- Email: [email protected]
Automatic phase-locked control of grating tiling
Optics and Lasers in Engineering 50 (2012) 262–267
Contents lists available at SciVerse ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Automatic phase-locked control of grating tiling Yuchuan Yang a,b,n, Xiao Wang b, Junwei Zhang b, Hui Luo a, Fuquan Li b, Xiaojun Huang b, Feng Jing b a b
College of Optic-electric Science and Engineering. National University of Defense Technology, Changsha 410073, China Research Center of Laser Fusion, CAEP, P.O. Box 919-988, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 February 2011 Received in revised form 28 July 2011 Accepted 23 August 2011 Available online 13 September 2011
The development of phased-array grating compressor is a crucial issue for the high-energy, ultra-short pulse petawatt-class lasers. Several systems have adopted the tiling-grating approach to meet the size requirements for the compression gratings. Grating tiling need to be precisely phased to ensure a transform-limited focal spot when focusing the high-energy laser pulses onto the target. Monochromatic grating automatic phasing and performance maintaining are experimentally achieved with a far-field CCD camera technique based on a two-tiling system. Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved.
Keywords: Tiling grating Phase locking Far-field monitor SPGA algorithm
1. Introduction The chirped-pulse amplification output stage of the OMEGA EP petawatt, multikilojoule, solid-state laser, presently under construction at the Laboratory for Laser Energetics, includes four 1 3 tiling-grating assemblies (TGAs) to compress the pulse before it is focused onto the target [1]. The proposed FIREX-I (Fast Ignition Realization Experiment) system at Osaka University contains a multiplexed tiling compressor [2]. TGAs were chosen to reach the required energy levels while staying within the size and damage-threshold limitations of currently available multilayer-dielectric diffraction gratings. Previous theoretical simulations and laser experiments demonstrated that the tiling approach can yield diffraction-limited focal spots [3]. Maintaining the grating–tiling mechanical stability is difficult in most thermal and vibrating environments. Within the framework of Pico2000 petawatt laser at LULI (Laboratoire pour I’Utilisation des Lasers Intense) and OMEGA EP (extended performance) petawatt laser at LLE (Laboratory for Laser Energetics), the monochromatic plane wave grating phasing with an accurate interferometric diagnostic was described [4 and 5]. Using this approach, the segmented mirror arrays have been aligned using interferometers by sensing tip, tilt, and piston. Since gratings are dispersive devices, tiling gratings exhibit three additional degrees of differential error: in-plane rotation (IPR), groove spacing, and lateral piston. These phase errors between grating tiling can be
n Corresponding author at: College of Optic-electric Science and Engineering. National University of Defense Technology, Changsha 410073, China. E-mail address: [email protected] (Y.C. Yang).
summarized as three-mirror terms and three-grating terms grouped according to their effect on the focal spot [3]. The mirror and grating terms are paired to compensate Y tip and in-plane rotation, lateral piston and longitudinal piston, and X tilt and groove spacing. For wavelength-scale errors, pairing reduces the number of control variables to three. By controlling tip, tilt, and piston among the tiles, the gratings can be properly phased. Bunkenburg et al. [6,7] have demonstrated the phase-locked control of the tiling-grating assemblies using a Mach–Zehnder interferometer. Qiao et al. [5] also demonstrated the interferometry for tiling automation, and firstly demonstrated of two large-aperture tiling-grating compressors, each consisting of four sets of tiling-grating assemblies in OMEGA EP using interferometric tiling technique [8]. Cotel et al. [3] proved that the measured far-field intensity distribution of the tiling small-scale gratings agrees well with the wavefront measured by an interferometer, but the automatic closed-loop tiling method based on the far-field method is not mentioned. Recently, Hideaki Habara et al. [2] utilized three-axis motion sensor to realize high-precision tiling-grating compressor for FIREX. The near-field interferometric tiling technique has successfully realized gratings automatic tiling. In the practical grating compressor, the high-power 1053 nm analogous laser source for collimating and adjusting tiling gratings hardly meets the narrow spectrum width requirement, so the reference arm in the interferometric system needs much more space making the optical path layout complicated, but the far-field technique can efficiently avoid these disadvantages. In this paper, we build a closed-loop system, consisted of a far-field CCD and PZT actuators, to realize automatic tiling and high-precision phase locking, the simple optical path layout and tiling process are more
0143-8166/$ - see front matter Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2011.08.004
Y.C. Yang et al. / Optics and Lasers in Engineering 50 (2012) 262–267
conveniently used in the practical situation. Section 2 describes the diffraction theory of a two-grating tiling system involving the angle errors and longitudinal piston error. According to the calculation results, some optimum algorithms could be used to realize the automatic alignment. Section 3 presents the far-field tiling technique based on the stochastic parallel gradient ascent (SPGA) algorithm, the automatic tiling process is developed and the results of tiling two mediate-aperture reflecting elements using this process are also reported.
