1. Introduction

Deformable mirror (DM) is the key component of an adaptive optics system (AOS). Among the DMs considered [1], bimorph (which is used in this paper to indicate bimorph DM) has the following advantages: larger stroke, higher damage threshold of laser power, and better correction of low-order aberrations with a given number of actuators. In addition, fabrication and application of bimorphs are relatively simple. Some institutions [2–4] have fabricated bimorphs independently, and encouragingly, most of them have obtained satisfying results in experiments.

The structure of layers may be the first problem to solve when designing a bimorph. At present, two-layer [Fig. 1(a)] and three-layer bimorphs [Fig. 1(b)] have been used in AOSs. With the increasing requirements on spatial resolution and actuator pitch, multilayer [n≥4, Fig. 1(c)] structures raise researchers’ interest, because it is known that the control electrodes on an extra layer enables a bimorph to correct more aberrations. A. V. Kudryashov [5] suggested that with the thickness of layers getting thinner and thinner, the disadvantage of reduced mirror sensitivity, caused by adding a new layer, doesn’t play a great role. That means it is feasible to improve the correcting capability of a bimorph by increasing the number of control layers. Then, what interests us is the difference in performance of multilayer (n≥4) bimorphs from that of bimorphs that have already been used.

 

Fig. 1. Various-layer bimorphs in cross section. (a) two-layer, (b) three-layer, (c) multilayer (n≥4). Light-colored layer indicates optical polished passive layer (e.g., glass); deep-colored layer indicates PZT.

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Response function is commonly used to study deformable mirrors. Since bimorph was envisaged in 1979, many studies have been done to calculate the response function [6–10]. These investigations, however, were based mostly on a single bimorph structure, like that in Fig. 1(a) or 1(b). E. M. Ellis [11] described a theoretical approach, which is suited for analyzing bimorphs of various layers. However, the comparison of these different structures was absent, and the final expression of the response function was not reported.

In this paper, numerical calculation is used to investigate the response function of various possible bimorph-type DMs, and a complete expression of response function is presented. The expression is used to compare sensitivity of a five-layer bimorph with that of two typical fewer-layer structures, to find out the advantages and disadvantages brought about by the increase in the number of layers.

2. Calculation of response function

As a complement for article [11], we present the complete expression of response functions for circular bimorphs with a clamped edge. Furthermore, the parameters used to construct the solution are described in detail.

A bimorph responds to a voltage distribution that is applied to it according to the biharmonic equation, which can be written as Eq. (1) in cylindrical polar coordinate:

where W(r,θ) is the displacement of the mirror surface and r is the normalized radius. V is the loading voltage distribution, ∇² is Laplacian, and Γ is a constant that is related to the geometrical, mechanical, and piezoelectric properties of the mirror.

The solution of Eq. (1) is not unique. Following Timoshenko [12], we can write it as a

where ϕ is the angular position of the applied load and w_n are given by

A_i, B_i, C_i, D_i(i≥0) are parameters of integration and determined by the boundary conditions.

 

Fig. 2. (a) Illustration of point voltage loading; (b) region of sectorial electrode; (c) electrodes distribution of a bimorph with 17 elements.

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To create a complete solution, the surface of the mirror is split into two regions: that within and that without of the radius at which the load M(ρ,ϕ) is applied [Fig. 2(a)], 0≤ρ≤1,0≤ϕ≤2π. The solution for the outer region is w′_n, which is distinguished from w_n for the inner region; both solutions are subject to the following four conditions:

(1) At the loading circle r=ρ, the displacement of the mirror surface and its first and second radial derivatives are continuous:

(2) At the loading circle r=ρ, the shearing force balance must be satisfied:

where $Q_r=-\frac{D}{a^3}\frac{\partial }{\partial r}\left(\frac{{\partial }^2{w}_n}{\partial r^2}+\frac{1}{r}\frac{\partial w_2}{\partial r}+\frac{1}{r^2}\frac{{\partial }^2{w}_n}{\partial {\theta }^2}\right)$, and a is the diameter of the bimorph. D is called bending stiffness, which relates the bending moment to the curvature of the bimorph. Kudryashov [10] derived the bending stiffness D for a bimorph structure like Fig. 1(a). By a similar method, D can be obtained for various bimorphs [11]. The parameter P is relevant to the extra moment induced by converse piezoelectric effect; thus in an intuitive manner, it can be described as a function P=f (V), where V is the amplitude of loading voltage V.

(3) A clamped outer boundary, which is the most practicable mounting arrangement in our experiment, is considered.

At the outer edge $r=1:{{w\prime }_n\mid }_{r=1}=0,{\frac{\partial {w\prime }_n}{\partial r}\mid }_{r=1}=0$.

(4) At the center of the bimorph plate r=0, the displacement of mirror surface is limited, so C_i=D_i=0,(i≥0).

