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Abstract
This report proposes an athermalization and achromatization method based on combined glasses and comprehensive distance weight to select and replace optical and housing tube materials quantitatively without multiple iterations. In addition, it presents a new achromatic and athermal condition of the replacement search method using combined glasses. It establishes an athermal glass map model combining the cluster center, tube materials, two combined lenses, and a rest equivalent lens to analyze the characteristics of the glass distribution. A cluster analysis method was introduced to analyze the distribution characteristics of the athermal glass map in the visible catalog. The athermal ability of the housing tube and the replacement of combined glass material are evaluated by distance weight in athermal glass map. A complex aerial multiple lenses system was designed using this method and maintained high imaging quality from –40 °C to 70 °C. This method can reduce the number of iterations for the selection of combined glass and significantly improves the optimization efficiency of athermalization.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The ambient temperature of the optical system installation and testing in the laboratory is generally approximately 20 °C. For cameras on satellites and airplanes, temperature variations under different working conditions are considerable [1,2]. The refractive index of optical glass is very sensitive to changes in temperature. The optical element expands or contracts with temperature changes, affecting the curvature radius and thickness of the optical element. The mechanical structure of the housing tube can also expand or contract with temperature changes, altering the optical element interval [3,4]. These temperature-dependent optical parameters can seriously degrade the image quality of optical systems. Therefore, it is necessary to match and optimize the thermal expansion coefficients and refractive index temperature coefficients of optical and mechanical materials to realize athermal design across a large environmental temperature range.
Athermalization approaches can be mainly divided into passive mechanical and optical methods. Passive athermal optical design utilizes thermal optical performance differences between several optical materials to eliminate the effects of temperature changes and realize athermal design. This method has the advantages of low weight, low cost, high reliability, and good environmental adaptability, without additional motion adjustment mechanisms, and is preferred for the athermalization of aeronautical and spaceborne optical systems [5–7]. For high target and complex optical system, it is always difficult to directly construct the initial structure of athermalization and achromatization of multiple lenses, including the glass material, the housing tube material and the optical power of each lens. For optical designers, they usually design the optical system under the conventional adjustment environment temperature at first, and strive to obtain better paraxial and non-paraxial imaging quality. Then the designer continuously optimizes the system by replacing glass and housing tube materials to achieve athermalization. Therefore, it is very important to select and replace glass materials efficiently and quantitatively for complex transmission optical systems with requirements of athermalization [8].
Deformable mirrors and diffractive optical elements have been used to reduce the defocus in the infrared waveband; however, these components are still not widely used in the field of visible light because of the complexity of machining and assembly [6–10]. An athermal chart has been developed in recent years. The early athermal glass chart method was limited to infrared systems and had difficulty determining the athermal conditions of the complicated visible light glass catalog. An athermalization and achromatism condition for three thin lenses of infrared system [11–13]. Stepan E. Ivanov indicated that the desired chromatic and thermo-optical characteristics can be synthesized using two-lenses components from two different optical materials [14]. However, the two or three-lenses components method is used in simple initial system construction and is not suitable for athermal optimization by glass substitution. A new graphical method was developed to obtain material combinations by redistributing the powers of the elements and selecting the material suitable for housing using an athermal glass map [15,16]. For the 1+∑ method, the initial system is equivalent to a double-lens system that consists of a single lens and an equivalent lens to enable the choice of optical materials. The chromatic weight and thermal weight were introduced to perform calculations and comparisons of the lenses through multiple iterations [17]. Unfortunately, this method could not realize the quantitative selection and replacement of tube materials. Glass material selection still requires multiple iterations. Sometimes, if the image quality of initial system is bad, the best glass replacement results cannot be obtained even after multiple iterations.
