Influence of Adhesive Bonding on the Surface Accuracy of Flat Optics: A Mechanistic Analysis and a Quantitative Approximation

Abstract

Surface accuracy is a crucial evaluation criterion for the life cycle performance of optical components. Throughout the assembly process, the optical surface undergoes deformation due to applied assembly stresses, causing the actual surface profile to deviate from the intended design. The key to the quantitative optimization of the adhesive bonding assembly process is to elucidate the quantitative coupling mechanism between assembly stress and optical surface deformation and to establish a corresponding quantitative relationship. To address this, a comprehensive study into the optical surface deformation of multi-point adhesive bonded flat optical components is presented. Firstly, the coupled influence mechanisms governing optical surface deformation are analyzed considering both the components properties and assembly parameters, and this mechanistic analysis includes dimensionless modeling, boundary conditions analysis, and reliable design requirement analysis. Secondly, based on the understanding of mechanisms, a quantitative approximation method is developed to predict the deformation patterns within the critical central region (50% aperture) of optical components subjected to multi-point bonding. Finally, the quantitative approximation for the assembly-induced surface deformation is experimentally validated. Through this research, the surface deviation during the multi-point adhesive bonding assembly of flat optical components can be effectively approximated, holding significant importance for further imaging quality prediction and assembly parameters optimization during assembly process, and facilitating a further performance improvement for optical instruments.

1. Introduction

Optical components are key parts of precision optical instruments. They convert the physical properties of light into measurable signals through the precise generation, manipulation, and detection of light, enabling high-precision perception of the physical world. This facilitates functions such as precision measurement, information imaging, and energy transfer and manipulation. The ideal surface shape of optical components is essential for optical design, which aims to precisely control the wavefront shape of light waves to achieve specific transformations, such as focusing (e.g., converting plane waves to spherical waves), collimation (e.g., converting spherical waves to plane waves), or imaging. However, during the manufacturing process, processing and assembly errors, along with residual stresses and assembly stresses, cause the actual surface of optical components to deviate from the ideal shape. This deviation distorts the optical path difference to varying degrees and finally reduces optical perception accuracy.

The surface accuracy of an optical component is initially established during its processing and assembly stages. Subsequently, it evolves throughout its life cycle under the influence of environmental factors such as vibrations and thermal loads. With the development of advanced machining technologies such as Single-Point Diamond Turning (SPDT) [1], Precision Glass Molding (PGM) [2,3,4], and Ion Beam Figuring (IBF) [5], the machining accuracy of precision optical components has reached unprecedented levels. For instance, by using ultra-precision equipment, vibration-free systems, accurate tool holders, and prescribed diamond tools, SPDT can achieve a nanometer-scale machining accuracy [6]. Despite these processing advances, assembly process can easily cause surface deviations in a much larger scale [7,8], such as sub-micrometer or even micrometer level. For instance, Lamontagne et al. [9] reduced centering errors between optics and barrel to approximately 5 μm using a threaded assembly method. Wang et al. [10] adopted numerical simulation to calculate the maximum deformation of a primary mirror in a dual-mirror reflective system connected by bolts, finding it to be approximately 0.7 micrometers after considering assembly errors. Similarly, Doyle et al. [11] observed that a 1 °C temperature change in an adhesive-bonded telescope caused a 10-micrometer change in the primary mirror’s radius of curvature, ultimately leading to a sub-micron change in the wavefront’s Peak-to-Valley (PV) value. Ma et al. [12] analyzed an optical lens assembled with RTV88 silicone adhesive and found surface deformations of 4 to 14 μm under thermal radiation at temperatures from −40 °C to +60 °C. Therefore, assembly process is critical to form the surface accuracy of optical components.

As mentioned above, adhesive bonding is a typical assembly method for precision optical components [12,13]. Compared to traditional mechanical methods like threads and pressure rings, modern adhesives offer improved strength and reduced curing stress, providing significant advantages, including simpler structures, smaller and more uniform assembly stresses, and higher reliability under environmental variations [14]. However, its limitations in high-precision optical applications are equally prominent: the bonding stress within the adhesive layer is inherently difficult to control precisely during assembly, and it further evolves dynamically throughout the entire service life. Such stress variations are driven by factors including stress relaxation of the viscoelastic adhesive, temperature fluctuations, and environmental vibrations, ultimately leading to unstable and uncontrollable optical accuracy—a core challenge that motivates the quantitative research on assembly-induced wavefront errors in this study. The selection process requires balancing multiple factors, such as adhesive strength, curing stress, thermal matching, and the stability of material parameters over time and temperature [15]. Moreover, the bond geometry of the adhesive layer should also be designed to further reduce stress concentration in the optical components [15,16].

Several researchers have investigated the optimization of adhesive bonding process to improve assembly quality. Bayar [14], Herbert [17], and Monti [18] studied methods to optimize adhesive layer thickness based on athermal bonding design to minimize thermal stress caused by different thermal expansions among the adhesive, optical components, and barrels. Feng et al. [19] modeled the curing shrinkage process as an equivalent to temperature reduction and used numerical simulations to study the influence of adhesive process parameters (e.g., dosage, position, and distribution) on the reflective surface shape. Jia et al. [20] tested the effects of adhesive layer thickness and bonding area on tensile and shear strength and found no significant correlation between them within the 0.1 to 0.3 mm thickness range. Guo et al. [21] found that for milled surfaces, shear strength increased with adhesive thickness (in the range from 30 to 100 μm), but this trend did not apply to other surface treatments, and they concluded that bonding strength depends on specific surface characteristics, such as peak density and nanoscale pores, rather than just roughness or wettability. Lee et al. [22] demonstrated that appropriate adhesives and joint geometries could achieve sub-micron positional accuracy, nearly an order of magnitude better than the typical 1 to 10 μm achieved with screw-based assembly. Maamar et al. [23] used numerical simulation to optimize adhesive layer thickness and the number of glue pads from the perspectives of thermo-elasticity, safety margins, and optical surface deformation. Xiong et al. established a theoretical model relating adhesive parameters to stress distribution within flat lens [24] and proposed a stress field uniformity evaluation method based on information entropy to evaluate the non-uniform stress distribution within lens [25]. Furthermore, a machine learning-based model was further created by Xiong et al. to build a reverse mapping from stress fields back to adhesive parameters for identification and future optimization [26].

In summary, current research on adhesive bonding for optics primarily focuses on optimizing adhesive layer thickness for thermal matching and using numerical simulations to predict assembly stress and deformation. It has been widely accepted that post-assembly surface accuracy is a key performance indicator, while the existing approaches on surface accuracy prediction rely heavily on case-by-case simulations rather than providing a general understanding; therefore, it may fail to reveal the general influence mechanism of bonding parameters on the optical surface accuracy. As a result, the design process becomes mainly reliant on experience, rather than being driven by a mechanistic approach. Existing studies on this topic have primarily focused on the characterization and prediction of internal stress fields in optical elements. For instance, our previous works [24,25,26] established theoretical frameworks to evaluate non-uniform internal stress distributions under multi-point bonding, revealing the mechanism of optical performance degradation caused by stress-induced refractive index variations. However, these existing models are generally based on the plane stress assumption, which ignores the axial stress component and thus fails to predict axial surface deformation—another critical factor affecting optical performance by altering the contact interface between the optical material and air.

