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Curr. Opt. Photon. 2024; 8(5): 472-483

Published online October 25, 2024 https://doi.org/10.3807/COPP.2024.8.5.472

Copyright © Optical Society of Korea.

Design of a Telephoto Optical System for SWIR Using Apochromatic and Athermal Method

Tae-Sik Ryu, Sung-Chan Park

Department of Physics, Dankook University, Cheonan 31116, Korea

Corresponding author: *scpark@dankook.ac.kr, ORCID 0000-0003-1932-5086

Received: August 26, 2024; Accepted: September 23, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents an intuitive method for selecting an optical material for achromatic and athermal design using the material selection index (MSI). In addition, in the case of a wide wavelength range such as a short-wave infrared (SWIR) waveband, we propose a new material selection method for apochromatic and athermal design by introducing the relative error of partial dispersion (REPD) and a first-order quantity redistribution method. To obtain a suitable material for effective apochromatic design, we first evaluate the REPDs of all lenses, deviated from that of an equivalent lens. Materials with a small REPD are then selected on a glass map to correct residual chromatic aberration while maintaining the existing MSI values to realize athermalization simultaneously. Using this proposed glass selection method, we successfully obtained an apochromatic and athermal telephoto system for SWIR that realizes stable performance over the specified temperature and wide waveband ranges.

Keywords: Aberrations, Apochromatization, Athermalization, Glass map

OCIS codes: (080.2740) Geometric optical design; (160.4670) Optical materials; (220.3620) Lens system design

The short-wave infrared (SWIR) wavelength from 0.9 µm to 1.7 µm uses light reflected from objects. With longer wavelengths than visible light, SWIR experiences less scattering and is advantageous in environments with turbulence, fog, haze, smoke, and clouds. Additionally, it can detect low levels of reflected light from long distances and recognize objects at night using reflected light. Recently, the development of electro-optics equipment for security and surveillance has been actively progressing, with research reported on replacing visible light images with SWIR images. However, due to the wider wavelength range of SWIR compared to visible light, stabilization to prevent wavelength changes in an optical system is required. Common optical materials are mainly developed for the visible wavelength range, and the lack of strong flint optical materials makes it challenging to effectively correct chromatic aberration in the SWIR waveband [13].

This study aims to design a surveillance telephoto optical system that uses the advantages of the SWIR waveband. Its long focal length makes it sensitive to temperature changes and object distance, so it requires a compensator [4]. However, in compact optical systems, performance stability issues can arise even with a compensator. Therefore, careful optical material arrangement is necessary [5]. We aim to design an optical system considering these issues. First, an initial solution is obtained using a two-group optical system, and then the achromatic and athermal optical material selection method is applied for each group using a glass map. Next, for groups where apochromatic design is feasible, optical material selection and a first-order quantity redistribution method are introduced to configure the initial apochromatic system. The SWIR telephoto optical system with a telephoto ratio of less than 0.5, configured through this process, was found to be stable across the wavelength range and advantageous for temperature compensation.

2.1. Achromatic and Athermal Conditions

Changes in the optical power of the lens due to variations in wavelength and temperature can be expressed using the chromatic power (ωi) and thermal power (γi) of the element material (Mi), as shown in Eqs. (1) and (2) [69]:

ωi=  1vi=  Δϕiϕi=  Δλni1niλ,
γi=  ϕiT1ϕi=  1ni1niTαi,

where ∆λ is the specified waveband, φi is the element optical power, vi is the Abbe number, ni is the refractive index at the reference wavelength, αi is the coefficient of thermal expansion (CTE) of the i-th lens material, and T is the temperature.

Longitudinal chromatic aberration arises from changes (∆fbch) in the back focal length (BFL) with wavelength and is expressed by Eq. (3). Thermal defocus (∆z′) is evaluated as the difference between the change (∆fbth) in the BFL with temperature and the change (∆Hb) in the flange back length (FBL) with temperature, as given in Eq. (4) [810]:

Δfb'-ch=  1ϕT2 i=1kωi'ϕi',
Δz' =  Δfb'-thΔHb' =  1ϕT 2 i=1kγi'ϕi'αhLΔT,

where φT is the total power and k is the total number of lens elements. In the above two equations, the primed parameters indicate that they are weighted by the ratio of the paraxial ray heights and are expressed as ϕi = (hi / h1)φi, ωi = (hi / h1)ωi, and γi = (hi / h1)γi, respectively. This implies that the air spacings between elements are included in Eqs. (3) and (4).

2.2. Method for Representing the Achromatic and Athermal Conditions on a Glass Map

In this study, an equivalent single lens is used to simplify an optical system with an arbitrary number of elements into a doublet system. Thus, an optical system with k elements can be recomposed into a doublet system composed of the specific j-th element Lj and an equivalent single lens Le. This equivalent single lens consists of the remaining k-1 elements. Therefore, in this separated doublet system composed of Lj and Le, the total optical power (φT), achromatic (∆fbth = 0), and athermal (∆z′ = 0) conditions are respectively given by [810]:

ϕT=   i=1kϕi'=  ϕj'+ϕe',
Δfb'-ch=  1ϕT2ωj'ϕj'+ωe'ϕe'=  0,
Δz' =  1ϕT2γj'ϕj'+γe'ϕe'αhLΔT=  0,

where ϕe' =   i=1kϕi 'ϕj', ωe' =   i=1k ωi 'ϕi ' ωj 'ϕj '/ ϕ e ', and γe' =   i=1k γi 'ϕi ' γj 'ϕj '/ ϕ e '.

By dividing the achromatic condition of Eq. (6) and the athermal condition of Eq. (7) in this doublet system by the ratio of the paraxial ray height (hi / h1), we can easily identify a specific lens location without weighting on a glass map. Additionally, dividing Eqs. (5), (6), and (7) by the total power (φT) leads to expressions for the achromatic and athermal conditions in a doublet system as follows [9, 10]:

pj+pe=  1,
ωjpj+ωe''pe=  0,
γjpj+γe''pe=  αh'',

where ωe'' =  ( h1 / hj )ωe', γe'' =  ( h1 / hj )γe', αh = (hk / hj)αh, pj = ϕj / φT, and pe = ϕe / φT. Thus, two parameters of pj and pe are the ratios of optical powers of Lj and Le with respect to the total power.

When an equivalent single lens is given, the point designated as Lc(ωc, γc) in Fig. 1 denotes the achromatic and athermal point of a specific lens, which we refer to as the aberration-corrected point for these two errors, or briefly, Lc(ωc, γc). By combining the achromatic condition of Eq. (9) and the athermal condition of Eq. (10) with the optical power equation of Eq. (8), we can rewrite the achromatic and athermal conditions as follows [10]:

Figure 1.Achromatic and athermal conditions on a glass map.

pjpe=  1ωe''ωcωe''ωc,
pjpe=  1γe''γcγe''+αh''γcαh,

where ωc = −ωe (pe / pj), and γc = −(γe + αh)(pe / pj) − αh. To represent the condition that simultaneously satisfies both achromatic and athermal requirements on a glass map, we can rearrange the above Eqs. (11) and (12), as follows [10]:

γc+αh''ωc  =  γe''+αh''ωe''.

Equation (13) holds because the left side uses Lc(ωc, γc) instead of Lj(ωj, γj). Thus, the difference in material properties between Lj and Le causes chromatic aberration and thermal defocus. Accordingly, it can be defined as an aberration factor (Af) using the sum of the relative error, as given in Eq. (14):

Af  ωcωjωc+γcγjγc.

In Eq. (13), the achromatic and athermal conditions require ideal material properties Lc(ωc, γc). The left side of this equation represents the slope of the line connecting housing material properties Mh(0, −αh) and Lc(ωc, γc), while the right side represents the slope of the line connecting Mh(0, −αh) and Le(ωe, γc). If Lc for a specific lens is not used, the equation does not hold. This can be represented on a glass map, and the aberration factor (Af) defined by Eq. (14) can also be depicted, as shown in Fig. 1.

2.3. Optical Material Selection Method for Achromatic and Athermal Design on a Glass Map

An optical system that does not satisfy achromatic and athermal conditions will have the lines of MhLj and MhLc, but they do not coincide, as shown in Fig. 1. The simplest method to align these lines as closely as possible is to change the material. To achieve this, it is necessary to select the most suitable material from the available materials (La) distributed on a glass map. In this process, a material selection index (MSI) is defined as the relative error between the material properties of Lc(ωc, γc) and La(ωa, γa), as given in Eq. (15). This MSI is used for material selection.

MSIωcωaωcWc+γcγaγcWtAf',

where Af is weighted aberration factor, that is Af(ωcωj)/ωcWc+(γcγj)/γcWt. In Eq. (15), the weighted aberration factor is represented using Wc for chromatic aberration weighting and Wt for thermal defocus weighting. For an optical system where chromatic aberration correction is more important, Wc is set higher, while for an optical system where thermal defocus correction is more important, Wt is set higher. This allows for identifying the weighted aberration factor according to the importance of aberration correction in an optical system. Additionally, by using the sum of the weighted relative errors of the material properties of Lc and La, we can select the material that is close to Lc. Dividing this by Af leads to the MSI of Eq. (15), which implies how much the available material can reduce existing aberration factors, being compared to the material used in a specific lens. Therefore, the MSI values for each available material on a glass map can be calculated and listed to select a reasonable material for aberration correction.

