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Automatic control method for freeform surface shapes

Dewen Cheng, Danyang Li, Cheng Yao, Hailong Chen, Cheng Chi, and Yongtian Wang

Open Access Open Access

Abstract

Designing freeform optics with high degrees of freedom can improve their optical performances; however, there are high requirements for controlling the surface shapes of such optics. Optical designers need to add constraints to the optimization process and make repeated adjustments to ensure the manufacturability of these shapes; this process is cumbersome and relies heavily on the experience of the designer. In this study, an automatic control method for freeform surface shapes is proposed. By adding an outer loop to the optimization process, the principal curvature and sag departure of the sampling points are gradually controlled during the optimization cycle based on the system requirements and surface evaluation results. The method was implemented in CODE V and successfully applied to a design example in freeform prism optics.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

A freeform surface (FFS) is a type of non-rotationally symmetric continuous surface that is used in a variety of optical systems, such as telescopic systems [1], spectrometers [2], lithography machines [3], head-mounted displays [4], and projectors [5]. Additionally, these surfaces are found in commercially available applications such as mobile phone lenses and augmented reality (AR) displays. Compared to spherical and aspherical surfaces, freeform surfaces provide more degrees of freedom for correcting aberrations, improving imaging quality, reducing the number of components, and compressing the sizes and weights of optical systems [6]. However, the mathematical complexity and non-rotational symmetry of freeform surfaces pose challenges for optimization and shape control in optical design.

Efficient ray-tracing and optimization algorithms determine the optimal parameters for freeform optic designs, and they employ a defined merit function for evaluating the designs [7]. The local minimum of the merit function and the corresponding design result can be obtained; however, such a design is not always the best in terms of manufacturability. The unrestricted use of too many freeform terms can result in unnecessarily large freeform departures and local curvatures, which increases the sensitivity of the assembly and produces difficulties in manufacturing and testing of the surface [8].

Single-point diamond turning is typically used to process optical surfaces or freeform surface molds for injection molding, and it can achieve a high-precision machining of optical freeform surfaces. However, certain local features of freeform surfaces, such as large local curvatures, different curvature centers, and curvature inversions can affect the precision of machining, and even make it impossible to machine the optical surface [9–12]. Optical plastic injection molding is a leading mass-produced method that reduces the production costs of FFS elements. The yield and precision of optical components, particularly ones with large effective apertures, are affected by the shapes of freeform surfaces [13]. For example, If the surface of an optical element has a violently fluctuating sag, the difficulty of processing its mold would be significantly increased, and if an optical surface has a drastic change or a very exaggerated principal curvature, the final injection molded lens would have reduced surface accuracy due to significant differences in shrinkage rates at different parts, and might even cannot be correctly injected. Thus, if the manufacturability of optical elements is considered and the shapes of freeform surfaces are controlled in the design process, the difficulty, costs and time in manufacturing can be reduced without sacrificing the optical performance.

In the process of optical design, there are some common methods for enhancing manufacturability through surface shape optimization. These include adopting aspheric surfaces with coefficients correlating to the slope and sag departure that reflect the manufacturability [14], and controlling the minimum incident ray angle on a surface to reduce the sensitivity of the surface shape error [15] or controlling the shape of an optical surface by introducing constraints, including the tangent of the surface [16], the degree of deviation from a reference surface [17], and the square sum of orthogonal polynomial coefficients [18]. The non-rotational symmetry of freeform surfaces determines that designers usually need to finely control them by constructing constraint with sampling the surfaces over a large grid of points in the effective aperture or larger range. This can be a tedious process, that potentially increases the computation and time of optimization iteration. In the optimization process, each iteration requires the derivatives of each variable to be calculated with respect to the image quality or constraints. The more complex the description of a freeform surface, the greater the computation required for an iteration and the slower the system convergence rate. Controlling many sampling points further slows down optimization speed. Additionally, designers need to adjust the constraints of surface shapes according to the results of each optimization loop to ensure that the optical surfaces gradually converge within a processable range; this requires considerable experience on the part of the designers.

In this study, an automatic control method for freeform surface shapes is proposed. An external loop was added to the optical optimization process, enabling the prior evaluation of the freeform surface shapes before each optimization and the construction of the corresponding constraints. The surface shape was controlled in each subsequent optimization cycle to achieve a smooth freeform surface shape with reduced processing difficulties. The second section introduces details regarding the principal curvatures of the surface and the method for controlling surface shapes, and the automatic control method realized in CODE V. The third section gives a design example for a freeform prism system applied to AR. Then, the prototype and experimental results are presented in the fourth section. Finally, the fifth section concludes the paper.