2. Analytic far-field parameters The diffraction grating phase consists of determining the phase errors between two adjacent gratings, which can be caused by relative translations and rotations, and then removing these phase errors using actuators. In the case of a two-grating configuration, a reduction in degrees of freedom can be realized by paired compensation, so tip (yx), tilt (yy), and longitudinal piston (Dz) need to be corrected (Fig. 1). A circular uniform beam (central wavelength l ¼1.053 mm) vertically lights up the gratings gap to form symmetrically straddling two segments, shown in Fig. 2. For this case, the electric field in the near-field plane can be written as 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > A expðjkðDz þ f yx x þ f yy yÞÞ x Z d=2; x2 þ y2 ra > < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x r d=2; x2 þ y2 ra ð1Þ Eðx,yÞ ¼ A > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > :0 x2 þy2 4 a where d is lateral translation (Dx) also called gratings gap, A is the average input wave amplitude, a is the incident beam radius, f is the focal length. Let f¼587 mm, a¼20 mm, and d ¼4 mm, the far-field intensity distribution is performed by fast Fourier transform operation in the case of differential subapertures piston of l/8, l/4, and l/2 (Fig. 3(a)), a grating differential tilt and tip of, respectively, yy¼4 mrad, 8 mrad, and 16 mrad (Fig. 3(b)) and yx ¼4 mrad, 8 mrad, and 16 mrad (Fig. 3(c)).
Fig. 1. Phased-array grating compressor scheme with five degrees of freedom between the two adjacent diffraction gratings G1 and G2 (Dx, Dz, yx, yy, yz).
¦Ñ
a ¦² 1
d
¦² 2
Fig. 2. Geometry of the beam aperture.
263
Δz=λ /8
λ /4
λ/2
θy=4μrad
8μrad
16μrad
θx=4μrad
8μrad
16μrad
Fig. 3. Theoretical diffraction patterns for a split circular subaperture with the piston and tilt/tip errors.
As piston is increased, the original peak continues to shift downward, and two peaks become equal at a physical step height of l/2 between the two halves of the circular subaperture. The change in tip between the subaperture corresponding to a peakto-valley phase is nearly twice of that in tilt, so tip is more effective on far-field pattern than tilt. Fig. 4 shows the Strehl ratio and encircled energy ratio in 0.5 diffraction limit (DL) around the peak intensity on the focal plane for tilt/tip-piston errors. The phase sensitivity is clear. When the two segments are in phase, the Strehl ratio and encircled energy ratio have the maximum value. As the angular and piston errors are increased, the values continue to shift downward, so searching for maximum will be done by proper algorithm. The stochastic parallel gradient ascent (SPGA) algorithm shall be applied to synchronal adjust the tilt/ tip-piston parameters of one segment maximizing the Strehl ratio or the encircled energy ratio in Section 3 [9,10]. In this discussion, we have implicitly assumed that the surfaces of the segments in question are perfect. In practice this is not the case, for grating, the fabrication errors result in the surface aberrations of approximately 50 nm rms. However, because the surface aberration are mainly in low-frequency domain and the subaperture diameter a¼20 mm is small compared with the grating size, it is reasonable to take the calculations in ideal condition. For now, we note that a proper algorithm based on the above ultimately results in a tiling-grating configuration that minimizes the tiling errors.
3. Monochromatic phasing experiments We have developed a mechanical system prototype to phase two medium-scale flat mirrors instead of diffraction gratings, so only three degrees of freedom between the two 220 mm 200 mm Al-coated flat mirrors are permitted. Each mirror reposes on two knee-joints and is fixed with nylon screws to reduce the vibrating effect. The lateral piston (Dx) is adjusted with manual translation stage and the longitudinal piston (Dz¼(AþB þC)/3) with three PZT translation stages (4 nm minimum displacement). The tip (yx¼(B C)/1.732r) and the tilt (yy¼(2A (B þC))/3r) are also controlled by three PZT translation stages for high resolution to achieve angular rotation less than
Y.C. Yang et al. / Optics and Lasers in Engineering 50 (2012) 262–267
0.8 0.4
0.7 0.6
0.3 0.5 0.4
0.2 -15
-10
-5
0 5 tilt (μrad)
10
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0.8 Strehl ratio
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0.5
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15
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10
15
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0.7 0.6
0.3 0.5
Encircled energy ratio in DL/2
264
0.2
0.4 -0.4
-0.2
0.0 0.2 piston (μm)
0.4
Fig. 4. Theoretical curve of the strehl ratio and encircled energy ratio in 0.5 diffraction limit (DL) around the peak intensity on the focal plane versus the tilt/tip-piston errors.
