Athermal and Apochromatic Design of Equivalent Two-Component Optical System in 3D Glass Diagram

Abstract

In the athermal and apochromatic design of optical systems, the distribution of lens’ optical powers and the selection of optical glass and structural materials are crucial. In this paper, an athermal and apochromatic design method is proposed for optical systems with a long focal length, large relative aperture, and wide spectrum. Firstly, a complex optical system composed of multiple lenses is equivalent to a two-component, single-lens system consisting of a replacement and an equivalent lens group. The optical glass for the replacement lens group is selected based on weight and the principle of material replacement in the 3D glass diagram, thus achieving an athermal and apochromatic design. Secondly, an athermal and apochromatic optical system with a focal length of 130 mm, an F-number of 2.0, a spectral range of 480 nm~800 nm, a field of view angle of 22°, and an operating temperature of −40 °C~+60 °C is designed. The modulation transfer function (MTF) at each field of view is greater than 0.6 at 50 lp/mm in the −40 °C~+60 °C temperature range, and the secondary spectrum aberration is 0.0056 mm, which is within the focal depth range of the optical system.

1. Introduction

With the advances in aerial optical remote sensing technology, the variety and functionality of aerial optical payloads have been enriched [1]. In particular, this is evident in the growing demand for aerial geographic surveying. In order to enhance the efficiency of geographic surveying work, higher demands are being placed on both the adaptability of optical payloads to different environmental illuminance levels in the survey area and on the mapping resolution. For example, the environmental illuminance levels vary from 10l ux at dawn and dusk to 10,000 lux at midday. To meet the high-resolution mapping requirements in areas with these varying levels, optical payloads need to possess characteristics such as a long focal length, large relative aperture, wide spectral range, chromatic aberration correction, and the suppression of secondary spectrum aberrations [2,3,4,5]. Additionally, due to the significant variations in flight altitude and the wide range of working environment temperatures, from −40 °C to +60 °C, aerial optical payloads require an athermal design to minimize heat effects.

Numerous studies have been conducted on the athermal design of visible-light optical systems [6,7,8,9,10,11,12,13,14,15,16,17,18,19], with the majority focusing on treating the optical system as a two-element, single-lens system. As a result, appropriate replacement lens materials are selected from two-dimensional glass maps to adjust the system parameters in order to achieve thermal insensitivity and correct chromatic aberrations. However, the secondary spectrum aberrations of the long-focus, large-relative aperture optical system have not been the subject of targeted corrections. In order to rectify the Petzval curvature aberration of a short-focus, wide-angle lens, Tae-Yeon Lim proposed the utilization of chromatic aberration coefficients, Petzval curvature coefficients, and thermal aberration coefficients to construct a three-dimensional glass map for the correction of chromatic aberrations, thermal aberrations, and the Petzval curvature of the lens. This proposed method is, however, primarily employed for the correction of Petzval curvature in short-focus, wide-angle lenses [20], and it is not applicable for the correction of secondary spectrum aberrations in telephoto optical systems. Zhiguang Ren proposed a novel design method for mid-telephoto optical systems based on graphic and analytical methods to simultaneously remedy chromatic aberrations, thermal aberrations, and a large secondary spectrum [1]. However, this method is not particularly applicable for optical systems with a long focal length, large relative aperture, and wide spectral range, where there are many types of optical glass. Therefore, the simultaneous correction of chromatic, secondary spectrum, and thermal aberrations in optical systems with a long focal length, large relative aperture, and wide spectral range poses an immensely challenging task. Additionally, achieving the design of apochromatic and athermal optical systems further adds to this complexity.

In this paper, a methodology for achieving an athermal and apochromatic design in the construction of optical systems with long focal lengths, large relative apertures, and wide spectral ranges is introduced. This methodology is based on equivalent two-element optical systems in a three-dimensional (3D) glass diagram and can be applied to optical systems composed of multiple materials and lenses. Subsequently, calculations are performed to determine the chromatic, secondary spectrum, and thermal power of the optical glass in the visible-light range, leading to the establishment of a 3D glass map for visible light applications. Finally, based on the parameters of the equivalent two-element optical system, suitable materials are selected from the 3D glass map to construct replacement lens groups. This approach effectively eliminates the chromatic, secondary spectrum, and thermal aberrations in order to achieve an athermal and apochromatic design for optical systems with a long focal length, large relative aperture, and wide spectral range.

2. Athermal and Apochromatic Theory

In practical applications, the Seidel aberration coefficient is commonly used to analyze the aberration characteristics of optical systems. However, it should be noted that in real optical systems, lenses have a certain thickness. Therefore, prior to conducting the analysis using the Seidel aberration coefficient, it is essential to convert the thick-lens optical system into an equivalent thin-lens optical system. Suppose an optical system consists of $k$ lenses. The refractive index of the i-th lens is $n_i$; the coefficient of the expansion of the optical glass is ${\alpha }_i$; the optical power is ${\phi }_i$; the chromatic power is ${\omega }_i$; the thermal power is ${\gamma }_i$; the secondary spectrum power is ${\rho }_i$; and the spectral range of the optical system is ${\lambda }_1~{\lambda }_2~{\lambda }_3$. Therefore, it is assumed that ${\lambda }_1<{\lambda }_2<{\lambda }_3$; the Abbe number of the optical lens material is ${\nu }_i=(n_{{\lambda }_2}-1)/(n_{{\lambda }_1}-n_{{\lambda }_3})$; $n_{{\lambda }_1},n_{{\lambda }_2},n_{{\lambda }_3}$ is the refractive index of the optical glass at the corresponding wavelength; $P_i$ is the partial dispersion coefficient of the optical glass; and $P_i=(n_{{\lambda }_2}-n_{{\lambda }_1})/(n_{{\lambda }_2}-n_{{\lambda }_3})$. The expressions for the chromatic power ${\omega }_i$, thermal power ${\gamma }_i$ [11,12], and secondary spectrum power ${\rho }_i$ are provided in the following:

