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Making use of the stretching effect of grazing incident laser beam, a novel method of wavefront correction was promoted. Without adding any extra beam expanding components, aberrations of wavefront could achieve satisfying correction by two grazing reflections along orthogonal directions on the deformable mirrors. The stretching effect expanded the beam size along grazing direction and the orientation dependent varying aberrations were well compensated as more actuators took effect in the correction process. Analysis showed that the fitting coefficient of all the first 30 order Zernike polynomials could be controlled within 5% by this method.
© 2013 Optical Society of America
1. Introduction
Since the deterioration of laser beam quality occurred as the output power increasing, high beam quality laser beam was difficult to obtain and sometimes even unavailable in high power applications. Adaptive optics (AO) provided an effective and convenient way to correct the wavefront distortion and improve beam quality [1–4]. In conventional AO systems, only one deformable mirror (DM) was employed to correct the wavefront of the incident beam [5–7], then multiple DMs were employed to provide more effective correction in aberration control [8,9]. In consideration of the power intensity tolerance of AO system in high power applications, the DM usually had a large caliber and sufficient thickness. For example, the DM used in National Ignition Facility (NIF) had a caliber of 400mm and thickness of 10-15mm [10,11]. To control such a large and comparatively thick DM, naturally, the discrete driven actuator arrays turned out to be a good choice. Interesting investigations had been carried out in recent years and quite a few actuators driven by different mechanisms were available. The mechanical driven actuator was the most original one [12], the piezoelectric driven actuator brought about more precise control [13,14]. Thermally deformable mirror, as a newly promoted concept, presented pleasing correcting effect in low order aberrations and brought down the cost a lot [15,16]. Other driving mechanisms like photothermal effect, magnetic liquids could be found in [17,18]. Nevertheless, no matter what the driven mechanism was, the correction ability of DM was closely connected to the distribution density of discrete actuators. Due to the properties of material and manufacturing techniques, the maximum available density of actuators were limited. In order to improve the correction ability of the DM at certain density of actuators, especially in the correction of aberrations containing high spatial frequencies, the input beam was normally enlarged by expanding lenses before correction [7]. More actuators covered by the input beam brought about more flexibility in control and better correction results as well. Other method using multiple DMs can be seen in [8,9], where the DMs were used in combination to produce a similar correction effect as one DM with denser actuators did.
Proceeding from the purpose to enlarge the input beam, the simple idea of grazing incidence occurred to us. This opinion was first employed to continuously control the fringe separation in the interferometry [19]. Benefitting from the grazing incidence, great improvements were achieved in last decades [20–23]. It also drew attention in the design of solid state laser where compactness and good beam quality were accomplished [24,25]. In this paper, the application of grazing incidence in wavefront correction was introduced. One distorted wavefront could get satisfying correction after two times grazing reflected on DMs along orthogonal directions. Without adding any extra components, this method brought about concise configuration as well as better system stability.
2. Grazing incidence wavefront correction system
When the input beam reflected on the DM at a large incidence angle, the beam size along the incident direction would be stretched, which was showed in Fig. 1. By changing the grazing incidence angle θ, the amplifying factor γ can be expressed as
Theoretically, as the incidence angle θ changing, the incident beam could be continuously stretched to infinite size as the incidence angle approaching 90°. In practical application, the amplifying factor within 10 could be reached as the incidence angle changed from 0° to 84.3°
Since the incidence angle was distinctly greater than 0°, the optical path difference (OPD) after reflection was dependent on the incidence angle. As showed in Fig. 2, the surface of DM could be modeled as steps. Two light beams were reflected on the step-shaped DM surface of which the difference in height was Δh.
The dash-lines AB and DG indicated the incident and reflected wavefront respectively. Obviously, the OPD Δφ between the two beams was |EG|-|CD|
Compared with vertical incidence condition, the OPD was reduced by the factor cosθ. Realization of the enlargement of incident beam sacrificed the modulation depth of DM. However, the modified range were among several micrometers which was small compared with the deformation range of DM. Thus, scarification within certain range in the modulation depth was acceptable in practical application.
Fig. 3. showed the configuration of twice grazing reflected correction system. The input beam was directed to be grazing reflected on DM1 and DM2 consecutively. A 4f optical system was employed to arrange the two reflection planes in conjugate positions. The grazing angles were θ1 and θ2 and the grazing directions were along X-axis and Y-axis respectively. The Hartmann Shack senser was employed to detected the output wavefront. After necessary data analysis, the control signals of DMs were generated and the whole close-loop AO system was established.
For explaining the stretching effect, diagrams of beam shapes under different incidence conditions were presented in Fig. 4, where the amplification factor in both directions was 3 times. Taking the central part of the input beam for analyzing, in Fig. 4(a), 4 actuators (painted black) mainly decided the correction result of that area. While under grazing incidence along X-direction, 8 actuators took effect according to the stretching along X-axis in Fig. 4(b). The X-directional varying spatial frequencies which would have not been corrected under vertical incidence would possibly been corrected now. Similar improvement of correction ability occurred along Y-axis as well when the grazing incidence was shifted to the orthogonal direction which was showed in Fig. 4(c).