2.4. Apochromatic Condition

Generally, achromatic refers to matching the focal lengths at the wavelengths of both ends. However, a difference in focal length occurs at other wavelengths within both ends. If an additional wavelength is specified to match the focal length, a more stable optical system can be achieved with respect to wavelength changes. An optical system that corrects chromatic aberration at these three wavelengths is called an apochromatic system [4].

As shown in Fig. 2, an achromatic optical system only matches the BFLs (fbS, fbL) for the wavelengths at both ends (λS, λL). For the BFL (fbJ) of a specific wavelength (λJ) within the wavelength range, residual chromatic aberration occurs. Therefore, when the system is achromatic, if we define the difference in BFL between the short wavelength and the specific wavelength as ∆fbchJ from Eq. (3), it can be expressed as follows:

Figure 2.Residual chromatic aberration in an achromatic optical system.

Δfb'-ch-J=  1ϕT2 i=1khih1nSnJnR1×hih1(ci1ci2)(nR1)                =  1ϕT2 i=1kPi(λS,λJ)ωi'ϕi' ,

where nR is the refractive index of the reference wavelength and Pi(λS, λJ) is the partial dispersion of the i-th lens material, and is defined as Pi(λS,λJ)=(nS nJ )/(nS nL ). Next, by introducing the concept of an equivalent single lens to simplify the above Eq. (16) into a doublet system, the residual chromatic aberration correction (i.e., apochromatic) condition can be expressed as follows:

Δfb'-ch-J=  1ϕT2Pj(λS , λJ )ωj'ϕj'+Pe(λS , λJ )ωe'ϕe'=  0,

where Pe(λS,λJ)= i=1k Pi( λ S , λ J )ωi'ϕi' P j( λ S , λ J ) ωj 'ϕj '/ i=1k ωi'ϕi'ωj 'ϕj '.

In Eq. (17), since the total optical power of an imaging optical system cannot be zero, the term inside the brackets must be zero for residual chromatic aberration correction to be possible. At this point, the chromatic power of Lj must be replaced with the aberration-corrected value ωc to achieve the achromatic condition. Additionally, by reorganizing Eq. (17) using the optical power ratio, the apochromatic condition can be rewritten as follows:

Pj(λS,λJ)=  ωe''peωcpjPe(λS,λJ).

In the above Eq. (18), note that the achromatic and athermal system should have the value of −ωe pe / ωc pj = 1. Therefore, the following Eq. (19) must hold true for residual chromatic aberration correction.

Pj(λS,λJ)=  Pe(λS,λJ).

The above Eq. (19) means that when the system is achromatic, the partial dispersion value of a specific lens must be the same as that of an equivalent single lens to achieve an apochromatic optical system.

2.5. Optical Material Selection Method for Apochromatic Design

We confirmed through Eq. (19) that when the achromatic condition is satisfied, the partial dispersion value of Lj should be the same as the partial dispersion value of Le to meet the apochromatic condition. Therefore, in an optical system where residual chromatic aberration has not been corrected, we use the relative error without taking the absolute value to determine whether the difference between the partial dispersion values of a specific lens and an equivalent single lens is positive (+) or negative (−), as shown in Eq. (20):

Relative error of Pe and PjPe(λS , λJ )Pj(λS , λJ )Pe(λS , λJ ).

Next, if a difference in partial dispersion is identified through Eq. (20), the material must be changed to an appropriate one that can reduce this difference. Therefore, we define the partial dispersion value of the available material as Pa(λS, λJ) and calculate the difference with Pe(λS, λJ), as shown in Eq. (21):

Relative error of Pe and PaPe(λS , λJ )Pa(λS , λJ )Pe(λS , λJ ).

By identifying the existing error through Eq. (20) and selecting a material that can reduce this error through Eq. (21), an optical system favorable for apochromatic conditions can be achieved. However, since it is nearly impossible to find a suitable material that has the same refractive index and chromatic power as the material currently in use while only reducing the error in partial dispersion, it is effective to choose materials of the same type (crown or flint) that can reduce residual chromatic aberration. Therefore, to achieve an apochromatic optical system, it is necessary to minimize the difference in partial dispersion through appropriate material selection and change. Then it is desirable to perform numerical redistribution of the first-order quantities to ensure that Pe(λS, λJ) and Pj(λS, λJ) match while maintaining the achromatic condition. According to the partial dispersion of an equivalent single lens given as Pe(λS, λJ) = i=1kPi(λS,λJ)ωi'ϕi'Pj(λS,λJ)ωj'ϕj'/ i=1kωi'ϕi'ωj'ϕj', Pe(λS, λJ) is strongly dependent on the first-order quantities of the lens elements that make up an equivalent single lens. By adjusting these parameters, the lenses can be rearranged to further reduce the difference with Pj(λS, λJ). During this process, the achromatic condition must be maintained, so −ωe pe / ωc pj = 1 must hold.

3.1. Design of a Telephoto Optical System

In this study, to design the structure of a telephoto optical system, we investigate the optical power arrangement of a two-group thin lens system according to the initial specifications [4].

Figure 3 illustrates the two-group (G1, G2) lens system placed in the air. Since they are in the form of a thin lens, the positions of the first principal plane and the second principal plane are the same in each group. Here, z1 is the distance between the second principal plane of the first group (G1) and the first principal plane of the second group (G2), z2 is the distance from the second principal plane of the second group (G2) to the image surface, and L is the total length from the first surface to the image surface (optical total track length). Additionally, we define the telephoto ratio (TR), as the ratio of the optical total track length (L) to the total focal length (fT) of an optical system, i.e., TR = L / fT. After that, the focal length of the first group (f1), the focal length of the second group (f2), and z2 can be expressed as Eqs. (22) to (24), respectively.

Figure 3.Two-group thin lens system in air.

f1' =  z1fT'(1TR)fT'+z1,
f2' =  f1'(TRfT'z1)fT'f1',
z2=  TRfT'z1.

Therefore, if only the total focal length (fT), optical total track length (L), and principal plane spacing (z1) between the groups are determined in the initial specifications of an optical system, the initial focal lengths (f1, f2) of each group and the image distance (z2) can be obtained through the above equations.

3.2. First-order Parameters Determination of Groups According to the Initial Specifications

The specifications of an optical system, such as the image height and operating temperature range, were determined by checking the catalog of an image sensor suitable for a surveillance camera in the in the SWIR waveband [11]. Due to the structure of the image sensor, the effective radius of the sensor entrance is 13.0 mm, and the distance from the entrance to the focal plane array (FPA) is confirmed to be 17.526 mm. Accordingly, in the optical design process, the system must be designed to ensure that all rays can pass through the sensor entrance, and the image distance must be at least 17.526 mm from the last surface of the lens. The total focal length and F-number of an optical system were determined to be 600 mm and F/5, classifying it as a super-telephoto type. While the telephoto ratio in typical telephoto optical systems was determined to be 0.6 to 0.9, in this study, the optical total track length was determined to achieve a compact optical system with a telephoto ratio of less than 0.5 [4]. Table 1 lists the specifications for a SWIR telephoto optical system.

TABLE 1 Target specifications for a short-wave infrared (SWIR) telephoto optical system

ParametersTarget Values
Sensor Type/Format/Pixel PitchInGaAs/1280×1024/10 µm
Wavelengths (µm)0.9–1.7 (SWIR)
Effective Focal Length (mm)600.0
F-number5.0
Image Height (mm)±8.20
Optical Total Track Length (mm)297.526 (including sensor structure)
MTF (@ 50 cycles/mm)More than 30 % (at all fields)
Operating Temperature (℃)−35~+60
Housing MaterialAL6061 (CTE = 23.4 × 10−6/℃)


In this optical design, we initially proceed with the structural design of a two-group thin lens system based on the initially calculated first-order values and aim to obtain an initial optical system by expanding each group into a two-group thick lens system. First, considering the lens arrangement and air spaces for each group, z1 was determined to be 160 mm. Next, when expanding from a two-group thin lens system to a two-group thick lens system, the first-order quantities of an optical system are maintained while being arranged with real lenses. Consequently, the optical total track length of an optical system inevitably becomes longer in a two-group thick lens system. Accordingly, when calculating the target specification TR using the optical total track length, including the structure of real lenses in each group and the sensor, it comes out to 0.496. However, in the initial structural design using a two-group thin lens system, it is necessary to reduce the optical total track length to obtain such a solution. Therefore, TR was determined to be 0.2 for a more compact optical system design.