2. Automatic control method for freeform surface shapes

2.1 Key mathematical indicators for control

There are many mathematical descriptions of freeform surfaces, including local descriptions, such as the radial basis function and non-uniform rational basis spline (NURBS), and global descriptions, such as the XY and Zernike polynomials. Each surface description has its pros and cons, such as high precision or high converge rate, but no matter which type of surface description is used, it is the final optimized surface shape that influences the processing difficulty. In Cartesian coordinate system, a smooth surface S in three-dimensional space can be expressed as:

$${\boldsymbol r}(x,y) = (x,y,z(x,y)). $$
Its unit normal vector is:
$${\boldsymbol n} = \frac{{({{\boldsymbol r}_x} \times {{\boldsymbol r}_y})}}{{\textrm{|}{{\boldsymbol r}_x} \times {{\boldsymbol r}_y}\textrm{|}}}, $$
where ${{\boldsymbol r}_x}$ and ${{\boldsymbol r}_y}$ denote the first-order partial derivatives of the surface parametric equation with respect to x and y, respectively. The unit normal vector n at any point on the surface has an infinite number of section planes. Each plane intersects the surface with a plane curve called a normal section line, which has a radius of curvature at the point. For freeform surfaces, the radii of curvature of the normal section lines on different section planes at one point are generally unequal. Among these, there are maximum and minimum radii of curvature, and their corresponding curvatures, which are denoted as ${k_1}$ and ${k_2}$, respectively, are the principal curvatures. The directions of these two curvatures have been mathematically proven to be perpendicular to each other [19]. Figure 1. is an illustration of the principal curvatures.

 figure: Fig. 1.

Fig. 1. Representation of the principal curvature, where n is the normal vector of a point on the surface, ${{\boldsymbol v}_1}$ and ${{\boldsymbol v}_2}$ are the tangent vectors of the normal section lines with the minimum and maximum radius of curvature at the point, and ${k_1}$ and ${k_2}$ are the principal curvatures of the point.

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According to Weingarten mapping [19], the principal curvature ${k_1}$ and ${k_2}$ at a certain point on a surface can be calculated using the following formulas:

$${k_1},{k_2} = H \pm \sqrt {{H^2} - K}, $$
$$H = \frac{{EN - 2FM + GL}}{{2({EG - {F^2}} )}}, $$
$$K = \frac{{LN - {M^2}}}{{EG - {H^2}}}, $$
where H is the average value of ${k_1}$ and ${k_2}$; K is the product of ${k_1}$ and ${k_2}$, that is, the Gaussian curvature; and E, F, and G are the coefficients of the first fundamental form:
$$E = {{\boldsymbol r}_x} \cdot {{\boldsymbol r}_x}\textrm{,}F = {{\boldsymbol r}_y} \cdot {{\boldsymbol r}_y}\textrm{,}G = {{\boldsymbol r}_x} \cdot {{\boldsymbol r}_y}, $$
and L, M, and N are the coefficients of the second fundamental form:
$$L = {{\boldsymbol r}_{xx}} \cdot {\boldsymbol n}\textrm{,}M = {{\boldsymbol r}_{xy}} \cdot {\boldsymbol n}\textrm{,}N = {{\boldsymbol r}_{yy}} \cdot {\boldsymbol n}, $$
where ${{\boldsymbol r}_{xx}}$, ${{\boldsymbol r}_{xy}}$, and ${{\boldsymbol r}_{yy}}$ are the second-order partial derivatives of the parametric equation of the surface [19].

Principal curvatures represent the local shape of a sampling point, which is an important feature of a surface, one of which corresponds to the steepest directions. The degree of surface bending can be constrained by controlling the principal curvatures. When both principal curvatures are greater than or less than zero, the curves corresponding to the two principal directions (i.e., the principal normal section line) bend in the same direction. In the vicinity of a point, the surface is located on the same side of the tangent plane; this point is an elliptic point. If the two principal curvatures have opposite signs, the normal section lines bend in opposite directions near the point. The surface is distributed on both sides of the tangent plane, and the point is a hyperbolic point.

Regarding optical surfaces with specific functions or processing requirements, elliptic points can be guaranteed in the effective apertures by controlling the positive and negative signs of the principal curvatures of each point to avoid saddle surfaces and hyperbolic paraboloids, which are more difficult to machine than other surfaces. According to the mathematical description of a surface, the partial derivative and normal vector of the parametric equation of the surface can be obtained. The principal curvatures ${k_1}$ and ${k_2}$ of each point on a surface can be calculated by Eq. (3)-(7). Therefore, the control method requires a freeform surface with second-order partial derivatives.

Regardless of the specific definition, the sag equation of a freeform surface can usually be divided into two parts: a base surface and sag departure [6]. In other words, a freeform surface can be obtained by offsetting each point in the sag direction on a base surface using relatively simple parameters. In general, the base sags of a freeform surface are much larger than the sag departures at the corresponding positions, indicating that the optical power is mainly provided by the base surface. While meeting the design requirements, minimizing the sag departures of the freeform surface is conducive to reducing the required fabrication time, which mitigates the shape errors associated with tool wear and temperature stabilization [8] and enables the measurement of optical surfaces through a wider variety of methods [18]. However, controlling only the sag departure of each point causes drastic fluctuations in the local surface shape when control requirements need to be met. Therefore, both principal curvatures and sag departures should be controlled simultaneously to accelerate convergence.

2.2 Automatic control strategy

When controlling a surface shape, it is first necessary to carry out rectangular array or circular array sampling according to the shape of the effective optical aperture and obtain a number of sampling points throughout the effective optical aperture. Theoretically, a surface shape can be controlled by constraining the principal curvatures and sag departures at all the sampling points. However, although the high degree of freedom of a freeform surface gives it a strong aberration correction ability, the number of constraints should not be greater than the number of variables when optimizing or solving the optical design. Otherwise, the system will be excessively constrained, and finding the optimal solution will be difficult. Therefore, the principal curvatures and sag departures at all sampling points cannot be included in the constraint terms. The optimization efficiency can be improved by reasonably reducing the constraint terms, solving this problem. The specific method is described as follows. In each round of optimization, all the sampling points are evaluated with the constraints of the principal curvature and sag departure, and the sampling points that do not meet the surface shape constraints are listed. These points are then sorted according to their actual principal curvatures or sag departure, and N points with the extreme values are selected (the value of N is preset according to the actual optimization effect); the most serious deviation from the constraints on the entire surface is found and strictly controlled. Owing to the influence of these strictly controlled sampling points, other sampling points can converge to the constraint range even if they are not directly controlled.

