Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
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.
Corresponding author: *scpark@dankook.ac.kr, ORCID 0000-0003-1932-5086
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 [1–3].
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.
Changes in the optical power of the lens due to variations in wavelength and temperature can be expressed using the chromatic power (ω_{i}) and thermal power (γ_{i}) of the element material (M_{i}), as shown in Eqs. (1) and (2) [6–9]:
where ∆λ is the specified waveband, φ_{i} is the element optical power, v_{i} is the Abbe number, n_{i} is the refractive index at the reference wavelength, α_{i} is the coefficient of thermal expansion (CTE) of the i-th lens material, and T is the temperature.
Longitudinal chromatic aberration arises from changes (∆
where φ_{T} 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
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 L_{j} and an equivalent single lens L_{e}. This equivalent single lens consists of the remaining k-1 elements. Therefore, in this separated doublet system composed of L_{j} and L_{e}, the total optical power (φ_{T}), achromatic (∆
where
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 (h_{i} / h_{1}), 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 (φ_{T}) leads to expressions for the achromatic and athermal conditions in a doublet system as follows [9, 10]:
where
When an equivalent single lens is given, the point designated as L_{c}(ω_{c}, γ_{c}) 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, L_{c}(ω_{c}, γ_{c}). 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]:
where ω_{c} = −
Equation (13) holds because the left side uses L_{c}(ω_{c}, γ_{c}) instead of L_{j}(ω_{j}, γ_{j}). Thus, the difference in material properties between L_{j} and L_{e} causes chromatic aberration and thermal defocus. Accordingly, it can be defined as an aberration factor (A_{f}) using the sum of the relative error, as given in Eq. (14):
In Eq. (13), the achromatic and athermal conditions require ideal material properties L_{c}(ω_{c}, γ_{c}). The left side of this equation represents the slope of the line connecting housing material properties M_{h}(0, −
An optical system that does not satisfy achromatic and athermal conditions will have the lines of M_{h} − L_{j} and M_{h} − L_{c}, 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 (L_{a}) distributed on a glass map. In this process, a material selection index (MSI) is defined as the relative error between the material properties of L_{c}(ω_{c}, γ_{c}) and L_{a}(ω_{a}, γ_{a}), as given in Eq. (15). This MSI is used for material selection.
where
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 (
where n_{R} is the refractive index of the reference wavelength and P_{i(λS, λJ)} is the partial dispersion of the i-th lens material, and is defined as
where
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 L_{j} must be replaced with the aberration-corrected value ω_{c} to achieve the achromatic condition. Additionally, by reorganizing Eq. (17) using the optical power ratio, the apochromatic condition can be rewritten as follows:
In the above Eq. (18), note that the achromatic and athermal system should have the value of −
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.
We confirmed through Eq. (19) that when the achromatic condition is satisfied, the partial dispersion value of L_{j} should be the same as the partial dispersion value of L_{e} 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):
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 P_{a(λS, λJ)} and calculate the difference with P_{e(λS, λJ)}, as shown in Eq. (21):
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 P_{e(λS, λJ)} and P_{j(λS, λJ)} match while maintaining the achromatic condition. According to the partial dispersion of an equivalent single lens given as P_{e(λS, λJ)} =
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, z_{1} is the distance between the second principal plane of the first group (G1) and the first principal plane of the second group (G2), z_{2} 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 (T_{R}), as the ratio of the optical total track length (L) to the total focal length (
Therefore, if only the total focal length (
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
Parameters | Target Values |
---|---|
Sensor Type/Format/Pixel Pitch | InGaAs/1280×1024/10 µm |
Wavelengths (µm) | 0.9–1.7 (SWIR) |
Effective Focal Length (mm) | 600.0 |
F-number | 5.0 |
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 Material | AL6061 (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, z_{1} 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 T_{R} 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, T_{R} 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 (T_{R}) and distance between principal planes (z_{1}) from Table 2 into Eqs. (22) to (24). This allows for the determination of the distance between the principal plane and the image surface (z_{2}), the focal length of the first group (
TABLE 2 Calculation of the first-order parameters in an initial two-group thin lens system
Parameters | Values |
---|---|
Telephoto Ratio (T_{R}) of Thin Lens System | 0.2 |
Distance Between Principal Planes (z_{1}) (mm) | 160.0 |
Distance from the Principal Plane to Image Surface (z_{2}) (mm) | −40.0 |
Effective Focal Length of Group 1 (f ′_{1}) (mm) | 150.0 |
Effective Focal Length of Group 2 (f ′_{2}) (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.