Finally, the initial first-order parameters for each group can be calculated by substituting the parameters of telephoto ratio (TR) and distance between principal planes (z1) from Table 2 into Eqs. (22) to (24). This allows for the determination of the distance between the principal plane and the image surface (z2), the focal length of the first group (f1), and the focal length of the second group (f2). By applying the calculated first-order parameters to a two-group thick lens system, the arrangement shown in Fig. 4 is obtained. The stop is positioned between the first and second groups to reduce the angle of the chief ray incident on the image surface, as shown in Fig. 4. The distance from the last surface of the second group to the image surface must be at least 17.526 mm due to the sensor structure.

Figure 4.Two-group optical system in air.

TABLE 2 Calculation of the first-order parameters in an initial two-group thin lens system

ParametersValues
Telephoto Ratio (TR) of Thin Lens System0.2
Distance Between Principal Planes (z1) (mm)160.0
Distance from the Principal Plane to Image Surface (z2) (mm)−40.0
Effective Focal Length of Group 1 (f1) (mm)150.0
Effective Focal Length of Group 2 (f2) (mm)13.333


The optical system being designed is a large-aperture system, and because the entrance pupil diameter (EPD) is very large, significant aberrations occur in the first group with positive optical power. Therefore, the refraction angles of the rays must be considered when the lenses are arranged. For these reasons, the principal planes of the first group tend to be positioned on the left. Additionally, while the second principal plane of the second group with positive optical power is located to the right of the image surface, the last surface of the real lens must be located to the left of the image plane. To achieve this structure, it is advantageous to arrange the optical power of the second group as negative power N (−) and positive power P (+) because the second principal plane will be located on the right. Also, placing negative lenses in areas with low paraxial ray height provides a favorable structure for Petzval sum correction.

Based on the calculated first-order parameters, the first and second groups are designed separately in this study using achromatic and athermal design methods for each group. In particular, the apochromatic method will be applied to the first group design with a large aperture. An initial telephoto optical system will then be built by combining the two groups.

3.3. Design of the First Group Lens

From the initial specifications, the first-order parameters for each group are obtained, and it is necessary to convert these into real lenses with the same first-order parameters. In this process, each group is independently converted into real lenses. First, as shown in Fig. 5, the first group is composed of six lenses by using multiple meniscus lenses to consider the refraction angles of the rays. This configuration is designed with an F/5 and a reference wavelength of 1.3 µm.

Figure 5.Optical layout of the first group.

Since a final optical system with the two combined groups has a large EPD, the diameter of the front lens in the first group becomes larger. Thus, the thicknesses of the first and third lenses with positive optical power increase. Therefore, the first and third lenses are fixed with NFK58, which is a Schott glass material with the highest transmittance in the SWIR wavelength range, and initial lens data is obtained experientially. To achieve the achromatic and athermal configuration of the first group, the thick lenses are converted to thin lenses, and the optical properties for each lens are listed in Table 3.

TABLE 3 Optical properties of the first group lens

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−6/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1NFK5812.2749−27.86130.00443915.0000
2NLAF218.7880−8.3744−0.00658914.0902
3NFK5812.2749−27.86130.00704614.0090
4NPK5113.1695−25.81780.00899012.9302
5NPSK53A16.8679−14.4508−0.01224910.7414
6NPSK53A16.8679−14.45080.0050378.5157


From Table 3, six cases can be classified according to the selection of a specific lens, and the values of Mh(0, −αh), Le(ωe , γe), and Lc(ωc, γc) can be calculated. After analyzing each case, it is confirmed that case 2, where the line MhLc exists near the available material, can reduce the MSI the most. Here, since the telephoto optical system will use a compensating lens, MSI analysis is conducted by setting Wc and Wt to 1.0 and 0.5 respectively, according to the importance of chromatic aberration correction rather than thermal defocus. As shown in Fig. 6(a) of case 2, the line MhLc differs from the line MhLj, and we need to reduce this difference with material selection and change. MSI values of the available materials were analyzed as shown in Table 4. An MSI value of 1 indicates the material of the currently used specific lens. An MSI value less than 1 signifies that the new material is more effective in aberration correction than the current material, while an MSI value greater than 1 indicates that it is less effective in aberration correction compared to the current material. Accordingly, if the material of the second lens, currently NLAF2, is changed to NSF10 with the smallest MSI and a similar refractive index while maintaining the first-order parameters, the glass map altered for achromatic and athermal design is as shown in Fig. 6(b).

Figure 6.Achromatic and athermal glass map analysis of case 2: (a) Initial first group, (b) first group after material selection and change.

TABLE 4 Material selection index (MSI) for each case of the first group lens

CaseRankMaterialMaterial Selection Index (MSI)
Case 21NSF100.6329
2NSF40.6361
3NSF140.6590
4NSF10.6629
5PSF690.6676


In that figure, the difference between Lc and Lj along the horizontal axis, i.e., chromatic power, is significantly reduced, and there is a slight reduction along the vertical axis. Consequently, the chromatic aberration is greatly reduced, and the thermal defocus has also decreased slightly. This shows that it is possible to achieve a material arrangement that considers both aberrations by simply selecting and changing materials. This approach results in a first group that is both achromatic and athermal. However, the targeted optical system is sensitive to large focal length changes owing to the wide wavelength range of SWIR. Therefore, we should achieve a more stable configuration to prevent wavelength variations by additionally implementing an apochromatic method. For apochromatic analysis, the achromatic condition must be satisfied: The chromatic power of the second lens used as a specific lens should be close to the ωc = 23.9616 (×10−3) of the aberration-corrected point. If not, these parameters can be matched to be the same with minimal change of first-order quantities. The changed optical properties are shown in Table 5.

TABLE 5 Optical properties of the first group lens

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−6/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1NFK5812.2749−27.86130.00447915.0000
2NSF1023.9430−10.4130−0.00662514.1043
3NFK5812.2749−27.86130.00693814.0291
4NPK5113.1695−25.81780.00893412.9741
5NPSK53A16.8679−14.4508−0.01209210.7511
6NPSK53A16.8679−14.45080.0050388.5389


This resulted in the first group being perfectly achromatic. At this point, we aim to verify the materials that satisfy the apochromatic condition, as shown in Table 6. Here, cases 1 and 3 were fixed with the highest-transmittance materials, and so they were excluded from the material change cases.

TABLE 6 Material selection index and relative error of partial dispersion for each case

CaseMaterialMaterial Selection Index (MSI)Relative Error from Pe (×10−3)
Case 2
NBAK41.7511.145
NPK52A1.7550.717
K5G201.625−0.340
NKZFS111.218−2.977
NSF101.000−169.818
Case 4SF570.774−7.959
SF60.75711.893
SF6G050.74812.297
SF56A0.76624.790
NPK511.000209.790
Case 5FK5HTI1.589−168.998
NFK51.595−169.302
NBK102.021−174.241
NKZFS22.694−188.186
NPSK53A1.000−296.679
Case 6PSF681.676283.226
SF571.581300.485
SF61.529314.262
SF6G051.536314.542
NPSK53A1.000465.400


Table 6 lists the relative errors for each material, calculated with reference to partial dispersion Pe, in order, along with the corresponding MSI values. To satisfy the apochromatic condition, the relative error between the partial dispersion Pj of the specific lens and the partial dispersion Pe of the equivalent single lens should be zero. However, Table 6 shows that there are significant differences between them. Therefore, it is necessary to select materials that can reduce these values while maintaining the existing MSI values to consider athermalization. For this reason, the materials in case 2 can make the relative error of partial dispersion small. Specifically, NKZFS11 was selected for its similarity to the NSF10 type along with its minimal change in MSI values. After replacing it with the selected material, first-order quantity redistribution was conducted to correct residual chromatic aberration. This design process ensured that the value of Af in case 2 was reduced. The layout of the first group built through this process is shown in Fig. 7. At this time, the first lens group was converted back to a thick lens system. Also, the optical properties of each lens are listed in Table 7, and the relative errors of partial dispersion are presented in Table 8.

Figure 7.. Optical layout of the first group obtained from the apochromatic and athermal design.

TABLE 7 Optical properties of the first group obtained from the first-order quantity redistribution for each case

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−6/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1NFK5812.2749−27.86130.00277815.0000
2NKZFS1123.9421−1.0765−0.00860714.3387
3NFK5812.2749−27.86130.00298313.7387
4NPK5113.1695−25.81780.00761314.6523
5NPSK53A16.8679−14.4508−0.00172713.5127
6NPSK53A16.8679−14.45080.00402413.0627


TABLE 8 Material selection index and relative error between both partial dispersions for case 2

CaseMaterialMaterial Selection Index (MSI)Relative Error from Pe (×10−3)
Case 2
NPK52A1.0943.683
K5G201.2462.629
NKZFS111.0000.000
PLAK351.199−3.131
NLAK221.421−5.625


3.4. Design of the Second Group Lens

Figure 8 illustrates the second group that is designed to have the same first-order parameters in Table 2.

Figure 8.Optical layout of the second group.