$${\omega }_i={\displaystyle \frac{1}{{\nu }_i}}=-{\displaystyle \frac{\Delta {\phi }_i}{{\phi }_i}}=-{\displaystyle \frac{\Delta \lambda }{n_i-1}}·{\displaystyle \frac{\partial n_i}{\partial \lambda }}$$

(1)

$${\gamma }_i={\displaystyle \frac{\partial {\phi }_i}{\partial T}}{\displaystyle \frac{1}{{\phi }_i}}={\displaystyle \frac{1}{n_i-1}}{\displaystyle \frac{\partial n_i}{\partial T}}-{\alpha }_i$$

(2)

$${\rho }_i={\displaystyle \frac{P_i}{{\nu }_i}}$$

(3)

For the apochromatic and athermal design of optical systems, the following optical power, chromatic power, secondary spectrum power, and thermal power must be met:

$${\displaystyle \sum _{i=1}^k\frac{h_i}{h_1}}{\phi }_i={\phi }_T$$

(4)

$${{\displaystyle \sum _{i=1}^k\left(\frac{h_i}{h_1}\right)}}^2{\phi }_i{\omega }_i=0$$

(5)

$${{\displaystyle \sum _{i=1}^k\left(\frac{h_i}{h_1}\right)}}^2{\phi }_i{\rho }_i=0$$

(6)

$${\displaystyle \frac{\partial {\phi }_T}{\partial T}}\cong {\displaystyle \frac{1}{{\phi }_T^2}}{\displaystyle \sum _{i=1}^k{(\frac{h_i}{h_1})}^2{\gamma }_i{\phi }_i}=-{\alpha }_{h}L$$

(7)

where ${\phi }_T$ is the optical system’s optical power; $h_i$ is the incident height of the paraxial ray at the i-th lens; ${\alpha }_h$ is the expansion coefficient of the tube material; and $L$ is the tube length, and for the purpose of simplifying the calculation process, the optical power, thermal power, secondary spectrum power, and thermal power are normalized to obtain Equations (8)–(11), which are as follows:

$${\displaystyle \sum _{i=1}^k{\phi }_i^{\text{'}}}={\phi }_T$$

(8)

$${\displaystyle \sum _{i=1}^k{\omega }_i^{\text{'}}{\phi }_i^{\text{'}}}=0$$

(9)

$${\displaystyle \sum _{i=1}^k{\rho }_i^{\text{'}}{\phi }_i^{\text{'}}}=0$$

(10)

$${\displaystyle \sum _{i=1}^k{\gamma }_i^{\text{'}}{\phi }_i^{\text{'}}}=-{\alpha }_{h}L{\phi }_T^2$$

(11)

In the above, ${\phi }_i^{\text{'}}=h_i/h_1·{\phi }_i$, ${\omega }_i^{\text{'}}=h_i/h_1·{\omega }_i$, ${\rho }_i^{\text{'}}=h_i/h_1·{\rho }_i$, and ${\gamma }_i^{\text{'}}=h_i/h_1·{\gamma }_i$.

To ensure an apochromatic and athermal design for an optical system, the system is simplified into two elementary groups composed of arbitrary multiple optical elements. One of these elements is selected as the replacement group to substitute for the glass material, while the remaining lenses are considered as an equivalent single-lens group. The optical design process involves achieving both apochromatic and athermal conditions by replacing the glass material of the replacement group lens. If the replaced glass material is unsuitable, it will be further replaced until the optical system satisfies both the apochromatic and athermal requirements. The j-th single lens $L_j$ should be chosen as the replacement group when the number of lenses in the optical system is $k$. The optical power of lens $L_j$ is ${\phi }_j^{\text{'}}$, the chromatic power is ${\omega }_j^{\text{'}}$, the secondary spectrum power is ${\rho }_j^{\text{'}}$, and the thermal power is ${\gamma }_j^{\text{'}}$. The remaining $k-1$ lenses form an equivalent single-lens group $L_e$, whose optical power is ${\phi }_e^{\text{'}}$, while the chromatic power is ${\omega }_e^{\text{'}}$, the secondary spectrum power is ${\rho }_e^{\text{'}}$, and the thermal power is ${\gamma }_e^{\text{'}}$. The coefficients satisfy the following conditions:

$${\phi }_e^{\text{'}}={\displaystyle \sum _{m=1}^K{\phi }_m^{\text{'}}}-{\phi }_j^{\text{'}}$$