The direct-gradient wavefront control algorithm [26,27] was applied in the system. At first the incident wavefront was meshed into discrete pixels. The influence of actuator located at (xa,ya) on each pixel (xi,yi) could be expressed as
Where θ was the grazing incidence angle, r was the influence radius of actuators, α and β were parameters measured from experiment. The influence on pixels induced by each actuator was set in one column, by conjugating all columns together, the influence matrix H was obtained. The control signal A was a column vector with elements equal to the number of actuators. The relationship between control signal and gradient vector W measured by wavefront senser can be expressed as
Where m was the pixels’ number of incident beam and n was the actuators’ number. As the pixels’ number m was greatly larger than the actuators’ number n, in most cases, the linear equations were inconsistent and had no exact solution. However, solution at the least-square sense still could be found as
Where H+ was the left inverse of H, A* was the least square solution to Eq. (5). If one laser beam with wavefront of W passed through the grazing reflected AO system with incidence angles of θx and θy, the control signal Ax* and residual wavefront error ex after grazing reflection along X-axis can be written as
After the second grazing reflection along Y-axis, the control signal Ay* was the function of both grazing angles
The residual error after two-sections correction can be written as the function of θx and θy3. Characteristics of grazing incidence wavefront correction system
Since the Zernike polynomials are orthogonal within the unit circle, the distortion of wavefront could be conveniently decomposed into the sums of Zernike polynomials. We supposed the aberration could be expressed by the first 30 order Zernike polynomials and investigated the correction effects on each Zernike polynomial. Parameters used in the following calculations were listed in Table 1.
Table 1. Parameters Used in Analysis
3.1 Orientation dependent characteristics of grazing incidence
At first, investigations were carried out under one reflection condition to reveal the characteristics of grazing incidence. Since this method was strongly orientation dependent, we picked out the 3th, 6th and 7th Zernike polynomials, which respectively represented radially, X-axis and Y-axis varying wavefront, to see the correction results after grazing reflection along X-direction.
The comparison of the three Zernike polynomials was listed in Fig. 5. The first column was the original wavefronts, the second column was the corrected wavefronts under vertical incidence condition and the last column was residual wavefronts after grazing reflection along X-axis at optimal incidence angle. The orientational characteristic was obvious, the 6th Zernike polynomial was well compensated under grazing incidence since its varying orientation was parallel to the grazing reflection direction, the radially varying 3th Zernike polynomial just obtained little improvement while improvement was barely observed in the correction of Y-axis varying 6th Zernike polynomial. In order to compare the correction results at different grazing angle, we defined the improvement factor along X-axis as
Where ex(θx) and ex(0°) were the residual wavefront errors defined in Eq. (8). RMS was the root mean square value of the residual errors. The relationship between improvement factor and grazing angle was demonstrated in Fig. 6.
As the grazing angle increased, the improvement factor of radially varying 3th Zernike polynomial deteriorated at first and then performed certain descent. The X-axis varying 6th Zernike polynomial experienced a clear improvement process and achieved the best correction at the largest grazing angle, while the Y-axis varying 7th Zernike polynomial was totally insensitive to the change of grazing angle.
From the comparison, we can see the strong orientation dependent characteristic of the grazing incidence AO system. Since the actuators were arranged in rectangular shape, the correction process was effective along certain direction and comparatively independent between orthogonal directions, thus there was potential to achieve satisfying correction by two times grazing correction in orthogonal directions.
3.2 Effectiveness after correction in orthogonal directions
The correction results of 3th, 6th and 7th Zernike polynomials were shown in Fig. 7. The three columns respectively presented the residual wavefront errors under vertical incidence, X-direction grazing incidence alone and both direction grazing incidence. The improvement in the control of residual error was obvious after both directions’ correction. Since the 6th and 7th Zernike polynomials were orientation dependent aberrations themselves, the wavefronts after correction were close to plane waves. The radially varying 3th Zernike polynomial also benefited from the two times’ stretching effect since more actuators took effect and more flexibilities were available in the direct-gradient algorithm.
As the grazing angle ranging from 0° to 81° in this calculation, an iterative program was applied to find out the optimal permutation of grazing angles to achieve the least RMS of residual wavefronts. In summary of the comparison among the three incidence conditions, the merit chosen to evaluate the correction was defined as the fitting coefficient ξ
Where Zj(x,y) was the jth order Zernike polynomial and e(x,y) was the residual wavefront error after correction. The fitting coefficient ξ measured the ratio of uncompensated aberration to the total aberration of original Zernike polynomial.
The fitting coefficients of the first 30 order Zernike polynomials in three different conditions were summarized in Fig. 8. It was obvious that one time grazing correction was not effective to all Zernike polynomials since it was strongly orientation dependent. While after grazing reflections along both directions, all the first 30 order Zernike polynomials were well corrected and the fitting coefficients were effectively controlled below 5%. The improvement was especially significant in the correction of some high order Zernike polynomials, such as the 23th,30th Zernike polynomials, where the residual distortions were not pleasing enough after conventional DM correction. Compared with the “Combinational-deformable-mirror” reported in [28], where two DMs were used in wavefront correction as well, the residual error after twice grazing incidence was better and the enhancement was even greater in corrections of high order Zernike polynomials.
Results of the analysis proved the possibility and effectiveness of the grazing incidence wavefront correction method. By simply stretching the size of the input beam along orthogonal directions, the correction ability of the AO system can be improved remarkably under certain actuators’ distribution density. Besides, none extra beam expanding components were needed which benefitted the stability of system as well.
4. Conclusion
In this paper, one simple and convenient method was promoted in the application of wavefront compensation. Under grazing incidence condition, the light beam reflected on the DM would be expanded by the stretching effect. At certain actuators’ distribution density, the correction effect could achieve great enhancement along the grazing direction. Since the enhancement in orthogonal directions are approximately independent, one distorted wavefront could be corrected in X and Y directions respectively by grazing reflection. The residual errors of all the first 30 order Zernike polynomials could be controlled below 5% after the correction and higher order aberration could achieve better compensation in this method.
Acknowledgments
Supported by the National Natural Science Foundation of China (No. 61275146), the Research Fund for the Doctoral Program of Higher Education of China (No. 20120002110066) and the Special Program of the Co-construction with Beijing Municipal Government of China (No. 20121000302).
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