By conditions (1) and (2), we establish relations of the parameters in Eq. (3) between the two regions. By conditions (3) and (4), these parameters are contacted with inner and outer boundaries. Thus, all the equations needed are constructed, and all the parameters can be derived as follows:

Substituting these parameters into Eq. (3), the w_n and w′_n are derived, and they construct the complete point-response function:

After simplification of n_w and w′_n, we get almost the same expressions as those presented in [12], where the response function was investigated on the same structure loaded with force. The differences in the results here consist of the parameters P and D [presented in Eq. (5)], which include all the information about the structure and loading voltage of bimorphs in our calculation.

The next stage in finding the response function is to integrate the point-response function on the region of an electrode. The usual case, which we shall consider, is that the electrode is bounded into a sectorial region [Fig. 2(b)], where ϕ ₁≤ϕ≤ϕ ₂ and ρ ₁≤ρ≤ρ ₂. Thus, the voltage distribution in Eq. (1) of this electrode can be described as $V(\rho ,\varphi )=VH(\rho -{\rho }_1,{\rho }_2-\rho ,\varphi -{\varphi }_1,{\varphi }_2-\varphi ),H(x_1,x_2,\dots )=\{\begin{array}{cc}1,& x_1>0\bigcup x_2>0\dots \\ 0,& \mathrm{others}\end{array}.$

Substituting Eq. (7) into the integral $W(r,\theta )={\int }_{{\varphi }_1}^{{\varphi }_2}{\int }_{{\rho }_1}^{{\rho }_2}{\nabla }^{2}V(\rho ,\varphi )W(r,\theta ;\rho ,\varphi )\rho d\rho d\varphi$, we can determine the profile of the mirror surface for the given electrode in series:

Incorporating Eqs. (8) and (9), we derive the final expression of the response function for a sectorial electrode:

where the series was truncated to n max terms.

Compared with the analysis presented in [11], this expression offers a more convenient form for calculating the response function.

3. Sensitivity comparison between various-layer bimorphs

The response function reflects the displacement of the mirror surface in response to an electrode. The amplitude of this response can also be described as sensitivity of a bimorph, and the sensitivity is influenced by many factors. Using Eq. (10), some of these factors can be found out for the bimorphs about which we are concerned.

Three types of laminate structures are considered in our study: Fig. 3(left) and Fig. 3(middle) represent the bimorphs that are commonly used at present, and Fig. 3(right) represents a promising bimorph structure that would be developed when more control electrodes are demanded. In Fig. 3, e1, e2, and e3 indicate the locations of control layers, and on each control layer electrodes can be distributed in an arbitrary format. g indicates a common electrode, which is grounded. The symbol ↑ indicates the direction of polarization. It is supposed that a positive voltage applied to a control electrode causes horizontal contraction of the PZT. Besides, the binding material and metal electrodes between layers are ignored in our numerical calculation.

 

Fig. 3. Cross section of three kinds of bimorphs with indication of control electrodes.

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3.1 Determination of the layer thickness for bimorphs

The first step of our analysis is to determine the thickness of each layer for the three kinds of bimorphs. In practice, the thickness of the PZT layer is determined by technology operation. So, it is feasible to give each PZT layer the same thickness, then to find out the influence of the passive layer thickness on the displacement of the mirror surface.

From Eq. (1) it is clear that Γ is a scaling factor to the displacement W(r,θ). To get this simple format of Eq. (1), factor Γ for the two-layer bimorph [Fig. 3(left)] is deduced according to [10]:

${\Gamma }_2\propto \left(\frac{d_{31}}{{t_1}^2}\right)\times \frac{6k\left(1+k\right)}{1+k^4+2k\left(2+3k+2k^2\right)},$

where Γ ₂ is the factor Γ for the two-layer bimorph, d ₃₁ is the piezoelectric constant, t ₁ is the thickness for each PZT layer, and k=t ₂/t ₁ is the thickness ratio of the passive layer to the PZT one.

Similarly, for the three-layer and five-layer bimorphs in Fig. 3, Γ can be obtained as follows after D and P in Eq. (5) are determined:

${\Gamma }_3\propto \left(\frac{d_{31}}{{t_1}^2}\right)\times \frac{12k\left(1+k\right)}{16+k^4+8k\left(4+3k+k^2\right)}{\Gamma }_5\propto \left(\frac{d_{31}}{t_1^2}\right)\times \frac{24k\left(4+k\right)}{256+k^2+16k\left(16+6k+k^2\right)}.$

In each of the above cases, elastic modulus of the two materials is equal, and d ₃₁/t ² ₁ is a scaling factor too, which shows that a thinner structure always bends more. The remaining factors of Γ are plotted in Fig. 4 as a function f(k) of the thickness ratio k.

 

Fig. 4. Variation of f(k) as a function of thickness ratio k for three kinds of bimorphs.

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In Fig. 4, the maximum f(k) of different structures appears at different values of k, and the following results are derived: the optimal thickness of the passive layer is 0.5t ₁. for a two-layer bimorph, t ₁ for a three-layer bimorph, and 2t ₁ for a five-layer one. That is to say, for a n -layer er (n≥2) bimorph, the passive layer should be determined based on the whole structure and should not always be treated as thin as possible.