If the controlled items are directly constrained to the target values in one optimization loop, they may lead to drastic changes in the freeform surface, errors in ray tracing, and failures in system convergence. Therefore, a damping coefficient α (α<1) can be added to limit the convergence rate, which results in the principal curvatures and sag departures of these N points being constrained as follows:

$$\begin{array}{l} - \alpha {|{{k_\textrm{1}}} |_n} \le {k_\textrm{1}}_{n + 1} \le \alpha {|{{k_\textrm{1}}} |_n}\\ - \alpha {|{{k_2}} |_n} \le {k_2}_{n + 1} \le \alpha {|{{k_2}} |_n}\\ - \alpha |sag{|_n} \le {sag _{n + 1}} \le \alpha |sag{|_n} \end{array}, $$
where n is the number of loops, sag is the sag departure, ${|{{k_1}} |_n}$ represents the absolute value of the principal curvature ${k_\textrm{1}}$ at one point in present rounds, which is an extreme value. ${k_\textrm{1}}_{n + 1}$ is the principal curvature ${k_\textrm{1}}$ at the sampling point in the subsequent round. The optimization constraint $- \alpha {|{{k_\textrm{1}}} |_n} \le {k_\textrm{1}}_{n + 1} \le \alpha {|{{k_\textrm{1}}} |_n}$ can reduce the degree of curvature of the point. Similarly, ${k_2}$ and sag can be gradually constrained. By controlling the principal curvatures and sag departures of the extreme points in this way, the surface shape in the entire effective optical aperture can be controlled.

Controlling the shapes of freeform surfaces limits their degrees of freedom. After each optimization loop, there are regular increases in the error function. However, if these increases are greater than a specified value ɛ, it means that the constraints are too strict for the system to converge. Then, it is necessary to reset the previous round of the system, increase the α value, and perform re-optimization to ensure that each round of optimization can control the surface shape without causing a significant decrease in image quality. The specified value ɛ depends on the different error function value of each system; and its change value and α can be set by the designer or changed according to a certain function law so that it is always less than 1, corresponding to the cycle always being convergent.

Two exit conditions can be used to avoid infinite loops: one is using the result of surface shape evaluation, in which the optimization ends if the constraint terms are less than the preset target values, and the other is adding a counter to calculate the total number of optimization trials, with the optimization being terminated if the counter exceeds a specified number i. If a system is still unable to converge or reach the target values after the i-th trial, the design specifications and surface-shape requirements of the system should be checked.

Figure 2 compares the traditional manual control optimization method and the proposed method for optical design. In the latter method, a grid of points over the effective optical aperture is first sampled, and the required parameters and exit conditions are set. Before each optimization loop, the principal curvatures ${k_1}$ and ${k_2}$ and the sag departure sag of the sampling points are calculated and sorted, and the former shapes are evaluated. If their values are within the target range, the surface shape meets the requirements, and the optimization is terminated. Otherwise, the shape constraint is applied. According to the principal curvatures and sag departures of the current extreme N points, the constraint code is generated to control the values of the next round. Then, it is verified whether the optimized error function is within the set range. If there is a significant increase, the system state before this optimization is revisited, and α is increased to control the surface shape more carefully; otherwise, the surface shape is evaluated, and the exit conditions are checked. As can be seen from the above description, the selection of some constraints and parameters, such as N, α and ɛ, depends on the experience of the designer and needs to be manually adjusted according to the actual situation. However, thanks to the automaticity of the optimization method, the designer does not need to manually interfere with these parameters in every optimization cycle, so the method can still improve the design efficiency.

 figure: Fig. 2.

Fig. 2. (a) Traditional manual control optimization method for surface shapes. (b) Proposed automatic control method for freeform surface shapes, in which an outer loop is added to automatically evaluate and constrain surface shapes.

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We implemented this method in CODE V and successfully applied it to the shape control of freeform surfaces in optical system designs. A design example using the proposed method is provided in the next section.

3. Design example

This section presents the design example that was used to verify the effectiveness of the surface-shape control method. This example was a freeform surface prism system applied to AR. In a previous study [20], a compact and lightweight optical see-through near-eye display was proposed, as shown in Fig. 3(a). It had serious stray light and required additional suppression measures to address issues such as the ghosting effect below the ideal illumination pattern caused by the total reflection on the surface S4. Therefore, in the design example of this study, prisms E1 and E2 were designed as one rhombus prism, as shown in Fig. 3(b). This eliminated stray light and simplified the system structure, which was conducive to processing and alignment. However, the removal of surfaces S3 and S4 reduced the optimization freedom of the system, raising the need for other surfaces to contribute more to providing the optical focal power and correcting aberrations. Consequently, the surface shape changed more sharply, increasing the difficulty in controlling the shapes of these surfaces. Additionally, the rhombus prism was immune to the structural parts of the fixed E2 in Fig. 3(a), which blocked the upper part of the field of view (FOV) when the human eye observed a real scene. As long as surface S1 is strictly controlled, the see-through FOV can be effectively expanded with the auxiliary lens extending upwards. This can reduce the adverse impacts on user activities in a real environment with AR glasses.

 figure: Fig. 3.