To address these issues, a glass replacement method for equivalent combined lenses to achieve passive athermalization in the visible band was developed in this study. More equivalent glass materials could be obtained by combining glasses to obtain new thermal and chromatic coefficients that do not actually exist in the optical glass catalog. Further, a new achromatic and athermal condition of the replacement search method using combined glasses was derived and an athermal glass map model combining the cluster center, tube materials, two combined lenses, and a rest equivalent lens was established. A cluster analysis method was introduced to analyze the distribution characteristics of the athermal glass map in the visible catalog. In addition, a quantitative selection and replacement method for housing tubes and glass materials was developed based on the comprehensive distance weight from an athermal glass map. As an example, a visible optical system was designed and optimized using our method to realize athermalization and achromatization from –40 °C to 70 °C. The results show that the proposed method can make the glass combination achieve the conditions of athermalization and achromatization, greatly reduce the number of iterations for the selection of combined glass, and significantly improve the optimization efficiency of athermalization.
2. Athermal and achromatic principles based on the replacement search method of combined glass
2.1 General athermal process based on quantitative glass replacement method
Before the athermalization design, the image quality and thermal adaptability were measured at different temperatures. If the image quality does not meet the requirement of an optical system over a wide temperature range, then athermalization and optimization must be performed. Figure 1 depicts the athermal process based on the quantitative glass replacement method. The specific steps are as follows, and the details of each step will be described later in our paper.
- (1) Achromatic and athermal conditions of the replacement search method with combined glass, as shown in Section 2.2.
- (2) Build the athermal distribution map of the glass catalog in a specific visible waveband. Determine the cluster center of the athermal glass map by clustering distribution algorithm, combining the cluster center, tube materials H, equivalent lens of the two combined lenses Lij, and equivalent lens of the remaining lenses Le, as shown in Section 3.2.
- (3) Evaluate the thermal characteristics of the housing and glass material, complete the material selection, and determine the best tube material quantitatively, as shown in Section 3.3.
- (4) Evaluate the thermal and achromatic properties of multiple visible light glass combinations and screen out the best pair of glass combinations. Screen out the visible glass and replace it quantitatively based on the equivalent glass distance weighting method, as shown in Section 3.4.
- (5) Redistribute the optical power of the optical elements and evaluate the image quality and temperature adaptability of the optical system to obtain excellent athermalization performance, as shown in Section 3.5.
2.2 Achromatic and athermal conditions
In an optical system, for the ith thin lens with power ${\phi _i}$, the chromatic power ${\omega _i}$ and thermal power ${\gamma _i}$ can be expressed as follows [11,12]:
2.3 Achromatic and athermal conditions of the replacement search method with combined glass
In the previous athermal design optimization process, the optical designer usually only replaced the glass material of a single lens and determined whether it satisfied the above conditions of achromatism and athermalization. If not, then it was necessary to perform multiple replacement rounds of a single lens to satisfy the achromatic and athermal conditions. However, owing to the discreteness of the chromatic power and thermal power on the glass map, only approximate solutions could be obtained, and it was difficult to achieve linear matching on the athermal map directly. Because of the complexity of the multi-round iterative replacement method and the lack of quantitative constraints for glass replacement, it was sometimes necessary to intervene in the type of glass replacement based on manual experience.
Therefore, a replacement method for the combined glass is proposed. The original single lens is changed into an equivalent glass based on a combination of two lenses. The optical characteristics of the composite lens glass can be calculated using the corresponding parameters of the two actual glass materials. More glass materials with different optical properties can be obtained by combining glass to obtain new thermal and chromatic coefficients that do not exist in the optical glass catalog. The selection variety of equivalent glass material and probability of near-perfect should match to achromatic and athermal conditions.