The first four lenses (first to fourth) are in the negatively powered group N (−), and the last two lenses (fifth and sixth) are in the positively powered group P (+). They are then placed using the available materials in the same manner as the first group design. To achieve the achromatic and athermal configuration of the second group, the thick lenses are converted to thin lenses, and the optical properties for each lens are listed in Table 9. Based on these properties, six cases can be classified according to the selection of a specific lens, and the values of Mh(0, −αh), Le(ωe , γe), and Lc(ωc , γc) can be calculated. After analyzing all cases, it is confirmed that the lines of MhLc in cases 4 and 5 are nearest to the available material distribution, as shown in Fig. 9.

Figure 9.Achromatic and athermal glass map analysis: (a) Case 4 and (b) case 5 of second group.

TABLE 9 Optical properties of the second group lens

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−6/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1NPSK53A12.2749−27.86130.00277815.0000
2NPK5123.9421−1.0765−0.00860714.3387
3NLAF212.2749−27.86130.00298313.7387
4NLAF213.1695−25.81780.00761314.6523
5SF5716.8679−14.4508−0.00172713.5127
6SF5716.8679−14.45080.00402413.0627


Table 10 presents MSI analyses for cases 4 and 5, which are the most advantageous in aberration correction among the six cases. Through this analysis, SF11 with a lower MSI and similar refractive index is selected as the fourth lens material for case 4. For the same reason, NPSK53A is selected as the fifth lens material for case 5. Next, by changing the previous glasses to the newly selected materials, the second group can reduce chromatic aberration and thermal defocus using the same principles outlined in the first group design. The glass map after the material change is shown in Fig. 10.

Figure 10.Achromatic and athermal glass map analysis after material selection and change: (a) Case 4 and (b) case 5 of second group.

As shown in Fig. 10(a) and 10(b), the difference between the two thermal powers is significantly reduced to be almost similar along the vertical axis. In addition, the chromatic power difference is slightly reduced along the horizontal axis. Here, it is difficult to use the apochromatic method for the second group in all cases due to the significant differences in material properties. However, the first group already has a more favorable configuration for apochromatic correction, and the second group has a lower ray height compared to that of the first group, which results in relatively less impact on aberration correction. Therefore, in the design of the second lens group, it is desirable to focus on material arrangements favorable for achromatic and athermal configurations. The layout, converted back to thick lenses, is shown in Fig. 11.

Figure 11.Optical layout of the second group obtained from the achromatic and athermal design.

To achieve the optical system arrangement as Fig. 4, the first and second groups designed separately were combined. The initial optical system configured through this process is shown in Fig. 12.

Figure 12.The initial optical system by combining two groups designed separately.

Finally, the optimization process is performed to meet the target specifications using the starting lens of Fig. 12. To meet the target specifications, the aperture and field size are increased to the F-number of F/5 and image size of ±8.2 mm. The high-order aberrations that arise during this process are corrected using eight aspherical surfaces on five lenses (6, 7, 9, 11, and 12). Finally, to perform the eighth lens as a compensator, sufficient air space was secured in front of and behind this lens. The optical system was also designed to have the minimum object distance and loose manufacturing tolerances. The specifications of the final designed optical system are listed in Table 11. The layout and optical performance analysis are shown in Fig. 13 to Fig. 15. This optical system fulfills the target specifications and yields an apochromatic and athermal configuration.

Figure 13.Layout of the final designed optical system.

Figure 15.Apochromatic and athermal design: (a) Longitudinal chromatic aberration graph and (b) modulation transfer functions (MTFs) from −35 ℃ to +60 ℃ of the final designed optical system.

In this study, we conducted achromatic and athermal analysis using a glass map, and then proposed a method for analyzing multiple optical materials and selecting the most suitable one by introducing an optical material selection index (MSI). In cases where stable performance is required in a wide wavelength range, we proposed a new optical material selection process for apochromatic design and a first-order quantity redistribution method. The application of these approaches to the design of a telephoto optical system for SWIR confirmed the usefulness of these methods. In this design process, we determined the first-order parameters for each group based on the initial specifications and performed the basic design with a real lens arrangement. In this step, we first selected a specific lens glass suitable for achromatic and athermal design in each group, and then applied the apochromatic design method.

Thus, each group designed independently was combined to form the initial telephoto optical system, which yielded an achromatic (apochromatic) and athermal design. Finally, with design optimization, we achieved the final optical system that meets the target specifications and secures stable optical performance in the operating environment.

In conclusion, this proposed design approach is expected to provide a useful means of determining the glasses for achromatic (apochromatic) and athermal designs over a specified temperature and extremely wide waveband ranges such as a SWIR system.

Figure 14.Modulation transfer function (MTF) chart at room temperature of the final designed optical system.

TABLE 10 Material selection index (MSI) for each case of the second group lens

CaseRankMaterialMaterial Selection Index (MSI)
Case 41PSF680.8464
2NZK70.8589
3SF110.8617
4NZK7A0.8623
5NKZFS20.8647
Case 5
4NPK510.6335
5NPK52A0.6342
6PPK530.7139
7NPSK53A0.7402


TABLE 11 Specifications of the final designed telephoto optical system for SWIR

ParametersTarget Values
Sensor Type/Format/Pixel PitchInGaAs/1280×1024/10 µm
Wavelengths (µm)0.9–1.7 (SWIR)
Effective Focal Length (mm)600.0
F-number5.0
Image Height (mm)±8.20
Field of View (deg.)±0.7804
Optical Total Track Length (mm)297.526 (including sensor structure)
MTF (@ 50 cycles/mm)More than 37.9 %
Relative Illumination (%)More than 98.9
Distortion (%)Less than 0.47
Operating Temperature (℃)−35~+60
Housing MaterialAL6061 (CTE = 23.4 × 10−6/℃)

The authors received no financial support for the research, authorship, and/or publication of this article.

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

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Article

Research Paper

Curr. Opt. Photon. 2024; 8(5): 472-483

Published online October 25, 2024 https://doi.org/10.3807/COPP.2024.8.5.472

Copyright © Optical Society of Korea.

Design of a Telephoto Optical System for SWIR Using Apochromatic and Athermal Method

Tae-Sik Ryu, Sung-Chan Park

Department of Physics, Dankook University, Cheonan 31116, Korea

Correspondence to:*scpark@dankook.ac.kr, ORCID 0000-0003-1932-5086

Received: August 26, 2024; Accepted: September 23, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents an intuitive method for selecting an optical material for achromatic and athermal design using the material selection index (MSI). In addition, in the case of a wide wavelength range such as a short-wave infrared (SWIR) waveband, we propose a new material selection method for apochromatic and athermal design by introducing the relative error of partial dispersion (REPD) and a first-order quantity redistribution method. To obtain a suitable material for effective apochromatic design, we first evaluate the REPDs of all lenses, deviated from that of an equivalent lens. Materials with a small REPD are then selected on a glass map to correct residual chromatic aberration while maintaining the existing MSI values to realize athermalization simultaneously. Using this proposed glass selection method, we successfully obtained an apochromatic and athermal telephoto system for SWIR that realizes stable performance over the specified temperature and wide waveband ranges.

Keywords: Aberrations, Apochromatization, Athermalization, Glass map

I. INTRODUCTION

The short-wave infrared (SWIR) wavelength from 0.9 µm to 1.7 µm uses light reflected from objects. With longer wavelengths than visible light, SWIR experiences less scattering and is advantageous in environments with turbulence, fog, haze, smoke, and clouds. Additionally, it can detect low levels of reflected light from long distances and recognize objects at night using reflected light. Recently, the development of electro-optics equipment for security and surveillance has been actively progressing, with research reported on replacing visible light images with SWIR images. However, due to the wider wavelength range of SWIR compared to visible light, stabilization to prevent wavelength changes in an optical system is required. Common optical materials are mainly developed for the visible wavelength range, and the lack of strong flint optical materials makes it challenging to effectively correct chromatic aberration in the SWIR waveband [13].

This study aims to design a surveillance telephoto optical system that uses the advantages of the SWIR waveband. Its long focal length makes it sensitive to temperature changes and object distance, so it requires a compensator [4]. However, in compact optical systems, performance stability issues can arise even with a compensator. Therefore, careful optical material arrangement is necessary [5]. We aim to design an optical system considering these issues. First, an initial solution is obtained using a two-group optical system, and then the achromatic and athermal optical material selection method is applied for each group using a glass map. Next, for groups where apochromatic design is feasible, optical material selection and a first-order quantity redistribution method are introduced to configure the initial apochromatic system. The SWIR telephoto optical system with a telephoto ratio of less than 0.5, configured through this process, was found to be stable across the wavelength range and advantageous for temperature compensation.

II. ACHROMATIC (APOCHROMATIC) AND ATHERMAL DESIGN METHODS

2.1. Achromatic and Athermal Conditions

Changes in the optical power of the lens due to variations in wavelength and temperature can be expressed using the chromatic power (ωi) and thermal power (γi) of the element material (Mi), as shown in Eqs. (1) and (2) [69]:

ωi=  1vi=  Δϕiϕi=  Δλni1niλ,
γi=  ϕiT1ϕi=  1ni1niTαi,

where ∆λ is the specified waveband, φi is the element optical power, vi is the Abbe number, ni is the refractive index at the reference wavelength, αi is the coefficient of thermal expansion (CTE) of the i-th lens material, and T is the temperature.