(12)

$${\omega }_e^{\text{'}}=\left({\displaystyle \sum _{m=1}^K{\omega }_m^{\text{'}}{\phi }_m^{\text{'}}-}{\omega }_j^{\text{'}}{\phi }_j^{\text{'}}\right)/{\phi }_e^{\text{'}}$$

(13)

$${\rho }_e^{\text{'}}=\left({\displaystyle \sum _{m=1}^K{\rho }_m^{\text{'}}{\phi }_m^{\text{'}}-}{\rho }_j^{\text{'}}{\phi }_j^{\text{'}}\right)/{\phi }_e^{\text{'}}$$

(14)

$${\gamma }_e^{\text{'}}=\left({\displaystyle \sum _{m=1}^K{\gamma }_m^{\text{'}}{\phi }_m^{\text{'}}-}{\gamma }_j^{\text{'}}{\phi }_j^{\text{'}}\right)/{\phi }_e^{\text{'}}$$

(15)

Therefore, the equivalent two-element optical system satisfies the following apochromatic and athermal conditions:

$${\phi }_j^{\text{'}}+{\phi }_e^{\text{'}}={\phi }_T$$

(16)

$${\omega }_j^{\text{'}}{\phi }_j^{\text{'}}+{\omega }_e^{\text{'}}{\phi }_e^{\text{'}}=0$$

(17)

$${\rho }_j^{\text{'}}{\phi }_j^{\text{'}}+{\rho }_e^{\text{'}}{\phi }_e^{\text{'}}=0$$

(18)

$${\gamma }_j^{\text{'}}{\phi }_j^{\text{'}}+{\gamma }_e^{\text{'}}{\phi }_e^{\text{'}}=-{\alpha }_{h}L{\phi }_T^2$$

(19)

In solving Equations (16) and (19), the optical power of the equivalent two-element optical system that satisfies both the apochromatic and athermal conditions is found to satisfy the following conditions:

$${\phi }_j^{\text{'}}=-\left({\gamma }_e^{\text{'}}+H_T\right){\phi }_T/\left({\gamma }_j^{\text{'}}-{\gamma }_e^{\text{'}}\right)$$

(20)

$${\phi }_e^{\text{'}}=\left({\gamma }_j^{\text{'}}+H_T\right){\phi }_T/\left({\gamma }_j^{\text{'}}-{\gamma }_e^{\text{'}}\right)$$

(21)

In the above equations, $H_T={\alpha }_{h}L{\phi }_T$. Substituting Equations (20) and (21) into Equation (17), we obtain the following:

$${\gamma }_j={\displaystyle \frac{{\gamma }_e^{"}+H_T^{\text{'}}}{{\omega }_e^{"}}}{\omega }_j-H_T^{\text{'}}$$

(22)

Substituting Equations (20) and (21) into Equation (18), we obtain the following:

$${\gamma }_j={\displaystyle \frac{{\gamma }_e^{"}+H_T^{\text{'}}}{{\rho }_e^{"}}}{\rho }_j-H_T^{\text{'}}$$

(23)

In the above equation, ${\gamma }_e^{"}=h_1/h_j·{\gamma }_e^{\text{'}}$, $w_e^{"}=h_1/h_j·w_e^{\text{'}}$, ${\rho }_e^{"}=h_1/h_j·{\rho }_e^{\text{'}}$, and $H_T^{\text{'}}=h_1/h_j·H_T$.

Based on Equations (22) and (23), Equation (24) can be constructed with the independent variable being the chromatic power $\omega$ and the dependent variable being the thermal power $\gamma$, and Equation (25) can be constructed with the independent variable being the secondary spectrum power $\rho$ and the dependent variable being the thermal power $\gamma$. A schematic diagram of a two-element optical system satisfying Equations (24) and (25) is shown in Figure 1.

$$\gamma ={\displaystyle \frac{{\gamma }_e^{"}+H_T^{\text{'}}}{{\omega }_e^{"}}}\omega -H_T^{\text{'}}$$

(24)

$$\gamma ={\displaystyle \frac{{\gamma }_e^{"}+H_T^{\text{'}}}{{\rho }_e^{"}}}\rho -H_T^{\text{'}}$$

(25)

As shown in Figure 1, when the two-element optical system satisfies the apochromatic and athermal conditions, the parameters ${\rho }_j$, ${\omega }_j$, and ${\gamma }_j$ of the replacement group in the optical system satisfy Equations (24) and (25). When Figure 1 is transformed into a 3D graph, it shows the line $L_e-L_j$ passing through points $L_e({\rho }_e^{"},{\omega }_e^{"},{\gamma }_e^{"})$, $L_j({\rho }_j,{\omega }_j,{\gamma }_j)$, and $H(0,0,-H_T^{\text{'}})$, as shown in Figure 2.