Because the existing achromatic and athermal theories are only applicable to single lens replacement, derivations of the achromatic and athermal constraints for the replacement of double-lenses combined glass are presented in this report. It is assumed that the optical system consists of K lenses (K > 3). The glasses of the two lenses that need to be replaced are called Li and Lj, the chromatic powers of the two lenses are called ωi and ωj, and the thermal powers of the two lenses are called γi and γj. When the composite glass of the two lenses is equivalent to a single lens called Lij, the optical power ϕij′, chromatic power ωij, and thermal power γij of the equivalent lens of two combined lenses are obtained as follows:
The remaining K–2 lenses can be equivalent to another single lens, Le. The optical power ϕe′, chromatic power ωe, and thermal power γe of the remaining equivalent lenses can be obtained as follows:
Then, the multilens optical system can be reconstructed into a lens Lij equivalent to the two replaced lenses and a lens Le equivalent to the remaining K–2 lenses. The achromatic and athermal conditions of the two equivalent lens systems can be derived as follows:
The optical powers of the two equivalent lenses can be obtained using Eqs. (9) and (10):
By substituting Eqs. (12) and (13) into Eq. (11), the achromatic and athermal conditions of the two equivalent lens systems can be rewritten as follows:
The athermal glass map of the two equivalent lens systems was plotted using Eq. (15), as shown in Fig. 2. The abscissa is the chromatic power, and the ordinate is the thermal power. This thermal–chromatic chart includes the equivalent lens of the two combined lenses Lij(ωij,γij), single equivalent lens of the remaining K–2 lenses ${L_e}({\omega _e},{\gamma _e})$, and housing $H(0, - {\alpha _h})$. Lij (ωij,γij) is the combination of the two replaced lenses, Li (ωi,γi) and Lj (ωj,γj). If Lij (ωij,γij), then ${L_e}({\omega _e},{\gamma _e})$ and $H(0, - {\alpha _h})$ satisfy the collinear condition, and the chromatic and thermal aberrations of the optical system can be corrected effectively.
3. Athermalization design example based on glass replacement method of equivalent combined lenses
3.1 Optical specifications and initial design
Figure 3 depicts an initial aerial optical system, including seven lenses, as an athermal example working in a large temperature range from –40 °C to +70 °C. The aperture was 50 mm, focal length was 200, the full field of view was 6°, and wavelength range was 550–750 nm. The initial mechanical material of the housing tube was aluminum with CTE αh = 22.50×10−6/°C. All seven lenses were spherical, and the total length was 236.818 mm.
Table 1 summarizes the optical properties of the lenses, including the glass material, chromatic power ω, thermal power γ, optical power ϕ, and paraxial ray height h.
Table 1. Optical properties of elements in the initial optical system
Figure 3 shows the polychromatic modulation transfer function (MTF) of the initial optical system, plotted against the spatial frequency at 50 cycles/mm. The initial optical system was optimized and designed at +20 °C, which was also the environmental temperature in the alignment room, and the average MTF of all fields was 0.747. However, the average MTFs at –40 °C and +70 °C drop sharply to almost zero, as shown in Figs. 4(a) and 4(c). Therefore, the optical quality is greatly affected by the environmental temperature without athermal optimization or design.
Fig. 4. MTF performance of the initial optical system at temperatures of (a) –40 °C, (b) +20 °C, and (c) +70 °C.
3.2 Establishment and analysis of visible glass catalog
To improve the success rate of non-thermal design, the replacement of glass material requires numerous optical properties of the glass material as supporting data. In the previous athermal optimization process, the selection and replacement of glass materials usually depended on experience, which has certain blindness and cannot be applied to our proposed replacement of combined glass. Any glass catalog that includes thermal and dispersion properties, such as Schott, OHARA, CDGM, can be modeled with athermal map based on thermal power and achromatic power of glasses. Schott catalog is widely used by optical designer and is applied to verify our approach as an example in our paper. Therefore, a new glass database was established, including the optical characteristic parameters of 103 types of visible light materials in the Schott glass catalog [18]. Figure 5 provides the Schott catalog of the glass material database which can be getting from Zemax for subsequent optimization design.
In Fig. 6, the materials in the visible glass catalog are represented as discrete points on an athermal glass map in the 550–750 nm range for this design example, as we shown in Dataset 1 [19]. Large variations in the thermal and chromatic optical power values of a single element may change the optical power value of the element considerably, eventually affecting the optical power value of the optical system, which may lead to undesirable lens surface shape and additional aberrations during subsequent redistribution of the optical power value. Therefore, when using an athermal glass map to replace glass material, glass material with a small effect on the total optical power of the element and optical system, that is, one that causes only minor differences in the chromatic focus Δω and thermal focus Δγ, should be chosen.
