 91 s respectively. In order to measure the instantaneous thermal lens during shot time, the OPCPA output beam was used as a probe. A single wavefront measurement was taken on-shot with no averaging being performed (see Figs. 2, 3, 4).
3.3. Thermal lens compensation
The effects of the decreasing positive thermal lens on the laser system can be mitigated by choosing a shot rate that exceeds the laser cool-down time (e.g. 15 and minutes in this case). However, since the negative thermal lens occurs only during shot, its value has to be known in advance and needs to be pre-compensated properly. This means that a variable lens element has to be introduced into the laser system, either by adjusting the lens separation in a telescope or by using a deformable mirror. Adjustment of telescope lenses can lead to beam misalignment and can pose problems when vacuum telescopes are used. Multi-actuator deformable mirrors however are complex and time consuming to implement and might be expensive when multiple mirrors are needed. Since the thermal lens only consists of spherical aberration we propose to use a single actuator deformable mirror. Our design uses an annulus to push onto the back surface of a flat mirror substrate, simply supported at the inner annulus to create a concave parabolic deformation within the encircled area of the inner annulus.
4. Mirror design
As in previous designs, a flat reflective substrate with radius a and thickness t is trapped between two concentric aluminum rings of different radii. The quality of the optic is important since the imposed parabolic curvature will not “wash out” preexisting flatness deviations. Opposing forces between the larger and smaller rings cause the flat substrate to bend into a paraboloid. The direction of the paraboloid (concave or convex) is determined by the orientation of the two rings. Since a concave paraboloid was needed for this application, the small ring was placed toward the incoming beam.
In our design, a flat reflective substrate with radius a and thickness t is trapped between two concentric aluminum rings of different radii. A ring pushing on a flat substrate with simple circular edge support will lead to a parabolic surface deformation inside the encircled area. This can clearly be seen from Eq. 2. For a given annular line force w, the deflected profile y(r) inside the ring diameter r 0 is given by :
is the center deflection, Mc=waC 1 is the moment at the center, E=82×109 Pa is the modulus of elasticity for BK7  and D=Et 3/(12(1-ν 2)). Furthermore, ν=0.206 is Poisson’s ratio , r (with 0≤r≤a) is the radius, and C 1 and C 2 are constants:
One should note that that Eqs. 2 and 3 are only valid for values of 0.2≤r0/a≤0.8. Below this limit the situation should be modeled using a point load. Once r 0/a exceeds 0.8 the ring radius r 0 approaches the radius a and these equations no longer apply .
COSMOSWorks®was used to analyze a quarter model of a 50.8 mm diameter 3 mm thick flat optic. The apparatus was designed to fit in a standard 3” optical mount and accommodate optics with thicknesses ranging from 3 mm to sub millimeter. Previous analysis work  on similar designs which included friction, slip, and separation led to a good understanding of the mechanical behavior and the necessary detail for an accurate FEA model. Prior work has also shown that FEA agrees very closely with the analytical expressions for this loading condition (see Fig. 6). Therefore, FEA is primarily used to determine the stresses within the apparatus and not as a proof of principle.
For this apparatus 6061-T6 aluminum bar stock was chosen for nearly all parts since it reduced the possibility that non-uniform internal stresses could develop during raw material fabrication which would affect the performance of the apparatus. An additional annealing and tempering process could be applied to the parts to further reduce this effect. Aluminum proved to be an excellent material for this application since precision, weight, and cost were driving design factors. Keeping the design lightweight allowed it to be mounted in a standard 3” mount since small commercial mounts tend to perform badly when holding heavy devices. Aluminum to glass junctions were not problematic so long as the apparatus was assembled with care.
The smaller stationary 32.5 mm OD ring is located on the front side of the optic and the 47.2 mm ID pusher ring is housed behind it (see Fig. 7).
