Synthetic system design method for off-axis stabilized zoom systems with a high zoom ratio

Xuemin Cheng; Xuemin Cheng; Xuemin Cheng; Hengzhi Ye; Hengzhi Ye; Qun Hao; Qun Hao

doi:10.1364/OE.420182

1. Introduction

Zoom systems play a significant role in several fields such as, medical screening, surveillance, and national defense construction [1–10]. A conventional zoom system can satisfy the application requirements of numerous fields. However, in certain applications, such as in robots or other moving carriers, an increasing attention has been paid to specific characteristics of the zoom system such as high zoom speed, high reliability, and miniaturization. Therefore, several researchers have designed zoom systems based on the focal length variable (FLV) opto-electronic components, which simplify the opto-mechanical system, and effectively improves the stability and zoom speed [5–10]. In stabilized zoom systems, the separation between the adjacent groups is fixed. Hence, the FLV elements must compensate for the image shift and accommodate variations in the focal length of the zoom system. Therefore, specific researches have been conducted on the zoom equation [5,6] and the primary configuration for the design of stabilized zoom systems. However, considering only the zooming capability of the zoom system is insufficient because the imaging performance is also essential, especially for the off-axis systems.

Refractive and off-axis reflective methods can be used to create stabilized zoom systems. Refractive methods employ a liquid lens or liquid crystal lens to alter the focal power [5–7], whereas off-axis reflective methods utilize deformable mirrors (DM) [8–10]. Compared to the refractive method, the off-axis reflective method has certain advantages, such as a large aperture, freedom from chromatic aberration, and the ease of compact miniaturization [11,12]. Moreover, deformable mirrors can be used to fit free-form surfaces, which provide optical designers a wide range of design options [8–10]. However, the design of off-axis optical systems has always been a challenge compared to the design of the rotationally symmetric systems. Zhong and Gross [13] presented a system design method for non-rotationally symmetric systems using Gaussian brackets and nodal aberration theory, through which the unobscured configuration with spherical surfaces can be directly obtained. Based on the nodal aberration theory, Cao [14] demonstrated the method of obtaining a configuration with conic surfaces for off-axis all-reflective systems by considering the spherical and aspherical (conic term induced) aberrations. These studies are applicable for aberration correction of the off-axis reflective system and the initial system establishment. However, while designing the zoom system, the system uses multiple-conjugate points and requires the balancing of the aberrations in the entire focal range. Especially, for the off-axis stabilized zoom systems, the reduced distances between each optical element are fixed during the changes in the system conjugate positions of the zooming procedure. It is necessary to rectify the aberration introduced by the asymmetric balance of the ray tracing paths inside the multi-conjugate positions of the system, and obtains the zoom ratio using the limited stroke range of the deformable mirror. Therefore, these invariants must satisfy the imaging requirements of different focal lengths to achieve the non-defocusing imaging and minimize the specific off-axis-induced aberrations. Therefore, it is necessary to design a balanced strategy for the aberration correction for complex optical systems such as off-axis stabilized zoom systems in the entire focal range. Moreover, we have assumed that the system exhibits a high zoom ratio, which is challenging for the stabilized zoom systems to achieve. Obtaining an optimal configuration is challenging because of these restrictions. Therefore, it is necessary to obtain the design strategy for an optimal configuration of off-axis stabilized zoom systems, which is especially significant for the complex system design.

In this study, a synthetic and automated method is proposed for an optimal configuration of the stabilized zoom system. To construct a multi-element stabilized zoom system that has a high zoom ratio and is free from the defocus, we establish a nonlinear merit function with selected constraints using the Gaussian brackets expressions [15–17]. Considering the multi-conjugate structure of the stabilized zoom system, two types of parameters are evaluated and controlled. The first parameters involve the zooming characteristics such as focal lengths, back focal lengths, defocus, and solutions for a high zoom ratio in the specific stroke ranges of FLV elements. The second parameter is the system aberration balance. Specific off-axis-induced aberrations of non-rotationally symmetric systems, especially astigmatism and distortion, are analytically characterized by discussing the conic expression according to the nodal aberration expressions at different multi-conjugate positions. Compared to the refractive stabilized zoom system, the off-axis reflective system possesses the asymmetric characteristics. Therefore, the full field aberration balance is analytically evaluated with nodal aberrations [18–20], and high-order surfaces are used. The results are discussed, which show that the increase in the zoom ratio will lead to the degradation of the image quality of system, which can be effectively alleviated by the rational use of conic surfaces. Subsequently, the controlling range for the optimal zooming solution is designed under the constraints of the multi-element stabilized zoom system, and the image aberration expressions are simultaneously set in the entire focal range. Therefore, the stabilized zoom system configuration design problem is transformed to retrieve an optimal solution with a nonlinear global merit function. The zooming merit function, image aberration merit function, and constraint condition of the focal power variation of FLV elements are combined into a global merit function to determine a small solution set. And the theory of Pareto Optimality [21] is applied to search for the optimization result in a set of optimal solutions, rather than a unique local solution, which is beneficial in realizing suitable configurations of the stabilized zoom system. Finally, an optimal off-axis stabilized zoom system configuration can be automatically retrieved with a high zoom ratio; at the same time, the obtained system can also be used as an initial point to achieve a higher performance through further optimization with higher-order surfaces such as free-form surfaces [22,23]. Therefore, this study provides the insight and guidance for the configuration design of off-axis stabilized zoom systems and other complex optical systems. The design method is described in detail in Section 2. The design of an off-axis three-mirror stabilized zoom system with two deformable mirrors and a zoom ratio of 10:1 is presented in Section 3. Finally, in Section 4, the advantages and shortcomings of the method are discussed along with the application prospects.

2. Design method

2.1 Aberration analysis of off-axis stabilized zoom systems with conic surfaces

In this section, the specific off-axis-induced aberrations of the off-axis stabilized zoom system is analytically analyzed. The conic surface is employed to balance the aberration in the system with a large field and pupil. Furthermore, the conic-surface system is a suitable infrastructure for the freeform-surface system. Then the discussion on the image quality of the stabilized zoom system configuration show that the increase in zoom ratio lead to the degradation of the image quality of system, which is alleviated by the rational use of conic surfaces. Improved results are obtained for a large zoom ratio of system.

According to the nodal aberration theory developed by Thompson [18–20], the wave aberrations of off-axis systems can be represented by vectors as follows:

(1)$$W = \mathop \sum \nolimits_j \mathop \sum \nolimits_p \mathop \sum \nolimits_n \mathop \sum \nolimits_m {({W_{klm}})_j}{[\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \sigma } }_j}} \right) \cdot \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \sigma } }_j}} \right)]^p}{(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } )^n} \times {[\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \sigma } }_j}} \right) \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } ]^m}$$

where $k = 2p + m$, and $l = 2n + m,\,{W_{klm}}$, denote the wave aberration coefficients of the on-axis system, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} $ denotes the normalized field vector, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \sigma } _j}$ denotes the field displacement vector of ${j^{\textrm{th}}}$ surface, and $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } $ denotes the normalized aperture vector.

Effective field vector ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} _{Ej}}$ of the non-rotationally symmetric system is given by

(2)$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} _{Ej}} = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} - {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \sigma } _j}$$

Then, the following constraints, which are based on the characteristics of stabilized zoom systems that can control the surface shape, are proposed for the designed systems. Initially, only tilted surfaces and a biased input field are employed to design the unobscured off-axis system. Therefore, we can obtain a fixed optical axis ray (OAR), which is defined as the chief ray of the center zero field of view during zooming. This is beneficial in stabilizing the position of the optical elements and image plane. Furthermore, ${\sigma _j}$ is zero for aspherical contributions [14,20]. Subsequently, we assume that the designed off-axis systems are symmetric about the $YOZ$ plane. Therefore, ${\sigma _{jx}}$ is zero for the spherical contributions. Then the displacement vectors are calculated by tracing the fixed optical axis ray [20]. The real ray-tracing model of a plane-symmetric system with three elements is shown in Fig. 1.