1 mrad. A–C are the respective displacement of three actuators and r is the radius of the PZT tripod drive. The experimental demonstration of the two-segment phasing with our opto-mechanical system and motion devices is achieved using a far-field CCD camera, which is shown in Fig. 5. The two-tiling configuration is installed in a double pass monitoring beam path, containing a far-field CCD camera at the end of beam path, with a He–Ne laser, operating at 632.8 nm. As shown in Fig. 5, the 40 cm-diameter collimated beam is transmitted through the M1 and directed to two adjacent tiles by the BS for the first time (Facula 1). The M2 is oriented to a small angle to retroreflect the left-hand and right-hand half beams on the two adjacent tiles at the same side again (Facula 2). Finally, the monitoring beams are transmitted through the M3 and directed to a 0.587 m-focal-length lens and a 10 microscope for focusing and zooming. Firstly, when the tiling-grating phasing is misaligned, the two individual focal spots are randomly distributed on the focal plane. Secondly, when the two focal spots move close enough to each other, the interference pattern appears. The automatic tiling is tested as the sums of the left phases (SLP) and of the right phases (SRP), a far-field CCD camera measures the SRP and SLP. By zeroing the difference between the SRP and SLP, the interference pattern is nearly transform-limited focal spot and the two parts are brought into coherent addition. According to the initial condition in Fig. 6(a), firstly the relative distance between the two focal spots should be reduced. One mirror is attached to control system, in Fig. 6(a), the upper focal spot is adjusted to match with the lower focal spot. Operating in this stage, the moving direction of each spot was real-time judged and corrected to reduce the relative distance, at the same time the
differential tip and tilt were gradually reduced to sub-milliradian. When the maximum light intensity was more than 1.2 times than that of individual focal spot, the operation was stopped. Then we judged the interference pattern appeared, for further minimizing the tiling errors, the SPGA algorithm was applied to realize the maximization of an objective function in the next stage. For comparison, Fig. 6(c) and (d) is the far-field pattern calculated by the theoretical simulations at initial state and after combination, the results (Fig. 6(a) and (b)) show that the far-field intensity to increase by N2 ¼4 as expected for fully coherent beam combinations is made by phase control algorithm. Because the entire optical path is affected by external vibration, the coherent combined focal spot is shaking on the focal plane, so the combined focal-spot position in Fig. 6(b) is not concordance with the lower focal-spot position. However, it is possible that, while the difference between SLP and SRP is nearly zero, still there may be misalignment shifts that can affect the combined beam properties. At this stage, the SPGA algorithm was still applied on tilt/tip-piston actuators to maintain the good tiling state. The steps for SPGA algorithm could be briefly described as follows [9]. The objective function J is defined as the encircled energy ratio detected by CCD camera. J¼ J(u)¼ J(u1,u2, y,un), where u is the PZT driving voltage generated by the iterative operation. The algorithm is implemented in finite iterations, each iteration cycle works as follows: (1) Constant disturbed voltage {duj} (j ¼1,y,n), where all 9duj9 are 0.01 V that are decided by actuator driving step (o5 nm) and we employ a Bernoulli distribution in the sign of {duj}. (2) Apply the control voltage with the positive perturbations and get the encircled energy ratio from the image surface,
Y.C. Yang et al. / Optics and Lasers in Engineering 50 (2012) 262–267
M1
L
L
He-Ne Laser
265
PZT actuators Capacity sensor
back
M2
Piezo driving configuration top view
BS M3
L
From CCD out Microscope Attenuating plate Far-field Camera M2
Driver A System computer
Driver signals
Driver B Driver C
BS 1 2 tiling mirror positive view M3
tiling mirror side view
Fig. 5. Layout of the double pass monitoring beam path and two Al-coated phased mirrors. M, mirror; BS, beam splitter; L, converging lens.
Fig. 6. Experimental far-field intensity for misaligned mirrors (a) and for correct aligned mirrors (b) and comparison with theoretical simulations (c, d).