These two aluminum rings are held concentric within a 12.7µm - 102 µm radial offset tolerance. The pusher ring presses on the back side of the optic and moves within a cylindrical bore with a radial gap tolerance of 12.7 µm -38.1 µm. A spherical bulge was machined onto the pusher body to allow for rotation about all three orthogonal axes. This rotational freedom prevents the pusher ring from binding inside the bore once a force is applied and allows the pusher ring to self-orient on the rear optic surface. A Diamond Motion Inc. screw-drive linear actuator with 104 nm step size and maximum force of 1300 N is held concentric to the apparatus axis within a 88.9 µm radial offset tolerance. The actuator presses on a spring (k=196×103 N/m) through an aluminum thimble which converts displacement into force at the pusher ring. This method of force transfer has two distinct advantages over a direct coupling method. First, having specified the actuator travel, the spring rate can be chosen to ensure the actuator and optic do not receive sufficient force to be permanently damaged. Second, the spring provides an effective reduction of motor travel which allows for extremely fine displacement control: this is an advantage at long focal lengths where small changes in displacement have dramatic effects on focal length. A 303 stainless disc and lubricated ball bearing are used to transfer the force from to spring to the pusher while reducing moment transfer from the spring to the pusher ring and allowing for rotational freedom of the pusher ring.
Radial and tangential (hoop) stress plots are shown with exaggerated displacement in Fig. 8. Compressive stress is shown in blue and tensile stress is red. Grey dominates where stresses are near zero. Figure 8(a) shows a tangential stress plot that is clipped to only show specific stress values, clearly indicating the stress balance in the optic despite the asymmetric (different diameter) ring restraints. Figure 8(b) shows this behavior in more detail. Stresses within the parabolic region are parallel to the optical surface whereas stresses outside the smaller ring (meaning outside the parabolic region) quickly deviate from parallel. The parallel stress profile is a result of the uniform moment through the optic in the parabolic region.
This deformable mirror concept has the potential to create a rapidly changing focusing optic. A coarse COSMOSWorks®modal analysis performed on the assembly without a spring and a 3 mm thick BK7 optic resulted in a fundamental mode of ≈3 kHz. However, this value will change depending on geometry, and the material choice for optic and hardware.
4.3. Design iterations
Two nearly identical devices were constructed where the only intended difference was the surface quality of the pusher and retainer rings. In one case the ring flatness exceeded the required flatness tolerances of 25.4 µm by nearly a factor of four, resulting in a flatness of 7.6 µm. The other case applied a flatness of 0.25 µm to parts that were otherwise identical to the first. The intention of this design variation was to determine if the optical surface was heavily influenced by the surface quality at the force transfer interface knowing from experience that a good quality optical surface can be created with standard machining tolerances.
5. Measurement of mirror performance
5.1. Interferometer measurements
Measurements on the performance of the deformable mirror assembly were performed with a FizCam 1500 interferometer from 4D Technology operating at a wavelength of 1054 nm and a beam diameter of 101.6 mm. In order to measure a large range of mirror deformations while using a planar transmission flat, the 30 mm clear aperture of the deformable mirror was masked down to 13.4 mm diameter. This technique samples only the low deformation area at the mirror center in order to avoid steep wavefront gradients which will cause problems as the wavefront curvature increases with shorter focal length. Interferometer data was taken for various motor positions until the measured wavefront curvature exceeded the capabilities of the analysis software. Stronger curvatures could in principle be measured if a curved reference were used.
Figure 9 shows a typical measurement taken with the FizCam interferometer. The data was then fitted to the first four Zernike polynomials which account for piston, tilt in the orthogonal propagation axes, and power. Lineouts from the resulting paraboloid were then fitted to the parabolic equation y=x 2/(4f), where f is the focal length of the measured parabola.
Figure 11 shows measured focal lengths versus motor travel (and calculated force) as well as the predicted focal lengths based on Eqs. 2 and 3 for a polished (red triangles) and unpolished (black squares) pusher surface.