Fig. 1. Real ray tracing of the plane-symmetric system.

Download Full Size | PDF

here, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over N} _j}$ denotes the unit normal vector of the surface, ${o_j}$ is the vertex of the surface, and ${S_j}$ is the ${j^{\textrm{th}}}$ surface. ${i_j}$ is the incident angle of OAR on ${j^{\textrm{th}}}$ surface, which is equal to the tilt angle ${\alpha _j}$ of the ${j^{\textrm{th}}}$ surface on ${o_j}$, and is also equal to the angle between ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over N} _j}$ and incident OAR.

Direction vector ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over N} _j}$ can be defined in the local coordinate system of each incident OAR as follows [20]:

(3)$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over N} _j} = ({0,SR{M_j},SR{N_j}} )$$

here, $SRM$ and $SRN$ denote the normalized direction cosines along the y and z directions, respectively. The tilt angles of each OAR can be expressed as

(4)$${\alpha _j} = \arcsin (SR{M_j})$$

The displacement vector can be derived in Eq. (5) [20]

(5)$$\sigma _{jy}^{sph} = \frac{{ - SR{M_j}}}{{{{\mathop u\limits^ - }_j} + {{\mathop h\limits^ - }_j}{c_j}}} = \frac{{ - \sin ({\alpha _j})}}{{{{\mathop u\limits^ - }_j} + {{\mathop h\limits^ - }_j}{c_j}}}$$

where $sph$ denotes the spherical contribution, ${\mathop h\nolimits^ - _j}$ denotes the chief ray height, ${\mathop u\nolimits^ - _j}$ denotes the chief ray incident angle, ${c_j}$ is the curvature of the ${j^{\textrm{th}}}$ surface.

Therefore, the primary wave aberration expansion including tilts and decenters can be stated as

(6)$$\begin{array}{l} W = \mathop \sum \limits_j {W_{040j}}{(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } )^2} + \mathop \sum \limits_j {W_{131j}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} }_{Ej}} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } } \right)\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } } \right)\\ \;\;\; + \frac{1}{2}\mathop \sum \limits_j {W_{222j}}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H}_{Ej}^2 \cdot {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } }^2}} \right) + \mathop \sum \limits_j {W_{220Mj}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} }_{Ej}} \cdot {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} }_{Ej}}} \right)\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } } \right)\\ \;\;\; + \mathop \sum \nolimits_j {W_{311j}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} }_{Ej}} \cdot {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} }_{Ej}}} \right)\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} }_{Ej}} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \rho } } \right) \end{array}$$

where ${W_{220Mj}} = {W_{220j}} + \frac{1}{2}{W_{222j}},\,{W_{040j}},\,{W_{131j}},\,{W_{222j}},\,{W_{220j}}$ can be obtained using the on-axis system paraxial ray-trace data since the field decenter vector has no impact on the primary aberration coefficients of the system.

2.1.1 Spherical aberration

Spherical aberration coefficient of the off-axis optical zoom system shown in Eq. (6) can be extracted as

(7)$${A_{spa}} = \mathop \sum \nolimits_j {W_{040j}}$$

where

(8)$${W_{040j}} = \frac{1}{8}({S_{{I} j}^{sph} + S_{{I} j}^{asph}} )$$

(9)$$\left\{ {\begin{array}{c} {S_{{I} j}^{sph} ={-} A_j^2 \cdot {h_j} \cdot \left( {\frac{{u{^{\prime}_j}}}{{n{^{\prime}_j}}} - \frac{{{u_j}}}{{{n_j}}}} \right)}\\ {S_{{I} j}^{asph} = c_j^3({n{^{\prime}_j} - {n_j}} ){k_j} \cdot h_j^4} \end{array}} \right.$$

and scalar term ${A_{spa}}$ represents the spherical aberration, ${S_I},\,{S_{III}},S_V$, are the Seidel sums of on-axis systems, ${h_j}$ denotes the marginal ray height, ${u_j}$ and $u{^{\prime}_j}$ denote marginal ray angles, $sph$ and $asph$ denote the spherical and aspherical contributions, respectively. $n{^{\prime}_j}$ is the refractive index after the ${j^{\textrm{th}}}$ surface, ${n_j}$ is the refractive index before the ${j^{\textrm{th}}}$ surface,${A_j} = {n_j}({{h_j}{c_j} + {u_j}} )$, and ${k_j}$ is the conic constant of the ${j^{\textrm{th}}}$ surface.

2.1.2 Astigmatism

Astigmatism coefficient of the off-axis optical zoom system shown in Eq. (6) can be extracted as

(10)$${A_{ast}} = \frac{1}{2}\mathop \sum \nolimits_j {W_{222j}} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} _{Ej}^2$$

where

(11)$${W_{\textrm{222}j}} = \frac{1}{2}({S_{{III} \;j}^{sph} + S_{{III} \;j}^{asph}} )$$

(12)$$\left\{ {\begin{array}{c} {S_{{III} \;j}^{sph} ={-} {\overline A _j}^2 \cdot {h_j} \cdot \left( {\frac{{u{^{\prime}_j}}}{{n{^{\prime}_j}}} - \frac{{{u_j}}}{{{n_j}}}} \right)}\\ {S_{{III} \;j}^{asph} = c_j^3({n{^{\prime}_j} - {n_j}} ){k_j} \cdot h_j^2 \cdot {\overline h _j}^2} \end{array}} \right.$$

and vector term ${A_{ast}}$ represents the astigmatism, ${\mathop A\limits^ - _j} = {n_j}\left( {{{\mathop h\limits^ - }_j}{c_j} + {{\mathop u\limits^ - }_j}} \right),\,\mathop u\limits^ - {^{\prime}_j}$ denotes chief ray angle.

2.1.3 Distortion

The distortion coefficient of an off-axis optical zoom system shown in Eq. (6) can be extracted as

(13)$${A_{dis}} = \mathop \sum \nolimits_j {W_{311j}} \cdot \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H}_{Ej} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H}_{Ej}} \right)\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} _{Ej}$$

where

(14)$${W_{\textrm{311}j}} = \frac{1}{2}({S_{V\;j}^{sph} + S_{V\;j}^{asph}} )$$

(15)$$\left\{ {\begin{array}{c} {S_{V\;j}^{sph} ={-} \left[ {\frac{{\overline {{A_j}} }}{{{A_j}}} Ж_j^2 {c_j}\left( {\frac{1}{{n{^{\prime}_j}}} - \frac{1}{{{n_j}}}} \right) + \frac{{\overline A _j}^3 }{{{A_j}}}{h_j}\left( {\frac{{u{^{\prime}_j}}}{{n{^{\prime}_j}}} - \frac{{{u_j}}}{{{n_j}}}} \right)} \right]}\\ {S_{v\;j}^{asph} = c_j^3({n{^{\prime}_j} - {n_j}} ){k_j} \cdot h_j \cdot {\overline h _j}^3} \end{array}} \right.$$

and vector term ${A_{dis}}$ represents the distortion, Lagrange invariant $Ж_j = {n_j}\left( {{h_j}{{\mathop u\limits^ - }_j} - {{\mathop h\limits^ - }_j}{u_j}} \right)$.