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.
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
Element | Material | Chromatic Power (×10^{−3}) | Thermal Power (×10^{−6}/℃) | Optical Power (mm^{−1}) | Paraxial Ray Height (mm) |
---|---|---|---|---|---|
1 | NFK58 | 12.2749 | −27.8613 | 0.004439 | 15.0000 |
2 | NLAF2 | 18.7880 | −8.3744 | −0.006589 | 14.0902 |
3 | NFK58 | 12.2749 | −27.8613 | 0.007046 | 14.0090 |
4 | NPK51 | 13.1695 | −25.8178 | 0.008990 | 12.9302 |
5 | NPSK53A | 16.8679 | −14.4508 | −0.012249 | 10.7414 |
6 | NPSK53A | 16.8679 | −14.4508 | 0.005037 | 8.5157 |
From Table 3, six cases can be classified according to the selection of a specific lens, and the values of M_{h}(0, −
TABLE 4 Material selection index (MSI) for each case of the first group lens
Case | Rank | Material | Material Selection Index (MSI) |
---|---|---|---|
Case 2 | 1 | NSF10 | 0.6329 |
2 | NSF4 | 0.6361 | |
3 | NSF14 | 0.6590 | |
4 | NSF1 | 0.6629 | |
5 | PSF69 | 0.6676 |
In that figure, the difference between L_{c} and L_{j} 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 ω_{c} = 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
Element | Material | Chromatic Power (×10^{−3}) | Thermal Power (×10^{−6}/℃) | Optical Power (mm^{−1}) | Paraxial Ray Height (mm) |
---|---|---|---|---|---|
1 | NFK58 | 12.2749 | −27.8613 | 0.004479 | 15.0000 |
2 | NSF10 | 23.9430 | −10.4130 | −0.006625 | 14.1043 |
3 | NFK58 | 12.2749 | −27.8613 | 0.006938 | 14.0291 |
4 | NPK51 | 13.1695 | −25.8178 | 0.008934 | 12.9741 |
5 | NPSK53A | 16.8679 | −14.4508 | −0.012092 | 10.7511 |
6 | NPSK53A | 16.8679 | −14.4508 | 0.005038 | 8.5389 |
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
Case | Material | Material Selection Index (MSI) | Relative Error from P_{e} (×10^{−3}) |
---|---|---|---|
Case 2 | ⁝ | ||
NBAK4 | 1.751 | 1.145 | |
NPK52A | 1.755 | 0.717 | |
K5G20 | 1.625 | −0.340 | |
NKZFS11 | 1.218 | −2.977 | |
⁝ | |||
NSF10 | 1.000 | −169.818 | |
Case 4 | SF57 | 0.774 | −7.959 |
SF6 | 0.757 | 11.893 | |
SF6G05 | 0.748 | 12.297 | |
SF56A | 0.766 | 24.790 | |
⁝ | |||
NPK51 | 1.000 | 209.790 | |
Case 5 | FK5HTI | 1.589 | −168.998 |
NFK5 | 1.595 | −169.302 | |
NBK10 | 2.021 | −174.241 | |
NKZFS2 | 2.694 | −188.186 | |
⁝ | |||
NPSK53A | 1.000 | −296.679 | |
Case 6 | PSF68 | 1.676 | 283.226 |
SF57 | 1.581 | 300.485 | |
SF6 | 1.529 | 314.262 | |
SF6G05 | 1.536 | 314.542 | |
⁝ | |||
NPSK53A | 1.000 | 465.400 |
Table 6 lists the relative errors for each material, calculated with reference to partial dispersion P_{e}, in order, along with the corresponding MSI values. To satisfy the apochromatic condition, the relative error between the partial dispersion P_{j} of the specific lens and the partial dispersion P_{e} 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
TABLE 7 Optical properties of the first group obtained from the first-order quantity redistribution for each case
Element | Material | Chromatic Power (×10^{−3}) | Thermal Power (×10^{−6}/℃) | Optical Power (mm^{−1}) | Paraxial Ray Height (mm) |
---|---|---|---|---|---|
1 | NFK58 | 12.2749 | −27.8613 | 0.002778 | 15.0000 |
2 | NKZFS11 | 23.9421 | −1.0765 | −0.008607 | 14.3387 |
3 | NFK58 | 12.2749 | −27.8613 | 0.002983 | 13.7387 |
4 | NPK51 | 13.1695 | −25.8178 | 0.007613 | 14.6523 |
5 | NPSK53A | 16.8679 | −14.4508 | −0.001727 | 13.5127 |
6 | NPSK53A | 16.8679 | −14.4508 | 0.004024 | 13.0627 |
TABLE 8 Material selection index and relative error between both partial dispersions for case 2
Case | Material | Material Selection Index (MSI) | Relative Error from P_{e} (×10^{−3}) |
---|---|---|---|
Case 2 | ⁝ | ||
NPK52A | 1.094 | 3.683 | |
K5G20 | 1.246 | 2.629 | |
NKZFS11 | 1.000 | 0.000 | |
PLAK35 | 1.199 | −3.131 | |
NLAK22 | 1.421 | −5.625 | |
⁝ |
Figure 8 illustrates the second group that is designed to have the same first-order parameters in Table 2.
