1. Introduction

Ambient temperature variations can impact the parameters and properties of the optical system, potentially leading to a degradation in image quality. Factors such as the refractive index of the lens material, lens shape, lens thickness, air gap and housing tube size are all influenced by temperature changes. In the absence of athermalization optimization in the design, the imaging performance may suffer due to defocus induced by varying temperatures. This issue is particularly critical for optical systems used in applications with large temperature ranges, such as automotive, military, and spacecraft applications. Consequently, athermalization of optical systems for different applications remains an active and vital area of research [1–4].

In addressing the aberrations caused by temperature changes, three primary approaches exist: electromechanical active, mechanical passive and optical passive methods. Among these, optical passive athermalization technology is preferred for most optical systems due to its simplicity, cost-effectiveness, and stability. This technology involves the strategic selection of optical and mechanical materials to create combinations that compensate for changes in focal length and focal plane position at different temperatures.

Most researchers optimize lens materials based on the athermal map by iteratively replacing glass materials graphically. The athermal map was introduced by Tamagawa, who also established the basic graphical method for athermalizing the lens system [5,6]. Lim proposed a new graphical method by redistributing element powers on the athermal glass map and also extended this map into a 3D glass chart to correct Petzval curvature [7,8]. Xie suggested achieving thermalization by transferring the lens system into a double-lens equivalent model on the athermal map [9]. Zhu proposed quantitatively selecting and replacing optical and housing tube materials based on combined glasses and comprehensive distance weight on the athermal map [10]. Recently, Li extended the graphical athermal method to catadioptric optical systems by optimizing both lens materials and the combination of mirror and housing materials [11]. Ren developed a method to resolve thermal aberrations and realize achromatic and apochromatic optical systems using a 3D glass chart [12].

Although the athermal map has been utilized thus far, it cannot intuitively indicate whether the glass combination of the optical system can achieve athermal and achromatic results. It needs to be checked whether the points of the doublets in the equivalent system and housing material lie on the line on the athermal map after some calculations [7,9,10].

In addition, methods that involve graphically replacing lens glasses have their limitations. The developed techniques often demand extensive manual effort and do not consistently yield successful outcomes, as observed by some researchers [10]. The achievement of good imaging quality in the optical system, after the replacement of one or two lens materials each time, may require some luck, as these methods solely focus on athermal and achromatic conditions. Moreover, each replacement necessitates optimization using optical software, typically requiring several iterations. While some researchers extend the athermal map into 3D to correct Petzval curvature or secondary spectrum aberration, these extensions cannot be applied simultaneously. Additionally, the quantitative measurement of lens power changes after glass replacement is not considered, potentially leading to significant alterations in lens shape and structure, ultimately compromising imaging performance. Most notably, this procedure cannot be developed into a reusable tool. For each new design, the same process must be repeated, lacking the efficiency.

While commercial optical software provides optimization tools like Hammer Optimizer in Zemax or Glass-Expert in CODE V, facilitating the athermalization of designs, achieving optimal designs for multi-lens systems is impeded by a time-consuming iterative process. This process, involving numerous iterations, each consisting of two consecutive steps—material replacement and imaging quality optimization—yields low efficiency. The hindrance primarily arises from the extensive search space, which encompasses the combined space of glass materials and optical element structures. At times, optimization outcomes may deviate, leading to unexpected changes in the optical structure and lens shape. Consequently, manual material replacement based on athermal conditions in a graphical athermal map remains a primary approach for many optical designers, as discussed above.

To address these issues, this report firstly introduces a novel athermal map to assess whether lens material combinations have the potential to achieve athermal and achromatic design. Subsequently, an automatic glass replacement method, named glass selection using simulated annealing with memory augmentation (GlaSAM), is presented. The GlaSAM method provides glass solutions for the initial structure based on a comprehensive objective function with five different characteristics, including thermal aberration, chromatic aberration, secondary spectrum aberration, Petzval curvature, and lens power deviation.