The 3D equation for the line $L_e-L_j$ passing through points $L_e({\rho }_e^{"},{\omega }_e^{"},{\gamma }_e^{"})$ and $L_j({\rho }_j,{\omega }_j,{\gamma }_j)$ is provided in Equation (26), with the line passing through point $H(0,0,-H_T^{\text{'}})$:

$${\displaystyle \frac{\rho -{\rho }_j}{{\rho }_e^{"}-{\rho }_j}}={\displaystyle \frac{\omega -{\omega }_j}{{\omega }_e^{"}-{\omega }_j}}={\displaystyle \frac{\gamma -{\gamma }_j}{{\gamma }_e^{"}-{\gamma }_j}}$$

(26)

For an apochromatic and athermalized optical system, the tube material $H_{tude}({\rho }_{tude},{\omega }_{tude},{\gamma }_{tude})$ is on the straight line $L_e-L_j$. When point $H_{tude}$ is entered into the straight line $L_e-L_j$, the equation of the straight line, as shown in Equation (27), can be calculated, and point $H_{tude}$, as shown in Equation (28), can also be calculated.

$${\displaystyle \frac{{\rho }_{tude}-{\rho }_j}{{\rho }_e^{"}-{\rho }_j}}={\displaystyle \frac{{\omega }_{tude}-{\omega }_j}{{\omega }_e^{"}-{\omega }_j}}={\displaystyle \frac{{\gamma }_{tude}-{\gamma }_j}{{\gamma }_e^{\text{'}}-{\gamma }_j}}$$

(27)

$$\begin{array}{l}{H}_{tude}({\rho }_{tude},{\omega }_{tude},{\gamma }_{tude})\\ =\left\{{\displaystyle \frac{{\rho }_j({\gamma }_{tude}-{\gamma }_j)-{\rho }_e^{"}(H_T^{\text{'}}+{\gamma }_j)}{{\gamma }_e^{"}-{\gamma }_j}},{\displaystyle \frac{{\omega }_j({\gamma }_{tude}-{\gamma }_j)-{\omega }_e^{"}(H_T^{\text{'}}+{\gamma }_j)}{{\gamma }_e^{"}-{\gamma }_j}},-H_T^{\text{'}}\right\}\end{array}$$

(28)

For the initial optical system structure with the selected tube material, the tube aberration coefficient $H(0,0,-H_T^{\text{'}})$ and the equivalent single-lens group $L_e({\rho }_e^{"},{\omega }_e^{"},{\gamma }_e^{"})$ form a straight line $H-L_e$, as shown in Equation (29). In general, the replacement lens group $L_j({\rho }_j,{\omega }_j,{\gamma }_j)$ is not on the line $H-L_e$. To eliminate chromatic, secondary spectrum, and thermal aberrations in an optical system simultaneously, assuming that the three aberration correction points $L_c({\rho }_c,{\omega }_c,{\gamma }_c)$ corresponding to the j-th lens are located on the straight line $H-L_e$, as shown in Equation (30), the values of ${\omega }_c$ that satisfy the chromatic aberration equation can be calculated using Equation (17). The values of ${\rho }_c$, ${\gamma }_c$ are calculated and shown as Equations (31) and (32). Therefore, when replacing the lens material $L_j$ in the 3D glass diagram with $L_i$, the lens material $L_i$ is selected so that the distance from $L_i$ to the aberration correction point $L_c$ and the distance from $L_i$ to the straight line $H-L_e$ are minimized. This allows $L_j$ to be as close as possible to the correction point $L_c$ and the straight line $H-L_e$, thereby achieving the simultaneous elimination of chromatic, secondary spectrum, and thermal aberrations through the optimization of the optical system.

$${\displaystyle \frac{\rho }{{\rho }_e^{"}}}={\displaystyle \frac{\omega }{{\omega }_e^{"}}}={\displaystyle \frac{\gamma +H_T^{\text{'}}}{{\gamma }_e^{"}+H_T^{\text{'}}}}$$

(29)

$${\omega }_c=-{\displaystyle \frac{{\omega }_e^{"}{\phi }_e^{\text{'}}}{{\phi }_j^{\text{'}}}}$$

(30)

$${\rho }_c={\displaystyle \frac{{\omega }_c}{{\omega }_e^{"}}}{\rho }_e^{"}$$

(31)

$${\gamma }_c={\displaystyle \frac{{\omega }_c}{{\omega }_e^{"}}}\left({\gamma }_e^{"}+H_T^{\text{'}}\right)-H_T^{\text{'}}$$

(32)

3. Athermal and Apochromatic Design Example

3.1. Structure and Parameters of Initial Optical System

To demonstrate the practicality of the aforementioned design approach, an optical lens with a focal length of 130 mm, an F-number of 2, a wavelength range spanning from 480 nm to 800 nm, and an operational temperature range between −40 °C and +60 °C was constructed. The optical lens parameters are presented in

Table 1. The design parameters of the optical system.

Effective Focal Length	130 mm
F-number	2
Field of view (FoV)	22°
Spectral range	480 nm–650 nm–800 nm
Operational temperature	−40 °C~+60 °C
Distortion	<2%
Depth of focus	±0.0052 mm
Housing material	$\mathrm{TC}4(\alpha =9.1\times {10}^{-6}/℃$)

; the optical system exhibits notable attributes, such as a large relative aperture, long focal length, wide wavelength range, and a large field of view. Figure 3. illustrates a schematic diagram of the optical system, which comprises ten lenses, including two double-convex lens groups. Specifically, lenses 1 and 2 possess aspheric surfaces of even order on their front side, whereas lens 3 features an aspheric surface on its rear side. The optical design of the system in question was created using the CODE V optical design software, which was also employed for the simulation of the optical system.