In this paper, a comprehensive study into the surface accuracy of multi-point adhesive bonded flat optical components is presented. Firstly, a mechanistic analysis about the coupled influence mechanisms governing surface deformation is carried out considering the interplay between the optical element’s own structural/material properties and the assembly parameters from multiple perspectives, including dimensionless modeling, boundary conditions analysis, and reliable design requirement analysis. Secondly, building on the understanding of mechanisms, a theoretical model and numerical simulations are employed to quantitatively approximate the optical surface deformation within the critical central region (50% aperture). Finally, the quantitative approximation for the assembly-induced surface deformation is experimentally validated. The overall framework is shown in Figure 1.

2. Mechanistic Analysis

A mechanistic analysis about the coupled influence mechanisms governing surface deformation is carried out considering the interplay between the optical element’s own structural/material properties and the assembly parameters from multiple perspectives, including dimensionless modeling, boundary conditions analysis, and reliable design requirement analysis.

2.1. A General Dimensionless Coupling Model

A general dimensionless model is built in this part to describe the coupled influence of assembly parameters and component properties on the optical surface deformation.

First of all, the boundary condition for a flat optical component subjected to multi-point bonding is depicted in Figure 2. A cylindrical coordinate system $(r,\theta ,z)$ is established with its origin O at the center of the optical component. The radius of the optical component is denoted by R and its thickness by H. The center points on the two symmetrical optical surfaces are designated as $P_1$ and $P_2$, respectively. For the description of boundary condition, the assembly states are defined by the bonding stress (denoted as $q_0$), the number of glue pads (denoted as K), and the size of each glue pads (denoted as its corresponding angular $2{\theta }_0$). For bonding stress $q_0$, it is considered as a positive value for tensile stress and negative for compressive stress. The magnitude of $q_0$ depends on various factors, including the material properties and geometric dimensions of both adhesive and its bonded components, as well as environmental loads such temperature and vibration. It should be noted that due to the symmetric nature from both the loads of assembly and the geometry of optical components, the axial deformation at points $P_1$ and $P_2$ will have the same magnitude but opposite directions (opposite signs). Therefore, in the subsequent discussion, the center point deformation for optical components will focus solely on point $P_1$ for analysis. For an arbitrary point on the optical surface, the coordinates $(r,\theta ,z)$ can be used to name the point, where z is the value in the ideal design model before any manufacturing errors. Since the ideal optical surface has a predetermined shape, the value of z can be uniquely determined once r and $\theta$ coordinates are known, which means that z is a known function of $(r,\theta )$ and it can be expressed as $z(r,\theta )$. Therefore, any point on the optical surface can be named after the coordinates $(r,\theta )$.

Based on this, a dimensionless model is developed as follows to provide a generalized description for the optical surface deformation under assembly stress:

$${\displaystyle \frac{u_z(r,\theta )}{u_{0,(q_0,K,{\theta }_0;H,R;E,\nu )}}}=f_{{\displaystyle \frac{q_0}{E}},K,{\displaystyle \frac{{\theta }_0}{2\pi }}{\displaystyle \frac{H}{R}},\nu }\left({\displaystyle \frac{r}{R}},{\displaystyle \frac{\theta }{2\pi }}\right)$$

(1)

where $u_z(r,\theta )$ is the axial deformation of points on optical surface with coordinate $(r,\theta )$, $u_0$ is the axial deformation the center point of optical surface, f is a function family to describe the distribution characteristic of normalized axial deformation. Furthermore, E is the Young’s modulus and $\nu$ is the Poisson’s ratio of the optical component, and other parameters have already been mentioned previously.

The left-hand side (LHS) of Equation (1) represents the axial deformation at an arbitrary point (denoted as $u_z(r,\theta )$ ) on the optical surface normalized by the deformation at the center point (denoted as $u_{0,(q_0,K,{\theta }_0;H,R;E,\nu )}$), and the footnote $(q_0,K,{\theta }_0;H,R;E,\nu )$ of $u_0$ represents the parameter lists that determine the value of $u_0$.

The right-hand side (RHS) of Equation (1) represents a generalized form for normalized axial deformation with all variables dimensionless, where f is the distribution function family of normalized axial deformation within optical surface. The form of function family f is determined by the normalized parameters including normalized assembly parameters ${\displaystyle \frac{q_0}{E}}$, K, and ${\displaystyle \frac{{\theta }_0}{2\pi }}$, normalized geometric parameter of component ${\displaystyle \frac{H}{R}}$, and the remaining material parameter, the dimensionless Poisson’s ratio $\nu$. The change of assembly parameters, geometric and material parameters will led to different parameter lists for function f and finally lead to different forms for function f.

The dimensional equivalence of RHS and LHS is guaranteed by the dimensionless parameters on both sides. To establish this dimensionless relationship, each variable on the LHS is normalized by known parameters with same dimension. The RHS is a function f with six key dimensionless parameters derived from the key influencing factors in the multi-point bonding of the optical component. Here are the six key parameters: $r/R$, $\theta /2\pi$, $q_0/E$, $\nu$, K, and $H/R$, which encompass geometric dimensions and material properties of optical component, and the assembly parameters. All parameters are dimensionless, which ensures the dimensional consistency between the LHS and RHS, regardless of the specific form of the function f. $({\displaystyle \frac{r}{R}},{\displaystyle \frac{\theta }{2\pi }})$ is the normalized coordinate to name a specific point on an optical surface. Therefore, the RHS consists of multiple dimensionless parameters about the assembly information and component information to construct a specific form for distribution function f.

To summarize, a dimensionless model is constructed to describe the coupling relations of the assembly situation and the component properties (geometry and material) on the formation of optical surface accuracy, in which the axial deformation at the center point on the optical surface is adopted as the dimension to normalize the axial deformation at each point on the optical surface, and the normalized distribution characteristic is described by a function family with normalized variable relating the assembly parameters and component property parameters.

2.2. Coupling from Boundary Condition

In this part, the boundary condition of multi-point bonded assembly is expanded into Fourier series to illustrate the coupling of assembly parameters. Based on the theory of elasticity, known boundary conditions are used to solve partial governing equations after converting the often discontinuous boundary conditions into a continuous functional form [24]. Therefore, the continuous form of boundary condition will directly influence the solution of partial functions and then influence the expression for optical surface deformation.

Here, the coupled effects of the assembly parameters K, ${\theta }_0$, and $q_0$ on the axial deformation of the optical surface are analyzed from the perspective of boundary condition. As can be seen from Figure 2, the main mechanical boundary condition experienced by the optical component subjected to multi-point bonding is the discontinuous radial load applied on its circumference. This mechanical boundary condition can be expressed as a discontinuous piece-wise function:

$${\sigma }_{rr}(r=R)=\left\{\begin{array}{cc}{q}_0,\hfill & \mathrm{for}|\theta -{\displaystyle \frac{2\pi i}{K}}|\le {\theta }_0\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.(i=0,1,\dots ,K-1)$$

(2)

where ${\sigma }_{rr}$ is the normal stress in the radial direction.