Therefore, numerous glass materials should be evenly distributed on both sides of the H–Le line to ensure more choices when replacing glass materials. Cluster analysis is a process of partitioning a set of data objects. Each subset is a cluster. The set of clusters resulting from a cluster analysis can be referred to as a clustering. As a data mining function, cluster analysis can be used as a standalone tool to gain insight into the distribution of data, to observe the characteristics of each cluster, and to focus on a particular set of clusters for further analysis [21]. The distance weight calculation method of discrete glass points relative to H–Le line happens to be similar to the distance error in the cluster analysis algorithm. A cluster analysis method is proposed to assess the distribution characteristics of athermal glass map in the visible catalog, where the clustering center of the catalog is calculated to serve as the basis to judge whether the visible light materials are evenly distributed on both sides of the H–Le line. The clustering center obtained from the algorithm can also be considered one of the factors in the subsequent selection of composite glass, reducing the number of calculations performed by the subsequent programs and greatly improving the selection and replacement efficiency and accuracy of optical glass.
The K-means approach is a common clustering algorithm, which has the advantages of a simple principle and effectiveness in judging the distribution characteristics of multivariate discrete points [20]. Therefore, the K-means algorithm was adopted to analyze the distribution of glass materials on the athermal glass map, and the steps for obtaining the clustering center are as follows.
- (1) Randomly select a data centroid from all visible light materials in the glass catalog as the initial clustering center.
- (2) Calculate the Euclidean distances between other glass materials and the initial clustering center.
The Schott catalog data set contains n objects in Euclidean space (n = 103). The objects are glass materials, and the Euclidean space is based on athermal map. The difference between object G and cluster center O, the representative of the cluster, can be measured by dist (G, O), which is the Euclidean distance between two points G and O. For each object, the distance from the object to the cluster center is squared, and the distances are summed [20]. The sum of squared error between all objects and cluster center O is defined as E$= \mathop \sum \nolimits_{i = 1}^n dist({{{\boldsymbol G}_i},{\boldsymbol O}} )$.
- (3) Recalculate the data of the clustering center and update the coordinates of the clustering center until the clustering center point no longer changes.
Comparing with computer database or remote sensing image field, the data amount of our glass catalog is relatively small, so the selection of the initial cluster center has little influence on the final cluster center solution. We compared the coordinates of the final cluster center point solved when N-BK7(12.994, −1.91), N-FK5(11.945, −11.68), N-SK5(13.478, 0.44) and N-SSK8(16.216, −3.16) glasses were used as the initial cluster center point. As shown in Fig. 6, all the coordinates of the final cluster center points obtained from the initial cluster center points of the four different glasses are (20.1041, −1.5246) by K-means algorithm. In practical application, since N-BK7 is close to the center without heat map and is one of the most commonly used glass materials in Schott glass catalog, and is preferred by optical designers, we can take N-BK7 as our initial clustering center by default., Thus, the visible materials in the glass catalog are uniformly distributed on the athermal glass map with these coordinates as the clustering center, which can represent the distribution characteristics of the athermal glass map and provide a foundation for subsequent glass replacement.
3.3 Quantitative selection of housing tube materials
On the athermal glass map of the equivalent glass, when the linear relationship between the composite glass Lij, equivalent single lens Le, and housing tube material H is satisfied, the athermal and achromatic conditions of the optical system are obtained. However, according to the composite glass Lij and equivalent single lens Le calculated based on the optical elements, it is difficult to obtain a suitable solution for subsistent-housing tube materials for practical engineering applications. Therefore, we comprehensively considered the replacement and selection of the housing tube material and glass material. Before replacing the glass material, the athermal ability of the housing tube material in the initial optical system was evaluated. Table 2 lists the thermal expansion coefficients of several materials with common mechanical structures.
Table 2. Thermal expansion coefficients of common housing tube materials
The athermal ability of the housing tube material was evaluated using the athermal glass map, with a nearly linear relationship between the H–Le line and composite glass Lij (ωij,γij). It is desired to select a housing tube material to ensure that numerous glass materials in the glass catalog are distributed on both sides of the H–Le line. The criteria for evaluating and selecting the housing tube materials can be quantified in two aspects: the distance between the H–Le line and the clustering center point of the glass catalog, and the position relationship between the H–Le line and composite glass Lij (ωij,γij).