Longitudinal chromatic aberration arises from changes (∆fbch) in the back focal length (BFL) with wavelength and is expressed by Eq. (3). Thermal defocus (∆z′) is evaluated as the difference between the change (∆fbth) in the BFL with temperature and the change (∆Hb) in the flange back length (FBL) with temperature, as given in Eq. (4) [810]:

Δfb'-ch=  1ϕT2 i=1kωi'ϕi',
Δz' =  Δfb'-thΔHb' =  1ϕT 2 i=1kγi'ϕi'αhLΔT,

where φT is the total power and k is the total number of lens elements. In the above two equations, the primed parameters indicate that they are weighted by the ratio of the paraxial ray heights and are expressed as ϕi = (hi / h1)φi, ωi = (hi / h1)ωi, and γi = (hi / h1)γi, respectively. This implies that the air spacings between elements are included in Eqs. (3) and (4).

2.2. Method for Representing the Achromatic and Athermal Conditions on a Glass Map

In this study, an equivalent single lens is used to simplify an optical system with an arbitrary number of elements into a doublet system. Thus, an optical system with k elements can be recomposed into a doublet system composed of the specific j-th element Lj and an equivalent single lens Le. This equivalent single lens consists of the remaining k-1 elements. Therefore, in this separated doublet system composed of Lj and Le, the total optical power (φT), achromatic (∆fbth = 0), and athermal (∆z′ = 0) conditions are respectively given by [810]:

ϕT=   i=1kϕi'=  ϕj'+ϕe',
Δfb'-ch=  1ϕT2ωj'ϕj'+ωe'ϕe'=  0,
Δz' =  1ϕT2γj'ϕj'+γe'ϕe'αhLΔT=  0,

where ϕe' =   i=1kϕi 'ϕj', ωe' =   i=1k ωi 'ϕi ' ωj 'ϕj '/ ϕ e ', and γe' =   i=1k γi 'ϕi ' γj 'ϕj '/ ϕ e '.

By dividing the achromatic condition of Eq. (6) and the athermal condition of Eq. (7) in this doublet system by the ratio of the paraxial ray height (hi / h1), we can easily identify a specific lens location without weighting on a glass map. Additionally, dividing Eqs. (5), (6), and (7) by the total power (φT) leads to expressions for the achromatic and athermal conditions in a doublet system as follows [9, 10]:

pj+pe=  1,
ωjpj+ωe''pe=  0,
γjpj+γe''pe=  αh'',

where ωe'' =  ( h1 / hj )ωe', γe'' =  ( h1 / hj )γe', αh = (hk / hj)αh, pj = ϕj / φT, and pe = ϕe / φT. Thus, two parameters of pj and pe are the ratios of optical powers of Lj and Le with respect to the total power.

When an equivalent single lens is given, the point designated as Lc(ωc, γc) in Fig. 1 denotes the achromatic and athermal point of a specific lens, which we refer to as the aberration-corrected point for these two errors, or briefly, Lc(ωc, γc). By combining the achromatic condition of Eq. (9) and the athermal condition of Eq. (10) with the optical power equation of Eq. (8), we can rewrite the achromatic and athermal conditions as follows [10]:

Figure 1. Achromatic and athermal conditions on a glass map.

pjpe=  1ωe''ωcωe''ωc,
pjpe=  1γe''γcγe''+αh''γcαh,

where ωc = −ωe (pe / pj), and γc = −(γe + αh)(pe / pj) − αh. To represent the condition that simultaneously satisfies both achromatic and athermal requirements on a glass map, we can rearrange the above Eqs. (11) and (12), as follows [10]:

γc+αh''ωc  =  γe''+αh''ωe''.

Equation (13) holds because the left side uses Lc(ωc, γc) instead of Lj(ωj, γj). Thus, the difference in material properties between Lj and Le causes chromatic aberration and thermal defocus. Accordingly, it can be defined as an aberration factor (Af) using the sum of the relative error, as given in Eq. (14):

Af  ωcωjωc+γcγjγc.

In Eq. (13), the achromatic and athermal conditions require ideal material properties Lc(ωc, γc). The left side of this equation represents the slope of the line connecting housing material properties Mh(0, −αh) and Lc(ωc, γc), while the right side represents the slope of the line connecting Mh(0, −αh) and Le(ωe, γc). If Lc for a specific lens is not used, the equation does not hold. This can be represented on a glass map, and the aberration factor (Af) defined by Eq. (14) can also be depicted, as shown in Fig. 1.

2.3. Optical Material Selection Method for Achromatic and Athermal Design on a Glass Map

An optical system that does not satisfy achromatic and athermal conditions will have the lines of MhLj and MhLc, but they do not coincide, as shown in Fig. 1. The simplest method to align these lines as closely as possible is to change the material. To achieve this, it is necessary to select the most suitable material from the available materials (La) distributed on a glass map. In this process, a material selection index (MSI) is defined as the relative error between the material properties of Lc(ωc, γc) and La(ωa, γa), as given in Eq. (15). This MSI is used for material selection.

MSIωcωaωcWc+γcγaγcWtAf',

where Af is weighted aberration factor, that is Af(ωcωj)/ωcWc+(γcγj)/γcWt. In Eq. (15), the weighted aberration factor is represented using Wc for chromatic aberration weighting and Wt for thermal defocus weighting. For an optical system where chromatic aberration correction is more important, Wc is set higher, while for an optical system where thermal defocus correction is more important, Wt is set higher. This allows for identifying the weighted aberration factor according to the importance of aberration correction in an optical system. Additionally, by using the sum of the weighted relative errors of the material properties of Lc and La, we can select the material that is close to Lc. Dividing this by Af leads to the MSI of Eq. (15), which implies how much the available material can reduce existing aberration factors, being compared to the material used in a specific lens. Therefore, the MSI values for each available material on a glass map can be calculated and listed to select a reasonable material for aberration correction.

2.4. Apochromatic Condition

Generally, achromatic refers to matching the focal lengths at the wavelengths of both ends. However, a difference in focal length occurs at other wavelengths within both ends. If an additional wavelength is specified to match the focal length, a more stable optical system can be achieved with respect to wavelength changes. An optical system that corrects chromatic aberration at these three wavelengths is called an apochromatic system [4].

As shown in Fig. 2, an achromatic optical system only matches the BFLs (fbS, fbL) for the wavelengths at both ends (λS, λL). For the BFL (fbJ) of a specific wavelength (λJ) within the wavelength range, residual chromatic aberration occurs. Therefore, when the system is achromatic, if we define the difference in BFL between the short wavelength and the specific wavelength as ∆fbchJ from Eq. (3), it can be expressed as follows:

Figure 2. Residual chromatic aberration in an achromatic optical system.

Δfb'-ch-J=  1ϕT2 i=1khih1nSnJnR1×hih1(ci1ci2)(nR1)                =  1ϕT2 i=1kPi(λS,λJ)ωi'ϕi' ,

where nR is the refractive index of the reference wavelength and Pi(λS, λJ) is the partial dispersion of the i-th lens material, and is defined as Pi(λS,λJ)=(nS nJ )/(nS nL ). Next, by introducing the concept of an equivalent single lens to simplify the above Eq. (16) into a doublet system, the residual chromatic aberration correction (i.e., apochromatic) condition can be expressed as follows:

Δfb'-ch-J=  1ϕT2Pj(λS , λJ )ωj'ϕj'+Pe(λS , λJ )ωe'ϕe'=  0,

where Pe(λS,λJ)= i=1k Pi( λ S , λ J )ωi'ϕi' P j( λ S , λ J ) ωj 'ϕj '/ i=1k ωi'ϕi'ωj 'ϕj '.

In Eq. (17), since the total optical power of an imaging optical system cannot be zero, the term inside the brackets must be zero for residual chromatic aberration correction to be possible. At this point, the chromatic power of Lj must be replaced with the aberration-corrected value ωc to achieve the achromatic condition. Additionally, by reorganizing Eq. (17) using the optical power ratio, the apochromatic condition can be rewritten as follows:

Pj(λS,λJ)=  ωe''peωcpjPe(λS,λJ).

In the above Eq. (18), note that the achromatic and athermal system should have the value of −ωe pe / ωc pj = 1. Therefore, the following Eq. (19) must hold true for residual chromatic aberration correction.

Pj(λS,λJ)=  Pe(λS,λJ).

The above Eq. (19) means that when the system is achromatic, the partial dispersion value of a specific lens must be the same as that of an equivalent single lens to achieve an apochromatic optical system.

2.5. Optical Material Selection Method for Apochromatic Design

We confirmed through Eq. (19) that when the achromatic condition is satisfied, the partial dispersion value of Lj should be the same as the partial dispersion value of Le to meet the apochromatic condition. Therefore, in an optical system where residual chromatic aberration has not been corrected, we use the relative error without taking the absolute value to determine whether the difference between the partial dispersion values of a specific lens and an equivalent single lens is positive (+) or negative (−), as shown in Eq. (20):

Relative error of Pe and PjPe(λS , λJ )Pj(λS , λJ )Pe(λS , λJ ).