Compared to current manual graphical glass replacement, the GlaSAM method provides a more comprehensive and automated approach, exploring a larger search space to identify numerous glass combinations rather than changing just one or two materials. After the glass replacement, only one additional optimization step is required for redistributing lens power in optical software. In contrast to current commercial optical software tools that optimize structure parameters for every new glass combination, the GlaSAM method is expected to be more efficient and stable.

2. New representation for athermal map

2.1 Achromatic and athermal conditions

For the $i\textrm{th}$ thin lens with power ${\phi _i}$ in an optical system, the chromatic power ${\omega _i}$ and thermal power ${\gamma _i}$ are given as follows [5,6]:

For the visible spectrum in the following discussion, the chromatic power ${\omega _i}$ can be represented the reciprocal of Abbe number ${\upsilon _i}$,given by ${\omega _i} = 1/{\upsilon _i}$.

And the total optical power ${\phi _T}$ for the optical system with $k$ thin lenses can be obtained as follows [7,13]:

Hence, we can derive the total differential based on Eq. (3) as follows:

Let’s consider the wavelength and temperature change. Variations in lens power due to these changes have a more significant impact on total power fluctuations compared to alterations in ray height in an optical system. Therefore, we can disregard the first term of the summation in Eq. (4), leading to the derivation of the achromatic condition Eq. (5) and the athermal condition Eq. (6) as follows [7]:

2.2 Composite achromatic and athermal map

In this segment, we will introduce another set of modified equations for athermal and achromatic conditions. Based on these modifications, a new composite achromatic and athermal map will be presented. By utilizing Eq. (3) and Eq. (6), we derive Eq. (7):

Let’s define the composite chromatic power $\omega _i^{\prime}$ and the composite thermal power $\gamma _i^{\prime}$ to take account into the sign of the lens power ${\phi _i}$.

By normalizing the weighted lens power $\phi _i^{\prime}$ through division by ${\phi _T}$, this transformation leads to the modification of Eq. (5) and Eq. (7), resulting in Eq. (10):

Then we have Eq. (11) and Eq. (12):

Equations (11) and (12) offer an alternative representation of achromatic and athermal conditions, accounting for individual lens power characteristics and housing material properties.

Now, turning our attention to the glass material aspect, we can create a chart with the abscissa axis representing the composite chromatic power $\omega ^{\prime}$ and the ordinate axis representing the composite thermal power $\gamma ^{\prime}$. For a given optical glass paired with a specific housing material, we can plot two points on an athermal chart, considering both positive and negative optical power.

In this report, the Schott glass catalog serves as an example for the glass library [14]. The glasses from the catalog are imported into the composite athermal chart shown in Fig. 1 as grey points, supposing the housing material is aluminum with ${\alpha _\textrm{h}}$=22.5 × 10⁻⁶/°C. The color rendering of the chart suggests that the athermal and achromatic design shares similarities with the additive color mixing principle.

 

Fig. 1. Composite athermal chart.

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Assuming a lens system comprises three or more types of materials, if the maximum area enclosed by the glass points in the athermal chart covers the origin point (0,0), there is a possibility of meeting the athermal and achromatic conditions. For instance, the combination of materials ABC can encircle the origin point, making it possible for an athermal design. Conversely, materials ACD, in which the origin point is not included, are not feasible. However, using ABCD is viable. Nevertheless, achieving a point within the covered glass area is a necessary but not a sufficient condition. If the power distributions do not satisfy Eq. (11), the final equivalent thermal-chromatic point E$(\omega _e^\mathrm{^{\prime}}$, $\gamma _e^\mathrm{^{\prime}})$ of the lens system may deviate from the origin. The distance between point E and the origin indicates the system's departure from achieving an athermal and achromatic design. In such a case, adjustments to either lens powers, materials, or both may be considered.

If the origin point is not encircled by the glass points used in the optical system, it requires the consideration for adjusting the materials, and a redistribution of optical powers may be necessary.

This newly proposed composite athermal map incorporating optical power sign allows designers to clearly visualize the results, and easily assess whether the glass material combination has the potential to meet the target.