J þ ¼J(u1 þ du1,y,un þ dun), then apply the control voltage with the negative perturbations and get the encircled energy ratio, J ¼J(u1 du1,y,un-dun). (3) Calculate the difference of encircled energy ratio dJ¼J 7J . (4) Update the driving voltage, uiþ1 ¼ui þ gdujdJ, i¼ 1, y, n, where g is inverse proportion to J. The open-loop experiments without driving actuators permitted the tiling mirror to drift as a function of time. The temperature, relaxation, and environmental vibration would typically change the OPD. In this OPD monitoring mode, Fig. 7(a) shows the peak light intensity strenuous change and the encircled energy ratio in the main-lobe (40-pixel-diameter) circle on CCD plane fluctuations between 0.3 and 0.55 over 1 h period. With the SPGA algorithm engaged over a same measurement period, the encircled energy ratio remains constant at
0.5870.01 (Fig. 7(d)); however, the peak light intensity has stepped change. Actually, the peak light intensity is easily influenced by extraneous factors, including air-turbulence, shaking of combined focal spot, detector noise and background noise, not proper for the objective function J, while the encircled energy ratio is less susceptible to these factors. With the peak ratio technique [11], the ratio of the two primary peaks formed in the far-field pattern as a function of the phase difference between the two apertures could be used to determine relative piston value (Fig. 8(a)). The SRP and SLP are individually described as exp(i2jR-mirror) and exp(-i2jL-mirror) in the double-pass beam path, the total phase difference is 2(jR-mirror þ jL-mirror), according to the Fourier transform calculation, the resulting piston (phase) differences for various piston steps are shown in Fig. 8(b). The corresponding piston errors after SPGA algorithm in 1 h are listed in Table 1.
Y.C. Yang et al. / Optics and Lasers in Engineering 50 (2012) 262–267
1.0 Encircled energy ratio
0.8 0.7 0.6 0.5 0.4 0.3 0.0
Normalized peak light intensity
0.6
0.9
0.2
0.4 0.6 time (hour)
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0.8 0.6 0.4 0.2 0.0 0.0
1.0
8
0.10
7
0.08
piston difference (λ)
main peak/secondary peak
Fig. 7. (a, b) Normalized peak light intensity and encircled energy drifts in the open loop because of thermal changes and environmental vibration. (c, d) Normalized peak light intensity and encircled energy are shown with the SPGA control algorithm engaged.
6 5 4 3 2
0.06 0.04 0.02 0.00
1 0.0
0.2
0.4 0.6 time (hour)
0.8
1.0
3
4 5 6 7 main peak/secondary peak
8
Fig. 8. (a) Ratio of the two primary peaks is shown with the algorithm engaged and holding steady over 1 h period. (b) Peak ratio curve is calibrated with a fitted exponential decay polynomial.
Table 1 Phasing results based on Fig. 8.
Min value Max value Mean value rms Value
Main peak/secondary peak
Piston error (nm)
3.344 5.973 4.857 0.46
13.5 54.9 28.4 7.045
automatic tiling process. System operation was continuously maintained for hour-long periods in the algorithm control mode. The realization of the fully automatic tiling and performance maintaining by the far-field monitor method should be very potential for the multi-array tiling configuration in practical application.
Acknowledgments 4. Conclusion In conclusion, we have demonstrated SPGA phase control algorithm of a tiling mirror for confirming the feasibility of grating tiling. The alignment errors were compensated by adjusting the PZT driving mirrors. The near-diffraction-limited focalspot performance was obtained for limited apertures by far-field
We are grateful valuable support from all the crew of XGIII in Research Center of Laser Fusion and Dr. Weijun Zhang provided the tiling frame. This work was supported by the National High Technology Research and Development Program of China (2009AA8044005) and the National Natural Science Foundation of China (Grant no. 11074225).
Y.C. Yang et al. / Optics and Lasers in Engineering 50 (2012) 262–267
References [1] Waxer LJ, Maywar DN, Kelly JH, Kessler TJ, Kruschwitz BE, Loucks SJ, et al. High-energy petawatt capability for the omega laser. Opt. Photonics News 2005;16:30. [2] Hideaki H, Guang X, Takahisa J, Ryosuke K, Kenji S, Kiyonobu S, et al. Pulse compression and beam focusing with segmented diffraction gratings in a high-power chirped-pulse amplification glass laser system. Opt. Lett. 2010;35:1783–5. [3] Cotel A, Castaing M, Pichon P, Le Blanc C,. Phased-array grating compression for high-energy chirped pulse amplification lasers. Opt. Express 2007;15(5):2742–52. [4] Le Blanc C, Felix C , Lagron JC, Forget N, Hollander P, Sautivet AM, et al. The Petawatt laser chain at LULI: from the diode-pumped front end to the new generation of compact compressor. In: Proceedings of the third international conference on inertial fusion sciences and applications (IFSA); 2003. p. 608 [Chapter X]. [5] Qiao J, Kalb A, Guardalben MJ, King G, Canning D, Kelly JH. Large-aperture grating tiling by interferometry for petawatt chirped-pulse-amplification systems. Opt. Express 2007;15(15):9562–74.