The image qualities of the stabilized zoom systems are discussed considering that the effective variation ranges of the focal power of FLV elements are consistent. Based on an off-axis three-mirror layout, the primary configurations of six stabilized zoom system sets are designed, and their RMS spot diameters take the average values of the wide-angle and telephoto ends.

It can be seen from Fig. 2 that the increase in zoom ratio leads to the degradation of the image quality of a stabilized zoom system, which can be alleviated effectively by the rational usage of a conic surface. The configuration data of stabilized zoom systems at an increasing zoom ratio (Fig. 2) are presented in Table 1, where ${d_i}$ is the thickness from the ${i^{\textrm{th}}}$ surface to the ${(i + 1)^{\textrm{th}}}$ surface, ${\alpha _i},\,{r_i}$, ${r_i}$, and are the tilt angle, radius and conic constant of the ${i^{\textrm{th}}}$ surface, respectively. W and T denote the wide-angle and telephoto ends, respectively. This table demonstrates that the first surface possesses a large conic constant, which is responsible for the heavy aberration balancing task, and is one of the directions for further optimization with higher-order surfaces such as free-form surfaces. Furthermore, with identical FLV elements and a single focal length, a high zoom ratio can lead to a large back focal length, which is one of the factors to be considered in the miniaturization of the stabilized zoom systems.

Fig. 2. Influence of the conic surface and zoom ratio on the image quality of a stabilized zoom system: the focal length ranges at zoom ratios 2, 5, and 10 are 30–60 mm, 30–150 mm, and 30–300 mm, respectively. The spherical surface denotes the systems consisting of only spherical surfaces, and the conic surface denotes the systems consisting of conic surfaces.

Download Full Size | PDF

Table 1. Configuration data of stabilized zoom systems shown in Fig. 2.

View Table | View all tables in this article

2.2 Zoom equation analysis of stabilized zoom systems with two FLV elements

We have analyzed the zoom equation of four-group stabilized systems in the previous work [5] by exploring the expression of Gaussian brackets formulated by Tanaka [16]. In this study, for practical purposes, we deduce the zoom equation of multi-element stabilized zoom systems with two FLV elements. This proves beneficial in achieving a stabilized zoom in classical optical systems such as the Ritchey-Chrétien system and three mirror anastigmatic system. Based on the zoom equation, the focal power of system can be deduced and optimized in limited range of the focal power of FLV elements.

For systems containing only conic surfaces, focal power ${\phi _i}$ of the ${i^{\textrm{th}}}$ element and reduced distance ${e_i}$ between the ${i^{\textrm{th}}}$ and ${(i + 1)^{\textrm{th}}}$ elements can be determined as follows:

(16)$${\phi _i} = ({{n_i} - {n_{i\textrm{ + 1}}}} ){c_i}$$

(17)$${e_i} = {d_i}/{n_{i\textrm{ + 1}}}$$

The paraxial characteristics associated with an optical system [15,17] can be described as follows:

(18)$$\left\{ \begin{array}{l} {}^i{A_j} = [{\phi_i}, - {e_i},{\phi_{i + 1}}, - {e_{i + 1}}, \cdots ,{\phi_{j - 1}}, - {e_{j - 1}}],\;\;\;{}^i{A_i} = 1\\ {}^i{B_j} = [\;\;\;\; - {e_i},{\phi_{i + 1}}, - {e_{i + 1}}, \cdots ,{\phi_{j - 1}}, - {e_{j - 1}}],\;\;\;{}^i{B_i} = 0\\ {}^i{C_j} = [{\phi_i}, - {e_i},{\phi_{i + 1}}, - {e_{i + 1}}, \cdots ,{\phi_{j - 1}}, - {e_{j - 1}},{\phi_j}],\;\;\;{}^i{C_i} = {\phi_i}\\ {}^i{D_j} = [\;\;\;\; - {e_i},{\phi_{i + 1}}, - {e_{i + 1}}, \cdots ,{\phi_{j - 1}}, - {e_{j - 1}},{\phi_j}],\;\;\;{}^i{D_i} = 1 \end{array} \right.$$

where ${}_{}^i{A_j},\,{}_{}^i{B_j},\,{}_{}^i{C_j}$, and $^i{D_j}$ are the generalized Gaussian constants between the ${i^{\textrm{th}}}$ and ${j^{\textrm{th}}}$ elements.

The relation between these expressions [15] can be represented as follow:

{}_{}^i{B_j} = \left\{ {\begin{array}{c} { - {}_{}^{i + 1}{A_j} \cdot {e_i} + {}_{}^{i + 1}{B_j},\;i < j}\\ {\;\;\;\;\;\;\;\;\;\;0\;\;\;\;\;\;\;\;\;\;\;,\;i = j} \end{array}} \right.

(19)$${}_{}^i{D_j} = \left\{ {\begin{array}{c} { - {}_{}^{i + 1}{C_j} \cdot {e_i} + {}_{}^{i + 1}{D_j},\;i < j}\\ {\;\;\;\;\;\;\;\;\;\;1\;\;\;\;\;\;\;\;\;\;\;,\;i = j} \end{array}} \right.$$

The paraxial ray tracing can be expressed as follows:

(20)$$\left( {\begin{array}{c} {{h_j}}\\ {{n_{j\textrm{ + 1}}}u{^{\prime}_j}} \end{array}} \right) = \left( {\begin{array}{cc} {{}_{}^i{A_j}}&{{}_{}^i{B_j}}\\ {{}_{}^i{C_j}}&{{}_{}^i{D_j}} \end{array}} \right)\left( {\begin{array}{c} {{h_i}}\\ {{n_i}{u_i}} \end{array}} \right)$$

(21)$$\left( {\begin{array}{c} {{h_j}}\\ {{n_j}{u_j}} \end{array}} \right) = \left( {\begin{array}{cc} {{}_{}^{i + 1}{A_j}}&{{}_{}^i{B_j}}\\ {{}_{}^{i + 1}{C_{j - 1}}}&{{}_{}^i{D_{j - 1}}} \end{array}} \right)\left( {\begin{array}{c} {{h_i}}\\ {{n_{i\textrm{ + 1}}}u{^{\prime}_i}} \end{array}} \right)$$

The system consists of n optical elements, where ${m^{\textrm{th}}}$ and ${n^{\textrm{th}}}$ are FLV elements. We consider the ${1^{\textrm{th}}}$ to ${(m - 1)^{\textrm{th}}}$ elements to be optical components with focal power ${\phi _{pre}}$. Similarly, we can obtain the second optical component with focal power ${\phi _{mid}}$, as shown in Fig. 3. $e{^{\prime}_0},\,e{^{\prime}_1},e{^{\prime}_2}$ and $e{^{\prime}_3}$ are the reduced distances between the consecutive principles in each optical component, while $e{^{\prime}_4}$ is the back focal length of the system. Therefore, the paraxial ray tracing can be deduced from Eq. (21) as follows:

(22)$$\left( {\begin{array}{c} {{h_m}}\\ {{n_m}{u_m}} \end{array}} \right) = \left( {\begin{array}{cc} {{}_{}^1{A_m}}&{{}_{}^0{B_m}}\\ {{}_{}^1{C_{m - 1}}}&{{}_{}^0{D_{m - 1}}} \end{array}} \right)\left( {\begin{array}{c} {{h_0}}\\ {{n_1}u{^{\prime}_0}} \end{array}} \right)$$

(23)$$\left( {\begin{array}{c} {{h_m}}\\ {{n_m}{u_m}} \end{array}} \right) = \left( {\begin{array}{cc} {{}_{}^1A{^{\prime}_2}}&{{}_{}^0B{^{\prime}_2}}\\ {{}_{}^1C{^{\prime}_1}}&{{}_{}^0D{^{\prime}_1}} \end{array}} \right)\left( {\begin{array}{c} {{h_0}}\\ {{n_1}u{^{\prime}_0}} \end{array}} \right)$$