Next, if a difference in partial dispersion is identified through Eq. (20), the material must be changed to an appropriate one that can reduce this difference. Therefore, we define the partial dispersion value of the available material as Pa(λS, λJ) and calculate the difference with Pe(λS, λJ), as shown in Eq. (21):

Relative error of Pe and PaPe(λS , λJ )Pa(λS , λJ )Pe(λS , λJ ).

By identifying the existing error through Eq. (20) and selecting a material that can reduce this error through Eq. (21), an optical system favorable for apochromatic conditions can be achieved. However, since it is nearly impossible to find a suitable material that has the same refractive index and chromatic power as the material currently in use while only reducing the error in partial dispersion, it is effective to choose materials of the same type (crown or flint) that can reduce residual chromatic aberration. Therefore, to achieve an apochromatic optical system, it is necessary to minimize the difference in partial dispersion through appropriate material selection and change. Then it is desirable to perform numerical redistribution of the first-order quantities to ensure that Pe(λS, λJ) and Pj(λS, λJ) match while maintaining the achromatic condition. According to the partial dispersion of an equivalent single lens given as Pe(λS, λJ) = i=1kPi(λS,λJ)ωi'ϕi'Pj(λS,λJ)ωj'ϕj'/ i=1kωi'ϕi'ωj'ϕj', Pe(λS, λJ) is strongly dependent on the first-order quantities of the lens elements that make up an equivalent single lens. By adjusting these parameters, the lenses can be rearranged to further reduce the difference with Pj(λS, λJ). During this process, the achromatic condition must be maintained, so −ωe pe / ωc pj = 1 must hold.

III. DESIGN EXAMPLE

3.1. Design of a Telephoto Optical System

In this study, to design the structure of a telephoto optical system, we investigate the optical power arrangement of a two-group thin lens system according to the initial specifications [4].

Figure 3 illustrates the two-group (G1, G2) lens system placed in the air. Since they are in the form of a thin lens, the positions of the first principal plane and the second principal plane are the same in each group. Here, z1 is the distance between the second principal plane of the first group (G1) and the first principal plane of the second group (G2), z2 is the distance from the second principal plane of the second group (G2) to the image surface, and L is the total length from the first surface to the image surface (optical total track length). Additionally, we define the telephoto ratio (TR), as the ratio of the optical total track length (L) to the total focal length (fT) of an optical system, i.e., TR = L / fT. After that, the focal length of the first group (f1), the focal length of the second group (f2), and z2 can be expressed as Eqs. (22) to (24), respectively.

Figure 3. Two-group thin lens system in air.

f1' =  z1fT'(1TR)fT'+z1,
f2' =  f1'(TRfT'z1)fT'f1',
z2=  TRfT'z1.

Therefore, if only the total focal length (fT), optical total track length (L), and principal plane spacing (z1) between the groups are determined in the initial specifications of an optical system, the initial focal lengths (f1, f2) of each group and the image distance (z2) can be obtained through the above equations.

3.2. First-order Parameters Determination of Groups According to the Initial Specifications

The specifications of an optical system, such as the image height and operating temperature range, were determined by checking the catalog of an image sensor suitable for a surveillance camera in the in the SWIR waveband [11]. Due to the structure of the image sensor, the effective radius of the sensor entrance is 13.0 mm, and the distance from the entrance to the focal plane array (FPA) is confirmed to be 17.526 mm. Accordingly, in the optical design process, the system must be designed to ensure that all rays can pass through the sensor entrance, and the image distance must be at least 17.526 mm from the last surface of the lens. The total focal length and F-number of an optical system were determined to be 600 mm and F/5, classifying it as a super-telephoto type. While the telephoto ratio in typical telephoto optical systems was determined to be 0.6 to 0.9, in this study, the optical total track length was determined to achieve a compact optical system with a telephoto ratio of less than 0.5 [4]. Table 1 lists the specifications for a SWIR telephoto optical system.

TABLE 1. Target specifications for a short-wave infrared (SWIR) telephoto optical system.

ParametersTarget Values
Sensor Type/Format/Pixel PitchInGaAs/1280×1024/10 µm
Wavelengths (µm)0.9–1.7 (SWIR)
Effective Focal Length (mm)600.0
F-number5.0
Image Height (mm)±8.20
Optical Total Track Length (mm)297.526 (including sensor structure)
MTF (@ 50 cycles/mm)More than 30 % (at all fields)
Operating Temperature (℃)−35~+60
Housing MaterialAL6061 (CTE = 23.4 × 10−6/℃)


In this optical design, we initially proceed with the structural design of a two-group thin lens system based on the initially calculated first-order values and aim to obtain an initial optical system by expanding each group into a two-group thick lens system. First, considering the lens arrangement and air spaces for each group, z1 was determined to be 160 mm. Next, when expanding from a two-group thin lens system to a two-group thick lens system, the first-order quantities of an optical system are maintained while being arranged with real lenses. Consequently, the optical total track length of an optical system inevitably becomes longer in a two-group thick lens system. Accordingly, when calculating the target specification TR using the optical total track length, including the structure of real lenses in each group and the sensor, it comes out to 0.496. However, in the initial structural design using a two-group thin lens system, it is necessary to reduce the optical total track length to obtain such a solution. Therefore, TR was determined to be 0.2 for a more compact optical system design.

Finally, the initial first-order parameters for each group can be calculated by substituting the parameters of telephoto ratio (TR) and distance between principal planes (z1) from Table 2 into Eqs. (22) to (24). This allows for the determination of the distance between the principal plane and the image surface (z2), the focal length of the first group (f1), and the focal length of the second group (f2). By applying the calculated first-order parameters to a two-group thick lens system, the arrangement shown in Fig. 4 is obtained. The stop is positioned between the first and second groups to reduce the angle of the chief ray incident on the image surface, as shown in Fig. 4. The distance from the last surface of the second group to the image surface must be at least 17.526 mm due to the sensor structure.

Figure 4. Two-group optical system in air.

TABLE 2. Calculation of the first-order parameters in an initial two-group thin lens system.

ParametersValues
Telephoto Ratio (TR) of Thin Lens System0.2
Distance Between Principal Planes (z1) (mm)160.0
Distance from the Principal Plane to Image Surface (z2) (mm)−40.0
Effective Focal Length of Group 1 (f1) (mm)150.0
Effective Focal Length of Group 2 (f2) (mm)13.333


The optical system being designed is a large-aperture system, and because the entrance pupil diameter (EPD) is very large, significant aberrations occur in the first group with positive optical power. Therefore, the refraction angles of the rays must be considered when the lenses are arranged. For these reasons, the principal planes of the first group tend to be positioned on the left. Additionally, while the second principal plane of the second group with positive optical power is located to the right of the image surface, the last surface of the real lens must be located to the left of the image plane. To achieve this structure, it is advantageous to arrange the optical power of the second group as negative power N (−) and positive power P (+) because the second principal plane will be located on the right. Also, placing negative lenses in areas with low paraxial ray height provides a favorable structure for Petzval sum correction.

Based on the calculated first-order parameters, the first and second groups are designed separately in this study using achromatic and athermal design methods for each group. In particular, the apochromatic method will be applied to the first group design with a large aperture. An initial telephoto optical system will then be built by combining the two groups.

3.3. Design of the First Group Lens

From the initial specifications, the first-order parameters for each group are obtained, and it is necessary to convert these into real lenses with the same first-order parameters. In this process, each group is independently converted into real lenses. First, as shown in Fig. 5, the first group is composed of six lenses by using multiple meniscus lenses to consider the refraction angles of the rays. This configuration is designed with an F/5 and a reference wavelength of 1.3 µm.

Figure 5. Optical layout of the first group.

Since a final optical system with the two combined groups has a large EPD, the diameter of the front lens in the first group becomes larger. Thus, the thicknesses of the first and third lenses with positive optical power increase. Therefore, the first and third lenses are fixed with NFK58, which is a Schott glass material with the highest transmittance in the SWIR wavelength range, and initial lens data is obtained experientially. To achieve the achromatic and athermal configuration of the first group, the thick lenses are converted to thin lenses, and the optical properties for each lens are listed in Table 3.

TABLE 3. Optical properties of the first group lens.