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[6] Bunkenburg J, Kessler TJ, Skulski W, Huang H. Phase-locked control of tilinggrating assemblies for chirped-pulse-amplified lasers using a Mach–Zehnder interferometer. Opt. Lett. 2006;31(10):1561–3. [7] Kessler TJ, Bunkenburg J, Huang H, Kozlov A, Meyerhofer DD. Phase-locked control of tiling-grating assemblies for chirped-pulse-amplified lasers using a Mach–Zehnder interferometer. Opt. Lett. 2004;29:635. [8] Qiao J, Kalb A, Nguyen T, Bunkenburg J, Canning D, Kelly JH. Demonstration of large-aperture tiling-grating compressors for high-energy, petawatt-class, chirped-pulse amplification systems. Opt. Lett. 2008;33(15):1684–6. [9] Vorontsov MA, Sivokon VP. Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction. J. Opt. Soc. Am. A 1998;15:2745. [10] Zhou P, Ma YX, Wang XL, Ma HT, Xu XJ, Liu ZJ. Coherent beam combination of three two-tone fiber amplifiers using stochastic parallel gradient descent algorithm. Opt. Lett. 2009;34:2939. [11] Chanan GA, Ohara C, Troy M. Phasing the mirror segments of the Keck telescopes II: the narrow-band phasing algorithm. Appl. Opt. 1998;39:4706.
Contents lists available at SciVerse ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Automatic phase-locked control of grating tiling Yuchuan Yang a,b,n, Xiao Wang b, Junwei Zhang b, Hui Luo a, Fuquan Li b, Xiaojun Huang b, Feng Jing b a b
College of Optic-electric Science and Engineering. National University of Defense Technology, Changsha 410073, China Research Center of Laser Fusion, CAEP, P.O. Box 919-988, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 February 2011 Received in revised form 28 July 2011 Accepted 23 August 2011 Available online 13 September 2011
The development of phased-array grating compressor is a crucial issue for the high-energy, ultra-short pulse petawatt-class lasers. Several systems have adopted the tiling-grating approach to meet the size requirements for the compression gratings. Grating tiling need to be precisely phased to ensure a transform-limited focal spot when focusing the high-energy laser pulses onto the target. Monochromatic grating automatic phasing and performance maintaining are experimentally achieved with a far-field CCD camera technique based on a two-tiling system. Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved.
Keywords: Tiling grating Phase locking Far-field monitor SPGA algorithm
1. Introduction The chirped-pulse amplification output stage of the OMEGA EP petawatt, multikilojoule, solid-state laser, presently under construction at the Laboratory for Laser Energetics, includes four 1 3 tiling-grating assemblies (TGAs) to compress the pulse before it is focused onto the target [1]. The proposed FIREX-I (Fast Ignition Realization Experiment) system at Osaka University contains a multiplexed tiling compressor [2]. TGAs were chosen to reach the required energy levels while staying within the size and damage-threshold limitations of currently available multilayer-dielectric diffraction gratings. Previous theoretical simulations and laser experiments demonstrated that the tiling approach can yield diffraction-limited focal spots [3]. Maintaining the grating–tiling mechanical stability is difficult in most thermal and vibrating environments. Within the framework of Pico2000 petawatt laser at LULI (Laboratoire pour I’Utilisation des Lasers Intense) and OMEGA EP (extended performance) petawatt laser at LLE (Laboratory for Laser Energetics), the monochromatic plane wave grating phasing with an accurate interferometric diagnostic was described [4 and 5]. Using this approach, the segmented mirror arrays have been aligned using interferometers by sensing tip, tilt, and piston. Since gratings are dispersive devices, tiling gratings exhibit three additional degrees of differential error: in-plane rotation (IPR), groove spacing, and lateral piston. These phase errors between grating tiling can be
n Corresponding author at: College of Optic-electric Science and Engineering. National University of Defense Technology, Changsha 410073, China. E-mail address: [email protected] (Y.C. Yang).