(24)$$\left( {\begin{array}{c} {{h_n}}\\ {{n_n}{u_n}} \end{array}} \right) = \left( {\begin{array}{cc} {{}_{}^{m + 1}{A_n}}&{{}_{}^m{B_n}}\\ {{}_{}^{m + 1}{C_{n - 1}}}&{{}_{}^m{D_{n - 1}}} \end{array}} \right)\left( {\begin{array}{c} {{h_m}}\\ {{n_{m\textrm{ + 1}}}u{^{\prime}_m}} \end{array}} \right)\;$$

(25)$$\left( {\begin{array}{c} {{h_n}}\\ {{n_n}{u_n}} \end{array}} \right) = \left( {\begin{array}{cc} {{}_{}^3A{^{\prime}_4}}&{{}_{}^2B{^{\prime}_4}}\\ {{}_{}^3C{^{\prime}_3}}&{{}_{}^2D{^{\prime}_3}} \end{array}} \right)\left( {\begin{array}{c} {{h_m}}\\ {{n_{m\textrm{ + 1}}}u{^{\prime}_m}} \end{array}} \right)\;$$

Fig. 3. Paraxial ray-tracing model of the multi-element stabilized zoom system.

Download Full Size | PDF

From Eqs. (22)–(25), we obtain

(26)$$\left\{ \begin{array}{l} {}^1{A_m} = 1 - {\phi_{pre}}e^{\prime}_1 = [{\phi_{pre}}, - e^{\prime}_1] = {}^1A^{\prime}_2\\ {}^0{B_m} = {\phi_{pre}}e^{\prime}_1e^{\prime}_0 - e^{\prime}_1 - e^{\prime}_0 = [ - e^{\prime}_0,{\phi_{pre}}, - e^{\prime}_1] = {}^0B^{\prime}_2\\ {}^1{C_{m - 1}} = {\phi_{pre}} = [{\phi_{pre}}] = {}^1C^{\prime}_1\\ {}^0{D_{m - 1}} = 1 - {\phi_{pre}}e^{\prime}_0 = [ - e^{\prime}_0,{\phi_{pre}}] = {}^0D^{\prime}_1 \end{array} \right.$$

(27)$$\left\{ \begin{array}{l} {}^{m + 1}{A_n} = 1 - {\phi_{mid}}e^{\prime}_3 = [{\phi_{mid}}, - e^{\prime}_3] = {}^3A^{\prime}_4\\ {}^m{B_n} = {\phi_{mid}}e^{\prime}_3e^{\prime}_2 - e^{\prime}_3 - e^{\prime}_2 = [ - e^{\prime}_2,{\phi_{mid}}, - e^{\prime}_3] = {}^2B^{\prime}_4\\ {}^{m + 1}{C_{n - 1}} = {\phi_{mid}} = [{\phi_{mid}}] = {}^3C^{\prime}_3\\ {}^m{D_{n - 1}} = 1 - {\phi_{mid}}e^{\prime}_2 = [ - e^{\prime}_2,{\phi_{mid}}] = {}^2D^{\prime}_3 \end{array} \right.$$

Therefore, the zoom equation can be represented by

(28)$$Z = {}_{}^1{A_{n + 1}} = [{{\phi_1}, - {e_1}, \cdots ,{\phi_m}, - {e_m}, \cdots ,{\phi_n}, - {e_n}} ]= \;\;[{{\phi_{pre}}, - e^{\prime}_1,{\phi_m}, - e^{\prime}_2,{\phi_{mid}}, - e^{\prime}_3,{\phi_4}, - e^{\prime}_4} ]= 0$$

The focal power equation of the zoom system can be expressed as

(29)$${}_{}^1{C_n} = [{{\phi_1}, - {e_1}, \cdots ,{\phi_m}, - {e_m}, \cdots ,{\phi_n}} ]= [{{\phi_{pre}}, - e^{\prime}_1,{\phi_m}, - e^{\prime}_2,{\phi_{mid}}, - e^{\prime}_3,{\phi_4}} ]= \Phi$$

Equations (26–29) can be used to obtain Eqs. (30) and (31), which can be used to calculate the focal power of the FLV elements corresponding to the change in system focal power $\mathrm{\Phi }$ as follows:

(30)$${\phi _m} = \frac{{{e_n}\mathrm{\Phi } - [{{\phi_{pre}}, - ({e^{\prime}_1 + e^{\prime}_2} ),{\phi_{mid}}, - e^{\prime}_3} ]}}{{{}_{}^1{A_m} \cdot {}_{}^m{B_n}}}$$

(31)$${\phi _n} = \frac{{{}_{}^1{A_m}}}{{{e_n}\mathrm{\Phi } \cdot {}_{}^m{B_n}}} + \frac{{{}_{}^m{B_n} - {e_n}[{ - e^{\prime}_2,{\phi_{mid}}} ]}}{{{e_n} \cdot {}_{}^m{B_n}}}$$

As the functions are monotonic, variations in ${\phi _m}$ and ${\phi _n}$ can be expressed as follows:

(32)$$\mathrm{\Delta }{\phi _m} = \frac{{{e_n}({\mathrm{\Gamma } - 1} ){\mathrm{\Phi }_L}}}{{{}_{}^1{A_m} \cdot {}_{}^m{B_n}}}$$

(33)$$\;\mathrm{\Delta }{\phi _n} = \frac{{{}_{}^1{A_m}({\mathrm{\Gamma } - 1} )}}{{{e_n}\mathrm{\Gamma }{\mathrm{\Phi }_L} \cdot {}_{}^m{B_n}}}$$

where $\mathrm{\Gamma }$ denotes the zoom ratio; ${\mathrm{\Phi }_L}$ represents the system focal power at the telephoto end; $\mathrm{\Delta }{\phi _m}$ and $\mathrm{\Delta }{\phi _n}$ are the variations in the focal power given by FLV elements.

2.2.1 Focal length

Focal length $f^{\prime}$ of an optical system with n elements can be represented using Eq. (34).

(34)$$f^{\prime} = \frac{1}{{{}_{}^1{C_n}}}$$

Therefore, we can obtain focal length representation ${A_{foc}}$, which can be used to control the focal length of the designed system, as shown in Eq. (35),

(35)$${A_{foc}} = \;\left|{\frac{1}{{{}_{}^1{C_n}}} - f} \right|$$

where f is the ideal focal length of the designed system.

2.2.2 Defocus

From Eqs. (19) and (20), back focal length ${e_n}$ of an optical system with n elements can be expressed in Eq. (36),

(36)$${e_n} = \frac{{ - {}_{}^1{A_n} \cdot {e_0} + {}_{}^1{B_n}}}{{ - {}_{}^1{C_n} \cdot {e_0} + {}_{}^1{D_n}}} = \frac{{{}_{}^0{B_n}}}{{{}_{}^0{D_n}}}. $$

From Eqs. (26), (27), (32), and (33), we can obtain expressions (37) and (38), which can be used to constrain the values of $\mathrm{\Delta }{\phi _m}$ and $\mathrm{\Delta }{\phi _n}$, and acquire a suitable back focal length of the designed zoom system.

(37)$${}_{}^m{B_n} = \sqrt {\frac{{{{(\mathrm{\Gamma } - 1)}^2}}}{{\mathrm{\Gamma } \cdot \mathrm{\Delta }{\phi _m}\mathrm{\Delta }{\phi _n}}}} $$

(38)$$\frac{{{e_n}}}{{{}_{}^1{A_m}}} = \frac{{e^{\prime}_4}}{{({1 - e^{\prime}_1{\phi_{pre}}} )}} = \frac{1}{{{\mathrm{\Phi }_L}}}\sqrt {\frac{{\mathrm{\Delta }{\phi _m}}}{{\mathrm{\Gamma } \cdot \mathrm{\Delta }{\phi _n}}}} $$

Therefore, we can obtain defocus representation ${A_{def}}$, which can be used to achieve non-defocus in a designed system, as shown in Eq. (39).