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 M_{h}(0, −
TABLE 9 Optical properties of the second group lens
Element | Material | Chromatic Power (×10^{−3}) | Thermal Power (×10^{−6}/℃) | Optical Power (mm^{−1}) | Paraxial Ray Height (mm) |
---|---|---|---|---|---|
1 | NPSK53A | 12.2749 | −27.8613 | 0.002778 | 15.0000 |
2 | NPK51 | 23.9421 | −1.0765 | −0.008607 | 14.3387 |
3 | NLAF2 | 12.2749 | −27.8613 | 0.002983 | 13.7387 |
4 | NLAF2 | 13.1695 | −25.8178 | 0.007613 | 14.6523 |
5 | SF57 | 16.8679 | −14.4508 | −0.001727 | 13.5127 |
6 | SF57 | 16.8679 | −14.4508 | 0.004024 | 13.0627 |
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.
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.
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.
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.
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.
TABLE 10 Material selection index (MSI) for each case of the second group lens
Case | Rank | Material | Material Selection Index (MSI) |
---|---|---|---|
Case 4 | 1 | PSF68 | 0.8464 |
2 | NZK7 | 0.8589 | |
3 | SF11 | 0.8617 | |
4 | NZK7A | 0.8623 | |
5 | NKZFS2 | 0.8647 | |
Case 5 | ⁝ | ||
4 | NPK51 | 0.6335 | |
5 | NPK52A | 0.6342 | |
6 | PPK53 | 0.7139 | |
7 | NPSK53A | 0.7402 |
TABLE 11 Specifications of the final designed telephoto optical system for SWIR
Parameters | Target Values |
---|---|
Sensor Type/Format/Pixel Pitch | InGaAs/1280×1024/10 µm |
Wavelengths (µm) | 0.9–1.7 (SWIR) |
Effective Focal Length (mm) | 600.0 |
F-number | 5.0 |
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 Material | AL6061 (CTE = 23.4 × 10^{−6}/℃) |
The authors received no financial support for the research, authorship, and/or publication of this article.
The authors declare no conflicts of interest.
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.
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.
Department of Physics, Dankook University, Cheonan 31116, Korea
Correspondence to:*scpark@dankook.ac.kr, ORCID 0000-0003-1932-5086
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
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 [1–3].
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.
Changes in the optical power of the lens due to variations in wavelength and temperature can be expressed using the chromatic power (ω_{i}) and thermal power (γ_{i}) of the element material (M_{i}), as shown in Eqs. (1) and (2) [6–9]:
where ∆λ is the specified waveband, φ_{i} is the element optical power, v_{i} is the Abbe number, n_{i} is the refractive index at the reference wavelength, α_{i} is the coefficient of thermal expansion (CTE) of the i-th lens material, and T is the temperature.