The specific steps of the housing tube material selection method are as follows.
- (1) The optical system includes seven lenses, so the number of combinations of equivalent single lenses is $C_7^2 = 21$. Therefore, calculate the vertical distance d1i from the combined glass Lij (ωij,γij) to the H–Le line and the vertical distance d2i from the cluster center to the H–Le line by using 21 groups of equivalent single lenses Le (ωe, γe) connected to four types of lens barrel materials.
- (3) Group the ratios of the two housing tube groups and add them together, yielding the comprehensive weights of the four types of housing tube materials:
Figure 7 presents the athermal maps of the four lens tube materials. Using the above calculation method, the comprehensive weight value range of each cylinder material was determined to be [0,42]. The closer the value is to 42, the more the athermal design requirements are met; on the contrary, the closer the value is to 0, the less the athermal design requirements are met. Table 3 presents the results of the comprehensive weight analysis, where the most suitable tube material for the athermal optical system is Invar, the second is TC4, and Al and magnesium alloy are far inferior to Invar and TC4.
Table 3. Comprehensive distance weights of four lens tube materials
In practical engineering, Invar is generally not the first choice for mechanical structural materials in optical systems with the shortcomings of high density, low strength, and hardness. As the second optional material for the housing tube, TC4 has a comprehensive weight close to that of Invar with a much lower density and is widely used in engineering, so we replaced Al with TC4 as the tube material.
3.4 Quantitative selection and replacement method for visible glass materials
After selecting the housing tube material, the combination of two glasses Lij for glass replacement should be selected from the 21 groups in the glass map, as shown in Fig. 8. The concept and method of realizing this step are similar to those of the tube material selection: the athermal maps should exhibit nearly linear relationships between the selected composite glass Lij (red dots) and the H–Le line (red lines). Simultaneously, numerous glass materials should be distributed on both sides of the H–Le line and Li. In addition, Li and Lj should not be far away from Lj; otherwise, the thermal and color powers of the Li and Lj glass may be changed considerably by glass replacement.
We propose the following algorithm to evaluate the thermal potential of each combined glass pair quantitatively.
Figure 9 presents the simulation results obtained using the distance weighting method. The algorithm selects the 13th group as the glass combination with the highest weight (corresponding to the L35 glass combination, i.e., the combination of lenses 3 and 5).
- (3) For the athermal map of the L35 combined glass group, calculate the vertical distance from L35 (ω35, γ35) to the H–Le line. Taking the projection point P as the center of the circle and the vertical distance as the radius of the circle, screen the materials in the visible light material database in the limited circle shown in Fig. 10. Form the random composite glasses again in the range of the primary screen circle and calculate the approximate equivalent points of each group.
- (4) Seven kinds of glasses were screened and combined in the circle. According to the glass map, the combined glass replacement $L_{35}^{\prime}$ is approximately the midpoint of the connection between $L_3^{\prime}$ and $L_5^{\prime}$. Therefore, divide the glass materials after preliminary screening into two groups on opposite sides of the H–Le line, and then regroup the glass combinations in the two groups. The distance between the midpoint of each glass combination and the projection point P is calculated and sorted, as shown in Table 4. The closer $L_{35}^{\prime}$ is to the projection point on the H–Le line, the more it meets the achromatic and athermal conditions on the chart of the glass map. The calculation results show that the distance between N-LAF21/N-BAF4 glass combination and projection point is the shortest, so they are selected as the best replacement glass combination.