Fig. 3. Layout of an optical system in (a) the previous study [20] and (b) the current study.

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The main prism has four optically effective surfaces that form a rhombus shape. Surface S1 not only supports transmission but also provides total reflection for the system, with the reuse of this optical surface effectively folding the light path. For better image quality, surface S1 is described as a freeform surface with more degrees of freedom. However, this also increase the design difficulty, because severe see-through distortion may easily be introduced if the surface shape is not well controlled. Therefore, in the previous study [20], a spherical surface was used for S1. In this study, the contradiction was solved using the proposed automatic control method. Four optical surfaces, including S1, were used as freeform surfaces to achieve a better optical performance.

In this design example, the system has a very large off-axis degree, so it is actually a tedious process to control the surface shape of the freeform surfaces while ensuring the imaging specifications, which requires the designer to control all constraints throughout the process, and the designer’s optimization strategy has a great impact on the optimization efficiency and the final optimization result. Therefore, it is necessary to use the automatic control method to improve the design efficiency.

XY polynomial is used to describe freeform surfaces. Theoretically, the proposed optimization method is suitable for freeform surface systems in other surface descriptions with second-order partial derivatives.$z(x,y)$ is the sag of the surface parallel to the z-axis. The freeform surface of XY polynomial can be mathematically represented as follows:

$$z(x,y) = \frac{{c({{x^2} + {y^2}} )}}{{1 + {{[{1 - (1 + k){c^2}({{x^2} + {y^2}} )} ]}^{1/2}}}} + \sum\limits_{j = 2}^N {{a_j}} {x^m}{y^n},\textrm{ }j = \frac{{{{(m + n)}^2} + m + 3n}}{2}\textrm{ + }1, $$
where c is the vertex curvature; k is the conic constant; ${a_j}$ is the coefficient of the higher-order term ${x^m}{y^n}$; and m and n, respectively, represent the order of x and y, which are nonnegative integers that satisfy $m + n \ge 1$. Considering the characteristics of an XY polynomial, it is easier to express a system with symmetry. If the coefficient of the odd term of x is set to 0, the surface is symmetric about the YOZ plane.

The parametric equation of a XY polynomial freeform surface can be expressed as:

$${\boldsymbol r}(x,y) = (x,y,\frac{{c({{x^2} + {y^2}} )}}{{1 + {{[{1 - (1 + k){c^2}({{x^2} + {y^2}} )} ]}^{1/2}}}} + \sum\limits_{j = 2}^N {{a_j}} {x^m}{y^n}). $$
The partial derivative can be written as follows:
$$\begin{array}{c} {{\boldsymbol r}_x}\textrm{ = }(1,0,\frac{{dz}}{{dx}}),{{\boldsymbol r}_y}\textrm{ = }(0,1,\frac{{dz}}{{dy}});\\ {{\boldsymbol r}_{xx}}\textrm{ = }(0,0,\frac{{{d^2}z}}{{d{x^2}}}),{{\boldsymbol r}_{xy}}\textrm{ = }(0,0,\frac{{{d^2}z}}{{dxdy}}),{{\boldsymbol r}_{yy}}\textrm{ = }(0,0,\frac{{{d^2}z}}{{d{y^2}}}). \end{array}$$
Assuming that ${R^2} = {x^2} + {y^2}$, $A = {[{1 - (1 + k){c^2}({{x^2} + {y^2}} )} ]^{1/2}}$; thus,
$$\left\{ \begin{array}{l} \frac{{dz}}{{dx}}\textrm{ = }\frac{{2cx}}{{1 + A}} + \frac{{{c^3}{R^2}(1 + k)x}}{{A{{(1 + A)}^2}}} + \sum\limits_{j = 2}^N {{a_j}} m{x^{m - 1}}{y^n}\\ \frac{{dz}}{{dy}}\textrm{ = }\frac{{2cy}}{{1 + A}} + \frac{{{c^3}{R^2}(1 + k)y}}{{A{{(1 + A)}^2}}} + \sum\limits_{j = 2}^N {{a_j}} n{x^m}{y^{n - 1}}\\ \frac{{{d^2}z}}{{d{x^2}}}\textrm{ = }\frac{{2c}}{{1 + A}} + \frac{{{c^3}(1 + k)(4{x^2} + {R^2}) }}{{A{{(1 + A)}^2}}} + \frac{{2{c^5}{R^2}{{(1 + k)}^2}{x^2}}}{{{A^2}{{(1 + A)}^3}}} + \frac{{{c^5}{R^2}{{(1 + k)}^2}{x^2}}}{{{A^3}{{(1 + A)}^2}}}\\ \textrm{ } + \sum\limits_{j = 4}^N {{a_j}} m(m - 1){x^{m - 2}}{y^n}\\ \frac{{{d^2}z}}{{dxdy}}\textrm{ = }\frac{{4{c^3}(1 + k)xy}}{{A{{(1 + A)}^2}}} + \frac{{2{c^5}{R^2}{{(1 + k)}^2}xy}}{{{A^2}{{(1 + A)}^3}}} + \frac{{{c^5}{R^2}{{(1 + k)}^2}xy}}{{{A^3}{{(1 + A)}^2}}}\\ \textrm{ } + \sum\limits_{j = 5}^N {{a_j}} mn{x^{m - 1}}{y^{n - 1}}\\ \frac{{{d^2}z}}{{d{y^2}}}\textrm{ = }\frac{{2c}}{{1 + A}} + \frac{{{c^3}(1 + k)(4{y^2} + {R^2}) }}{{A{{(1 + A)}^2}}} + \frac{{2{c^5}{R^2}{{(1 + k)}^2}{y^2}}}{{{A^2}{{(1 + A)}^3}}} + \frac{{{c^5}{R^2}{{(1 + k)}^2}{y^2}}}{{{A^3}{{(1 + A)}^2}}}\\ \textrm{ } + \sum\limits_{j = 4}^N {{a_j}} n(n - 1){x^m}{y^{n - 2}} \end{array} \right.$$
The principal curvatures of each point on a surface can be obtained using Eq. (2)–(7).