Longitudinal chromatic aberration arises from changes (∆
where φ_{T} 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
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 L_{j} and an equivalent single lens L_{e}. This equivalent single lens consists of the remaining k-1 elements. Therefore, in this separated doublet system composed of L_{j} and L_{e}, the total optical power (φ_{T}), achromatic (∆
where
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 (h_{i} / h_{1}), 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 (φ_{T}) leads to expressions for the achromatic and athermal conditions in a doublet system as follows [9, 10]:
where
When an equivalent single lens is given, the point designated as L_{c}(ω_{c}, γ_{c}) 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, L_{c}(ω_{c}, γ_{c}). 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]:
where ω_{c} = −
Equation (13) holds because the left side uses L_{c}(ω_{c}, γ_{c}) instead of L_{j}(ω_{j}, γ_{j}). Thus, the difference in material properties between L_{j} and L_{e} causes chromatic aberration and thermal defocus. Accordingly, it can be defined as an aberration factor (A_{f}) using the sum of the relative error, as given in Eq. (14):
In Eq. (13), the achromatic and athermal conditions require ideal material properties L_{c}(ω_{c}, γ_{c}). The left side of this equation represents the slope of the line connecting housing material properties M_{h}(0, −
An optical system that does not satisfy achromatic and athermal conditions will have the lines of M_{h} − L_{j} and M_{h} − L_{c}, 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 (L_{a}) distributed on a glass map. In this process, a material selection index (MSI) is defined as the relative error between the material properties of L_{c}(ω_{c}, γ_{c}) and L_{a}(ω_{a}, γ_{a}), as given in Eq. (15). This MSI is used for material selection.
where
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 (
where n_{R} is the refractive index of the reference wavelength and P_{i(λS, λJ)} is the partial dispersion of the i-th lens material, and is defined as
where
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 L_{j} must be replaced with the aberration-corrected value ω_{c} to achieve the achromatic condition. Additionally, by reorganizing Eq. (17) using the optical power ratio, the apochromatic condition can be rewritten as follows:
In the above Eq. (18), note that the achromatic and athermal system should have the value of −
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.
We confirmed through Eq. (19) that when the achromatic condition is satisfied, the partial dispersion value of L_{j} should be the same as the partial dispersion value of L_{e} 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):
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 P_{a(λS, λJ)} and calculate the difference with P_{e(λS, λJ)}, as shown in Eq. (21):
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 P_{e(λS, λJ)} and P_{j(λS, λJ)} match while maintaining the achromatic condition. According to the partial dispersion of an equivalent single lens given as P_{e(λS, λJ)} =
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, z_{1} is the distance between the second principal plane of the first group (G1) and the first principal plane of the second group (G2), z_{2} 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 (T_{R}), as the ratio of the optical total track length (L) to the total focal length (
Therefore, if only the total focal length (
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.
Parameters | Target Values |
---|---|
Sensor Type/Format/Pixel Pitch | InGaAs/1280×1024/10 µm |
Wavelengths (µm) | 0.9–1.7 (SWIR) |
Effective Focal Length (mm) | 600.0 |
F-number | 5.0 |
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 Material | AL6061 (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, z_{1} 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 T_{R} 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, T_{R} 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 (T_{R}) and distance between principal planes (z_{1}) from Table 2 into Eqs. (22) to (24). This allows for the determination of the distance between the principal plane and the image surface (z_{2}), the focal length of the first group (
TABLE 2. Calculation of the first-order parameters in an initial two-group thin lens system.
Parameters | Values |
---|---|
Telephoto Ratio (T_{R}) of Thin Lens System | 0.2 |
Distance Between Principal Planes (z_{1}) (mm) | 160.0 |
Distance from the Principal Plane to Image Surface (z_{2}) (mm) | −40.0 |
Effective Focal Length of Group 1 (f ′_{1}) (mm) | 150.0 |
Effective Focal Length of Group 2 (f ′_{2}) (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.
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.
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.
Element | Material | Chromatic Power (×10^{−3}) | Thermal Power (×10^{−6}/℃) | Optical Power (mm^{−1}) | Paraxial Ray Height (mm) |
---|---|---|---|---|---|
1 | NFK58 | 12.2749 | −27.8613 | 0.004439 | 15.0000 |
2 | NLAF2 | 18.7880 | −8.3744 | −0.006589 | 14.0902 |
3 | NFK58 | 12.2749 | −27.8613 | 0.007046 | 14.0090 |
4 | NPK51 | 13.1695 | −25.8178 | 0.008990 | 12.9302 |
5 | NPSK53A | 16.8679 | −14.4508 | −0.012249 | 10.7414 |
6 | NPSK53A | 16.8679 | −14.4508 | 0.005037 | 8.5157 |
From Table 3, six cases can be classified according to the selection of a specific lens, and the values of M_{h}(0, −
TABLE 4. Material selection index (MSI) for each case of the first group lens.