Based on this chart, designers can observe that the goal of achieving athermal and achromatic design is to select lens material combinations that bring the equivalent point close to the origin point.

3. Glass selection using simulated annealing with memory augmentation

3.1 Simulated annealing algorithm introduction

Now we would present an automated approach for glass selection to attain athermalization and achromatism designs, departing from traditional graphical methods.

Initially, the lens powers in the system remain unchanged, and the algorithm selects suitable glass materials to replace part or all of the initial lens materials, ensuring the satisfaction of specific equations. Once the optimal lens selections are identified, a slight redistribution of lens power provides the system with the potential to achieve optimal performance.

The glass selection process involves making discrete choices from glass libraries, constituting a combinatorial optimization problem. This optimization approach involves determining the best solution from a finite set of possibilities, which entails handling discrete and interconnected decision variables. Its applications span various domains, including logistics, scheduling, and network design [15].

To perform the optimization, we employ the simulated annealing (SA) algorithm, a stochastic optimization approach well-suited for finding global minima or maxima of an objective function [15]. It is inspired by the annealing process in solids, where higher temperatures lead to increased randomness. As the temperature decreases, internal energy diminishes, guiding particles toward a more ordered state. In practical applications, this involves an initial phase of randomly generating solutions, even with lower performance. During this phase, there is a higher acceptance rate for suboptimal solutions as the best solution. Over iterations, this rate gradually decreases. Termination conditions, such as reaching a specified iteration limit, mark the end of the optimization process.

Next, we will introduce the objective function and filters. Subsequently, we will outline the procedures for utilizing SA in the optimization of glass selection.

3.2 Objective function

Generally, to achieve good performance, an optical design not only needs to address primary achromatic and athermal aberrations but also should satisfy the Petzval sum condition [13],as Eq. (13):

For improved chromatic performance, the lens design should also satisfy the apochromatic equation as Eq. (14):

Then our goal is to select new lens materials and identify the best combinations that come closest to meeting all these conditions. This involves calculating the weighted sum of the residuals of these aberrations, with the formula on the left-hand sides of Eqs. (5), (7), (13), and (14).

Here, we can express the weighted sum of aberration functions for the glass solution s as:

After substituting the expression for aberration residuals, the function$g({\boldsymbol s} )$ becomes Eq. (16):

As glass selections may sometimes reduce $g({\boldsymbol s} )$,the optical power of the same lens structure may deviate significantly from the original when using new glasses. Consequently, the optical structure might not exhibit good performance with the new glass combination. To ensure the stability of the original optical structure with the new glass solution, an additional penalty term is included in the objective function to limit power deviations.

Then the final objective function $f({\boldsymbol s} )$ is set as:

The significance of aberration weights lies in their role in prioritizing aberrations, influencing the outcomes of the glass selections based on their importance. Variations in weights lead to diverse outputs for the chosen glasses while ensuring a balanced consideration of different aberrations. This balance can be calibrated according to the permissible tolerances for aberrations in the design system. Given that ideal aberrations are zero, setting a reference (denoted as r) for each residual provides a measure for controlling aberration magnitudes. Then each aberration weight is defined as:

This method for setting up weights also standardizes the dimensional units of each aberration while enabling the distinction of correction requirements for different aberrations. Additionally, we can modify the weight by adjusting τ based on our intentions after obtaining the current solution result.

During the optimization, the optical system can be normalized and initialized with the effective focal length (EFL) = 1 mm. This ensures that the weights in the formula remain relatively stable and serve as a good reference for different applications.

For applications where athermal requirements are not very strict, and it's acceptable to compromise on image quality to some extent, especially for extreme temperature conditions, we can reduce the weight assigned to the thermal aberration.

However, finding appropriate weights may still require some trial and error. The easiest way to adjust the weights is based on the running results. If a specific type of aberration is not satisfactory, simply improving the weights can address the issue.