By applying a Fourier series expansion to the above boundary condition, a continuous form can be further obtained as follows:

$${\sigma }_{rr}(r=R)={\displaystyle \frac{K{\theta }_0{q}_0}{\pi }}+\sum _{n=1}^{\infty }{\displaystyle \frac{2q_0}{n\pi }}sin\left(nK{\theta }_0\right)cos\left(nK\theta \right)$$

(3)

As can be seen in Equation (3), the three assembly parameters, including the number of glue pads K, bonding stress $q_0$, and the size of glue pads $2{\theta }_0$, are combined together to get a new set of three dependent parameters to determine the optical surface deformation, that is, $K{\theta }_0$, $q_0$, and K. In fact, this new set of parameters are related to the total loads on a single glue pads, the stress on a single glue pad, and the number of glue pads, respectively. It is obvious that the number of glue pads K and the adhesive stress on each glue pads $q_0$ are dependent parameters to influence optical surface deformation, but the size of glue pads $2{\theta }_0$ works with number of glue pads K to form the third dependent parameter $K{\theta }_0$ in the continuous form.

2.3. Coupling from Reliable Design Process

Despite the aforementioned coupling relation between K and ${\theta }_0$ from the boundary condition, there is also a trade-off between K and ${\theta }_0$ during design process. To guarantee the reliability of adhesive bonding in high-acceleration service environments, a minimum bonding area $Q_{min}$ is required to guarantee the adhesive strength, which is given by the formula:

$$Q_{min}={\displaystyle \frac{W{a}_g{f}_s}{J}}$$

(4)

where W is the weight of the optical component, $a_g$ is the worst-case acceleration, $f_s$ is the safety factor, and J is the adhesive strength.

For a flat optical component, the weight W can be described as

$$W={\displaystyle \frac{\pi D^{2}H\rho }{4}}$$

(5)

where D is the diameter, H is the thickness, and $\rho$ is the density of the material.

Alternatively, assume that the adhesive occupies the whole thickness of the edge, and the adhesive area $Q_{min}$ can be expressed into the form with assembly parameters and the geometrical parameters of the optical component:

$$Q_{min}={\left(K{\theta }_{0}DH\right)}_{min}$$

(6)

By replacing Equations (5) and (6) into Equation (4), a trade-off between K and ${\theta }_0$ during the design process can be constructed a follows:

$${\left(K{\theta }_0\right)}_{min}={\displaystyle \frac{\pi D\rho a_g{f}_s}{4J}}$$

(7)

For a specific scenario, parameters are considered as follows:

• Adhesive: 704 silicone rubber.
• Optical component material: N-BK7 glass.
• Shear strength of adhesive J: 704 silicone rubber as 1.1 MPa.
• Diameter of optical component D: 120 mm.
• Density of optical component $\rho$: N-BK7 glass as 2.5 g/cm³.
• The worst-case acceleration, $a_g$: 30 G, $G=9.8{\mathrm{m}/\mathrm{s}}^2$.
• Safety factor $f_s$: 2.

Let ${\tilde{\theta }}_0$ be the angle in degrees corresponding to the angle ${\theta }_0$ in radians (here, ${\tilde{\theta }}_0$ is adopted to distinguish between radian and degree measures when both are present, and in the following part, only ${\theta }_0$ will be used but with specific units explicitly stated), where the conversion is ${\theta }_0=\pi {\tilde{\theta }}_0/180$. Then, adopting the expression in Equation (7), the minimum value required for the product of K and the angle in degrees ${\tilde{\theta }}_0$ in the above scenario can be calculated as follows:

$${\left(K{\tilde{\theta }}_0\right)}_{min}={\displaystyle \frac{180·D\rho a_g{f}_s}{4J}}\approx {55}^{\circ }$$

(8)

To conclude, the above three perspectives, including dimensionless modeling, boundary condition analysis, and reliable design requirement analysis, illustrate the inherent coupled relations when designing the bonding parameters to control optical surface deformation. A generalized design sequence is formed as follows: (1) control the bonding stress $q_0$ separately, such as selecting an adhesive with low curing stress and designing the bondline thickness to reduce thermal stress; (2) control the product of K and ${\theta }_0$ to achieve both reliable assembly and precision assembly. A more detailed suggestion will be further provided at the end of Section 3.

3. Quantitative Approximation

This section proposes an approximation method to quantitatively describe the optical surface deformation of flat optical components subjected to multi-point adhesive bonding. Firstly, a theoretical model based on the theory of elasticity is developed to predict the deformation at the center point of optical surface. Secondly, numerical simulations are carried out to verify the theoretical solution for the center point and futher analyze the distribution pattern within the critical 50% aperture.

3.1. Simplification of the Model

To facilitate the analysis of the bonding process for planar optical elements, several reasonable simplifications are adopted in this study for the theoretical modeling process, as detailed below: (1) uniform multi-point glue pads are employed, with consistent spacing between adjacent pads and identical bonding area dimensions for each pad; (2) the bonding stress is assumed to be uniform, both within individual glue pads and across different glue pads. Additionally, to simplify the characterization of surface deformation, the uniformity of surface deformation within the 50% aperture of the planar optical element is first validated. Based on this validation, a single representative value is selected to quantify the surface deformation within the 50% aperture range.

3.2. Theoretical Model for Center Point Deformation

In this part, a theoretical model based on the theory of elasticity is proposed to predict the center point deformation for multi-point bonded flat optics, and a circumferential uniform bonding circumstances is studied in advance to provide essential prerequisites.

3.2.1. Circumferential Uniform Bonding

A theoretical model is developed here to calculate the surface deformation of a circumferential uniform adhesive bonded flat optical component, providing a fundamental basis for subsequent Section 3.2.2 for multi-point bonding.

A flat optical component subjected to circumferential uniform bonding is depicted in Figure 3. The boundary conditions can be described as follows:

(1)

On the edge surface at $r=R$, the optical component is subjected to a uniformly distributed normal stress $q_0$ (where tension is positive and compression is negative), while shear stress is zero.

(2)

On the optical surface/surfaces typically located at $z=\pm H/2$, they are free surfaces, which means the normal stresses on both surfaces are zero.

The above two boundary conditions can be expressed by the following three equations:

$$\begin{array}{cc}\hfill {\sigma }_{rr}{|}_{r=R}& =q_0\hfill \\ \hfill {\sigma }_{rz}{|}_{r=R}& =0\hfill \\ \hfill {\sigma }_{zz}{|}_{z=\pm H/2}& =0\hfill \end{array}$$

(9)

where ${\sigma }_{rr}$ is the radial normal stress, ${\sigma }_{rz}$ is the shear stress in the $rz$-plane, and ${\sigma }_{zz}$ is the axial normal stress.

The geometry of THE optical component and its boundary condition with continuous and uniform circumferential loads makes this case for solving a deformation field within optical components as a three-dimensional axisymmetric problem. The stress, strain, and displacement fields are all independent of the angular coordinate $\theta$ and are functions of only r and z. For such a three-dimensional axisymmetric stress problems, Love introduced a displacement potential function, $\Phi (r,z)$, which is a biharmonic function. By finding an appropriate form for $\Phi (r,z)$ that satisfies specific boundary conditions, analytical expressions for the stress, strain, and displacement components throughout the entire domain can all be derived. This approach allows for the deformation calculation at the center point $P_1$ on optical surface when subjected to a circumferential adhesive bonding.