Case | Rank | Material | Material Selection Index (MSI) |
---|---|---|---|
Case 2 | 1 | NSF10 | 0.6329 |
2 | NSF4 | 0.6361 | |
3 | NSF14 | 0.6590 | |
4 | NSF1 | 0.6629 | |
5 | PSF69 | 0.6676 |
In that figure, the difference between L_{c} and L_{j} 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 ω_{c} = 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.
Element | Material | Chromatic Power (×10^{−3}) | Thermal Power (×10^{−6}/℃) | Optical Power (mm^{−1}) | Paraxial Ray Height (mm) |
---|---|---|---|---|---|
1 | NFK58 | 12.2749 | −27.8613 | 0.004479 | 15.0000 |
2 | NSF10 | 23.9430 | −10.4130 | −0.006625 | 14.1043 |
3 | NFK58 | 12.2749 | −27.8613 | 0.006938 | 14.0291 |
4 | NPK51 | 13.1695 | −25.8178 | 0.008934 | 12.9741 |
5 | NPSK53A | 16.8679 | −14.4508 | −0.012092 | 10.7511 |
6 | NPSK53A | 16.8679 | −14.4508 | 0.005038 | 8.5389 |
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.
Case | Material | Material Selection Index (MSI) | Relative Error from P_{e} (×10^{−3}) |
---|---|---|---|
Case 2 | ⁝ | ||
NBAK4 | 1.751 | 1.145 | |
NPK52A | 1.755 | 0.717 | |
K5G20 | 1.625 | −0.340 | |
NKZFS11 | 1.218 | −2.977 | |
⁝ | |||
NSF10 | 1.000 | −169.818 | |
Case 4 | SF57 | 0.774 | −7.959 |
SF6 | 0.757 | 11.893 | |
SF6G05 | 0.748 | 12.297 | |
SF56A | 0.766 | 24.790 | |
⁝ | |||
NPK51 | 1.000 | 209.790 | |
Case 5 | FK5HTI | 1.589 | −168.998 |
NFK5 | 1.595 | −169.302 | |
NBK10 | 2.021 | −174.241 | |
NKZFS2 | 2.694 | −188.186 | |
⁝ | |||
NPSK53A | 1.000 | −296.679 | |
Case 6 | PSF68 | 1.676 | 283.226 |
SF57 | 1.581 | 300.485 | |
SF6 | 1.529 | 314.262 | |
SF6G05 | 1.536 | 314.542 | |
⁝ | |||
NPSK53A | 1.000 | 465.400 |
Table 6 lists the relative errors for each material, calculated with reference to partial dispersion P_{e}, in order, along with the corresponding MSI values. To satisfy the apochromatic condition, the relative error between the partial dispersion P_{j} of the specific lens and the partial dispersion P_{e} 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
TABLE 7. Optical properties of the first group obtained from the first-order quantity redistribution for each case.
Element | Material | Chromatic Power (×10^{−3}) | Thermal Power (×10^{−6}/℃) | Optical Power (mm^{−1}) | Paraxial Ray Height (mm) |
---|---|---|---|---|---|
1 | NFK58 | 12.2749 | −27.8613 | 0.002778 | 15.0000 |
2 | NKZFS11 | 23.9421 | −1.0765 | −0.008607 | 14.3387 |
3 | NFK58 | 12.2749 | −27.8613 | 0.002983 | 13.7387 |
4 | NPK51 | 13.1695 | −25.8178 | 0.007613 | 14.6523 |
5 | NPSK53A | 16.8679 | −14.4508 | −0.001727 | 13.5127 |
6 | NPSK53A | 16.8679 | −14.4508 | 0.004024 | 13.0627 |
TABLE 8. Material selection index and relative error between both partial dispersions for case 2.
Case | Material | Material Selection Index (MSI) | Relative Error from P_{e} (×10^{−3}) |
---|---|---|---|
Case 2 | ⁝ | ||
NPK52A | 1.094 | 3.683 | |
K5G20 | 1.246 | 2.629 | |
NKZFS11 | 1.000 | 0.000 | |
PLAK35 | 1.199 | −3.131 | |
NLAK22 | 1.421 | −5.625 | |
⁝ |
Figure 8 illustrates the second group that is designed to have the same first-order parameters in Table 2.
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 M_{h}(0, −
TABLE 9. Optical properties of the second group lens.