(39)$${A_{def}} = \;\left|{\frac{{{}_{}^0{B_n}}}{{{}_{}^0{D_n} \cdot {}_{}^1{A_m}}} - \frac{1}{{{\mathrm{\Phi }_L}}}\sqrt {\frac{{\mathrm{\Delta }{\phi_m}}}{{\mathrm{\Gamma } \cdot \mathrm{\Delta }{\phi_n}}}} } \right|$$

Based on the above analysis, we can construct the zoom equation for a multi-element stabilized zoom system with two FLV elements that possesses a high zoom ratio and is free from defocus.

2.3 Configuration design for off-axis stabilized zoom systems

Based on the analysis of the zoom equation in Section 2.2, zooming merit function F is established, and a series of configurations of stabilized zoom systems that exhibit high zoom ratios and are free from defocus are retrieved. Furthermore, according to the aberration analysis of off-axis stabilized zoom systems in Section 2.1, we can establish a nonlinear global merit function L using aberration representations ${A_{spa}},\,{A_{ast}},{A_{dis}}$, and zooming merit function F. Merit function L is used to directly evaluate the imaging performance and zooming capability using the variables and invariants of the off-axis stabilized zoom system. Therefore, the configuration design work can be transformed into an investigation for an optimal solution using nonlinear global merit function L [4,24]. The aberrations that characterize the off-axis system, capability of the zoom system (for example, maximum zoom ratio), and first-order properties (for example, a fixed back focal length of the zoom system) are regarded as the performance metrics for the evaluation.

To represent the overall zooming performance of the designed zoom system, a sufficient number of zoom ratio points are uniformly selected in the zoom range. Zooming merit function F can be expressed as follows:

(40)$$\begin{array}{l} F = F({{e_j},{c_i},c_m^l,c_n^l} )= \frac{1}{M}\mathop \sum \limits_{l = 1}^M [{\nu_1^l \cdot A_{foc}^l + \nu_2^l \cdot A_{def}^l} ]\\ j = 1,2, \cdots ,n;\;\;\;i = 1, \cdots ,m - 1,m + 1, \cdots ,n - 1 \end{array}$$

where M is the number of sampled zoom ratio points, l denotes the ${l^{\textrm{th}}}$ sampled zoom ratio point, and ${\nu _i}$ denotes the weight of the corresponding term.

For a system with a fixed focal length (for example, the zoom system at the wide-angle end), the sum of the absolute value of the aberration coefficients can be described as a function of $E(H )$:

(41)$$E(H )= \nu _4|{A_{spa}} |+ \nu _5|{|{A_{ast}} |{|_1} + \nu_6} ||{A_{dis}} |{|_1}$$

here, $||{\;\;} |{|_1}$ denotes the 1-norm.

To represent the overall imaging performance of the designed zoom system, a sufficient number of field-of-view points are uniformly selected in full field of view (FOV). The full-field performance can be evaluated using the following image aberration merit function:

(42)$$E = \frac{1}{N}\mathop \sum \nolimits_{k = 1}^N E({{H_k}} )$$

where N is the number of sampled field points, and ${H_k}$ is the ${k^{\textrm{th}}}$ sampled field.

Therefore, we can obtain nonlinear global merit function L, which can be used to evaluate the overall performance of the off-axis stabilized zoom system under several sampled zoom ratio points as follows in Eq. (43).

(43)$$\begin{array}{l} L = L({H_k^l,{e_j},{\alpha_j},{c_i},{k_i},c_m^l,k_m^l,c_n^l,k_n^l} )\\ \;\;\;\; = \frac{1}{M}\mathop \sum \limits_{l = 1}^M \left[ {\nu_1^l \cdot A_{foc}^l + \nu_2^l \cdot A_{def}^l\; + \frac{{\nu_3^l}}{N}\mathop \sum \limits_{k = 1}^N {E^l}({H_k^l} )} \right];\\ j = 1,\; 2, \cdots ,n;\;\;\;i = 1, \cdots ,m - 1,m + 1, \cdots ,n - 1 \end{array}$$

Equation (43) shows that the configuration design problem given is a multi-objective optimization problem. As the zooming properties of the zoom system is a priority, it is recommended that the weight coefficients $\nu _1$ and $\nu _2$ for zooming properties be set to 1, and the weight coefficient $\nu _3$ for image quality be set to 0.5. Moreover, the weight coefficients of spherical aberration, astigmatism and distortion are recommended to be 1, 0.5 and 0.1. In this study, we employ a global optimization algorithm (for example, a genetic algorithm) to retrieve an optimal configuration, and the primary steps are displayed in Fig. 4.

Fig. 4. Flowchart of the design process.

Download Full Size | PDF

Therefore, regardless of the optimal solution, we can obtain the suboptimal solution as a configuration of the off-axis stabilized zoom system. The weights of each representation of nonlinear global merit function L are used to achieve specific requirements of the designed system (for example, sacrificing the imaging performance of the system at the wide-angle end to achieve a high resolution at the telephoto end), as the zooming capability is represented by representations ${A_{foc}}$ and ${A_{def}}$, and the imaging performance is represented by representations ${A_{spa}},\,{A_{ast}}$, and ${A_{dis}}$.

In this way, zooming merit function F consists of $2({n + l - 1} )$ decision variables and $2l$ objective function, and we can obtain a set of optimal solutions instead of a unique solution according to the theory of Pareto Optimality [21]. The Pareto optimization solutions of F can be described as group A in Fig. 5. Group B denotes the Pareto optimization solutions of F with the constraint condition of the focal power variation of FLV elements given by Eq. (37). Additionally, global merit function L consisting of zooming merit function F, image aberration merit function E and constraint condition of FLV elements has $4({n + l - 1} )$ decision variables and $5l$ objective function, and its Pareto optimization solutions is presented as group C in Fig. 5.

Fig. 5. Sketch map of Pareto optimization solutions for different merit function constructions.

Download Full Size | PDF

Based on the consideration that the effective variation ranges of the focal power of FLV elements are consistent, the variations of the focal power used in the off-axis stabilized zoom system and their image qualities are shown in Fig. 6. Using the off-axis three-mirror layout, the primary configurations of six stabilized zoom system groups are designed. The zoom ratios are 10 and the focal length ranges are 10–100 mm. The RMS spot diameters take the average values of the wide-angle and telephoto ends.

Fig. 6. Influence of the merit function constructions on $\mathrm{\Delta }{\phi _1},\,\; \mathrm{\Delta }{\phi _2}$, and the image quality of a stabilized zoom system; all groups adopt zooming merit function F to design the primary configurations that satisfy the zoom requirement. The constraint condition of the focal power variation of FLV elements is introduced in group B. The constraint conditions of FLV elements and image aberration merit function E are employed in group C. $\mathrm{\Delta }{\phi _1}$ and$\; \mathrm{\Delta }{\phi _2}$ denote the variations in the focal power of FLV-element-1 and FLV-element-2, respectively.