According to the theory of elasticity, the displacement in three directions and stress components in six directions can be expressed in terms of the Love displacement function $\Phi (r,z)$ as follows:

$$\begin{array}{cc}\hfill 2G{u}_r& ={\displaystyle \frac{{\partial }^2\Phi }{\partial r\partial z}}\hfill \\ \hfill u_{\theta }& =0\hfill \\ \hfill 2G{u}_z& =2(1-\nu ){\nabla }^2\Phi -{\displaystyle \frac{{\partial }^2\Phi }{\partial z^2}}\hfill \\ \hfill {\sigma }_{rr}& ={\displaystyle \frac{\partial }{\partial z}}\left(\nu {\nabla }^2\Phi -{\displaystyle \frac{{\partial }^2\Phi }{\partial r^2}}\right)\hfill \\ \hfill {\sigma }_{\theta \theta }& ={\displaystyle \frac{\partial }{\partial z}}\left(\nu {\nabla }^2\Phi -{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \Phi }{\partial r}}\right)\hfill \\ \hfill {\sigma }_{zz}& ={\displaystyle \frac{\partial }{\partial z}}\left((2-\nu ){\nabla }^2\Phi -{\displaystyle \frac{{\partial }^2\Phi }{\partial z^2}}\right)\hfill \\ \hfill {\sigma }_{r\theta }& =0,{\sigma }_{\theta z}=0\hfill \\ \hfill {\sigma }_{rz}& ={\displaystyle \frac{\partial }{\partial r}}\left((1-\nu ){\nabla }^2\Phi -{\displaystyle \frac{{\partial }^2\Phi }{\partial z^2}}\right)\hfill \end{array}$$

(10)

where G is the shear modulus and $\nu$ is Poisson’s ratio, and the proposed Love function $\Phi (r,z)$ must satisfy the biharmonic equation:

$${\nabla }^2{\nabla }^2\Phi (r,z)=0$$

(11)

Based on the specified boundary conditions expressed in Equation (9), and considering that the uniform tensile/compressive assembly stress on the edge surface at $r=R$ is independent of z as mentioned previously, and stress/displacement values must remain finite at the center point (Point $P_1$ with $r=0$, $\theta =0$, and $z=H/2$ ), and the following forms for harmonic functions ${\varphi }_0$ and ${\varphi }_3$ that constitute the general solution are proposed:

$$\begin{array}{cc}\hfill {\varphi }_0(r,z)& =A(r^2-2z^2)\hfill \\ \hfill {\varphi }_3(r,z)& =Bz\hfill \end{array}$$

(12)

where ${\displaystyle \frac{\partial \Phi }{\partial z}}={\varphi }_0+z{\varphi }_3$, and A and B are unknown coefficients to be determined by the known boundary condition.

Substituting Equation (12) into the expressions for stress and displacement in Equation (10), the following explicit formula for stresses and displacements with unknown coefficients A and B are derived

$$\begin{array}{cc}\hfill 2G{u}_r& =2Ar\hfill \\ \hfill 2G{u}_z& =-\left[4A+2(1-2\nu )B\right]z\hfill \\ \hfill {\sigma }_{rr}& =2A-2\nu B\hfill \\ \hfill {\sigma }_{\theta \theta }& =2A-2\nu B\hfill \\ \hfill {\sigma }_{zz}& =-4A-2(1-\nu )B\hfill \\ \hfill {\sigma }_{rz}& =0\hfill \end{array}$$

(13)

By comparing the known boundary conditions in Equation (9) with the general form in Equation (13), the value for constants A and B can be determined as follows:

$$\begin{array}{cc}\hfill A& ={\displaystyle \frac{1-\nu }{2(1+\nu )}}{q}_0\hfill \\ \hfill B& =-{\displaystyle \frac{1}{1+\nu }}{q}_0\hfill \end{array}$$

(14)

Substituting the determined constants A and B back into the expression for the axial displacement $u_z$ in Equation (13), the displacement at any point within the flat optical components under a uniform circumferential assembly stress is derived as follows:

$$u_z=-{\displaystyle \frac{1}{G}}{\displaystyle \frac{\nu q_{0}z}{1+\nu }}$$

(15)

Specifically, the deformation at the center points of optical surfaces ($P_1$ and $P_2$) is

$$u_z(r=0,\theta =0,z=\pm H/2)=\mp {\displaystyle \frac{\nu q_{0}H}{2G(1+\nu )}}$$

(16)

To summarize, the deformation at center points of optical surfaces ($P_1$ and $P_2$) is calculated theoretically based on the theory of elasticity, and the result provides basis for the deformation calculation of multi-point bonding in the following part.

3.2.2. Multi-Point Adhesive Bonding

A theoretical model for the center point deformation of multi-point bonded flat optic is developed in this part based on Section 3.2.1. One thing to mention is that, at first, the authors tried to develop a more complicated theoretical solution for the deformation in the full three-dimensional field within multi-bond optics, but struggled with convergence difficulties which become particularly severe when employing complex function-based functions (including Bessel functions). Therefore, a simplified model is compromised and employed here to approximate the center point deformation of the optical surface.

Since the glue pads in the multi-point adhesive assembly are of identical size and are equally spaced, the central axis (as shown in Figure 2 as $\overrightarrow{P_1{P}_2}$) is not only the central axis of geometry but also that of the stress boundary. Therefore, the deformation direction of center point $P_1$ will only be in the axial direction (z), regardless of whether the component is bonded at discrete multi-point bonding or continuously circumferential bonding. This allows the center point displacement to be treated as a scalar quantity, where the sign indicates the direction along the positive or negative z-axis. Furthermore, by applying the principle of superposition from theory of elasticity, the total displacement at the center point can be calculated by the scalar summation of the effects from each individual load on the edge surface.

As illustrated in Figure 2, for a multi-point bonding scenario, the symmetry of both geometry and the loading implies that any unit-radian segment of a circumferential load contributes equally to the deformation at the center point. Therefore, the deformation at the center point for an optic with K discrete bonding points can be calculated by scaling the result from a uniform circumferential load by the ratio of the total bonding arc length to the full circumference. This relationship is expressed as

$$\begin{array}{cc}\hfill u_z(P_1,K)& =\left(-{\displaystyle \frac{\nu q_{0}H}{2G(1+\nu )}}\right)\times {\displaystyle \frac{K·2{\theta }_0}{2\pi }}\hfill \\ \hfill & =-{\displaystyle \frac{K\nu {\theta }_0{q}_{0}H}{2\pi G(1+\nu )}}\hfill \end{array}$$

(17)

where $u_z(P_1,K)$ represents the axial deformation at center point $P_1$ when subjected to K-point adhesive bonding.

Adopting the relation between shear modulus G and Young’s modulus E, $G=E/\left[2\right(1+\nu \left)\right]$, the center point deformation for K-point adhesive bonding in terms of Young’s modulus is derived as follows:

$$u_z(P_1,K)=-{\displaystyle \frac{K\nu {\theta }_0{q}_{0}H}{\pi E}}$$

(18)

where K is the number of glue pads in the multi-point assembly, and the angular size of each glue pads is denoted by $2{\theta }_0$ (measured in radians).

To conclude, the center point deformation for a flat optical component subjected to multi-point bonding is yielded theoretically. In the following part, numerical simulations will be carried out to verify the theoretical results and further analyze the distribution characteristics within the critical 50% aperture.

3.3. Numerical Verification for Center Point Deformation and Further Analysis Within 50% Aperture

Numerical simulation methods are employed in this part for both the verification of quantitative approximation at center point and for a further distribution analysis within the critical 50% aperture.