An addition, the animated GIF in Fig. 10 depicts the changing 3-D spherical wavefront as the stepper motor continuously pushes onto the back side of the mirror substrate. It can be seen that the measured focal lengths are the same for both pusher surfaces and both show good agreement with the analytical expressions above (solid line). Furthermore, focal lengths on the order of 100 m all the way down to 5 m were measured. This 5 m focal length measurement is limited by the interferometer when flat transmission flats are used.
Figure 12(a) depicts the contour plot of the residual wavefront deviation from Fig. 9 after subtraction of the first four Zernike terms. The PV and RMS in this case are 0.04 and 0.008 waves compared to 0.02 and 0.005 waves in the undeformed state. A slight astigmatism is visible. Figure 12(b) shows a plot of the residual RMS wavefront error versus focal length for the polished (red triangles) and unpolished (black squares) pusher rings.
It can be seen that the RMS error triples for long focal lengths compared to the undeformed state. This is due to the fact that initially the contact rings and the mirror substrate are not perfectly parallel yet. As the force on the pusher increases (resulting in shorter focal length) these freely moving parts “settle” into the correct position and the RMS error steadily decreases until f=10 m for the polished surface and f=30 m for the unpolished surface. Beyond these minima the RMS error increases, possibly due to measurement errors (because a planar transmission flat is used), material stresses or coating stresses.
5.2. Measurements of laser focal spot quality
In order to demonstrate the feasibility of a single actuator deformable mirror for on-shot thermal lens compensation focal spot sizes were measured for four different cases: The cw alignment beam, the 10 Hz OPCPA beam, the uncompensated rod shot using the 16 mm and 25 mm diameter amplifiers, and a compensated rod shot using the deformable mirrors. A 12 Bit Basler 102F CCD camera with a 10× microscope objective from Olympus was used to measure the beam size in the focal plane. The pixel calibration was preformed by measuring the width of a glass needle of known size in the focal spot plane.
Figure 13(a) shows the focal spot quality for a cw alignment beam that is injected into the amplifier chain after the OPCPA system.Without adaptive optics (meaning multi actuator mirrors) the aberrations in the system limit the Strehl Ratio to 0.39 (see Table 1). Figure 13(b) displays the focal spot for the 10 Hz OPCPA beam traversing the amplifier chain without amplification. The overall spot quality is slightly worse than the cw alignment beam by itself due to aberrations stemming from the OPCPA front end as well as chromatic aberrations in the amplifier chain. In order to measure the on-shot focal spot quality, the rod amplifier input and output was heavily attenuated using NG filters from Schott. Figure 13(c) depicts the focus quality for an uncompensated rod shot using the 16 mm and 25 mm rod amplifiers on the same shot. Due to the instantaneous thermal lens the focus is heavily distorted leading to a Strehl Ratio of 0.03. This aberration can be compensated by using the deformable mirrors as retro-reflectors in the double passed rod amplifiers. After measuring the on-shot thermal lens (see Figs. 2 and 3) the mirrors are manually driven to an equal but opposite curvature. Using a spring for force transfer in the deformable mirror assembly allows for fine control of the desired mirror curvature and leads to good spherical aberration correction within a few iterations (see Fig. 13(d)). It can be seen from Table 1 that the FWHM is on the order of the unamplified OPCPA beam and that the Strehl Ratio has only decreased from 0.23 to 0.21.
We have presented a simple method for compensating spherical aberration in an optical system by using the single actuator deformable mirror described above. This mirror design can be implemented for concave mirror shapes as well as convex deformations . This has applications for on-shot thermal lens compensation as well as thermal lens compensation for high repetition rate systems. The mechanical assembly has been carefully modeled using FEA and the results show good agreement with the analytical expressions as well as the interferometer data. Focal lengths from hundreds of meters down to the meter scale have been demonstrated experimentally. A quantitative comparison between polished and unpolished pusher surfaces has been performed. Both types yield the same focal length for equal motor travels but the polished surface exhibits a roughly two times better RMS wavefront quality. Depending on the desired application it has to be decided if the improved performance of a polished pusher and retainer warrants the additional cost.
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energys National Nuclear Security Administration under contract DE-AC04-94AL85000.
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