Download Full Size | PDF

It can be seen from Fig. 6 that the usage of the constraint condition of FLV elements can effectively reduce the requirement of the focal power variation; the focal power variations of FLV elements are less than 0.01 in groups B and C. Additionally, the employment of image aberration merit function E can significantly improve the image quality; the RMS spot diameters are less than $0.2\; \textrm{mm}$ in group C. The configuration data of stabilized zoom systems given in Fig. 6 are listed in Table 2, which demonstrates that the focal power of the first surface (FLV-element-1) varies significantly and passes through the zero point in the process of system zooming, and thus the FLV-element-1 considers a large zooming task. This is one of the key factors that improve the zoom ratio of a stabilized zoom system. Furthermore, appropriate tilt angles and the rational use of conic surface are constructive in improving the image quality of the stabilized zoom system, and the operation depends on image aberration merit function E.

Table 2. Configuration data of stabilized zoom systems given in Fig. 6

View Table | View all tables in this article

3. Experimental design results

To demonstrate the proposed design method, we designed using an off-axis three-mirror layout, which was a stabilized zoom system with two deformable mirrors (assumed a maximum stroke of 100 micrometer the same as the VISIONICA’s product [25] specifications), and zoom ratio $\mathrm{\Gamma }$ was 10. The primary mirror (PM) and the tertiary mirror (TM) were designated as deformable mirror 1 (DM1) and deformable mirror 2 (DM2), respectively. The design specifications of this optical system are given in Table 3, where FOVs ranged from −5°–5° and 0°–5° in the x and y directions, respectively, at the wide-angle end; FOVs ranged from −1.5811°–1.5811° and 0°–1.5811° in the x and y directions, respectively, at the medium-angle end; FOVs ranged from −0.5°–0.5° and 0°–0.5° in the x and y directions, respectively, at the telephoto end.

Table 3. Specifications of the stabilized zoom system.

View Table | View all tables in this article

Because of the plane-symmetry characteristic of the off-axis stabilized zoom system, the imaging performance over a half of a full field was sufficient to represent the system. As shown in Eq. (46), 8 × 8 + 2 × 6 × 6 uniform sampled FOV points were used to establish nonlinear global merit function L. $W,\,M$ and T denote the wide-angle, medium-angle, and telephoto ends, respectively.

(44)$$\left\{ \begin{array}{l} H_k^W\; = [\frac{1}{7}({x_1} - 1)\;\;\;\;\frac{1}{7}({y_1} - 1)]\\ H_k^M = H_k^T\;\; = [\frac{1}{5}({x_2} - 1)\;\;\;\;\frac{1}{5}({y_2} - 1)]\\ {x_1} = 1,2,3,4,5,6,7,8\;;\;\;{y_1} = 1,2,3,4,5,6,7,8\\ {x_2} = 1,2,3,4,5,6;\;\;{y_2} = 1,2,3,4,5,6 \end{array} \right.$$

The initial marginal and chief ray data are defined in Table 4.

Table 4. Initial ray data defined in the entrance pupil.

View Table | View all tables in this article

Here, the focal power ranged from $- 0.005\;\textrm{m}{\textrm{m}^{ - 1}}$ to $0.005\;\textrm{m}{\textrm{m}^{ - 1}}$ and $- 0.0\textrm{15}\;\textrm{m}{\textrm{m}^{ - 1}}$ to $- 0.005\;\textrm{m}{\textrm{m}^{ - 1}}$ in PM and TM, respectively. Therefore, from Eqs. (37) and (38), the constraints on zooming capability can be expressed as

(45)$$\left\{ {\begin{array}{c} {{}_{}^1{B_3} \ge \sqrt {\frac{{\textrm{81}}}{{\textrm{10} \cdot \mathrm{\Delta }{\phi_1} \cdot \mathrm{\Delta }{\phi_3}}}} \approx 2\textrm{85}}\\ {{d_3} = \textrm{280} \cdot \sqrt {\frac{{\mathrm{\Delta }\phi ^{\prime}_1}}{{\textrm{10} \cdot \mathrm{\Delta }\phi^{\prime}_3}}} } \end{array}} \right.$$

where $\mathrm{\Delta }{\phi _1} = 0.01$ and $\mathrm{\Delta }{\phi _3} = 0.01$ are the theoretical variations of the focal power given by DM1 and DM2, respectively; $\mathrm{\Delta }\phi {^{\prime}_1}$ and $\mathrm{\Delta }\phi {^{\prime}_3}$ denote the variations used in the stabilized zoom system.

As the off-axis three-mirror layout can favor a large field of view, when the aperture is set on the secondary mirror (SM), reduced distance from the entrance pupil to the primary mirror ${d_0}$ is given by Eq. (46) as

(46)$${d_0} = \frac{{{d_1}}}{{1 - 2{c_1}{d_1}}}$$

As mentioned in section 2.3, a genetic algorithm was employed to optimize nonlinear global merit function L. To obtain an enhanced configuration with no obscurations, the boundary values were adjusted several times. The boundary values of the parameters and the final optimization solutions are listed in Table 5.

Table 5. Boundary values of parameters and optimization solutions.

View Table | View all tables in this article

Through Eqs. (30) and (31), the relationship among $ {\Phi ,}\,{\phi _1}$, and ${\phi _3}$ can be seen in Fig. 7. This demonstrated that the variation in ${\phi _1}$ and ${\phi _3}$ was monotonous during zooming. We calculated the values of ${\phi _1}$ and ${\phi _3}$ at a sampled focal length. The results were consistent with the data from the designed system. Additionally, the length of the focal power range of the stabilized zoom system is $\mathrm{\Delta } = ({1/28 - 1/280} )$, and the zoom sensitivity of the system is evaluated at 1/500. It means that the minimum change in focal power of the zoom system is $\mathrm{\Delta }/500.\,\phi $ and $\phi ^{\prime}$ denote the focal power of deformable mirror before and after the minimum effective zoom, respectively, and $\mathrm{\Delta }\phi $ denotes the change in focal power of deformable mirror. Table 6 shows that the required surface control accuracies of the DM1 and DM2, respectively, when the zoom sensitivity of the stabilized zoom system is evaluated at 1/500.

Fig. 7. Relationship among designed system focal power $\mathrm{\Phi }$, DM1 focal power ${\phi _1}$, and DM2 focal power ${\phi _3}$. PM (DM1): primary mirror, and TM (DM2): tertiary mirror.

Download Full Size | PDF

Table 6. Requirements for deformable mirror while the system zoom sensitivity is 1/500.

View Table | View all tables in this article

Figures 8(a)–(c) present the optical layouts of the unobscured stabilized zoom system at the sampled focal length points. Figures 8(d)–(f) show the root mean square (RMS) spot diameters of the system at different zoom ratios, and the configuration had a balanced imaging performance over the full zoom range. The distortion grids are shown in Figs. 9(a)–(c), where the maximum relative distortions were 1.77%, 0.65%, and 1.11% at the wide-angle, medium-angle, and telephoto ends, respectively. The astigmatisms over the full field are shown in Figs. 9(d)–(f). Because of the no obscuration layout at boundary conditions, the nodal points of astigmatism aberration were not in the field of view at the wide-angle and medium-angle ends. However, these nodes were symmetrically distributed around the center of the working field of view. Moreover, two nodal points of astigmatism can be practically seen at the telephoto end. Figures 10(a)–(c) show the modulation transfer function (MTF) plots of the stabilized zoom system at three zoom ratios, and the MTF plots were nearly diffraction-limited. Furthermore, simulation experiments were performed using the 2D image simulation function in CODE V to achieve the three designed focal length of the stabilized zoom system. The results are shown in Figs. 11(a)–(c); the image semi-diagonals were 1.1402 mm, 3.5734 mm, and 11.3990 mm at the wide-angle, medium-angle, and telephoto ends, respectively. Therefore, the zoom ratios of the zoom system were 3.1340 (3.5734 mm/1.1402 mm, the theoretical value was $\sqrt {10} \approx 3.1623$) and 9.9974 (11.3990 mm/1.1402 mm). These results demonstrated that we can obtain an effective configuration of the off-axis stabilized zoom system using the proposed method. Furthermore, if a higher performance system is required, the achieved configuration of the off-axis stabilized zoom system can be further optimized with higher-order surfaces, such as free-form surfaces.