3.3.1. Simulation Setup

A flat optical component with a diameter of 120 mm, a thickness of 10 mm, and made of N-BK7 glass is adopted as the simulation model here. The specific simulation parameters are detailed in

Table 1. Summary of simulation parameters and boundary conditions.

Terms	Parameters	Values
Simulation model	Software	Abaqus 6.14
Element in mesh	Analysis type	Static, general
	Element type	C3D8R
Material	N-BK7 glass	Young’s modulus: 82 GPa
		Poisson’s ratio: 0.206
Boundary conditions	Stress boundary	Pressure: $q_0\in \{0.10,0.25,0.50\}$ MPa
	Constrained DOFs	Center of lens, U1 = 0, U2 = 0, UR3 = 0

.

The way to analyze deformation pattern within aperture is illustrated in Figure 4. The method involves introducing a horizontal cutting plane into the 3D surface deformation map and then generating a top-down view. Specifically, the process is as follows: firstly, a cutting plane is defined at a specific deformation value of interest, denoted as $z=z_0$. This cutting plane partitions the surface into regions where the deformation is either greater than or less than this threshold value. A top-down view is then generated, as shown in Figure 4b):

(1)

Regions with deformation values greater than $z_0$ (i.e., those above the plane) are shown in color;

(2)

Regions with deformation values less than $z_0$ (i.e., those below the plane) are obscured by a semi-transparent gray overlay representing the cutting plane.

Additionally, a dashed circle is drawn on this top-down view to indicate the boundary of the central 50% aperture, which is the most critical region for optical performance. By observing whether this circle is entirely within the gray-shaded area or if it intersects with the colored regions, one can quickly determine the relationship between the deformation magnitudes within the critical 50% aperture and the deformation value $z_0$ of the chosen cutting plane.

3.3.2. Verification for Center Point on Optical Surface

The effectiveness of the quantitative approximation for center point deformation in Section 3.2.2 is assessed by varying the bonding parameters and comparing the simulation results against the approximation formula in Equation (19). Simulations were carried out using the following set of parameters, in which the number of glue pads $K\in \{2,3,6\}$, size of glue pads $2{\theta }_0\in \{{10}^{\circ },{20}^{\circ }\}$, and bonding stress $q_0$ is set as 0.5 MPa. The simulation results and the comparisons are shown in

Table 2. Comparison of FEM simulation results and quantitative approximation results, which are identical.

No.	Dimensional Parameters		Assembly Parameters			Center Point Deformation (mm)
	R (mm)	H (mm)	K	$\mathbf{2}{\mathbf{\theta }}_{\mathbf{0}}$ (°)	${\mathbf{q}}_{\mathbf{0}}$ (MPa)	Results (FEM and Theoretical)
A1	60	10	6	20	0.5	−4.19 × 10−6
A2	60	10	2	20	0.5	−1.40 × 10−6
A3	60	10	3	20	0.5	−2.09 × 10−6
A4	60	10	6	10	0.5	−2.09 × 10−6
A5	60	10	3	10	0.5	−1.05 × 10−6
A6	60	10	2	10	0.5	−6.98 × 10−7

.

For each combination of these assembly parameters, the deformation of optical component was computed via abaqus (version 6.14). A direct comparison between these numerical results and the values predicted by the quantitative approximation formula are presented in Table 2. Table 2 shows a good consistency between the approximation and the simulation results for the deformation at the center point of the optical surface, and as such, the validity of the quantitative approximation was confirmed.

3.3.3. Further Analysis Within 50% Aperture

Six groups of simulations were further conducted, covering variations in the number of glue pads $K\in \{3,6\}$, the bonding stress $q_0\in \{0.10,0.25,0.50\}$ MPa, and the angular size of the glue pads $2{\theta }_0\in \{{10}^{\circ },{20}^{\circ }\}$, as shown in

Table 3. Simulation groups for analysis of the distribution pattern within the 50% aperture.

No.	Assembly Parameters
	Number of Glue Pads (K)	Size of Glue Pads ($\mathbf{2}{\mathit{\theta }}_{\mathbf{0}}$) (°)	Bonding Stress (${\mathit{q}}_{\mathbf{0}}$) (MPa)
B1	3	10	0.50
B2	3	20	0.50
B3	6	20	0.10
B4	6	20	0.25
B5	6	10	0.50
B6	6	20	0.50

. The distribution pattern of deformation on the optical surface is analyzed to reveal the characteristics within the critical 50% aperture.

Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 illustrate the distribution of optical surface deformation under various assembly conditions, specifically, the number of glue pads is $K\in \{3,6\}$, the bonding stress is $q_0\in \{0.10,0.25,0.50\}$ MPa, and the angular size of glue pads is $2{\theta }_0\in \{{10}^{\circ },{20}^{\circ }\}$. Figure 5a, Figure 6a, Figure 7a, Figure 8a, Figure 9a and Figure 10a show a 3D contour plot of the optical surface deformation under varies assembly parameters; Figure Figure 5b, Figure 6b, Figure 7b, Figure 8b, Figure 9b and Figure 10b and Figure Figure 5c, Figure 6c, Figure 7c, Figure 8c, Figure 9c and Figure 10c show the vertical views of the 3D contour plot intersected with a cutting plane $z=z_0$, where $z_0$ is set to $110\%·u_0$ and $90\%·u_0$, respectively. Here, $u_0=u_z(P_1,K)$, which is the axial deformation at the center point $P_1$ on optical surface at each bonding circumstance. By comparing the relative position of the colored regions with respect to the 50% aperture dashed line in Figure Figure 5b, Figure 6b, Figure 7b, Figure 8b, Figure 9b and Figure 10b and Figure Figure 5c, Figure 6c, Figure 7c, Figure 8c, Figure 9c and Figure 10c, it is found that the magnitude of deformation in the 50% aperture is within the $\pm 10\%$ relative deviation compared with the central point deformation $u_0=u_z(P_1,K)$. The colored regions in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 indicate these areas with deformation greater than $z_0$, while the gray areas represent regions with deformation less than $z_0$.

The results in Figure Figure 5b, Figure 6b, Figure 7b, Figure 8b, Figure 9b and Figure 10b demonstrate that increasing the number of glue pads significantly improves the uniformity of the normalized deformation $u_z/u_0$ within 50% aperture. When the number of glue pads reaches 6, the normalized deformation $u_z/u_0$ within the 50% aperture satisfies $90\%<u_z/u_0\le 110\%$, which means that the relative deformation deviation within this region is no more than $\pm 10\%$ compared to that $u_0$ at the center point $P_1$. Variations in the size $2{\theta }_0$ and bonding stress $q_0$ do not have a significant impact on the uniformity of normalized deformation within the 50% aperture. Furthermore, it can be noticed that outside the 50% aperture, particularly near the boundary where the bonding stresses are applied, the optical surface deformation is closely related to the local boundary loads. In contrast, within the 50% aperture, the surface is more significantly affected by the superposition of loads with more glue pads, resulting in a more uniform stress distribution.

To summarize, the distribution of optical surface deformation within the 50% aperture relative to the displacement at the center point is quite uniform with no more than $\pm 10\%$ deviations from the deformation $u_0$ at the center point $P_1$ when the number of glue pads reaches $K=6$, and the uniformity can be further improved with the number increases in glue pads. Therefore, the quantitative approximation for surface deformation at the center point can serve as a useful indicator to characterize the overall surface deformation within this critical 50% aperture. This conclusion can be further adopted to guide the design and optimization process for multi-point bonding when doing the trade-off between the number of glue pads K and the size of glue pads $2{\theta }_0$.