summarized as three-mirror terms and three-grating terms grouped according to their effect on the focal spot [3]. The mirror and grating terms are paired to compensate Y tip and in-plane rotation, lateral piston and longitudinal piston, and X tilt and groove spacing. For wavelength-scale errors, pairing reduces the number of control variables to three. By controlling tip, tilt, and piston among the tiles, the gratings can be properly phased. Bunkenburg et al. [6,7] have demonstrated the phase-locked control of the tiling-grating assemblies using a Mach–Zehnder interferometer. Qiao et al. [5] also demonstrated the interferometry for tiling automation, and firstly demonstrated of two large-aperture tiling-grating compressors, each consisting of four sets of tiling-grating assemblies in OMEGA EP using interferometric tiling technique [8]. Cotel et al. [3] proved that the measured far-field intensity distribution of the tiling small-scale gratings agrees well with the wavefront measured by an interferometer, but the automatic closed-loop tiling method based on the far-field method is not mentioned. Recently, Hideaki Habara et al. [2] utilized three-axis motion sensor to realize high-precision tiling-grating compressor for FIREX. The near-field interferometric tiling technique has successfully realized gratings automatic tiling. In the practical grating compressor, the high-power 1053 nm analogous laser source for collimating and adjusting tiling gratings hardly meets the narrow spectrum width requirement, so the reference arm in the interferometric system needs much more space making the optical path layout complicated, but the far-field technique can efficiently avoid these disadvantages. In this paper, we build a closed-loop system, consisted of a far-field CCD and PZT actuators, to realize automatic tiling and high-precision phase locking, the simple optical path layout and tiling process are more
0143-8166/$ - see front matter Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2011.08.004
Y.C. Yang et al. / Optics and Lasers in Engineering 50 (2012) 262–267
conveniently used in the practical situation. Section 2 describes the diffraction theory of a two-grating tiling system involving the angle errors and longitudinal piston error. According to the calculation results, some optimum algorithms could be used to realize the automatic alignment. Section 3 presents the far-field tiling technique based on the stochastic parallel gradient ascent (SPGA) algorithm, the automatic tiling process is developed and the results of tiling two mediate-aperture reflecting elements using this process are also reported.
2. Analytic far-field parameters The diffraction grating phase consists of determining the phase errors between two adjacent gratings, which can be caused by relative translations and rotations, and then removing these phase errors using actuators. In the case of a two-grating configuration, a reduction in degrees of freedom can be realized by paired compensation, so tip (yx), tilt (yy), and longitudinal piston (Dz) need to be corrected (Fig. 1). A circular uniform beam (central wavelength l ¼1.053 mm) vertically lights up the gratings gap to form symmetrically straddling two segments, shown in Fig. 2. For this case, the electric field in the near-field plane can be written as 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > A expðjkðDz þ f yx x þ f yy yÞÞ x Z d=2; x2 þ y2 ra > < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x r d=2; x2 þ y2 ra ð1Þ Eðx,yÞ ¼ A > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > :0 x2 þy2 4 a where d is lateral translation (Dx) also called gratings gap, A is the average input wave amplitude, a is the incident beam radius, f is the focal length. Let f¼587 mm, a¼20 mm, and d ¼4 mm, the far-field intensity distribution is performed by fast Fourier transform operation in the case of differential subapertures piston of l/8, l/4, and l/2 (Fig. 3(a)), a grating differential tilt and tip of, respectively, yy¼4 mrad, 8 mrad, and 16 mrad (Fig. 3(b)) and yx ¼4 mrad, 8 mrad, and 16 mrad (Fig. 3(c)).
Fig. 1. Phased-array grating compressor scheme with five degrees of freedom between the two adjacent diffraction gratings G1 and G2 (Dx, Dz, yx, yy, yz).
¦Ñ
a ¦² 1
d
¦² 2
Fig. 2. Geometry of the beam aperture.
263
Δz=λ /8
λ /4
λ/2
θy=4μrad
8μrad
16μrad
θx=4μrad
8μrad
16μrad
Fig. 3. Theoretical diffraction patterns for a split circular subaperture with the piston and tilt/tip errors.