3.3 Filtering criteria for reducing the search space

In a larger search space, exploring all possible solutions becomes a challenge for SA. Thorough exploration may demand more iterations and parameter tuning, consequently requiring additional time to achieve similar quality solutions.

Consider a scenario with $N$ lens pieces and $M$ glass material candidates, resulting in ${M^N}$ permutations. For instance, with 100 glass types and a lens system comprising 8 pieces, the total combinations reach 10¹⁶, representing a vast search space.

To reduce the search space, we can opt to maintain the materials of certain elements in the optical system while only altering the rest. Additionally, not all available glass types are suitable; some may be expensive, contain environmentally unfriendly elements (e.g., Pb), or exhibit low transmission.

Moreover, for specific lenses, maintaining stability in power during material replacement is crucial. Deviations in refractive index could lead to significant geometric differences, worsening other aberrations. If the initial system is achromatic and performs well at room temperature, the optimized solution's Abbe number shouldn't deviate significantly from the original value.

To address these considerations, filters are applied to limit glass candidates for optimization. The following lines represent the filtering process, with a fictional code snippet demonstrating the filtering criteria:

combined_filter = nd_filter & vd_filter & price_filter & eco_filter & other_filter

filtered_glass = import_glass[combined_filter]

3.4 Algorithm flow chart

Now let’s dive into the SA algorithm. During the annealing process, the comparison between the current best solution and the new solution, generated after a random perturbation, is facilitated by an acceptance probability term. Following the Metropolis principle [15], the acceptance probability ${P_t}$ is intricately linked to the simulated temperature t, set as:

Applying the Metropolis principle increases the likelihood of accepting suboptimal solutions at higher temperatures, while predominantly superior solutions are accepted at lower temperatures.

However, the classical SA algorithm, following the Metropolis principle, may encounter inefficiencies as accepting a worse solution could result in the simultaneous rejection of a potentially better solution [16]. To address this limitation, the GlaSAM method enhances SA algorithm by introducing a memory feature [17]. The memory, with a capacity defined as memory_size, stores the best solutions. The algorithm then randomly selects a new solution either from the memory or generates a new solution randomly. The improved algorithm with the memory feature has additional benefits, allowing us to obtain several top solutions. Since we don't include all aberrations in our optimization, a slightly worse solution here might lead to better overall performance for the lens system.

In our specific application, the process of generating a new solution involves randomly replacing one lens glass with another chosen randomly from a set of lens glass candidates. The GlaSAM algorithm is designed to enhance the exploration of solutions from various regions of the solution space and improve the exploitation of promising solutions stored in the memory.

The flowchart of the GlaSAM method is illustrated in Fig. 2. The procedure commences with input parameters and the importation of glass libraries for filtering, followed by algorithm initialization.

 

Fig. 2. Flowchart of the GlaSAM method.

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Then two nested loops are established: the outer loop, responsible for setting the temperature ${t_k}$ at the $k\textrm{th}$ iteration and determining the maximum allowed iterations, denoted as iterations, and an internal loop within it for regional search. Within each internal cycle, a new solution is randomly generated, as explained earlier. If the new solution ${\boldsymbol s^{\prime}}$ demonstrates superior performance, it replaces the current best solution ${\boldsymbol s}$. In cases where the new solution does not surpass the existing best solution, it still has a chance of acceptance based on the introduced Metropolis rules. This process iterates until the internal cycle limit, denoted as internal_cycle, is reached.

Upon completion of each internal cycle, the temperature ${t_k}$ is lowered by the $cooling\_rate$, initiating another iteration. This process continues until the maximum iteration count is reached. The optimal solutions, along with the memory storing additional high-performing solutions, are then outputted. This marks the conclusion of the glass selection process, achieving an optimal lens material combination through the GlaSAM method (see Supplement 1).