Based on the previous finding, the deformation of an optical surface subjected to multi-point adhesive bonding is primarily governed by three key parameters: the number of adhesive points K, the total bonding areas (proportional to the product $K{\theta }_0$), and the bonding stress $q_0$. Based on these relationships, a recommended design sequence is further proposed based on the previous studies and illustrated for the determination of bonding parameters:

(1)

First, determine the allowable stress $q_0$, which can be optimized by the geometrical design of optics/adhesive layer/barrel, and it can also be further optimized by the selection of their material properties;

(2)

Second, determine the minimum bonding area needed by applying reliability-based design principles (such as safety factors) to the worst-case acceleration load, and determine the maximum bonding area allowable by predicting the assembly outcomes (such as surface deformation, and stress distribution), and then determine a proper value for $K{\theta }_0$;

(3)

Finally, optimize the number of glue pads K. The increase in glue pads significantly enhances the deformation uniformity on optical surface; therefore, it is recommended to design more glue pads on the premise that each individual adhesive joint has adequate strength.

4. Experimental Verification

4.1. Experimental Setup

The basic idea of the experiment is to quantitatively simulate the loads induced by multi-point adhesive bonding surround optics. This was accomplished using a multi-point quantitative loading device, previously developed by the authors, to apply a maximum of six-point calibrated loads to an optical component. The induced deformation on the optical surface is then measured using a Tyggo interferometer (Tyggo, Chengdu, China, CS100-LS, with $\lambda =632.8$ nm and a $\Phi =100$ mm aperture).

The experiments consist of multiple magnitude of loads, number of loads, and loading area to simulate the corresponding varying bonding stresses, numbers of glue pads, and bonding size, respectively. The measurement for optical surface is carried out for each set of parameters. The general experimental process is illustrated in Figure 11. First, the optical component to be tested is selected, and a series of simulated loading cases are designed based on its dimensions and the desired assembly parameters (number of loading points, loading area, and load magnitude). Second, loading cases are then applied to the optical component one by one using the quantitative loading device that the authors already developed. Once the loads are applied to a stable condition, the optical surface profile is then measured with the interferometer. Third, change the loads and repeat the surface measurement until all loading cases are finished.

Figure 12 shows a photograph of the experimental setup, the most essential equipment, and instruments included as follows.

(1)

An optical component integrated with the multi-point quantitative loading device.

(2)

A 5-DOF(degree of freedom) stage for the alignment of quantitative loading device.

(3)

A linear guide rail to position the 5-DOF stage.

(4)

A customized fixture for vertically mounting the quantitative loading device onto the 5-DOF stage.

(5)

An interferometer for surface measurement.

(6)

An optical platform for vibration isolation to position all the equipment and components above.

The essential assembly process for the whole experimental system involved mounting the optical component along with the quantitative loading device onto the 5-DOF stage using customized fixtures. The 5-DOF stage utilized in the experiments comprises five degrees of freedom (DOFs), specifically, one linear guide rail (translational DOF), one shear lifting stage (translational DOF), and one 3-DOF stage (including one pitch DOF, one yaw DOF, and one translational DOF). The detailed performance indicators are as follows: the linear guide rail (translational DOF) has a stroke of 1000 mm and a positioning accuracy of 5 μm; the shear lifting stage (translational DOF) has a stroke of 60 mm and a positioning accuracy of 5 μm; for the 3-DOF stage, the pitch DOF features a rotation range of 10° and a rotation accuracy of 0.1°, the yaw DOF has a rotation range of 10° and a rotation accuracy of 0.1°, and the translational DOF possesses a stroke of 6.5 mm and a positioning accuracy of 10 μm.

This entire assembly is positioned on the optical platform in the measurement path of the interferometer. The 5-DOF stage is used to precisely adjust the position of the quantitative loading device and the optical component. The fine alignment between the measured optical surface and the interferometer is a critical step for obtaining clear and uniform interference fringes required for further accurate measurement.

4.2. Experimental Process and Results

The experimental workflow consists of three primary adjustment steps followed by a measurement step:

(1)

Loading configuration setup: adjust the number and contact area of the loading heads on the multi-point loading device to set the desired number of load points K and loading area $2{\theta }_0$.

(2)

Loads magnitude setup: adjust the force bolts on the quantitative loading device to apply specific, calibrated loads.

(3)

Fine alignment before measurement: adjust the 5-DOF stage to position the optical surface, optimizing the alignment between optical surface and the interferometer to obtain uniform and clear interference fringes for a better surface profile measurement.

Once these adjustments were completed, the interferometer is used to measure the surface profile of the optical component under specific load conditions. To ensure data reliability, each measurement was repeated multiple times (10 measurements each) and take the average of multiple measurements. All experimental loading cases and their following sequences are listed in

Table 4. Experimental sequence and results; $\Delta Z_r$ is the relative deformation compared with the previous experiment.

No.	Loading Parameters				Surface Profile Within 50% Aperture ($\times {10}^{-4}$)
	K	$\mathbf{2}{\mathbf{\theta }}_{\mathbf{0}}$ ( ° )	$\mathit{F}/\mathit{N}$ (N)	${\mathit{q}}_{\mathit{0}}$ (MPa)	${\mathit{Z}}_{\mathbf{Mean}}$	${\mathit{Z}}_{\mathbf{0}}$	${\mathit{Z}}_{\mathit{r}}$	$\Delta {\mathit{Z}}_{\mathit{r}}$
G1:1	6	10	10	0.10	−0.5720	1.6213	2.1932	–
G1:2			20	0.20	−0.3336	1.8214	2.1550	−0.0382
G1:3			50	0.50	−0.4250	1.7617	2.1867	0.0317
G1:4			100	1.00	−1.0128	1.1259	2.1387	−0.048
G2:1	3	10	10	0.10	−0.4482	1.6856	2.1337	–
G2:2			20	0.20	−0.4396	1.7263	2.1659	0.0322
G2:3			50	0.50	−0.8700	1.3136	2.1836	0.0177
G2:4			100	1.00	−0.4880	1.6858	2.1738	−0.0098
G3:1	6	20	10	0.05	−0.7766	1.3182	2.0948	–
G3:2			20	0.10	−0.7272	1.3238	2.0510	−0.0438
G3:3			50	0.25	−0.5660	1.5393	2.1053	0.0543
G3:4			100	0.50	−0.5273	1.5559	2.0832	−0.0221
G3:5			150	0.75	−0.4919	1.5651	2.0570	−0.0262
Loads removed							2.3377	0.2807

, including 3 groups with a total of 13 subgroups. This loading magnitude for each groups is increased little by little, following an increasing sequence to minimize the influence of any short-term residual stresses within the optical component that might not be fully released immediately after a load change, thereby improving the reliability of the measurement results. Once the full range of loads for a group is finished, the number of loading points or loading area are then changed to begin the next experimental group. The change of number and size of glue pads is achieved by the change of number and size of customized loading head, and the change of loads is achieved by tightening and loosening the screw in the quantitative loading device. The details for the three groups of experiments are summarized below.

