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A new deformable mirror influence function based on a Gaussian function is introduced to analyze the fitting capability of a deformable mirror. The modified expressions for both azimuthal and radial directions are presented based on the analysis of the residual error between a measured influence function and a Gaussian influence function. With a simplex search method, we further compare the fitting capability of our proposed influence function to fit the data produced by a Zygo interferometer with that of a Gaussian influence function. The result indicates that the modified Gaussian influence function provides much better performance in data fitting.
©2008 Optical Society of America
1. Introduction
A deformable mirror (DM) is applied extensively in a variety of study areas as one of the key elements for adaptive optical wavefront correction, and the exact, corrected residual errors of a wavefront are of great importance. With the utilization of the linearity superposition principle, the main error of a DM is derived from the influence function (IF), which describes the shape to which the mirror will deform when forces are applied by actuators; therefore, constructing an influence function as close as possible to the virtual expression is crucial to the design of an optics system.
A Gaussian influence function (GIF), regarded as a good simulation method, has been employed in most research works [1–3], and can be expressed as
where ω is the coupling coefficient that is the influence at the center of the neighboring actuator and normally varies from 5% to 15%, which is confirmed by fitting the IF belonging to DMs of apertures 70 to 150 mm; the actuators number 19 to 61, and the k4 glass or quartz thickness ranges from 3 to 5mm; d0 is the actuator spacing; and α is the Gaussian index and normally varies from 1.5 to 2.5.
It is noted that the GIF does not possess the actual IF features, in either the azimuthal or radial directions, as can be seen in Figs. 1 and 2; the shape of the actual IF varies with the arrangement of actuators rather than with a circular structure used in a GIF. In the radial direction, as analyzed below, there is a collapse at the surrounding actuators as well. In general, the GIF is not an exact model, so we need a better model that can precisely characterize the actual IF.
As alternatives, discrete data produced by either finite element analysis or interferometer [4–6] are adapted, but these schemes are restricted in a certain DM. Although Menikoff and Arnold researched theoretically the influence function of an active mirror [7–9], they failed to extract the proper expression. Van Dam also briefly mentioned a double Gaussian function to fit the actual IFs of a membrane DM [10]. In this paper, a modified Gaussian influence function (MGIF) based on a GIF is introduced, and with a simplex search method [11], an expression closer to the virtual IF is obtained.
The remainder of this paper is organized as follows. The difference between the GIF and the actual IF is first presented in Section 2, the expression of the MGIF and its components are derived in Section 3, the simplex search method used to determine the parameters of MGIF is briefly introduced in Section 4, the result of comparison to a GIF is shown in Section 5, and conclusions are drawn in Section 6.
2. Difference between the GIF and the actual IF
IFs produced by (a) the GIF and the ZYGO interferometer of (b) a 45-element hexagonal arrangement DM and (c) a 32-element square arrangement DM are illustrated in Fig. 1. Both of the DMs are produced by the Institute of Optics and Electronics at the Chinese Academy of Sciences. The residual error between (a) and (b) of Fig. 1 is depicted in Fig. 2, where we can see six blue regions distributed periodically around the center; in other words, instead of a constant constructed by the GIF, the amplitudes of the actual IF differ periodically in the azimuthal direction. The period depends on the arrangement of the actuators; for a hexagonal arrangement of the DM, the period is 6; for a square arrangement DM, the period is 4. We also realize that there is a collapse in the radial direction, and the azimuthal periodicity accounts for the mismatch of the two curves illustrated in Fig. 2(b), implying that the azimuthal expression should be altered prior to the radial one so that curves in the diverse azimuthal direction turn out to be identical to facilitate the radial modification.
Fig. 1. IF produced by (a) GIF and ZYGO interferometer of (b) hexagonal arrangement DM and (c) square arrangement DM.
3. Modified Gaussian influence function
As mentioned above, the deviation between the GIF and the actual IF are mainly from the azimuthal direction and the radial direction. Therefore, a more exact IF can be achieved by some modification with respect to these two directions. Here, we take a hexagonal arrangement DM as an example to derive the new IF formulation.
3.1 Azimuthal modification
It is already known that the actual IF is a periodic structure in the azimuthal direction, and we owe this period to the change of actuator spacing d, which is considered to be a constant in the GIF. In order to find out the changing rule of d, we study the relationship between d and the angle. Figure 3 is the cutaway view of the IF of a hexagonal arrangement DM measured by a Zygo interferometer. Figure 4 shows d̄/d as a function of the angle in the cross-section of r=d̄ depicted in Fig. 3, which seems to be a normalized cosine function with an average of 1 and a period of 6. It can be expressed as
where d̄ is the mean actuator spacing, i.e., the average between d=d0/cos (π/6) and d0 or, d̄=1.0773d0; for a square arrangement, 6 is substituted by 4 and d̄=1.2073d0. In these formulas, d0 is the distance between center actuator and the nearest actuator and λ is the amplitude of the period, which depicts the azimuthel fluctuation caused by discrete neighboring actuators and ranges from -0.01 to -0.02.