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−6/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1NFK5812.2749−27.86130.00443915.0000
2NLAF218.7880−8.3744−0.00658914.0902
3NFK5812.2749−27.86130.00704614.0090
4NPK5113.1695−25.81780.00899012.9302
5NPSK53A16.8679−14.4508−0.01224910.7414
6NPSK53A16.8679−14.45080.0050378.5157


From Table 3, six cases can be classified according to the selection of a specific lens, and the values of Mh(0, −αh), Le(ωe , γe), and Lc(ωc, γc) can be calculated. After analyzing each case, it is confirmed that case 2, where the line MhLc exists near the available material, can reduce the MSI the most. Here, since the telephoto optical system will use a compensating lens, MSI analysis is conducted by setting Wc and Wt to 1.0 and 0.5 respectively, according to the importance of chromatic aberration correction rather than thermal defocus. As shown in Fig. 6(a) of case 2, the line MhLc differs from the line MhLj, and we need to reduce this difference with material selection and change. MSI values of the available materials were analyzed as shown in Table 4. An MSI value of 1 indicates the material of the currently used specific lens. An MSI value less than 1 signifies that the new material is more effective in aberration correction than the current material, while an MSI value greater than 1 indicates that it is less effective in aberration correction compared to the current material. Accordingly, if the material of the second lens, currently NLAF2, is changed to NSF10 with the smallest MSI and a similar refractive index while maintaining the first-order parameters, the glass map altered for achromatic and athermal design is as shown in Fig. 6(b).

Figure 6. Achromatic and athermal glass map analysis of case 2: (a) Initial first group, (b) first group after material selection and change.

TABLE 4. Material selection index (MSI) for each case of the first group lens.

CaseRankMaterialMaterial Selection Index (MSI)
Case 21NSF100.6329
2NSF40.6361
3NSF140.6590
4NSF10.6629
5PSF690.6676


In that figure, the difference between Lc and Lj along the horizontal axis, i.e., chromatic power, is significantly reduced, and there is a slight reduction along the vertical axis. Consequently, the chromatic aberration is greatly reduced, and the thermal defocus has also decreased slightly. This shows that it is possible to achieve a material arrangement that considers both aberrations by simply selecting and changing materials. This approach results in a first group that is both achromatic and athermal. However, the targeted optical system is sensitive to large focal length changes owing to the wide wavelength range of SWIR. Therefore, we should achieve a more stable configuration to prevent wavelength variations by additionally implementing an apochromatic method. For apochromatic analysis, the achromatic condition must be satisfied: The chromatic power of the second lens used as a specific lens should be close to the ωc = 23.9616 (×10−3) of the aberration-corrected point. If not, these parameters can be matched to be the same with minimal change of first-order quantities. The changed optical properties are shown in Table 5.

TABLE 5. Optical properties of the first group lens.

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−6/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1NFK5812.2749−27.86130.00447915.0000
2NSF1023.9430−10.4130−0.00662514.1043
3NFK5812.2749−27.86130.00693814.0291
4NPK5113.1695−25.81780.00893412.9741
5NPSK53A16.8679−14.4508−0.01209210.7511
6NPSK53A16.8679−14.45080.0050388.5389


This resulted in the first group being perfectly achromatic. At this point, we aim to verify the materials that satisfy the apochromatic condition, as shown in Table 6. Here, cases 1 and 3 were fixed with the highest-transmittance materials, and so they were excluded from the material change cases.

TABLE 6. Material selection index and relative error of partial dispersion for each case.

CaseMaterialMaterial Selection Index (MSI)Relative Error from Pe (×10−3)
Case 2
NBAK41.7511.145
NPK52A1.7550.717
K5G201.625−0.340
NKZFS111.218−2.977
NSF101.000−169.818
Case 4SF570.774−7.959
SF60.75711.893
SF6G050.74812.297
SF56A0.76624.790
NPK511.000209.790
Case 5FK5HTI1.589−168.998
NFK51.595−169.302
NBK102.021−174.241
NKZFS22.694−188.186
NPSK53A1.000−296.679
Case 6PSF681.676283.226
SF571.581300.485
SF61.529314.262
SF6G051.536314.542
NPSK53A1.000465.400


Table 6 lists the relative errors for each material, calculated with reference to partial dispersion Pe, in order, along with the corresponding MSI values. To satisfy the apochromatic condition, the relative error between the partial dispersion Pj of the specific lens and the partial dispersion Pe of the equivalent single lens should be zero. However, Table 6 shows that there are significant differences between them. Therefore, it is necessary to select materials that can reduce these values while maintaining the existing MSI values to consider athermalization. For this reason, the materials in case 2 can make the relative error of partial dispersion small. Specifically, NKZFS11 was selected for its similarity to the NSF10 type along with its minimal change in MSI values. After replacing it with the selected material, first-order quantity redistribution was conducted to correct residual chromatic aberration. This design process ensured that the value of Af in case 2 was reduced. The layout of the first group built through this process is shown in Fig. 7. At this time, the first lens group was converted back to a thick lens system. Also, the optical properties of each lens are listed in Table 7, and the relative errors of partial dispersion are presented in Table 8.

Figure 7. . Optical layout of the first group obtained from the apochromatic and athermal design.

TABLE 7. Optical properties of the first group obtained from the first-order quantity redistribution for each case.

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−6/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1NFK5812.2749−27.86130.00277815.0000
2NKZFS1123.9421−1.0765−0.00860714.3387
3NFK5812.2749−27.86130.00298313.7387
4NPK5113.1695−25.81780.00761314.6523
5NPSK53A16.8679−14.4508−0.00172713.5127
6NPSK53A16.8679−14.45080.00402413.0627


TABLE 8. Material selection index and relative error between both partial dispersions for case 2.

CaseMaterialMaterial Selection Index (MSI)Relative Error from Pe (×10−3)
Case 2
NPK52A1.0943.683
K5G201.2462.629
NKZFS111.0000.000
PLAK351.199−3.131
NLAK221.421−5.625


3.4. Design of the Second Group Lens

Figure 8 illustrates the second group that is designed to have the same first-order parameters in Table 2.

Figure 8. Optical layout of the second group.

The first four lenses (first to fourth) are in the negatively powered group N (−), and the last two lenses (fifth and sixth) are in the positively powered group P (+). They are then placed using the available materials in the same manner as the first group design. To achieve the achromatic and athermal configuration of the second group, the thick lenses are converted to thin lenses, and the optical properties for each lens are listed in Table 9. Based on these properties, six cases can be classified according to the selection of a specific lens, and the values of Mh(0, −αh), Le(ωe , γe), and Lc(ωc , γc) can be calculated. After analyzing all cases, it is confirmed that the lines of MhLc in cases 4 and 5 are nearest to the available material distribution, as shown in Fig. 9.

Figure 9. Achromatic and athermal glass map analysis: (a) Case 4 and (b) case 5 of second group.

TABLE 9. Optical properties of the second group lens.

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−6/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1NPSK53A12.2749−27.86130.00277815.0000
2NPK5123.9421−1.0765−0.00860714.3387
3NLAF212.2749−27.86130.00298313.7387
4NLAF213.1695−25.81780.00761314.6523
5SF5716.8679−14.4508−0.00172713.5127
6SF5716.8679−14.45080.00402413.0627


Table 10 presents MSI analyses for cases 4 and 5, which are the most advantageous in aberration correction among the six cases. Through this analysis, SF11 with a lower MSI and similar refractive index is selected as the fourth lens material for case 4. For the same reason, NPSK53A is selected as the fifth lens material for case 5. Next, by changing the previous glasses to the newly selected materials, the second group can reduce chromatic aberration and thermal defocus using the same principles outlined in the first group design. The glass map after the material change is shown in Fig. 10.

Figure 10. Achromatic and athermal glass map analysis after material selection and change: (a) Case 4 and (b) case 5 of second group.

As shown in Fig. 10(a) and 10(b), the difference between the two thermal powers is significantly reduced to be almost similar along the vertical axis. In addition, the chromatic power difference is slightly reduced along the horizontal axis. Here, it is difficult to use the apochromatic method for the second group in all cases due to the significant differences in material properties. However, the first group already has a more favorable configuration for apochromatic correction, and the second group has a lower ray height compared to that of the first group, which results in relatively less impact on aberration correction. Therefore, in the design of the second lens group, it is desirable to focus on material arrangements favorable for achromatic and athermal configurations. The layout, converted back to thick lenses, is shown in Fig. 11.

Figure 11. Optical layout of the second group obtained from the achromatic and athermal design.

To achieve the optical system arrangement as Fig. 4, the first and second groups designed separately were combined. The initial optical system configured through this process is shown in Fig. 12.

Figure 12. The initial optical system by combining two groups designed separately.

Finally, the optimization process is performed to meet the target specifications using the starting lens of Fig. 12. To meet the target specifications, the aperture and field size are increased to the F-number of F/5 and image size of ±8.2 mm. The high-order aberrations that arise during this process are corrected using eight aspherical surfaces on five lenses (6, 7, 9, 11, and 12). Finally, to perform the eighth lens as a compensator, sufficient air space was secured in front of and behind this lens. The optical system was also designed to have the minimum object distance and loose manufacturing tolerances. The specifications of the final designed optical system are listed in Table 11. The layout and optical performance analysis are shown in Fig. 13 to Fig. 15. This optical system fulfills the target specifications and yields an apochromatic and athermal configuration.

Figure 13. Layout of the final designed optical system.

Figure 15. Apochromatic and athermal design: (a) Longitudinal chromatic aberration graph and (b) modulation transfer functions (MTFs) from −35 ℃ to +60 ℃ of the final designed optical system.