• Group 1 (four subgroups, denoted by G1: 1–4):Corresponds to assembly parameters with number of glue pads $K=6$ and the size of glue pads as $2{\theta }_0={10}^{\circ }$. The simulated adhesive stresses were $q_0\in \{0.10,0.20,0.50,1.00\}$ MPa. The area of each loading head is 100 mm², and the applying loads on loading heads are gradually increasing from 10 N, 20 N, 50 N, to 100 N, respectively.
• Group 2 (four subgroups, denoted by G2: 1–4): Corresponds to assembly parameters with number of glue pads $K=3$. All other parameters (size ${\theta }_0$, bonding stresses $q_0$, and its corresponding loads) are identical to Group 1.
• Group 3 (five subgroups, denoted by G3: 1–5): Corresponds to assembly parameters with number of glue pads $K=6$ and a larger size of glue pads $2{\theta }_0={20}^{\circ }$. The simulated bonding stresses are $q_0\in \{0.05,0.10,0.25,0.50,0.75\}$ MPa. The area of each loading head is 200 mm², and the applying loads on loading heads are gradually increasing from 10 N, 20 N, 50 N, 100 N, and 150 N, respectively.

Figure 13 shows the measurement interface of the interferometer for the unloaded state. It is important to note a key experimental limitation. The full aperture of the optical surface to be measured is of 120 mm, whereas the maximum measurement aperture of interferometer is 100 mm. Consequently, the full aperture of the test optic could not be measured. Furthermore, physical constraints of the 5-DOF stage prevented perfect concentric alignment between the central axis of optical component and the reference axis of the interferometer. As a result, the raw measurement data (as seen in Figure 13) cover an area slightly smaller than the full optic and exhibit a minor center offset.

To mitigate the influence of this offset in the subsequent analysis, all measured surface maps underwent a post-processing translation step to move the geometric center of the optic to the center of the data map. While this processing results in a final analyzed aperture that is slightly smaller than the full 100 mm measurement aperture, it remains more than sufficient to include the central 50% aperture region, which is the focus in this study.

Figure 14, Figure 15 and Figure 16 present the measured optical surface profiles for the 13 experimental load conditions. For each case, two contours are shown: (a) the raw measurement data captured within the full aperture of interferometer, and (b) the post-progressing data for optical surface profile within the central 50% aperture after applying a translation correction. The term translation correction refers to a data post-processing step that compensates for the unavoidable misalignment between the geometric center of the test optic and the aperture of measurement. This numerical shift ensures that the center of the measuring data be coincident with the geometrical center of the optical component and provides convenience for the subsequent efficient data analysis.

4.3. Results Analysis

Due to a complex combined influence of processing errors and residual stress on the optical surface accuracy, as well as the alignment error of quantitative loading device that occurs during the loading process, the experimental results are not as clear and straightforward as the simulated ones. When the assembly stresses are applied, the deformation field induced by assembly stress will undergo vector superposition with processing errors and residual stress, which means that the idea flat optical surface is definitely not a ideal flat one at the beginning and there are also three-dimensional residual stresses inside the optics. Therefore, the actual optical surface deformation under assembly stress is much more complicated than the ideal circumstance mentioned before. The problem is further complicated by the fact that in any given interferometric measurement, the rigid-body displacement of the surface under test, introduced by variations in the experimental loading process, cannot be determined.

To make it possible to verify the effectiveness of quantitative approximation, in this study, the experimental data is further processed to obtain a better understanding for the deformation of optical surface under adhesive bonding. Two indicators, the relative surface position $Z_r$ and the relative surface deformation $\Delta Z_r$, are proposed to characterize the surface position within 50% aperture.

$$\begin{array}{c}\hfill Z_r(K,{\theta }_0,q_0)=-[Z_{\mathrm{Mean}}(K,{\theta }_0,q_0)-Z_0(K,{\theta }_0,q_0)]\\ \hfill \Delta Z_r=Z_r(K^2,{\theta }_0^2,q_0^2)-Z_r(K^1,{\theta }_0^1,q_0^1)\end{array}$$

(19)

where $Z_0$ is the z coordinate measured for the center point on optical surface, $Z_{\mathrm{Mean}}$ is the mean value of z coordinates for all measured points within 50% aperture. A negative sign is added to make the positive value corresponding to the convex shape and negative to the concave shape. The superscripts in $K^1,{\theta }_0^1,q_0^1$ and $K^2,{\theta }_0^2,q_0^2$ represent two cases with different assembly parameters. In general, $Z_r$ will evaluate the relative surface position of the 50% aperture, and $\Delta Z_r$ will characterize the change of relative position between two loading cases so as the describe the deformation of optical surface. The relative surface position $Z_r$ mitigates the influence of rigid-body translation in the z-axis (piston error) that might be introduced during the experimental process, such as when changing loading heads or adjusting the load magnitude or fine alignment for clear fringes, providing a more accurate characterization and comparison of the surface shape deformation under different load parameters.

Table 4 lists the relative position of optical surface $Z_r$ for the 13 subgroups of experiments corresponding to the contour map shown in Figure 14, Figure 15 and Figure 16. Figure 17 plots relative position $Z_r$ versus the applied load for the 3 experimental groups and 13 subgroups. As can be seen in Table 4 and Figure 14, Figure 15 and Figure 16, the surface position does not change monotonically as the load is initially increased. This behavior is attributed to the interaction between the externally applied assembly stress and the pre-existing, non-uniform residual stresses within the optical component. Only after the applied assembly loads surpass a certain threshold does its effect become dominant. Across all experimental groups, a consistent trend emerges when the load exceeds 50 N that the deformation direction of optical surface becomes regular, which indicates the deformation is now primarily driven by the applied assembly stresses and the influence of residual stress is significantly diminished. One thing to mention is that the final surface, after the removal of all assembly load, is different from the initial optical surface before loading. This circumstances can be reasonably explained in that initially the optical surface is influenced by the residual stress which is highly non-uniform and irregular in contrast to the assembly stress field. However, during the loading process, the residual stress is partly released and partly reorganized into a more uniform manner, thereby significantly reducing the influence of residual stress on the optical surface profile.

Comparison with Quantitative Approximation

Table 5. Comparisons between the quantitative approximations and the experimental results.

Property	G1	G2	G3
Loads range for the calculation/N	10–100	50–100	50–150
Increment of stress/MPa	0.9	0.5	0.5
Surface deformation $\Delta Z_r$ ($\times {10}^{-6}$ mm)	5.45	0.98	4.83
Experimental deformation per 0.5 MPa ($\times {10}^{-6}$ mm)	−3.028	−0.980	−4.830
Quantitative approximation per 0.5 MPa ($\times {10}^{-6}$ mm)	−2.090	−1.050	−4.190
Error ($\times {10}^{-6}$ mm)	−0.938	0.070	−0.640
Relative error	44.5%	6.7%	15.0%

presents the comparison between the processed experimental results and the predictions from the quantitative approximation by considering the regime where the applied assembly loads are sufficient to induce a monotonic change in the optical surface profile to reduce the non-linear influence of residual stress. The analysis considers only the monotonic response regime, rather than the entire range starting from the initial load. To be specific:

• Group G1 (First Loading): As this was the first time the optic was loaded, the residual stress is the primary stress component in the optical component prior to loading. To demonstrate the effect of residual stress, the calculation of deformation within loads range is chosen from 10 N to 100 N for this group.
• Groups G2 and G3 (Subsequent Loadings): These experiments were performed after the optic had already undergone significant loading in G1, left the optic in a state of heightened residual assembly stress that was difficult to release quickly. To minimize this influence, the analysis for these two groups uses only the data from the stable monotonic deformation regime.