4. Athermalization design example based on GlaSAM method

4.1 Initial optical design and performance

As an illustration of athermal design within a broad temperature range (-40°C to +70°C) using the GlaSAM method, Fig. 3 showcases the initial optical structure of a telephoto lens featuring eight spherical lenses [18](see Supplement 1). The design is with the aperture 28.9 mm, EFL 100 mm, the full field of view 12°, and a wavelength range of 486-656 nm. The housing tube is constructed from aluminum with CTE ${\alpha _\textrm{h}}$=22.5 × 10⁻⁶/°C. The total system length of 132.347 mm.

 

Fig. 3. Layout of initial design.

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Table 1 provides detailed optical properties of the lenses, including the glass material, chromatic power $\omega $, thermal power $\gamma $,optical power $\phi $, and paraxial ray height $h$.

Table 1. Optical properties of lenses in the initial optical design

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From Table 1, we can see that this optical system is designed with two types of glass: N-PK51 and N-LAK7. N-PK51 functions as the positive lens material, while N-LAK7 serves as the negative lens material.

Utilizing the data provided in Table 1 and considering the CTE of the housing material${\alpha _\textrm{h}}$=22.5 × 10⁻⁶/°C, we can calculate the composite chromatic power$\omega ^{\prime}$=-0.6 × 10⁻³ and the composite thermal power$\gamma ^{\prime}$=-8.72 × 10⁻⁶ /°C using Eqs. (11) and (12). Subsequently, the equivalent thermal and chromatic point E$({\omega^{\prime}\textrm{,}\gamma^{\prime}} )$and the corresponding glass materials can be plotted on the composite athermal chart in Fig. 4. In the chart, $\omega ^{\prime}$and$\gamma ^{\prime}$are abbreviated as chromatic power and thermal power, respectively.

 

Fig. 4. Lens glasses and equivalent point of initial design in the composite athermal chart.

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Given that the equivalent point E is positioned on the vertical axis and away from the origin point, it indicates that this optical system is achromatic but lacks athermal optimization.

Figure 5(b) illustrates the Modulation Transfer Function (MTF) performance of the initial optical design, which is optimized and designed at +20°C. The performance is excellent and closely approaches the diffraction limit at this temperature. However, as depicted in Figs. 5(a) and 5(c), the MTF experiences a significant drop close to zero, even for 20 cycles/mm, at temperatures of -40°C or +70°C. This decline can be attributed to the pronounced thermal defocus effect. Consequently, the optical quality undergoes substantial degradation due to environmental temperature variations.

 

Fig. 5. MTF performance of initial optical design at (a) − 40°C, (b) + 20°C, and (c) + 70°C.

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This observation aligns with the findings in Fig. 4, emphasizing the challenge of simultaneously achieving both achromatism and athermalization with just two glass materials, unless the line connecting these two material points coincidentally passes through the origin point.

4.2 Parameters setup for glass selection optimization

Firstly, we establish filters to eliminate unwanted glass types and reduce the search space.

For nd_filter, we set the maximum deviation for refractive index as$dn = 0.3$, which is a very loose filter to retain more candidates.

For vd_filter, the maximum deviation for Abbe number is set as$dv = 20$.

Price_filter is set to 100, higher than the highest price (36.7 in the Schott glass library), ensuring no filtering based on cost to maximize the candidate pool.

Eco_filter is set to 1, including only environmentally friendly glasses. Special glasses are labeled as 0 and are subsequently excluded by this filter.

Based on these filters and rules, we identify 36 glasses for N-PK51 and 63 glasses for N-LAK7, including the original glass. The glass candidate numbers for the optical system are ${N}$ = (36,63,63,36,36,63,36,63), allowing each lens material to be replaced. The total glass selection combinations amount to 2.6 × 10¹³.

Regarding setting each aberration targets, designers could define them based on theoretical calculations or input specifications for specific aberrations. Here, we use initial design performance data as a reference. The residuals of each aberration after normalized with EFL = 1 mm are calculated as:

R_th= -77.75 × 10⁻⁶, R_ch= -0.00 × 10⁻³, R_pch= 0.01 × 10⁻³, R_ptz= -0.0476.