Element | Material | Chromatic Power (×10^{−3}) | Thermal Power (×10^{−6}/℃) | Optical Power (mm^{−1}) | Paraxial Ray Height (mm) |
---|---|---|---|---|---|
1 | NPSK53A | 12.2749 | −27.8613 | 0.002778 | 15.0000 |
2 | NPK51 | 23.9421 | −1.0765 | −0.008607 | 14.3387 |
3 | NLAF2 | 12.2749 | −27.8613 | 0.002983 | 13.7387 |
4 | NLAF2 | 13.1695 | −25.8178 | 0.007613 | 14.6523 |
5 | SF57 | 16.8679 | −14.4508 | −0.001727 | 13.5127 |
6 | SF57 | 16.8679 | −14.4508 | 0.004024 | 13.0627 |
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.
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.
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.
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.
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.
TABLE 10. Material selection index (MSI) for each case of the second group lens.
Case | Rank | Material | Material Selection Index (MSI) |
---|---|---|---|
Case 4 | 1 | PSF68 | 0.8464 |
2 | NZK7 | 0.8589 | |
3 | SF11 | 0.8617 | |
4 | NZK7A | 0.8623 | |
5 | NKZFS2 | 0.8647 | |
Case 5 | ⁝ | ||
4 | NPK51 | 0.6335 | |
5 | NPK52A | 0.6342 | |
6 | PPK53 | 0.7139 | |
7 | NPSK53A | 0.7402 |
TABLE 11. Specifications of the final designed telephoto optical system for SWIR.
Parameters | Target Values |
---|---|
Sensor Type/Format/Pixel Pitch | InGaAs/1280×1024/10 µm |
Wavelengths (µm) | 0.9–1.7 (SWIR) |
Effective Focal Length (mm) | 600.0 |
F-number | 5.0 |
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 Material | AL6061 (CTE = 23.4 × 10^{−6}/℃) |
The authors received no financial support for the research, authorship, and/or publication of this article.
The authors declare no conflicts of interest.
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.
TABLE 1 Target specifications for a short-wave infrared (SWIR) telephoto optical system
Parameters | Target Values |
---|---|
Sensor Type/Format/Pixel Pitch | InGaAs/1280×1024/10 µm |
Wavelengths (µm) | 0.9–1.7 (SWIR) |
Effective Focal Length (mm) | 600.0 |
F-number | 5.0 |
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 Material | AL6061 (CTE = 23.4 × 10^{−6}/℃) |
TABLE 2 Calculation of the first-order parameters in an initial two-group thin lens system
Parameters | Values |
---|---|
Telephoto Ratio (T_{R}) of Thin Lens System | 0.2 |
Distance Between Principal Planes (z_{1}) (mm) | 160.0 |
Distance from the Principal Plane to Image Surface (z_{2}) (mm) | −40.0 |
Effective Focal Length of Group 1 (f ′_{1}) (mm) | 150.0 |
Effective Focal Length of Group 2 (f ′_{2}) (mm) | 13.333 |
TABLE 3 Optical properties of the first group lens
Element | Material | Chromatic Power (×10^{−3}) | Thermal Power (×10^{−6}/℃) | Optical Power (mm^{−1}) | Paraxial Ray Height (mm) |
---|---|---|---|---|---|
1 | NFK58 | 12.2749 | −27.8613 | 0.004439 | 15.0000 |
2 | NLAF2 | 18.7880 | −8.3744 | −0.006589 | 14.0902 |
3 | NFK58 | 12.2749 | −27.8613 | 0.007046 | 14.0090 |
4 | NPK51 | 13.1695 | −25.8178 | 0.008990 | 12.9302 |
5 | NPSK53A | 16.8679 | −14.4508 | −0.012249 | 10.7414 |
6 | NPSK53A | 16.8679 | −14.4508 | 0.005037 | 8.5157 |
TABLE 4 Material selection index (MSI) for each case of the first group lens