  –

  For G2, the deformation is calculated using the difference between the surface shapes measured from 50 N to 100 N.

  –

  For G3, the deformation is calculated using the difference between the surface shapes measured from 50 N to 150 N.

Results in Table 5 and Figure 17 reveal a relative discrepancy for the G1 experiment, where the measured deformation resulted in an error of 44.5% relative to the quantitative approximation. In contrast, the errors for groups G2 and G3 were much smaller, at 6.7% and 15%, respectively, and the effectiveness of the proposed quantitative approximation is verified. The large error in G1 is likely attributable to the initial, less regular and less non-uniform residual stress field within the optic. The smaller errors observed in G2 and G3 suggest that the high-load cycle from the G1 experiment acted as a form of “mechanical settling,” forcing the stress distribution within the component change into a more regular and uniform state that more closely resembles the idealized multi-point loading condition, thereby significantly reducing the confounding influence of the initial residual stress.

On thing to mention here is that the experimental verification only applied the compressive loads at the maximum value of 150 N, and limited the applying bonding stress within 1 MPa, therefore the magnitude of relative deformation is only about nanometer in both experiments and quantitative approximations. The reason for this is that the quantitative loading device at the current version can only safely apply a maximum of around 150 N compressive forces by fastening the screw to press the spring, and larger loads might cause the spring to fly out and injure operator. In the practical adhesive bonding assembly process, the bonding stress can reach tens of megapascals [23,27]; therefore, the surface deformation should reach tens of nanometer under the practical situations according to the quantitative approximation method proposed in this paper and also consistent with the values mentioned in these papers.

5. Conclusions and Prospects

5.1. Conclusions

This paper presented a comprehensive study on the surface deformation of flat optical components caused by multi-point adhesive bonding. A general understanding is provided based on mechanistic analysis and quantitative approximation for the optical surface deformation. The mechanistic analysis illustrates the coupled relations between assembly parameters and the geometrical/material property of optical components on the formation of optical surface accuracy. On this basis, a quantitative approximation method for the deformation of center point and 50% aperture on the optical surface when subjected to a multi-point adhesive bonding is developed based on theory of elasticity and numerical simulation, and a design sequence for the attribution of assembly parameters is recommended. The approximation method is further verified by experiments with a relative error of no more than 15% during the repeated loading process. Several conclusions can be drawn as follows.

(1)

The optical surface deformation for the optical component when subjected to multi-point adhesive bonding is normalized by the deformation at center point to form a more generalized dimensionless model to describe the distribution of deformation within aperture with all dimensionless parameters containing both assembly information and component information.

(2)

The three independent assembly parameters governing the deformation of optical surface subjected to multi-point bonding is clarified by boundary condition analysis and reliable design requirement analysis, that is, the number of glue points K, the magnitude of bonding stress $q_0$, and the combined parameter $K{\theta }_0$ (which is proportional to the total bonding areas).

(3)

A quantitative approximation method is proposed based on the theory of elasticity and numerical simulations to characterize the optical surface deformation within the critical 50% aperture using the center point deformation. It is found that the increase in the number of glue pads will increase the uniformity within the aperture, while the change of size and bonding stress of glue pads do not show significant influence. When the number of glue pads reaches six, the deformation within 50% aperture normalized by that of center point is quite uniform with a maximum of ±10% relative deviation from the center point.

(4)

A design sequence is further recommended for the determination of bonding parameters:

• Determine the allowable stress $q_0$, which can be optimized by the geometrical design of optics/adhesive layer/barrel, and it can also be further optimized by the selection of their material properties;
• Determine the minimum bonding area needed by applying reliability-based design principles (such as safety factors) to the worst-case acceleration load, and determine the maximum bonding area allowable by predicting the assembly outcomes(such as surface deformation, stress distribution), and then determine a proper value for $K{\theta }_0$;
• Optimize the number of glue pads K. The increase in glue pads significantly enhances the deformation uniformity on optical surface; therefore, it is recommended to design more glue pads on the premise that each individual adhesive joint has adequate strength.

(5)

Three groups and thirteen subgroups of experiments are carried out to verify the efficiency of the quantitative approximation method. The experimental result of Group 1 is about 44.5% relative error compared with the quantitative approximation, this is most likely due to the effect of the non-uniform and non-regular residual stress after processing. For Group 2 and Group 3, the relative error is significantly reduced to 15% or less, and this is because the residual stress was released during the loading process in Group 1 and was reorganized into a more uniform and regular manner forced by the assembly loads. Therefore, the impact of non-uniform residual stress on stress distribution induced by assembly process is significantly reduced and the effectiveness of the quantitative approximation is verified.

5.2. Future Work and Prospects

In this paper, a general understanding for the coupled relations between assembly parameters, property of optical component, and the optical surface deformation are investigated by a mechanistic analysis and a quantitative approximation. This study also highlights several avenues for future research.

(1)

A more general three-dimensional analytical solution for the optical surface deformation within the full aperture for a better understanding. The authors spent a long time on this research but the convergence problem is hard to solve when doing the final unknown coefficient determination by adopting Bessel functions, so the general expression for the deformation within full aperture still has not been solved yet.

(2)

Extend the results to non-uniform multi-point adhesive bonding condition and to concave and convex optics. In the real assembly process, the assembly parameters in practical varies from the intended ideal uniform condition, and the surface shape of optics varies a lot. Further work should address this realistic problem to achieve a closer agreement between the predicted and actual outcomes. To address this more comprehensively, we will also incorporate systematic sensitivity analyses for curvature factors of curved optical elements (spherical, aspheric, etc.) and non-uniform bonding conditions (e.g., adhesive layer thickness variations, curing shrinkage, and sequential differences in multi-point bonding). Finite element method (FEM) simulations will be adopted to investigate how these factors alter stress transmission paths, edge constraint effects, and asymmetric stress field distributions, thereby expanding the geometric and practical applicability of the proposed model.

(3)

Quantify the interaction between residual stress and assembly stress. The experiments in this paper confirmed a significant coupling between the initial residual stress and the applied assembly stress, this interaction was not quantitatively modeled here. Future research should focus on developing methods to quantify the residual stress and studying the relaxation and redistribution of residual stress during the assembly process. Building on this, we will supplement high-stress-level loading tests using typical optical adhesives (e.g., RTV rubber and UV-curable epoxy) to cover stress ranges more relevant to engineering practice. Additionally, accelerated aging experiments will be conducted to evaluate creep and stress relaxation effects at the bond interface induced by environmental factors such as temperature cycling and vibration fatigue. These efforts aim to establish a coupled stress–time–environment constitutive relationship, which will be integrated into the existing model to enhance its engineering reliability. We will also systematically review and incorporate relevant research findings [28] to deepen the understanding of temperature–stress coupling effects on optomechanical assemblies.

(4)

Building upon the established forward prediction model, inverse algorithms will be developed to optimize the topology of optics and assembly parameters, thereby enabling the active minimization of wavefront error during the design phase. Future research will also incorporate Pareto frontier analysis or a multi-objective optimization framework based on response surface methodology and systematically analyze the trade-offs between deformation uniformity, bonding area, manufacturing costs, and bonding reliability. This will provide a more comprehensive and comparable decision-making basis for the optimal design of multi-point bonding processes.