The overall system specifications are listed in Table 1. A 0.71-in organic light emitting diode (OLED) display with a resolution of 1920 × 1080 px and pixel size of 8.2 µm was selected. Considering the importance of a large eyebox and eye relief (ERF) in vision devices, we set eyebox to 8 × 6 mm and ERF to 19 mm. During the optimization process, the other parameters were adjusted constantly. After much consideration, a diagonal FOV of 43° was finally determined. The distortion could be corrected using digital correction methods for the original image source; therefore, no requirements were specified. The modulation transfer function (MTF) was selected to evaluate the overall image sharpness [13]. The MTF value should be greater than 10% at half the Nyquist frequency of the display throughout the FOV.

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Table 1. Specifications of the optical system

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First, the freeform prism system was optimized according to the design method and strategy mentioned in a previous paper [4], which included the addition of the structural and total internal reflection (TIR) constraints. Structural constraints were used to ensure that each surface could form an effective prism shape and that the light in all the FOVs was not blocked or escaped early from the surface. The TIR constraints were set to ensure that the light would be completely reflected on some internal surfaces of the prism instead of exiting the prism and resulting in a reduced light efficiency. Additionally, it was necessary to control factors such as the aberration, image height and chief ray angle. Accordingly, the primary optimization of the virtual image light path was completed. To demonstrate the effect of the subsequent surface control methods, the surface shape was not controlled.

The sag of each surface of the main prism after preliminary optimization is shown in Fig. 4; S2 and S3 were relatively flat in the effective optical aperture of each surface, which was attributed to them being only used as reflective surfaces and not being easily deformed during optimization. The sag of S1 and S4 varied greatly at a fast rate. The principal curvatures of the surfaces are shown in Fig. 5. The curvature of S1 varied from −0.185 to 0.223, which was large for a surface with an aperture of 36 × 22 mm. This resulted in a large sag at the edge of the clear aperture that caused large and uneven optical power in the see-through optical path and destroyed the see-through effect. The curvature of S4 varied from −0.156 to 0.432. In some local areas of the surface, the principal curvatures ${k_1}$ and ${k_2}$ have very large difference, which makes it more difficult to process. Although its aperture was smaller than that of S1, it was still 20 × 12 mm; therefore, an automatic control method for the surface shape was required to ensure the machinability and see-through effect of the system.

 figure: Fig. 4.

Fig. 4. Sag of each surface of the main prism: (a) S1, (b) S2, (c) S3, (d) S4.

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 figure: Fig. 5.

Fig. 5. Principal curvatures of surface S1 and S4 of the main prism: the principal curvatures (a) ${k_1}$ of S1, (b) ${k_2}$ of S1, (c) ${k_1}$ of S4, and (d) ${k_2}$ of S4.

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Figure 6(a) shows the polychromatic MTF curve of the selected fields for the virtual image light path. It was evaluated at half of the Nyquist frequency (approximately 30 lp/mm) with an exit pupil of 4 mm. The MTF curves of nine fields, including the center and marginal fields, were higher than 0.13 at 30 lp/mm. The RMS spot diameters are presented in Fig. 6(b). The average RMS spot diameter of the full field of view was only 27.8 µm when the eyebox was 8 × 6 mm.

 figure: Fig. 6.

Fig. 6. Optical performance of virtual image light path after preliminary optimization. (a) At 30 lp/mm, the MTF values are higher than 0.135 for all fields with the 4-mm exit pupil. (b) RMS spot diameters for the full eyebox.

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Regarding the see-through light path, the two surfaces S2’ and S3’ of the auxiliary lens picked up the surfaces S2 and S3 of the main prism, respectively, forming a whole system after being glued together. The optical power introduced by surface S1 was compensated by the front surface S5 of the auxiliary lens. The surface coefficients of S5 were set as variables, and the distortion and aberration of the see-through light path were optimized.

The MTF plot and the distortion grid of the see-through light path within the range corresponding to the FOV of the virtual image are shown in Fig. 7. Owing to the large curvature of surface S1, it was difficult to perform corrections using only surface S5. At the edge field F9, the MTF at 60 lp/mm was only 0.13, and the distortion reached 4.1%. In a wider see-through range, even if the external light could enter the human eye without obstruction, there was still a large distortion and degradation that was not conducive to the observation of the real scene.

 figure: Fig. 7.

Fig. 7. Optical performance of the see-through light path with the uncontrolled surface S1. (a) At 60 lp/mm, the MTF values are higher than 0.13 for all the fields with the 4-mm exit pupil. (b) The distortion grids.