We initially estimate the defocus length by calculating the power deviation$\textrm{d}{\phi _T}$, which is the product of${R_{th}}$and the temperature change. In the normalized system working at -40°C, $\textrm{d}{\phi _T}$is determined to be 4.6 × 10⁻³mm^-1. For the original EFL = 100 mm system, the defocus distance induced by temperature difference is around 0.46 mm. This value is considered too high and leads to poor performance in extreme temperatures. After analyzing the defocus effect using the through-focus MTF chart, we observe that reducing the defocus to one-tenth of the current value shows no obvious MTF drop. Therefore, we set the residual reference${r_{th}}$=7.8 × 10⁻⁶. Furthermore, we could also adapt the residual reference according to the depth of focus of the lens system as a systematic approach, especially if no special specifications are requested.

The MTF of the initial structure at 20°C is close to the diffraction limit, indicating excellent chromatic and apochromatic residuals as well as a favorable Petzval sum. We can set some relaxed aberration residual references for chromatic performance, specifically, r_ch =0.05 × 10⁻³ and r_pch= 0.05 × 10⁻³.

The chromatic residual R_chrepresents the optical power difference between F and C spectral lines, while R_pchcharacterizes the difference for C and d spectral lines, as Schott provides the partial dispersion ratio P_d,C in the glass database. In our example, apochromatism is narrowly defined to minimize the secondary spectrum aberration for the working spectrum. For an optical system with a normalized focal length of 1 mm, it is acceptable for the focal power difference to be $0.05 \times {10^{ - 3}}$ for F/C/d spectral lines. This means that for a focal length of 100 mm, the chromatic aberration is around 5 μm.

For the Petzval Sum, we set the residual reference${r_{ptz}} = 0.05$,similar to the initial value.

The adjustment factors${\tau _{th}}$, ${\tau _{ch}}$, ${\tau _{pch}}$are set to 1, while${\tau _{ptz}}$is set to 0.1, taking into account the system's small field.

The weight factor to limit power variation is set as W_power= 2, promoting optical system stability after introducing a new glass solution.

Then the GlaSAM algorithm is initialized with input parameters: t₀= 10000, cooling_rate = 0.002, iterations = 5000, internal_cycle =5, and memory_size =5. The initial solution is set to a random selection from candidates, which is better for global optimization.

A high initial temperature${t_0}$and a low cooling rate are also advantageous for global optimization, as per the Metropolis principle. This principle suggests that the likelihood of accepting suboptimal solutions is higher, facilitating escape from local optima. It's beneficial to set these two parameters reasonably so that the final temperature approaches zero after reaching the maximum number of iterations.

A higher iteration count for the outer loop is necessary for a more comprehensive search. Having fewer internal cycles compared to the total iterations allows the algorithm to focus more on global exploration, avoid premature convergence to local optima, and conserve computational resources and time. Therefore, we assign internal_cycle as 5 to prioritize global exploration and optimize running time.

Setting memory_size to 5 allows for the examination of the top five solutions in terms of optical performance.

It's important to note that these parameter settings are provided as an illustrative example to demonstrate the GlaSAM method and may not necessarily represent the optimal values for all scenarios.

After initialization, the algorithm is executed.

4.3 Final results

The GlaSAM algorithm took 9 minutes to run on my computer (CPU: i7-13700KF 3.40 GHz) to generate the top five solutions, as presented in Table 2. The progression of objective function values over iterations is illustrated in Fig. 6, which shows that the values decrease and converge to approximately 10.

 

Fig. 6. Iterative objective function values.

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Table 2. Top five solutions calculated by GlaSAM method

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All these five solutions share the same material combinations for the first seven lenses, with only the eighth lens using a different material. After the initial optical system using a new glass solution is slightly optimized for power redistribution in the optical software, all five solutions exhibit good performance, maintaining MTF values above 0.6 at 100 cycles/mm across various fields from -40°C to +70°C. Among them, solution s3 stands out as the best-performing option, leading us to choose it as the final design.

Figure 7 shows the layout of the final design after athermalization. Table 3 provides detailed optical properties of the lenses in the final optical design (see Supplement 1).