As piston is increased, the original peak continues to shift downward, and two peaks become equal at a physical step height of l/2 between the two halves of the circular subaperture. The change in tip between the subaperture corresponding to a peakto-valley phase is nearly twice of that in tilt, so tip is more effective on far-field pattern than tilt. Fig. 4 shows the Strehl ratio and encircled energy ratio in 0.5 diffraction limit (DL) around the peak intensity on the focal plane for tilt/tip-piston errors. The phase sensitivity is clear. When the two segments are in phase, the Strehl ratio and encircled energy ratio have the maximum value. As the angular and piston errors are increased, the values continue to shift downward, so searching for maximum will be done by proper algorithm. The stochastic parallel gradient ascent (SPGA) algorithm shall be applied to synchronal adjust the tilt/ tip-piston parameters of one segment maximizing the Strehl ratio or the encircled energy ratio in Section 3 [9,10]. In this discussion, we have implicitly assumed that the surfaces of the segments in question are perfect. In practice this is not the case, for grating, the fabrication errors result in the surface aberrations of approximately 50 nm rms. However, because the surface aberration are mainly in low-frequency domain and the subaperture diameter a¼20 mm is small compared with the grating size, it is reasonable to take the calculations in ideal condition. For now, we note that a proper algorithm based on the above ultimately results in a tiling-grating configuration that minimizes the tiling errors.
3. Monochromatic phasing experiments We have developed a mechanical system prototype to phase two medium-scale flat mirrors instead of diffraction gratings, so only three degrees of freedom between the two 220 mm 200 mm Al-coated flat mirrors are permitted. Each mirror reposes on two knee-joints and is fixed with nylon screws to reduce the vibrating effect. The lateral piston (Dx) is adjusted with manual translation stage and the longitudinal piston (Dz¼(AþB þC)/3) with three PZT translation stages (4 nm minimum displacement). The tip (yx¼(B C)/1.732r) and the tilt (yy¼(2A (B þC))/3r) are also controlled by three PZT translation stages for high resolution to achieve angular rotation less than
Y.C. Yang et al. / Optics and Lasers in Engineering 50 (2012) 262–267
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Fig. 4. Theoretical curve of the strehl ratio and encircled energy ratio in 0.5 diffraction limit (DL) around the peak intensity on the focal plane versus the tilt/tip-piston errors.
1 mrad. A–C are the respective displacement of three actuators and r is the radius of the PZT tripod drive. The experimental demonstration of the two-segment phasing with our opto-mechanical system and motion devices is achieved using a far-field CCD camera, which is shown in Fig. 5. The two-tiling configuration is installed in a double pass monitoring beam path, containing a far-field CCD camera at the end of beam path, with a He–Ne laser, operating at 632.8 nm. As shown in Fig. 5, the 40 cm-diameter collimated beam is transmitted through the M1 and directed to two adjacent tiles by the BS for the first time (Facula 1). The M2 is oriented to a small angle to retroreflect the left-hand and right-hand half beams on the two adjacent tiles at the same side again (Facula 2). Finally, the monitoring beams are transmitted through the M3 and directed to a 0.587 m-focal-length lens and a 10 microscope for focusing and zooming. Firstly, when the tiling-grating phasing is misaligned, the two individual focal spots are randomly distributed on the focal plane. Secondly, when the two focal spots move close enough to each other, the interference pattern appears. The automatic tiling is tested as the sums of the left phases (SLP) and of the right phases (SRP), a far-field CCD camera measures the SRP and SLP. By zeroing the difference between the SRP and SLP, the interference pattern is nearly transform-limited focal spot and the two parts are brought into coherent addition. According to the initial condition in Fig. 6(a), firstly the relative distance between the two focal spots should be reduced. One mirror is attached to control system, in Fig. 6(a), the upper focal spot is adjusted to match with the lower focal spot. Operating in this stage, the moving direction of each spot was real-time judged and corrected to reduce the relative distance, at the same time the
differential tip and tilt were gradually reduced to sub-milliradian. When the maximum light intensity was more than 1.2 times than that of individual focal spot, the operation was stopped. Then we judged the interference pattern appeared, for further minimizing the tiling errors, the SPGA algorithm was applied to realize the maximization of an objective function in the next stage. For comparison, Fig. 6(c) and (d) is the far-field pattern calculated by the theoretical simulations at initial state and after combination, the results (Fig. 6(a) and (b)) show that the far-field intensity to increase by N2 ¼4 as expected for fully coherent beam combinations is made by phase control algorithm. Because the entire optical path is affected by external vibration, the coherent combined focal spot is shaking on the focal plane, so the combined focal-spot position in Fig. 6(b) is not concordance with the lower focal-spot position. However, it is possible that, while the difference between SLP and SRP is nearly zero, still there may be misalignment shifts that can affect the combined beam properties. At this stage, the SPGA algorithm was still applied on tilt/tip-piston actuators to maintain the good tiling state. The steps for SPGA algorithm could be briefly described as follows [9]. The objective function J is defined as the encircled energy ratio detected by CCD camera. J¼ J(u)¼ J(u1,u2, y,un), where u is the PZT driving voltage generated by the iterative operation. The algorithm is implemented in finite iterations, each iteration cycle works as follows: (1) Constant disturbed voltage {duj} (j ¼1,y,n), where all 9duj9 are 0.01 V that are decided by actuator driving step (o5 nm) and we employ a Bernoulli distribution in the sign of {duj}. (2) Apply the control voltage with the positive perturbations and get the encircled energy ratio from the image surface,
Y.C. Yang et al. / Optics and Lasers in Engineering 50 (2012) 262–267
M1
L
L
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PZT actuators Capacity sensor
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Piezo driving configuration top view
BS M3
L
From CCD out Microscope Attenuating plate Far-field Camera M2
Driver A System computer
Driver signals
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BS 1 2 tiling mirror positive view M3
tiling mirror side view
Fig. 5. Layout of the double pass monitoring beam path and two Al-coated phased mirrors. M, mirror; BS, beam splitter; L, converging lens.