Fig. 8. Optical layouts of the stabilized zoom system: (a) wide-angle, (b) medium-angle, and (c) telephoto. Here, PM (DM1), TM (DM2), and SM (Stop) denote the primary mirror with deformable mirror 1, tertiary mirror with deformable mirror 2, and secondary mirror with stop, respectively. The RMS spot diameter is as follows: (d) wide-angle, (e) medium-angle, and (f) telephoto. PM: primary mirror, TM: tertiary mirror, SM: secondary mirror.

Download Full Size | PDF

Fig. 9. Distortions: (a) wide-angle, (b) medium-angle, and (c) telephoto; astigmatism: (d) wide-angle, (e) medium-angle, and (f) telephoto.

Download Full Size | PDF

Fig. 10. MTF: (a) wide-angle, (b) medium-angle, and (c) telephoto.

Download Full Size | PDF

Fig. 11. Simulation results of the zooming capability of the system: Image at the (a) wide-angle end, (b) medium-angle end, and (c) telephoto end.

Download Full Size | PDF

4. Discussion and conclusions

In this study, a universal synthetic system design method was developed for off-axis stabilized zoom systems by establishing nonlinear global merit function L based on the zoom equations of the Gaussian brackets and nodal aberration theory. According to the analysis of the first-order properties and ray tracing characteristics of multi-element stabilized zoom systems with two FLV elements, representations ${A_{foc}}$ and ${A_{def}}$ were used to construct a zoom system equation without defocusing in a specified focal length range; at the same time, the primary aberrations of off-axis systems were presented analytically by representations ${A_{spa}},\,{A_{ast}}$, and ${A_{dis}}$, which were effectively employed in the merit function. Thus, this nonlinear global merit function L are used to evaluate the overall performance of the off-axis stabilized zoom systems and the superiority of the solutions is analyzed in the parameter space of optimization. The experimental design results show that the off-axis stabilized zoom system with two deformable mirrors can be designed automatically, with an eligible overall performance in the specific focal range, and a high zoom ratio of 10:1.

In this way, we construct a semi-empirical mathematical model based on nodal aberration theory and nonlinear zoom equation for maintaining the stability of focal length and image plane drift. It breaks through the limitation of the off-axis astigmatism in the entire focal range of the off-axis reflection system, and satisfies the requirement of high zoom ratio for fast optical zoom. Despite its conceptual simplicity, our proposed synthetic system design method proved to be an effective and promising tool that can be used in the off-axis reflection system stabilized zoom system design process as well as obtain an appropriate initial point for complex optical systems of higher performance. Additionally, this study provides the insight and guidance for the design of aberration balanced strategy for off-axis stabilized zoom systems in the entire focal range. The off-axis stabilized zoom systems with a large field of view range and low F-numbers are yet to be researched, further improvements to this method can be implemented in future studies.

Funding

National Natural Science Foundation of China (51735002, 61527826); National Key Research and Development Program of China (2017YFC1403602); Major Scientific and Technological Innovation Project of Shandong Provincial Key Research and Development Program (2019JZZY020708); Shenzhen Science and Technology Innovation Program (JCYJ20160428182026575, JCYJ20170412171011187).

Acknowledgments

The authors would also like to thank the anonymous reviewers for their valuable suggestions.

Disclosures

The authors declare no conflicts of interest.

References

1. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component type,” Appl. Opt. 21(12), 2174–2183 (1982). [CrossRef]

2. F. Yi-Chin, T. Cheng-Mu, and C. Cheng-Lun, “A study of optical design and optimization of zoom optics with liquid lenses through modified genetic algorithm,” Opt. Express 19(17), 16291–16302 (2011). [CrossRef]

3. A. Mikš and J. Novák, “Method of first-order analysis of a three-element two-conjugate zoom lens,” Appl. Opt. 56(18), 5301–5306 (2017). [CrossRef]

4. Z. Fan, S. Wei, Z. Zhu, Y. Yan, Y. Mo, L. Yan, and D. Ma, “Globally optimal first-order design of zoom systems with fixed foci as well as high zoom ratio,” Opt. Express 27(26), 38180–38190 (2019). [CrossRef]

5. Q. Hao, X. Cheng, and K. Du, “Four-group stabilized zoom lens design of two focal-length-variable elements,” Opt. Express 21(6), 7758–7767 (2013). [CrossRef]

6. H. Li, X. Cheng, and Q. Hao, “An electrically tunable zoom system using liquid lenses,” Sensors 16(1), 45 (2015). [CrossRef]

7. S. H. Jo and S. C. Park, “Design and analysis of an 8x four-group zoom system using focus tunable lenses,” Opt. Express 26(10), 13370–13382 (2018). [CrossRef]

8. H. T. Hsieh, H. C. Wei, M. H. Lin, W. Y. Hsu, Y. C. Cheng, and G. D. J. Su, “Thin autofocus camera module by a large-stroke micromachined deformable mirror,” Opt. Express 18(11), 11097–11104 (2010). [CrossRef]

9. Y. H. Lin, Y. L. Liu, and G. D. J. Su, “Optical zoom module based on two deformable mirrors for mobile device applications,” Appl. Opt. 51(11), 1804–1810 (2012). [CrossRef]

10. H. Zhao, X. Fan, G. Zou, Z. Pang, W. Wang, G. Ren, Y. Du, and Y. Su, “All-reflective optical bifocal zooming system without moving elements based on deformable mirror for space camera application,” Appl. Opt. 52(6), 1192–1210 (2013). [CrossRef]

11. D. Korsch, “Design and optimization technique for three-mirror telescopes,” Appl. Opt. 19(21), 3640–3645 (1980). [CrossRef]

12. Q. Meng, H. Wang, W. Liang, Z. Yan, and B. Wang, “Design of off-axis three-mirror systems with ultrawide field of view based on an expansion process of surface freeform and field of view,” Appl. Opt. 58(3), 609–615 (2019). [CrossRef]

13. Y. Zhong and H. Gross, “Initial system design method for non-rotationally symmetric systems based on Gaussian brackets and Nodal aberration theory,” Opt. Express 25(9), 10016–10030 (2017). [CrossRef]

14. C. Cao, S. Liao, Z. Liao, Y. Bai, and Z. Fan, “Initial configuration design method for off-axis reflective optical systems using nodal aberration theory and genetic algorithm,” Opt. Eng. 58(10), 1 (2019). [CrossRef]

15. M. Herzberger, “Gaussian optics and Gaussian brackets,” J. Opt. Soc. Am. 33(12), 651–655 (1943). [CrossRef]

16. K. Tanaka, “II Paraxial theory in optical design in terms of Gaussian Brackets,” Prog. Opt. 23, 63–111 (1986). [CrossRef]

17. X. Yuan and X. Cheng, “Lens design based on lens form parameters using Gaussian brackets,” Proc. SPIE 9272, 92721L (2014). [CrossRef]

18. K. Thompson, Aberration Fields in Tilted and Decentered Optical Systems (University of Arizona, 1980).

19. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005). [CrossRef]

20. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009). [CrossRef]

21. N. Srinivas and Kalyanmoy Deb, “Muiltiobjective optimization using nondominated sorting in genetic algorithms,” Evol. Comput. 2(3), 221–248 (1994). [CrossRef]

22. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending nodal aberration theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express 20(18), 20139–20155 (2012). [CrossRef]