 

Fig. 7. Layout of final design after athermalization.

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Table 3. Optical properties of lenses in the final optical design

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Utilizing the data provided in Table 3, we can calculate the composite chromatic power$\omega ^{\prime}$=-0.26 × 10⁻³ and the composite thermal power$\gamma ^{\prime}$=0.22 × 10⁻⁶ /°C .The equivalent point E$({\omega^{\prime},\gamma^{\prime}} )$ and the corresponding glass materials can be plotted on the composite athermal chart in Fig. 8.

 

Fig. 8. Lens glasses and equivalent point of final design in the composite athermal chart.

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The outstanding performance of the athermalized final design is illustrated in Fig. 9, portraying its MTF across diverse environmental temperatures. At 20°C, the on-axis MTF is 0.681 for the frequency of 100 cycles/mm. In the 1.0 field, the tangential and sagittal MTF values reach 0.643 and 0.682, respectively, approaching the diffractive limit of 0.740. What distinguishes this design is its remarkable stability, maintaining consistently high MTF values under extreme temperature conditions of -40°C or +70°C.

 

Fig. 9. MTF performance of athermalized final design at (a) −40°C, (b) + 20°C, and (c) + 70°C.

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The assessment of image quality across varying temperatures demonstrates that the GlaSAM method can effectively athermalize the optical system while maintaining excellent optical performance.

5. Discussion and conclusion

It should be noted that final optimization and image quality assessment rely on standard optical design software that utilizes material libraries based on comprehensive models and measurement results. A simplified glass model with fixed values for thermal power and chromatic power to represent glass properties is insufficient for final performance evaluation; it is idealized for glass selection during the design.

For thermal properties, slight variations in the refractive index and CTE of glass, along with the CTE of housing material, can occur across different environmental temperatures. And the derivative of the refractive index with respect to the wavelength in optical glasses is not always constant. Therefore, thermal power and chromatic power based on these parameters would also exhibit minor variations. After glass selection, additional fine-tuning for lens shapes, thicknesses, and air gaps needs to be performed using reliable glass libraries in standard optical software.

Besides, the proposed GlaSAM method can extend the working spectrum beyond the typical 486-656 nm, even into the non-visible spectrum. However, when operating in a broader visible spectrum or even outside the visible spectrum, the chromatic power represented by the reciprocal of the Abbe number may not provide high accuracy. For a more accurate representation of glass properties, the chromatic power should be calculated by its definition based on the working spectrum.

The GlaSAM method, when applied to specific cases, might require more iterations to achieve optimal results. Additionally, it may not always deliver perfect corrections for all aberrations. Apochromatic correction, in general, presents a significant challenge, especially for a broad spectrum like g/F/d/C correction. In our example, despite attempting to minimize secondary spectrum, a small residual aberration remains. By devoting more computational resources and setting a higher weight on it, apochromatic aberration could have chance to be further improved, potentially at the expense of other aberrations.

While our example demonstrates a system with eight lenses, the GlaSAM method is applicable to designs with varying numbers of elements. For designs with fewer lenses, the overall search space is reduced, thereby increasing the likelihood of reaching the global optimum. Conversely, designs with more lenses possess a larger search space, typically necessitating more iterations for convergence to a satisfactory solution. To maintain search efficiency, we can either fix the materials of certain lenses (special materials may be preferred in some cases) or implement a stringent filter to refine the pool of glass candidates.

In a summary, we introduce a novel composite athermal map aimed to assist designers in visualizing selected glasses and assessing the feasibility of achieving athermalization and achromatism in optical systems. Utilizing this athermal map, we propose an automatic GlaSAM method for athermalization optimization, which effectively addresses various aberration conditions, including athermal, achromatic, apochromatic, and Petzval sum conditions simultaneously. Compared to the current manual approach of graphically replacing glass for one or two lenses at a time, our method significantly enhances the search capabilities for more intricate and promising combinations. The demonstrated example of athermalization highlights the utility, effectiveness, and efficiency of the GlaSAM method.