Table 2. The parameters of the lenses in the initial optical system.

Element	Material	$\mathit{\omega }(\times {10}^{-2})$	$\mathit{\rho }(\times {10}^{-2})$	$\mathit{\gamma }(\times {10}^{-6})$	$\mathit{\phi }(\mathit{m}{\mathit{m}}^{-1})$	$\mathit{h}(\mathit{m}\mathit{m})$
1	H-ZLAF90	3.93	1.95	−4.70	0.0123	32.50
2	H-QK1	1.50	0.92	−8.15	−0.0113	25.78
3	H-ZF4A	3.53	1.79	−8.75	−0.0167	23.38
4	H-ZK20	1.85	1.04	−3.59	0.0226	28.91
5	TF3	2.27	1.28	5.42	−0.0158	27.28
6	H-LAK54	1.94	1.12	2.53	0.0133	27.66
7	H-ZF6	3.63	1.83	−7.64	−0.0153	25.78
8	H-LAK53A	1.91	1.10	−1.46	0.0270	26.35
9	H-ZF4A	3.53	1.79	−8.75	−0.0261	21.53
10	H-ZF88	5.57	2.63	−4.20	0.0103	18.13

summarizes the optical properties of the lenses, including the glass material, chromatic power $\omega$, secondary spectrum power $\rho$, thermal power $\gamma$, optical power $\phi$, and paraxial ray height $h$.

The initial optical system structure exhibits a modulation transfer function (MTF) curve at 50 lp/mm, as illustrated in Figure 4 and Figure 5. The center field of view transfer function is measured to be 0.6, while the average transfer functions in the meridian and sagittal directions at the edge of the field of view are found to be 0.37 under a working temperature of +20 °C. However, when operating at −40 °C or +60 °C, all fields of view experience an average transfer function close to zero, indicating the optical system’s significant susceptibility to ambient temperature variations, which necessitates an athermal design.

The longitudinal spherical aberration curve of the initial optical system is depicted in Figure 6. The axial chromatic aberration $l_{FS}^{\text{'}}$ between the 800 nm and 400 nm spectral bands of the optical system is considerable, with a minimum value of 0.1 mm at 0.85x aperture. However, it should be noted that the axial chromatic aberration value significantly exceeds the system depth of focus.

3.2. Three-Dimensional Glass Map Construction

Correcting the chromatic, secondary spectrum, and thermal aberrations is relatively complex for optical systems with a large number of lenses, such as the one described in this paper, which has 10 optical lenses, because these systems have a wide spectral range and require a large relative aperture and long focal length. This often requires the designer to have extensive, relevant experience. In this paper, to quantitatively address the correction of the chromatic, secondary spectrum, and thermal aberrations in optical system design, the properties of the chromatic, secondary spectrum, and thermal powers of visible-light glass materials are leveraged to establish a 3D glass map. Secondly, based on Equations (12)–(15), the multi-lens optical system is the equivalent of a two-element optical system. For this system, where one element consists of a single lens which becomes the replacement lens group and the remaining lenses form an equivalent lens group, the equivalent two-element lens group and the optical system tube material parameters can be effectively quantified in the 3D glass map. Ultimately, through the selection of the optical glass for the replacement lens group $L_j$, the optical system can be designed to meet the conditions of achromatic, secondary spectrum, and thermal aberration suppression listed in Equations (4)–(7), thereby achieving the goal of an apochromatic and athermal design.

In this paper, the CDGM glass catalog is used. After removing some optical glass with the same properties but different brand names, the optical glass list is that shown in

Table 3. The CDGM glass catalog for apochromatic and athermal design.