23. A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018). [CrossRef]

24. F. E. Sahin, “Open-source optimization algorithms for optical design,” Optik 178, 1016–1022 (2019). [CrossRef]

25. http://www.visionica.biz/uflex-eng.htm (accessed on Jan. 17, 2021)

Parameter	Zoom ratio: 2		Zoom ratio: 5		Zoom ratio: 10
Parameter	spherical	conic	spherical	conic	spherical	conic
$d_{1}$ / $mm$	−142.9516		−196.6297		−159.2854
$d_{2}$ / $mm$	239.7245		179.6226		180.4619
$d_{3}$ / $mm$	−152.0983		−167.5541		−178.1928
$α_{1}$ /deg	12.0583		10.6382		10.2878
$α_{2}$ /deg	−10.6341		−10.5620		−11.7742
$α_{3}$ /deg	5.0420		5.0094		5.1528
$r_{2}$ / $mm$	−213.2659		−177.1967		−184.4922
$k_{2}$	—	−26.1916	—	−27.0275	—	−74.1102
$r_{1}^{W}$ / $mm$	768.8867		606.4325		436.5856
$k_{1}^{W}$	—	−80.0000	—	−80.0000	—	80.0000
$r_{3}^{W}$ / $mm$	−207.8862		-202.1899		−209.6693
$k_{3}^{W}$	—	−0.2752	—	−0.3121	—	−0.6033
$r_{1}^{T}$ / $mm$	−1976.2527		−811.5136		−551.7690
$k_{1}^{T}$	—	−80.0000	—	−80.0000	—	−50.0128
$r_{3}^{T}$ / $mm$	−214.1439		−223.0145		−277.2772
$k_{3}^{T}$	—	−0.3954	—	−1.4625	—	−17.7398

Parameter	A₁	A₂	A₃	B₁	B₂	C
$d_{1}$ / $mm$	−206.1591	−199.0040	−168.7878	−172.5412	−173.8623	−227.0623
$d_{2}$ / $mm$	109.2138	143.7374	74.6616	232.7862	155.5987	219.8072
$d_{3}$ / $mm$	−38.0676	−106.4537	−40.4774	−58.4240	−58.8979	−49.3097
$α_{1}$ / $deg$	15.1677	10.6977	15.7590	12.8032	12.9346	10.5044
$α_{2}$ / $deg$	−15.5938	−15.1287	−15.0510	−12.4944	−12.5209	−18.7916
$α_{3}$ / $deg$	12.8481	10.5432	12.0090	12.4794	12.4372	10.0201
$r_{2}$ / $mm$	460.6387	−224.8762	392.0383	−409.6076	−183.0597	−631.9667
$k_{2}$	0.0000	0.0000	0.0000	0.0000	0.0000	−19.4386
$r_{1}^{W}$ / $mm$	132.6882	143.0274	104.5531	320.4172	393.1886	376.6068
$k_{1}^{W}$	0.0000	0.0000	0.0000	0.0000	0.0000	−12.0730
$r_{3}^{W}$ / $mm$	−78.3025	−144.1661	−82.9968	−100.4879	−93.6971	−87.8143
$k_{3}^{W}$	0.0000	0.0000	0.0000	0.0000	0.0000	−0.5742
$r_{1}^{T}$ / $mm$	−2999.4422	−984.2725	−1672.5018	−775.4528	−592.3717	−1006.0879
$k_{1}^{T}$	0.0000	0.0000	0.0000	0.0000	0.0000	−21.3598
$r_{3}^{T}$ / $mm$	−136.2344	−160.5181	−171.0545	−115.4825	−105.7878	−101.1448
$k_{3}^{T}$	0.0000	0.0000	0.0000	0.0000	0.0000	−0.8648

Parameter	Zoom 1 (wide-angle)	Zoom 2 (medium-angle)	Zoom 3 (telephoto)
Wavelength range/μm	8-12
Focal length/mm	28.0000	88.5438	280.0000
F-number	4.00	4.43	5.00
FOV/deg	10.0000 $\times$ 5.0000	3.1622 $\times$ 1.5811	1.0000 $\times$ 0.5000

Parameter	Marginal ray		Chief ray
Parameter	Ray height/ $mm$	Ray angle/ $rad$	Ray height/ $mm$	Ray angle/ $rad$
Wide-angle	3.5000	0.0000	0.0000	8.7266 $\times 10^{- 2}$
Medium-angle	10.0000	0.0000	0.0000	2.7596 $\times 10^{- 2}$
Telephoto	28.0000	0.0000	0.0000	8.7266 $\times 10^{- 3}$

Parameter	Range	Solution
$d_{1}$ /mm	[−250, −80]	−196.9698
$d_{2}$ /mm	[80, 250]	95.0967
$α_{1}$ /deg	[7, 20]	7.8609
$α_{2} / deg$	[−20, −10]	−13.8555
$α_{3}$ /deg	[5, 10]	5.6116
$c_{2}$ / $m m^{- 1}$	[−0.02, 0]	−3.8778×10⁻³
$k_{2}$	[−50, 50]	23.7732
$c_{1}^{W}$ / $m m^{- 1}$	[−0.0025, 0.0025]	2.2949×10⁻³
$k_{1}^{W}$	[−50, 50]	−50.0000
$c_{3}^{W}$ / $m m^{- 1}$	[−0.0075, -0.0025]	−7.3529×10⁻³
$k_{3}^{W}$	[−50, 50]	−0.6706
$c_{1}^{M} / m m^{- 1}$	[−0.0025, 0.0025]	−6.3236×10⁻⁴
$k_{1}^{M}$	[−50, 50]	50.0000
$c_{3}^{M}$ / $m m^{- 1}$	[−0.0075, −0.0025]	-6.6921×10⁻³
$k_{3}^{M}$	[−50, 50]	-0.8166
$c_{1}^{T}$ / $m m^{- 1}$	[−0.0025, 0.0025]	−1.5570×10⁻³
$k_{1}^{T}$	[−50, 50]	6.7007
$c_{3}^{T}$ / $m m^{- 1}$	[−0.0075, −0.0025]	−4.6000×10⁻³
$k_{3}^{T}$	[−50, 50]	3.0539

Synthetic system design method for off-axis stabilized zoom systems with a high zoom ratio

Abstract

1. Introduction

2. Design method

2.1 Aberration analysis of off-axis stabilized zoom systems with conic surfaces

2.1.1 Spherical aberration

2.1.2 Astigmatism

2.1.3 Distortion

2.2 Zoom equation analysis of stabilized zoom systems with two FLV elements

2.2.1 Focal length

2.2.2 Defocus

2.3 Configuration design for off-axis stabilized zoom systems

3. Experimental design results

4. Discussion and conclusions

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (11)

Tables (6)

Equations (47)

Optics Express

Parameter		$ϕ$	$ϕ^{'}$	$Δ ϕ$	Surface control accuracy
PM (DM1)	Wide-angle	0.4590×10⁻²	0.4440×10⁻²	−0.0150×10⁻²	$RMS shape error \leq 120 nm$
	Medium-angle	−0.1265×10⁻²	−0.1280×10⁻²	−0.0015×10⁻²
	Telephoto	−0.3115×10⁻²	−0.3099×10⁻²	0.0016×10⁻²
TM (DM2)	Wide-angle	−1.4706×10⁻²	−1.4692×10⁻²	0.0014×10⁻²	$RMS shape error \leq 50 nm$
	Medium-angle	−1.3384×10⁻²	−1.3371×10⁻²	0.0013×10⁻²
	Telephoto	−0.9203×10⁻²	−0.9314×10⁻²	−0.0111×10⁻²