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Subsequently, the automatic control method for surface shapes was introduced. In the system, the aperture shape was rectangular, and a rectangular array was used to evenly sample an 11 × 11 grid of points. The initial value of α was 0.95, which increased in increments of 0.01. Based on the surface shape and processing requirements of S1 and S4, the exit conditions were set as follows:

$$\left\{ {\begin{array}{{c}} { - 0.03 < {\textrm{k}_1}_{\textrm{ S1}} < 0.03}\\ { - 0.03 < {\textrm{k}_2}_{\textrm{ S1}} < 0.03} \end{array}} \right., $$
$$\left\{ {\begin{array}{{c}} { - 0.1 < {\textrm{k}_1}_{\textrm{ S4}} < 0.1}\\ { - 0.1 < {\textrm{k}_2}_{\textrm{ S4}} < 0.1} \end{array}} \right.$$
Surface S1 had more stringent exit conditions than S4 did because of its larger effective optical aperture and its contribution to the see-through light path.

Figure 8 shows the variation curves of the error function and indices controlled automatically with the number of optimization trials. Controlling the surface shape resulted in significant reductions in the maximum absolute values of the curvatures and sag departures and an increase in the smoothness of the surface shape. The error function gradually increased from 101.7 to 121.6, owing to the limitation in the degrees of freedom caused by controlling the surface shape. However, considering the improved smoothness of the surface shape, which increased the accuracy of the processed surface and supported its applicability in the actual manufacturing of the optical system, the increased error was excusable.

 figure: Fig. 8.

Fig. 8. Variation curves of the error function; the maximum absolute value of the principal curvatures and sag departures of S1 and S4 with the number of the optimization trials.

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The maximum sag departure of S4 decreased significantly in the first 10 optimizations and was stable at approximately 4 mm. After a period of stability, the maximum sag departure of S1 continued to decrease with the stricter control of the curvature, finally becoming 1.44 mm. In the 27th optimization, a better solution was found for the optical system, resulting in the error function significantly decreasing. In the following optimization, the error function followed an increasing trend. The maximum absolute values of the principal curvature and sag departure showed decreasing trends. With the increase in α, the decreasing trend gradually slowed down and finally reached the exit condition. Overall, the optimization ended at the 59th trial.

Finally, the system that exited the automatic control process after 59th trial was used with adjustment of the weights of the sampled fields and design of the auxiliary lens. The layout of the final system is illustrated in Fig. 9. The polychromatic MTF curve and RMS spot diameter of the virtual image light path are shown in Fig. 10. At 30 lp/mm, the MTF of the marginal field decreased from 0.13 to 0.11 but it was still higher than 0.1. The average RMS under the full eyebox of 8 × 6 mm increased from 27.8 to 28.6 µm, which was within the acceptable range. Figure 11 demonstrates the distortion grids of the virtual image light path. The distortion is up to 8% at the marginal FOV. The maximum chromatic aberration of the final system is calculated to be 0.42 mm, which is about 5.1 pixels, according to the image heights of different wavelengths at the same FOV. The electronic correction method of chromatic aberration is similar to that of the distortion, which could be processed by correcting the distortion of each RGB channel separately, so that each sub-pixel of the microdisplay is located at the ideal virtual image height.

 figure: Fig. 9.

Fig. 9. Final optical layout of the optical see-through near-eye display.

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 figure: Fig. 10.

Fig. 10. Optical performance of the virtual image light path after controlling the surface shape. (a) At 30 lp/mm, MTF values are higher than 0.11 for all the fields with the 4-mm exit pupil. (b) RMS spot diameter for the eyebox.

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 figure: Fig. 11.

Fig. 11. The distortion grids of the virtual image light path after controlling the surface shape.

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The final surface shapes of S1 and S4 are shown in the sag diagram in Fig. 12. Comparing Fig. 5(a) and 5(d), the shapes of both the surfaces are smoother. The principal curvature distributions are shown in Fig. 13. The principal curvature range of S1 was reduced from −0.185 ∼ 0.223 to −0.0299 ∼ 0.0181, and the principal curvature range of S4 was reduced from −0.156 ∼ 0.432 to −0.1 ∼ 0.1, greatly improving the machinability of the surface. Furthermore, the well-controlled surface shape of S1 also realized a large see-through field. After the auxiliary lens was glued, the see-through FOV of the system reached 60 × 40°, which provided users with a better AR experience. The MTF curve evaluated by the 4-mm exit pupil and the distortion grid of the see-through light path are shown in Fig. 14. There was only a slight pillow distortion less than 1.4%.

 figure: Fig. 12.

Fig. 12. Sag of each surface of the main prism after controlling the surface shape: (a) S1 and (b) S4.

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 figure: Fig. 13.

Fig. 13. Principal curvatures of surfaces S1 and S4 of the main prism after controlling the surface shape: principal curvatures (a) ${k_1}$ of S1; (b) ${k_2}$ of S1; (c) ${k_1}$ of S4; and (d) ${k_2}$ of S4.

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 figure: Fig. 14.

Fig. 14. Optical performance of the see-through light path within the range corresponding to the FOV of the virtual image. (a) At 60 lp/mm, MTF values are higher than 0.4 for all the fields with the 4-mm exit pupil. (b) The distortion grids.