Fig. 6. Experimental far-field intensity for misaligned mirrors (a) and for correct aligned mirrors (b) and comparison with theoretical simulations (c, d).
J þ ¼J(u1 þ du1,y,un þ dun), then apply the control voltage with the negative perturbations and get the encircled energy ratio, J ¼J(u1 du1,y,un-dun). (3) Calculate the difference of encircled energy ratio dJ¼J 7J . (4) Update the driving voltage, uiþ1 ¼ui þ gdujdJ, i¼ 1, y, n, where g is inverse proportion to J. The open-loop experiments without driving actuators permitted the tiling mirror to drift as a function of time. The temperature, relaxation, and environmental vibration would typically change the OPD. In this OPD monitoring mode, Fig. 7(a) shows the peak light intensity strenuous change and the encircled energy ratio in the main-lobe (40-pixel-diameter) circle on CCD plane fluctuations between 0.3 and 0.55 over 1 h period. With the SPGA algorithm engaged over a same measurement period, the encircled energy ratio remains constant at
0.5870.01 (Fig. 7(d)); however, the peak light intensity has stepped change. Actually, the peak light intensity is easily influenced by extraneous factors, including air-turbulence, shaking of combined focal spot, detector noise and background noise, not proper for the objective function J, while the encircled energy ratio is less susceptible to these factors. With the peak ratio technique [11], the ratio of the two primary peaks formed in the far-field pattern as a function of the phase difference between the two apertures could be used to determine relative piston value (Fig. 8(a)). The SRP and SLP are individually described as exp(i2jR-mirror) and exp(-i2jL-mirror) in the double-pass beam path, the total phase difference is 2(jR-mirror þ jL-mirror), according to the Fourier transform calculation, the resulting piston (phase) differences for various piston steps are shown in Fig. 8(b). The corresponding piston errors after SPGA algorithm in 1 h are listed in Table 1.
Y.C. Yang et al. / Optics and Lasers in Engineering 50 (2012) 262–267
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Fig. 7. (a, b) Normalized peak light intensity and encircled energy drifts in the open loop because of thermal changes and environmental vibration. (c, d) Normalized peak light intensity and encircled energy are shown with the SPGA control algorithm engaged.
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Fig. 8. (a) Ratio of the two primary peaks is shown with the algorithm engaged and holding steady over 1 h period. (b) Peak ratio curve is calibrated with a fitted exponential decay polynomial.
Table 1 Phasing results based on Fig. 8.
Min value Max value Mean value rms Value
Main peak/secondary peak
Piston error (nm)
3.344 5.973 4.857 0.46
13.5 54.9 28.4 7.045
automatic tiling process. System operation was continuously maintained for hour-long periods in the algorithm control mode. The realization of the fully automatic tiling and performance maintaining by the far-field monitor method should be very potential for the multi-array tiling configuration in practical application.
Acknowledgments 4. Conclusion In conclusion, we have demonstrated SPGA phase control algorithm of a tiling mirror for confirming the feasibility of grating tiling. The alignment errors were compensated by adjusting the PZT driving mirrors. The near-diffraction-limited focalspot performance was obtained for limited apertures by far-field
We are grateful valuable support from all the crew of XGIII in Research Center of Laser Fusion and Dr. Weijun Zhang provided the tiling frame. This work was supported by the National High Technology Research and Development Program of China (2009AA8044005) and the National Natural Science Foundation of China (Grant no. 11074225).
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