As a result, the actuator spacing changes as
So we have the azimuthal modification, and the new GIF can be expressed as
By means of the new GIF, the residual error between the new GIF and the actual IF is shown in the Fig. 5(a) 2D-map and Fig. 5(b) cross-sections. Figure 5(a) indicates that the period existing in Fig. 2 disappears, and the residual error, as shown in Fig. 5(b), mainly is in the radial direction, so that it is possible to use a function of the radius to compensate for the residual error.
Fig. 5. Residual error after modification in angle direction; (a) is the 2-D map and (b) is the cross-section.
3.2 Radial modification
As implied in Fig. 5, the force of the surrounding actuators not only causes the mirror to decline as the Gaussian function, but also results in a quick collapse around the circle with radii d=d̄. In fact, the principle behind the surrounding actuators and the center actuator is the same; therefore, it is reasonable for the collapse to be a symmetrical Gaussian function. After azimuthal modification, the amplitude of collapse, once different in various azimuthal directions, is modulated to be identical. So a Gaussian function of offset d̄ can be used to compensate for the residual error. Two more parameters are needed to describe this Gaussian function: the amplitude (β) representing the depth pulled by the surrounding actuators and the width (γ), somewhat similar to d̄ of the center actuator referring the influence area of the surrounding actuators. The value of β and γ ranges from 0.01 to 0.06 and from 3 to 10, respectively, and together they all describe the radial change, which not only declines as a Gaussian function but also contains a Gaussian-like collapse of depth β and width γ at the distance of d̄.
Combining the azimuthal modification, the new IF can be expressed as
where ω is the coupling coefficient; α is Gaussian index; d̄ is the mean actuator spacing; λ is the amplitude of period; β is the radial correction amplitude; γ is the radial correction width; and the number 6 refers to the hexagonal arrangement DM, and the number 4 refers to the square arrangement DM.
Fig. 6. Residual error after modification in angle and radial direction; (a) is the 2-D map and (b) is the cross-section.
We also note that the actual coupling coefficient of ω+β is obtained at the region of the nearest actuator for this new formulation, and when λ=0 and β=0, the MGIF degenerates to a GIF.
After modification both in the azimuthal direction and the radial direction, the residual error is shown in Fig. 6, which indicates that the difference of peak–valley greatly reduces from the range of ±0.03 to ±0.01.
4. Parameters determination
Considering more parameters and complex relationship between parameters in the MGIF, for which conventional methods such as least square cannot be adapted, we choose the simplex search method to seek the best parameters for the MGIF to fit the known data. The simplex search method has the simplest graphics in a specific dimension space. For example, in two dimensions, a simplex is a triangle; in three dimensions, it is a pyramid. In a word, if n is the length of a searched object, the simplex in the n-dimension is characterized by the n+1 distinct vectors that are its vertices, and the simplex search method searches the extreme through the simplex. The simplex search method is a direct search method that does not employ numerical or analytic gradients by function values only. At each step of the search, a new point in or near the current simplex is generated. The function value at the new point is compared with the function’s values at the vertices of the simplex and, usually, one of the vertices is replaced by the new point, giving a new simplex. This step is repeated until the diameter of the simplex is less than the specified tolerance.
Initial values are needed for this method, and they are specified in Table 1.
Table 1. Initial Values Specified for the Simplex Search Method
The optimization object is the root-mean-square (RMS) of the residual error, defined as follows
where M and N represent the IF’s sample size of the x, y direction, I res is the residual error, and Īres is the average of residual error. The suitable value for parameters of MGIF can be determined by searching the minimal RMS value.
5. Comparisons of GIF and MGIF
Using the simplex search method and the criterion in Section 4, the fitting error for different position actuators is compared and shown in Fig. 7, where the arrangement of a 45-element DM is placed on the left side and the fitting error of the GIF (blue color) and the MGIF (brown color) is set on the right side. Obviously, the MGIF has an overall smaller residual error, and the fitting error decreases as it approaches the center actuators. It is also noted that the variation of the MGIF fitting error for actuators at different locations is smaller than that of the GIF, confirming that the MGIF is applicable for edge actuators as well.
Fig. 7. actuators arrangement and comparison of fitting capability using GIF and MGIF for different position actuators.
Fig. 8. Comparisons of the fitting capability of the square arrangement deformable mirror using GIF and MGIF. (a) and (b) IF calculated by Gaussian function and MGIF, respectively, (c) and (d) 2-D map of the residual error.
A deformable mirror of a square arrangement is also considered. Figure 8 shows the result. The IF generated by the MGIF in Fig. 8(b) evidently shows square geometry compared with Fig. 8(a). The residual error between Fig. 8(b) and Fig. 1(c) (RMS 0.0122) shown in Fig. 8(d) indicates that the model error is greatly reduced compared to that in Fig. 8(c) (RMS 0.0189) fitted by the GIF. Basally, the MGIF has the ability to characterize the actual IF with a fitting capability better than that of the GIF.
6. Conclusions
A MGIF is demonstrated in this paper. With the simplex search method, we have fitted the data produced by the Zygo interferometer, drawing the conclusion that, with azimuthal and radial modification, the MGIF is capable of characterizing the actual IF of various position actuators with better fitting capability.
Acknowledgments
This work was supported by the 863 High-Tech Project of China.
References and links
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