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4. Prototype and experimental results

Based on the application scenario and cost considerations, a 0.49-in microdisplay was adopted to realize the prototype used to verify the above design example. The FOV of the system was 38°, and the distortion of the virtual image light path was less than 2%. Both the main rhombus prism and auxiliary lens were machined using injection molding. The prototype is shown in Fig. 15. The main prism and auxiliary lens were glued with ultraviolet glue; and the holder was fixed to the microdisplay, main rhombus prism, and auxiliary lens.

 figure: Fig. 15.

Fig. 15. Overall appearance of the prototype.

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The surface shapes of S1 and S4 were measured using a Panasonic’s Ultrahigh Accurate 3-D Profilometer (UA3P). The peak-to-valley (PV) values of S1 and S4 were 2.4491 and 2.5512 µm, respectively. This implied that the method could realize the control of freeform surface shapes.

Simple tests were performed to demonstrate the image quality of the virtual and see-through light path of the optical see-through head-mounted display. A commercial camera was used to simulate the human eye and record the projected image transmitted through the system. The stop aperture and optical axis of the camera were matched to the eyebox and viewing axis of the optical module, respectively.

The image resolution test chart shown in Fig. 16(a) was displayed on the microdisplay. The image shown in Fig. 16(b) indicates the good virtual image quality of the optical system. Figure 17 shows the fusion of a virtual bird and a real potted plant, demonstrating the potential application of this module in AR.

 figure: Fig. 16.

Fig. 16. Testing results of the optical prototype. (a) The input image displayed in the micro-display when testing the performance of the system. (b) The output image captured by the camera at the eyebox of the system.

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 figure: Fig. 17.

Fig. 17. The result of the fusion of a virtual bird and a real potted plant.

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5. Conclusion

Surface shape control is crucial in the design and optimization of freeform optical systems. In this study, an automatic control method for freeform surface shapes has been proposed and implemented. Based on the principal curvature and sag departure of the surfaces, constraints were constructed in the locally optimized outer loop to limit the surface shape, and the desired effect was achieved. The slight decrease in image quality during the process was acceptable, considering that the resulting smoothness of the surface shape could reduce difficulties in processing and improve the accuracy of the shape. Automatic control methods can reduce the need for manual involvement and the dependence on the designer’s experience, improve the work efficiency and reduce the workload of designers. The design results of these methods may differ from the results provided by a designer, whose performance depends on their experience and the time they spent. Finally, an example based on freeform prism optics has been presented and verified. The results showed that the automatic control method could improve the manufacturability of freeform surfaces at the expense of limited optical performance.

Funding

National Key Research and Development Program of China (2021YFB2802100); Young Elite Scientist Sponsorship Program by CAST (2019QNRC001); Beijing Municipal Science & Technology Commission, Administrative Commission of Zhongguancun Science Park (Z221100006722011).

Acknowledgments

We would like to thank Synopsys for providing the educational license of CODE V.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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10. C. Zhaohan, L. Jinpeng, X. Yongjun, and M. Xianglong, “Adaptive point design algorithm to generate toolpath in fast tool servo ultra-precision machining of freeform surface,” in Ninth Symposium on Novel Photoelectronic Detection Technology and Applications, W. Liu, H. Xu, and J. Chu, eds. (SPIE, 2023), p. 359.

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13. Q. F. Wang, D. W. Cheng, Y. T. Wang, H. Hua, and G. F. Jin, “Design, tolerance, and fabrication of an optical see-through head-mounted display with free-form surface elements,” Appl. Opt. 52(7), C88–C99 (2013). [CrossRef]  

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15. Y. T. Deng, G. F. Jin, and J. Zhu, “Design method for freeform reflective-imaging systems with low surface-figure-error sensitivity,” Chin. Opt. Lett. 17(9), 092201 (2019). [CrossRef]  

16. J. Nishikawa and M. Nishiyama, “Design of ultrashort throw lenses with catadioptric relays for curved screens,” in Current Developments in Lens Design and Optical Engineering XXII (SPIE, 2021), p. 1181404.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (17)
Fig. 1.
Fig. 1. Representation of the principal curvature, where n is the normal vector of a point on the surface, ${{\boldsymbol v}_1}$ and ${{\boldsymbol v}_2}$ are the tangent vectors of the normal section lines with the minimum and maximum radius of curvature at the point, and ${k_1}$ and ${k_2}$ are the principal curvatures of the point.

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Fig. 2.
Fig. 2. (a) Traditional manual control optimization method for surface shapes. (b) Proposed automatic control method for freeform surface shapes, in which an outer loop is added to automatically evaluate and constrain surface shapes.

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Fig. 3.
Fig. 3. Layout of an optical system in (a) the previous study [20] and (b) the current study.

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Fig. 4.
Fig. 4. Sag of each surface of the main prism: (a) S1, (b) S2, (c) S3, (d) S4.

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Fig. 5.
Fig. 5. Principal curvatures of surface S1 and S4 of the main prism: the principal curvatures (a) ${k_1}$ of S1, (b) ${k_2}$ of S1, (c) ${k_1}$ of S4, and (d) ${k_2}$ of S4.

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Fig. 6.
Fig. 6. Optical performance of virtual image light path after preliminary optimization. (a) At 30 lp/mm, the MTF values are higher than 0.135 for all fields with the 4-mm exit pupil. (b) RMS spot diameters for the full eyebox.