3.2 Analysis on the influence factors of sensitivity of various bimorphs

In this part, relations between sensitivity and the following three parameters are discussed: voltage amplitude V, area of an electrode S, and the thickness of the PZT layer t ₁. With these parameters enlarged twice one by one, the response function of electrode number 1 in Fig. 2(c) is calculated according to Eq. (10). Simultaneously, the electrode varies its location to different control layers, indicated as e1, e2, or e3 [Fig. 3], to find out the influence of the position where a control electrode is situated.

The initial values of these parameters are given as: V=150v, S=25π (mm), 45678987-09876541=0.4mm, a=30mm (diameter of bimorph), and t ₂ for each structure is determined according to the optimized results presented above. The displacements in response to electrode number 1 in various situations are presented in Fig. 5. When this electrode is discussed in a definite control layer, all the electrodes on other control layers are grounded.

 

Fig. 5. Cross profile of the response function of electrode number 1 in Fig. 2(c) at different control layer for various-layer bimorphs.

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It is shown in Fig. 5(a) that, for the two-layer bimorph, the displacement of the mirror surface is directly proportional to the loading voltage V, inversely proportional to the square of the thickness of PZT layer t ₁, and increased about 1.4 times when the area of electrode doubles. For the three-layer bimorph [Fig. 5(b)], the above proportional relations are still validated when electrode number 1 locates at the control layer e1. But when the electrode is moved to e2, the displacement vanishes, which means any electrodes in this control layer would play no role at all. Before proceeding on to the explanation for this, let’s see another problem that appears in Fig. 5(c). For the five-layer bimorph, the displacement varies according to the proportional relations at the position of e1 and e2 but presents as a negative value at e3.

To find out the reason for these unexpected results, the concept of median plane is introduced. Median plane, widely used in thin plate deformation theory, refers to a surface on which material deformation is absent when the plate itself bends. For a n -layer (n≥2) bimorph in Fig. 1(d), the location of the median plane can be calculated by a simple expression:

where Δ is the distance from the bottom surface to the median plane, and t_j and E_j are the thickness and elastic modulus of material for layer j, respectively. According to Eq. (11), median planes for the three bimorph structures are calculated and marked out in Fig. 6 with a dot-dash line.

 

Fig. 6. Schematic plan of the position of the median plane for various bimorphs.

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For the two-layer bimorph [Fig. 6(a)], t ₁=0.4mm, t ₂=0.5t ₁, and Δ=0.3mm. For the three-layer bimorph [Fig. 6(b)], t ₁=0.4mm, t ₂=t ₁, Δ=0.6mm, and the median plane is situated right in the middle of the second PZT layer controlled by e2. That means sensitivity of this layer nearly equals zero and the three-layer structure is equivalent to the two-layer bimorph. Therefore, the optimized value of t ₂ is not suitable here, and a thicker passive layer of 0.6mmis considered. Thus, median plane is moved to 1 mm below e2 [Fig. 6(c)]. Response functions of electrode number 1 for this new structure are plotted in Fig. 7(a). Although the linear summation of the displacements caused by e1 and e2 here is smaller than the value caused by e1 in Fig. 5(b), the electrodes distributed on e2 would enable the bimorph to correct more aberrations. From Fig. 6(d) it is clear that loading a positive voltage on e3 would result in a concave deformation on the five-layer bimorph surface, because the middle of this PZT layer is beyond the median plane. If the polarity of voltage is adjusted to negative for e3, it would contribute a positive part [Fig. 7(b)] to the total deformation. Therefore, the polarity of loading voltage on each electrode should be determined by the relative location of the PZT layer to the median plane.

 

Fig. 7. Cross profile of the response function of electrode number 1 in Fig. 2(c), calculated with adjusted thickness of the passive layer.

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From Fig. 7(b), the influence of the position where an electrode is situated can also be suggested. We mention that the displacement of the five-layer bimorph surface in response to e1 is larger than that to e2, although the number of PZT layers controlled by e2 is two. This can be explained by the different distances of control layers from the median plane: the farther electrode contributes the larger deformation. This conclusion is consistent with the results in [5], which were derived by a different approach of the finite elements method.

4. Conclusion

In this article, a complete expression of response function, which is suitable for analyzing various possible bimorph-type deformable mirrors, is presented. The expression is used to compare the performance of a five-layer bimorph with two- and three-layer structures. It is shown that the newly added layer would enable the five-layer bimorph to correct more aberrations, but the displacement of the mirror surface reduces with the increase in the number of layers. Moreover, we found that the displacement is closely related to the distance of the control layer from the median plane. As a conclusion, the farther electrode contributes the larger deformation.

Several factors that influence the displacement for a fixed electrode position have also been investigated. It is found that the value of response function is directly proportional to the loading voltage, inversely proportional to the square of the thickness of the PZT layer, and increases about 1.4 times when the area of the electrode doubles.