- (5) The purpose of the athermal glass map method is to optimize three points, L35 (ω35, γ35), Le (ωe, γe), and H (0, –αh), so that they obey a linear relationship; that is, under ideal conditions, the replaced L35 should be moved vertically to the H–Le line. Therefore, the projection point of ${L_{35}}$ onto the H–Le line is the equivalent point of ideal combined glass, so the glass material that can replace L3 and L5 should have thermal and chromatic powers similar to those of the individual materials, to avoid excessive influence on the total optical power of the optical system. Because the optical power values of L3 and L5 are 0.0134 and 0.0192, respectively, the change in the optical power value of L5 has a greater impact on the optical system. L5 should have smaller $\Delta \omega $ $\Delta \gamma $ and $\Delta \gamma $ values when the glass material is replaced. Therefore, change the glass material selected for L3 from N-BaF10 to N-BaF4, and change that selected for L5 from N-BaF10 to N-LaF21, as shown in Fig. 11.
- (6) Because the refractive indices of the system components change after replacing the glass material, redistribute and optimize the optical power value of the optical system to ensure that $L_{35}^{\prime}$, H, $L_e^{\prime}$ approximately satisfy a linear relationship after replacing the glass material.
Table 4. Distance between midpoint of glass combination and projection point
3.5 Final athermal optical system
Firstly, the single lens glass replacement method proposed by Na Xie is used to replace and optimize the glass of the initial optical system as a comparation. The structure of the system is grouped by comparison among the comprehensive weights, and the initial system is equivalent to a double-lens system that consists of a single lens and an equivalent lens in order to have the choice of optical materials [17]. However, this system does not satisfy the image quality requirements. Hence, several rounds of glass replacement are needed. Three rounds of glassre placement and the comprehensive weights are shown in Table 5. In round 1, the lens 4 has the highest weights 0.6562, and the glass is replaced from N-LLF1 to N-BK10. The average MTF of all fields at –40 °C, +20°C and +70 °C are 0.201, 0.181 and 0.133 respectively. In round 2, the lens 1 has the highest weights 0.6754, and the glass is replaced from N-SK10 to N-SSK8. The average MTF of all fields at –40 °C, +20°C and +70 °C are 0.355, 0.326 and 0.196 respectively. In round 3, the lens 7 has the highest weights 0.6014, and the glass is replaced from N-SK10 to N-LLF1. After three rounds of glass replacement and optimization, the average MTF of all fields at –40 °C, +20°C and +70 °C are 0.399, 0.556 and 0.284 respectively, which are still much lower than diffraction limit and need more iterations of glass replacement and optimization.
Table 5. Three rounds of comprehensive weight and single glass replacement
Figure 12 depicts the optimized athermal optical system structure, and Table 6 lists the characteristic parameters of the optical components. The curvature radius, thickness, and air interval of the lens change little between the initial lens and athermalization lens, and the optical power of the lens also differs only slightly.
Table 6. Characteristic parameters of the optical components after athermalization
The image quality of the optimized optical system was evaluated in the temperature range from –40 °C to 70 °C. Figure 13 presents the polychromatic MTF of the athermal optical system, plotted against the spatial frequency at 50 cycles/mm. The average MTF of all fields is greater than 0.7, at both –40 °C and 70 °C. The image quality evaluation and temperature adaptability analysis results of the athermalized optical system demonstrate that the athermal principle and method of the proposed combined glass replacement approach can quantitatively screen out the combined glass materials satisfying the athermal and achromatic conditions.
Fig. 13. MTF performance of the athermal optical system at temperatures of (a) –40 °C, (b) +20 °C, and (c) +70 °C.
4. Conclusion
In this study, a lens athermalization and achromatization method was developed involving the quantitative replacement of combined glasses on an athermal visible glass map. Further, a new achromatic and athermal condition for the replacement search method using combined glasses was derived. An athermal glass map was built to convert optical parameters into graphical characteristics in the visible band. The housing tube and glass materials were quantitatively selected and replaced by a comprehensive distance weight based on the athermal glass map. A visible aerial optical system was designed as an example using our method to maintain sufficient image quality over a large temperature range. The results indicate that the quantitative replacement method involving combined glasses on an athermal visible glass map is an effective means of achieving athermalization and achromatization and has the advantage of high efficiency in significantly reducing the number of iterative cycles required for athermal design and glass replacement.
Funding
National Natural Science Foundation of China (41974210, 6210031355).
Disclosures
The authors declare no conflicts of interest.
Data availability
The raw data from this work is available as Dataset 1 [19].
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