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Fig. 7.
Fig. 7. Optical performance of the see-through light path with the uncontrolled surface S1. (a) At 60 lp/mm, the MTF values are higher than 0.13 for all the fields with the 4-mm exit pupil. (b) The distortion grids.

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Fig. 8.
Fig. 8. Variation curves of the error function; the maximum absolute value of the principal curvatures and sag departures of S1 and S4 with the number of the optimization trials.

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Fig. 9.
Fig. 9. Final optical layout of the optical see-through near-eye display.

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Fig. 10.
Fig. 10. Optical performance of the virtual image light path after controlling the surface shape. (a) At 30 lp/mm, MTF values are higher than 0.11 for all the fields with the 4-mm exit pupil. (b) RMS spot diameter for the eyebox.

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Fig. 11.
Fig. 11. The distortion grids of the virtual image light path after controlling the surface shape.

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Fig. 12.
Fig. 12. Sag of each surface of the main prism after controlling the surface shape: (a) S1 and (b) S4.

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Fig. 13.
Fig. 13. Principal curvatures of surfaces S1 and S4 of the main prism after controlling the surface shape: principal curvatures (a) ${k_1}$ of S1; (b) ${k_2}$ of S1; (c) ${k_1}$ of S4; and (d) ${k_2}$ of S4.

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Fig. 14.
Fig. 14. Optical performance of the see-through light path within the range corresponding to the FOV of the virtual image. (a) At 60 lp/mm, MTF values are higher than 0.4 for all the fields with the 4-mm exit pupil. (b) The distortion grids.

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Fig. 15.
Fig. 15. Overall appearance of the prototype.

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Fig. 16.
Fig. 16. Testing results of the optical prototype. (a) The input image displayed in the micro-display when testing the performance of the system. (b) The output image captured by the camera at the eyebox of the system.

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Fig. 17.
Fig. 17. The result of the fusion of a virtual bird and a real potted plant.

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Tables (1)

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Table 1. Specifications of the optical system

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Equations (14)

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r ( x , y ) = ( x , y , z ( x , y ) ) .
n = ( r x × r y ) | r x × r y | ,
k 1 , k 2 = H ± H 2 − K ,
H = E N − 2 F M + G L 2 ( E G − F 2 ) ,
K = L N − M 2 E G − H 2 ,
E = r x ⋅ r x , F = r y ⋅ r y , G = r x ⋅ r y ,
L = r x x ⋅ n , M = r x y ⋅ n , N = r y y ⋅ n ,
− α | k 1 | n ≤ k 1 n + 1 ≤ α | k 1 | n − α | k 2 | n ≤ k 2 n + 1 ≤ α | k 2 | n − α | s a g | n ≤ s a g n + 1 ≤ α | s a g | n ,
z ( x , y ) = c ( x 2 + y 2 ) 1 + [ 1 − ( 1 + k ) c 2 ( x 2 + y 2 ) ] 1 / 2 + ∑ j = 2 N a j x m y n ,   j = ( m + n ) 2 + m + 3 n 2  +  1 ,
r ( x , y ) = ( x , y , c ( x 2 + y 2 ) 1 + [ 1 − ( 1 + k ) c 2 ( x 2 + y 2 ) ] 1 / 2 + ∑ j = 2 N a j x m y n ) .
r x  =  ( 1 , 0 , d z d x ) , r y  =  ( 0 , 1 , d z d y ) ; r x x  =  ( 0 , 0 , d 2 z d x 2 ) , r x y  =  ( 0 , 0 , d 2 z d x d y ) , r y y  =  ( 0 , 0 , d 2 z d y 2 ) .
{ d z d x  =  2 c x 1 + A + c 3 R 2 ( 1 + k ) x A ( 1 + A ) 2 + ∑ j = 2 N a j m x m − 1 y n d z d y  =  2 c y 1 + A + c 3 R 2 ( 1 + k ) y A ( 1 + A ) 2 + ∑ j = 2 N a j n x m y n − 1 d 2 z d x 2  =  2 c 1 + A + c 3 ( 1 + k ) ( 4 x 2 + R 2 ) A ( 1 + A ) 2 + 2 c 5 R 2 ( 1 + k ) 2 x 2 A 2 ( 1 + A ) 3 + c 5 R 2 ( 1 + k ) 2 x 2 A 3 ( 1 + A ) 2   + ∑ j = 4 N a j m ( m − 1 ) x m − 2 y n d 2 z d x d y  =  4 c 3 ( 1 + k ) x y A ( 1 + A ) 2 + 2 c 5 R 2 ( 1 + k ) 2 x y A 2 ( 1 + A ) 3 + c 5 R 2 ( 1 + k ) 2 x y A 3 ( 1 + A ) 2   + ∑ j = 5 N a j m n x m − 1 y n − 1 d 2 z d y 2  =  2 c 1 + A + c 3 ( 1 + k ) ( 4 y 2 + R 2 ) A ( 1 + A ) 2 + 2 c 5 R 2 ( 1 + k ) 2 y 2 A 2 ( 1 + A ) 3 + c 5 R 2 ( 1 + k ) 2 y 2 A 3 ( 1 + A ) 2   + ∑ j = 4 N a j n ( n − 1 ) x m y n − 2
{ − 0.03 < k 1  S1 < 0.03 − 0.03 < k 2  S1 < 0.03 ,
{ − 0.1 < k 1  S4 < 0.1 − 0.1 < k 2  S4 < 0.1
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