Serial Number	Brand of Glass	Serial Number	Brand of Glass	Serial Number	Brand of Glass	Serial Number	Brand of Glass	Serial Number	Brand of Glass	Serial Number	Brand of Glass	Serial Number	Brand of Glass
1	H-FK61	31	H-BaK7	61	H-LaK8A	91	F5	121	H-ZF4A	151	H-ZF71	181	H-ZLaF50D
2	H-FK71	32	H-BaK8	62	H-LaK10	92	F6	122	H-ZF4AGT	152	H-ZF71GT	182	H-ZLaF51
3	H-FK95N	33	H-ZK1	63	H-LaK11	93	F7	123	ZF4	153	H-ZF72A	183	H-ZLaF52A
4	H-QK1	34	H-ZK2	64	H-LaK12	94	H-F13	124	H-ZF5	154	H-ZF72AGT	184	H-ZLaF52
5	H-QK3L	35	H-ZK3	65	H-LaK50A	95	F13	125	H-ZF6	155	H-ZF73	185	H-ZLaF53B
6	H-K1	36	H-ZK3A	66	H-LaK51A	96	H-F51	126	ZF6	156	H-ZF73GT	186	H-ZLaF53BGT
7	H-K2	37	H-ZK4	67	H-LaK52	97	H-BaF2	127	H-ZF7LA	157	H-ZF88	187	H-ZLaF55D
8	H-K3	38	H-ZK5	68	H-LaK53B	98	H-BaF3	128	H-ZF7LAGT	158	H-ZF88GT	188	H-ZLaF55C
9	K4A	39	H-ZK6	69	H-LaK53A	99	H-BaF4	129	ZF7	159	H-LaF1	189	H-ZLaF56B
10	H-K5	40	H-ZK7A	70	H-LaK54	100	H-BaF5	130	ZF7L	160	H-LaF2	190	H-ZLaF66
11	H-K6	41	H-ZK7	71	H-LaK59A	101	H-BaF6	131	ZF7LGT	161	H-LaF3B	191	H-ZLaF66GT
12	H-K7	42	H-ZK8	72	H-LaK61	102	H-BaF7	132	H-ZF10	162	H-LaF4	192	H-ZLaF68C
13	H-K8	43	H-ZK9B	73	H-KF6	103	H-BaF8	133	ZF10	163	H-LaF4GT	193	H-ZLaF68N
14	H-K9L	44	H-ZK9A	74	H-QF1	104	H-ZBaF1	134	H-ZF11	164	H-LaFL5	194	H-ZLaF68B
15	H-K10	45	H-ZK10	75	H-QF3	105	H-ZBaF3	135	H-ZF12	165	H-LaF6LA	195	H-ZLaF69
16	H-K11	46	H-ZK10L	76	H-QF6A	106	H-ZBaF4	136	ZF12	166	H-LaF7	196	H-ZLaF69A
17	H-K12	47	H-ZK11	77	H-QF8	107	H-ZBaF5	137	H-ZF13	167	H-LaF10LA	197	H-ZLaF71
18	H-K50	48	H-ZK14	78	H-QF14	108	H-ZBaF16	138	H-ZF13GT	168	H-LaF50B	198	H-ZLaF71AGT
19	H-K51	49	H-ZK20	79	H-QF50	109	ZBaF17	139	ZF13	169	H-LaF51	199	H-ZLaF73
20	H-ZPK1A	50	H-ZK21	80	H-QF50A	110	H-ZBaF20	140	H-ZF39	170	H-LaF52	200	H-ZLaF75A
21	H-ZPK2A	51	H-ZK50	81	QF50	111	H-ZBaF21	141	H-ZF50	171	H-LaF53	201	H-ZLaF75B
22	H-ZPK3	52	H-ZK50GT	82	H-QF56	112	H-ZBaF50	142	ZF50	172	H-LaF54	202	H-ZLaF76
23	H-ZPK5	53	H-LaK1	83	H-F1	113	ZBaF51	143	ZF51	173	H-LaF55	203	H-ZLaF76A
24	H-ZPK7	54	H-LaK2A	84	H-F2	114	H-ZBaF52	144	H-ZF52	174	H-LaF62	204	H-ZLaF78B
25	H-BaK1	55	H-LaK3	85	F2	115	H-ZF1	145	H-ZF52GT	175	H-ZLaF1	205	H-ZLaF89L
26	H-BaK2	56	H-LaK4L	86	H-F3	116	H-ZF1A	146	H-ZF52TT	176	H-ZLaF2A	206	H-ZLaF90
27	H-BaK3	57	H-LaK5A	87	F3	117	H-ZF2	147	H-ZF52A	177	H-ZLaF3	207	H-ZLaF92
28	H-BaK4	58	H-LaK6A	88	H-F4	118	ZF2	148	ZF52	178	H-ZLaF4LA	208	H-TF3L
29	H-BaK5	59	H-LaK7A	89	F4	119	H-ZF3	149	H-ZF62	179	H-ZLaF4LB	209	TF3
30	H-BaK6	60	H-LaK8B	90	H-F5	120	ZF3	150	H-ZF62GT	180	H-ZLaF50E	210	H-TF5

. The chromatic, secondary spectrum, and thermal power of each glass are calculated separately, and a 3D glass map is constructed, as shown in Figure 7, with the red points representing K-series optical glass (K-glass) and the blue points representing F-series optical glass (F-glass).

3.3. The Options for the Replacement Lens $L_j$ and Material Selection

In order to improve the correction of the chromatic, secondary spectrum, and thermal aberrations of the optical system and to more efficiently select suitable replacement lens groups $L_j$ from the multi-lens optical system while considering the aberration characteristics of the optical system, the selection of the replacement lens group $L_j$ was based on the following principles:

(1)

From Figure 7, it can be observed that the thermal power $\gamma$ distribution range of the K-series optical glass is greater than that of the F-series optical glass. When using K-series optical glass for material replacement, there is a more significant potential for thermal correction.

(2)

Equations (4)–(7) indicate that the optical power of each lens in the optical system has a significant influence on the chromatic, secondary spectrum, and thermal aberrations. If a lens has a larger optical power, then it makes a greater contribution to these aberrations. Therefore, replacing the material of such lenses presents a greater potential for correcting these three types of aberration.

(3)

In an optical system with these three types of uncorrected aberration, the larger the volume of the polyhedra formed by the optical materials in the 3D glass diagram, the smaller the optical focus of the corresponding lens for each material. This makes it easier to correct these aberrations in the optical system. At the same time, the smaller the higher-order aberration corresponding to each lens is. Consequently, in a two-component optical system, the greater the distance from the replacement lens group $L_j$ to the straight line $H-L_e$, the greater the potential for selecting that lens for material replacement to correct the three types of aberration.