IV. CONCLUSION

In this study, we conducted achromatic and athermal analysis using a glass map, and then proposed a method for analyzing multiple optical materials and selecting the most suitable one by introducing an optical material selection index (MSI). In cases where stable performance is required in a wide wavelength range, we proposed a new optical material selection process for apochromatic design and a first-order quantity redistribution method. The application of these approaches to the design of a telephoto optical system for SWIR confirmed the usefulness of these methods. In this design process, we determined the first-order parameters for each group based on the initial specifications and performed the basic design with a real lens arrangement. In this step, we first selected a specific lens glass suitable for achromatic and athermal design in each group, and then applied the apochromatic design method.

Thus, each group designed independently was combined to form the initial telephoto optical system, which yielded an achromatic (apochromatic) and athermal design. Finally, with design optimization, we achieved the final optical system that meets the target specifications and secures stable optical performance in the operating environment.

In conclusion, this proposed design approach is expected to provide a useful means of determining the glasses for achromatic (apochromatic) and athermal designs over a specified temperature and extremely wide waveband ranges such as a SWIR system.

Figure 14. Modulation transfer function (MTF) chart at room temperature of the final designed optical system.

TABLE 10. Material selection index (MSI) for each case of the second group lens.

CaseRankMaterialMaterial Selection Index (MSI)
Case 41PSF680.8464
2NZK70.8589
3SF110.8617
4NZK7A0.8623
5NKZFS20.8647
Case 5
4NPK510.6335
5NPK52A0.6342
6PPK530.7139
7NPSK53A0.7402


TABLE 11. Specifications of the final designed telephoto optical system for SWIR.

ParametersTarget Values
Sensor Type/Format/Pixel PitchInGaAs/1280×1024/10 µm
Wavelengths (µm)0.9–1.7 (SWIR)
Effective Focal Length (mm)600.0
F-number5.0
Image Height (mm)±8.20
Field of View (deg.)±0.7804
Optical Total Track Length (mm)297.526 (including sensor structure)
MTF (@ 50 cycles/mm)More than 37.9 %
Relative Illumination (%)More than 98.9
Distortion (%)Less than 0.47
Operating Temperature (℃)−35~+60
Housing MaterialAL6061 (CTE = 23.4 × 10−6/℃)

FUNDING

The authors received no financial support for the research, authorship, and/or publication of this article.

DISCLOSURES

The authors declare no conflicts of interest.

DATA AVAILABILITY

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

Fig 1.

Figure 1.Achromatic and athermal conditions on a glass map.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 2.

Figure 2.Residual chromatic aberration in an achromatic optical system.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 3.

Figure 3.Two-group thin lens system in air.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 4.

Figure 4.Two-group optical system in air.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 5.

Figure 5.Optical layout of the first group.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 6.

Figure 6.Achromatic and athermal glass map analysis of case 2: (a) Initial first group, (b) first group after material selection and change.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 7.

Figure 7.. Optical layout of the first group obtained from the apochromatic and athermal design.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 8.

Figure 8.Optical layout of the second group.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 9.

Figure 9.Achromatic and athermal glass map analysis: (a) Case 4 and (b) case 5 of second group.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 10.

Figure 10.Achromatic and athermal glass map analysis after material selection and change: (a) Case 4 and (b) case 5 of second group.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 11.

Figure 11.Optical layout of the second group obtained from the achromatic and athermal design.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 12.

Figure 12.The initial optical system by combining two groups designed separately.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 13.

Figure 13.Layout of the final designed optical system.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 14.

Figure 14.Modulation transfer function (MTF) chart at room temperature of the final designed optical system.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

Fig 15.

Figure 15.Apochromatic and athermal design: (a) Longitudinal chromatic aberration graph and (b) modulation transfer functions (MTFs) from −35 ℃ to +60 ℃ of the final designed optical system.
Current Optics and Photonics 2024; 8: 472-483https://doi.org/10.3807/COPP.2024.8.5.472

TABLE 1 Target specifications for a short-wave infrared (SWIR) telephoto optical system

ParametersTarget Values
Sensor Type/Format/Pixel PitchInGaAs/1280×1024/10 µm
Wavelengths (µm)0.9–1.7 (SWIR)
Effective Focal Length (mm)600.0
F-number5.0
Image Height (mm)±8.20
Optical Total Track Length (mm)297.526 (including sensor structure)
MTF (@ 50 cycles/mm)More than 30 % (at all fields)
Operating Temperature (℃)−35~+60
Housing MaterialAL6061 (CTE = 23.4 × 10−6/℃)

TABLE 2 Calculation of the first-order parameters in an initial two-group thin lens system

ParametersValues
Telephoto Ratio (TR) of Thin Lens System0.2
Distance Between Principal Planes (z1) (mm)160.0
Distance from the Principal Plane to Image Surface (z2) (mm)−40.0
Effective Focal Length of Group 1 (f1) (mm)150.0
Effective Focal Length of Group 2 (f2) (mm)13.333

TABLE 3 Optical properties of the first group lens

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−6/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1NFK5812.2749−27.86130.00443915.0000
2NLAF218.7880−8.3744−0.00658914.0902
3NFK5812.2749−27.86130.00704614.0090
4NPK5113.1695−25.81780.00899012.9302
5NPSK53A16.8679−14.4508−0.01224910.7414
6NPSK53A16.8679−14.45080.0050378.5157

TABLE 4 Material selection index (MSI) for each case of the first group lens

CaseRankMaterialMaterial Selection Index (MSI)
Case 21NSF100.6329
2NSF40.6361
3NSF140.6590
4NSF10.6629
5PSF690.6676

TABLE 5 Optical properties of the first group lens

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−6/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1NFK5812.2749−27.86130.00447915.0000
2NSF1023.9430−10.4130−0.00662514.1043
3NFK5812.2749−27.86130.00693814.0291
4NPK5113.1695−25.81780.00893412.9741
5NPSK53A16.8679−14.4508−0.01209210.7511
6NPSK53A16.8679−14.45080.0050388.5389

TABLE 6 Material selection index and relative error of partial dispersion for each case

CaseMaterialMaterial Selection Index (MSI)Relative Error from Pe (×10−3)
Case 2
NBAK41.7511.145
NPK52A1.7550.717
K5G201.625−0.340
NKZFS111.218−2.977
NSF101.000−169.818
Case 4SF570.774−7.959
SF60.75711.893
SF6G050.74812.297
SF56A0.76624.790
NPK511.000209.790
Case 5FK5HTI1.589−168.998
NFK51.595−169.302
NBK102.021−174.241
NKZFS22.694−188.186
NPSK53A1.000−296.679
Case 6PSF681.676283.226
SF571.581300.485
SF61.529314.262
SF6G051.536314.542
NPSK53A1.000465.400

TABLE 7 Optical properties of the first group obtained from the first-order quantity redistribution for each case

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−6/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1NFK5812.2749−27.86130.00277815.0000
2NKZFS1123.9421−1.0765−0.00860714.3387
3NFK5812.2749−27.86130.00298313.7387
4NPK5113.1695−25.81780.00761314.6523
5NPSK53A16.8679−14.4508−0.00172713.5127
6NPSK53A16.8679−14.45080.00402413.0627

TABLE 8 Material selection index and relative error between both partial dispersions for case 2

CaseMaterialMaterial Selection Index (MSI)Relative Error from Pe (×10−3)
Case 2
NPK52A1.0943.683
K5G201.2462.629
NKZFS111.0000.000
PLAK351.199−3.131
NLAK221.421−5.625

TABLE 9 Optical properties of the second group lens

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−6/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1NPSK53A12.2749−27.86130.00277815.0000
2NPK5123.9421−1.0765−0.00860714.3387
3NLAF212.2749−27.86130.00298313.7387
4NLAF213.1695−25.81780.00761314.6523
5SF5716.8679−14.4508−0.00172713.5127
6SF5716.8679−14.45080.00402413.0627

TABLE 10 Material selection index (MSI) for each case of the second group lens

CaseRankMaterialMaterial Selection Index (MSI)
Case 41PSF680.8464
2NZK70.8589
3SF110.8617
4NZK7A0.8623
5NKZFS20.8647
Case 5
4NPK510.6335
5NPK52A0.6342
6PPK530.7139
7NPSK53A0.7402

TABLE 11 Specifications of the final designed telephoto optical system for SWIR

ParametersTarget Values
Sensor Type/Format/Pixel PitchInGaAs/1280×1024/10 µm
Wavelengths (µm)0.9–1.7 (SWIR)
Effective Focal Length (mm)600.0
F-number5.0
Image Height (mm)±8.20
Field of View (deg.)±0.7804
Optical Total Track Length (mm)297.526 (including sensor structure)
MTF (@ 50 cycles/mm)More than 37.9 %
Relative Illumination (%)More than 98.9
Distortion (%)Less than 0.47
Operating Temperature (℃)−35~+60
Housing MaterialAL6061 (CTE = 23.4 × 10−6/℃)

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