(4)

For a system with uncorrected aberrations, based on Equations (30)–(32), the correction point $L_c$, which simultaneously eliminates three types of aberrations, can be calculated for the different replacement lens group $L_j$. It is observed that during the three types of aberration correction process, the greater the distance between the replacement lens group $L_j$ and the aberration correction point $L_c$, the greater the range of variation in the chromatic power $\omega$, secondary spectrum power $\rho$, and thermal power $\gamma$ of the replacement lens group $L_j$ and the equivalent lens group $L_e$ in the three-dimensional glass map. Furthermore, in relation to these correction processes, it is also observed that the greater the range of variation in the focal power of the various lenses within the optical system and the optical materials, the greater the potential of the replacement lens group.

In accordance with the selection principles of the replacement lens group $L_j$, in this paper, the selection weight $W_j$ for the replacement lens group $L_j$ is defined as follows:

$$W_j=W_g\left(d_{ju}+abs({\phi }_{ju})+d_{jcu}\right)$$

(33)

In the above equation, $d_{ju}$ is the normalized distance from each lens in the optical system to the straight line $H-L_e$; ${\phi }_{ju}$ is the normalized optical power of each lens in the optical system; and $d_{jcu}$ is the normalized distance from each lens in the optical system to the corresponding three aberration correction points $L_c$. $W_g$ is the weight of the optical glass, and $W_K$ is the normalized thermal power of the K-series of optical glass in the 3D glass map; meanwhile, $W_F$ is the normalized thermal power of the F-series of optical glass in the 3D glass map; ${\gamma }_K$ is the distribution width of the thermal power of the K-series of optical glass in the 3D glass map; and ${\gamma }_F$ is the distribution width of the thermal power of the F-series of optical glass in the 3D glass map. The expressions for each power are as follows:

$${\gamma }_K=\max ({\gamma }_{mK})-\min ({\gamma }_{mK})$$

(34)

$${\gamma }_F=\max ({\gamma }_{nF})-\min ({\gamma }_{nF})$$

(35)

$$W_K={\gamma }_K/({\gamma }_K+{\gamma }_F)$$

(36)

$$W_K={\gamma }_K/({\gamma }_K+{\gamma }_F)$$

(37)

$mK\in \left[1,72\right]$ represents the serial number of the K-series optical glass, and $nF\in \left[73,210\right]$ represents the serial number of the F-series optical glass. If the lens material is from the K-series optical glass, then the optical glass weight $W_g$ equals $W_K$. Similarly, if the lens material is from the F-series optical glass, then the optical glass weight $W_g$ equals $W_F$.

After selecting the replacement lens group $L_j$, the most suitable optical glass $L_i$ is chosen in the 3D glass map to replace the material of $L_j$, thereby correcting the chromatic, secondary spectrum, and thermal aberrations of the optical system. As shown in Figure 8, the distance from the optical glass $L_i$ to the straight line $H-L_e$ in the 3D glass map is defined as $d_i$, and the distance from the optical glass $L_i$ to the correction point $L_c$ is defined as $d_{ic}$. The optical glass selection of the replacement lens group $L_j$ observes the following principle. For each optical glass $L_i$ in the 3D glass map, a smaller $d_i+d_{ic}$ indicates easier correction of the chromatic, secondary spectrum, and thermal aberrations. Therefore, when selecting replacement materials, it is important to prioritize the materials that are closer to both the straight line $H-L_e$ and the correction point $L_c$. However, due to the optical system being composed of multiple lens elements and various materials, the material with the smallest $d_i+d_{ic}$ value may not be conducive to correcting other aberrations such as spherical aberrations and comas. Therefore, this paper selects the four materials with the smallest $d_i+d_{ic}$ values and replaces the material of lens $L_j$ with each of them to compare their respective aberration correction effects at different temperatures. This process aims to select an appropriate optical glass for replacing $L_j$.

3.4. Optimization of Optical Systems

For the initial optical system, the weighting parameters $W_j$ for selecting the replacement lens group $L_j$, shown in

Table 4. The selection weights for each lens in the initial optical system for the replacement lens group $L_j$.

Lens Serial Number	1	2	3	4	5	6	7	8	9	10
$W_j$	0.54	0.61	0.40	1.32	0.32	0.91	0.37	0.98	0.48	0.69

, are calculated using Equation (33). Among them, the maximum weight for lens 4 is 1.32, indicating that lens 4 is selected as the replacement lens group $L_j$. The 3D glass map is illustrated in Figure 9. At this point, there are four potential materials that could replace this lens group, as listed in

Table 5. The optical glasses available for selection for replacing lens 4 in the initial optical system.

Serial Number	1	2	3	4
Available glass	H-ZPK5	H-FK61	H-FK71	H-FK95N

. Each of these materials is used to replace lens 4 and optimize the optical system. In Figure 10, the MTF is illustrated when the optical material H-ZPK5 is selected for lens 4. The selection of the optical material H-ZPK5 for lens 4 resulted in the highest MTF of the optical systems across all temperatures. The temperature span at which the MTF of each field of view at the spatial frequency of 50 lp/mm exceeded 0.2 is approximately 25 °C, and the three types of aberration correction exhibited greater efficacy.

After replacing lens 4 with H-ZPK5, the selection weight $W_j$ of the replacement lens group $L_j$ for the optical system is calculated, presented in

Table 6. The weighted selection of each lens as a replacement lens group $L_j$ after replacing the material of lens 4.

Lens Serial Number	1	2	3	4	5	6	7	8	9	10
$W_j$	0.55	0.48	0.32	0.50	0.41	0.77	0.30	0.79	0.42	0.52

. The maximum weight is found to be 0.79 for lens 8; thus, it is chosen as the replacement lens group $L_j$. The 3D glass diagram is illustrated in Figure 11. At this point, four types of materials are available for replacing this lens group, as detailed in

Table 7. The optical glasses available for selection for replacing lens 8 in the optical system.

Serial Number	1	2	3	4
Available glass	H-ZPK3	H-QK3L	H-K11	H-ZPK2A

. Each of these four optical glasses is used to replace lens 8 and optimize the optical system. As shown in Figure 12, the selection of the optical material H-ZPK2A for lens 8 results in the highest transfer function of the optical system at all temperatures. Furthermore, the transfer function of each field of view at the spatial frequency of 50 lp/mm is greater than 0.2 within the temperature range of −40 °C to +60 °C. In particular, the three types of aberration correction are evident.

After replacing the optical glass of lens 8, the selection weight $W_j$ of the replacement lens group $L_j$ is calculated for the optical system, presented in

Table 8. The weighted selection of each lens for replacement lens group $L_j$ after replacing the material of lens 8.

Lens Serial Number	1	2	3	4	5	6	7	8	9	10
$W_j$	0.24	1.43	0.38	0.57	0.37	0.55	0.33	0.75	0.45	0.38

. At this time, lens 2 has the highest weight at 1.43; thus, it is chosen as the replacement lens group $L_j$. The 3D glass diagram is depicted in Figure 13. There are four types of material available for replacing this lens group, as detailed in

Table 9. The optical glasses available for selection for replacing lens 2 in the optical system.

Serial Number	1	2	3	4
Available glass	H-ZPK5	H-FK61	H-FK71	H-FK95N

. Each of these four optical glasses is used to replace lens 2 and optimize the optical system.

When using the H-FK61 optical glass for lens 2, as shown in Figure 14 and Figure 15, a higher MTF at various temperatures within a working temperature range from −40 °C to +60 °C is the result, and all field-of-view MTF values are greater than 0.6, indicating good imaging quality. The optical system simultaneously corrects the axial chromatic aberration at 480 nm and 800 nm wavelengths, as depicted in Figure 16. This results in a secondary spectrum aberration value (SSAV) of 0.0056 mm, which is smaller than the depth of focus of the optical system, thereby enabling high-quality imaging. The configuration of the optical system within the 3D glass diagram when lens 8 is designated as the replacement lens group is illustrated in Figure 17. This configuration requires the replacement lens group $L_j$, the equivalent lens group $L_e$, the lens tube $H$, and the aberration correction point $L_c$ to be approximately coextensive. At this juncture, the optical system is in alignment with the stipulations of Equations (4)–(7), thereby enabling the simultaneous rectification of the chromatic, secondary spectrum, and thermal aberrations. The parameters of each lens element within the optical system are listed in

Table 10. The parameters of lenses in the optimal optical system.

Element	Material	$\mathit{\omega }(\times {10}^{-2})$	$\mathit{\rho }(\times {10}^{-2})$	$\mathit{\gamma }(\times {10}^{-6})$	$\mathit{\phi }(\mathit{m}{\mathit{m}}^{-1})$	$\mathit{h}(\mathit{m}\mathit{m})$
1	H-ZLAF90	3.93	19.52	−4.70	0.0079	32.50
2	H-FK61	1.23	7.18	−25.78	−0.0117	24.99
3	H-ZF4A	3.53	17.87	−8.75	−0.0154	26.86
4	H-ZPK5	1.46	8.34	−24.21	0.0149	32.30
5	TF3	2.27	12.81	0.54	−0.0051	32.30
6	H-LAK54	1.94	11.15	2.53	0.0113	34.09
7	H-ZF6	3.63	18.28	−7.64	−0.0106	31.52
8	H-ZPK2A	1.53	8.98	−15.77	0.0198	32.39
9	H-ZF4A	3.53	17.87	−8.75	−0.0225	27.72
10	H-ZF88	5.57	26.28	−4.20	0.0103	30.23

.

4. Conclusions

In this paper, we have proposed a method for the apochromatic and athermal design of long-focal length, large-relative aperture, wide-spectral range optical systems. Firstly, this method simplifies a complex optical system composed of multiple lenses into an equivalent two-element, single-lens system and displays it in a 3D glass diagram based on the principles of chromatic aberration correction, secondary spectrum aberration suppression, and thermal aberration elimination. Subsequently, via the selection of replacement lens groups according to their weight and applying the principle of replacing lens materials, the most effective lenses for correcting chromatic, secondary spectrum, and thermal aberrations within the optical system are identified, and their optical glasses are replaced. This process enables an apochromatic and athermal design for long-focal length, large-relative aperture, wide-spectral range optical systems to be achieved. After replacing the types of optical glasses in the 3D glass diagram, this proposed method also extends to enabling an athermal design for optical systems operating within infrared or ultraviolet spectral ranges.