Case | Rank | Material | Material Selection Index (MSI) |
---|---|---|---|
Case 2 | 1 | NSF10 | 0.6329 |
2 | NSF4 | 0.6361 | |
3 | NSF14 | 0.6590 | |
4 | NSF1 | 0.6629 | |
5 | PSF69 | 0.6676 |
TABLE 5 Optical properties of the first group lens
Element | Material | Chromatic Power (×10^{−3}) | Thermal Power (×10^{−6}/℃) | Optical Power (mm^{−1}) | Paraxial Ray Height (mm) |
---|---|---|---|---|---|
1 | NFK58 | 12.2749 | −27.8613 | 0.004479 | 15.0000 |
2 | NSF10 | 23.9430 | −10.4130 | −0.006625 | 14.1043 |
3 | NFK58 | 12.2749 | −27.8613 | 0.006938 | 14.0291 |
4 | NPK51 | 13.1695 | −25.8178 | 0.008934 | 12.9741 |
5 | NPSK53A | 16.8679 | −14.4508 | −0.012092 | 10.7511 |
6 | NPSK53A | 16.8679 | −14.4508 | 0.005038 | 8.5389 |
TABLE 6 Material selection index and relative error of partial dispersion for each case
Case | Material | Material Selection Index (MSI) | Relative Error from P_{e} (×10^{−3}) |
---|---|---|---|
Case 2 | ⁝ | ||
NBAK4 | 1.751 | 1.145 | |
NPK52A | 1.755 | 0.717 | |
K5G20 | 1.625 | −0.340 | |
NKZFS11 | 1.218 | −2.977 | |
⁝ | |||
NSF10 | 1.000 | −169.818 | |
Case 4 | SF57 | 0.774 | −7.959 |
SF6 | 0.757 | 11.893 | |
SF6G05 | 0.748 | 12.297 | |
SF56A | 0.766 | 24.790 | |
⁝ | |||
NPK51 | 1.000 | 209.790 | |
Case 5 | FK5HTI | 1.589 | −168.998 |
NFK5 | 1.595 | −169.302 | |
NBK10 | 2.021 | −174.241 | |
NKZFS2 | 2.694 | −188.186 | |
⁝ | |||
NPSK53A | 1.000 | −296.679 | |
Case 6 | PSF68 | 1.676 | 283.226 |
SF57 | 1.581 | 300.485 | |
SF6 | 1.529 | 314.262 | |
SF6G05 | 1.536 | 314.542 | |
⁝ | |||
NPSK53A | 1.000 | 465.400 |
TABLE 7 Optical properties of the first group obtained from the first-order quantity redistribution for each case
Element | Material | Chromatic Power (×10^{−3}) | Thermal Power (×10^{−6}/℃) | Optical Power (mm^{−1}) | Paraxial Ray Height (mm) |
---|---|---|---|---|---|
1 | NFK58 | 12.2749 | −27.8613 | 0.002778 | 15.0000 |
2 | NKZFS11 | 23.9421 | −1.0765 | −0.008607 | 14.3387 |
3 | NFK58 | 12.2749 | −27.8613 | 0.002983 | 13.7387 |
4 | NPK51 | 13.1695 | −25.8178 | 0.007613 | 14.6523 |
5 | NPSK53A | 16.8679 | −14.4508 | −0.001727 | 13.5127 |
6 | NPSK53A | 16.8679 | −14.4508 | 0.004024 | 13.0627 |
TABLE 8 Material selection index and relative error between both partial dispersions for case 2
Case | Material | Material Selection Index (MSI) | Relative Error from P_{e} (×10^{−3}) |
---|---|---|---|
Case 2 | ⁝ | ||
NPK52A | 1.094 | 3.683 | |
K5G20 | 1.246 | 2.629 | |
NKZFS11 | 1.000 | 0.000 | |
PLAK35 | 1.199 | −3.131 | |
NLAK22 | 1.421 | −5.625 | |
⁝ |
TABLE 9 Optical properties of the second group lens
Element | Material | Chromatic Power (×10^{−3}) | Thermal Power (×10^{−6}/℃) | Optical Power (mm^{−1}) | Paraxial Ray Height (mm) |
---|---|---|---|---|---|
1 | NPSK53A | 12.2749 | −27.8613 | 0.002778 | 15.0000 |
2 | NPK51 | 23.9421 | −1.0765 | −0.008607 | 14.3387 |
3 | NLAF2 | 12.2749 | −27.8613 | 0.002983 | 13.7387 |
4 | NLAF2 | 13.1695 | −25.8178 | 0.007613 | 14.6523 |
5 | SF57 | 16.8679 | −14.4508 | −0.001727 | 13.5127 |
6 | SF57 | 16.8679 | −14.4508 | 0.004024 | 13.0627 |
TABLE 10 Material selection index (MSI) for each case of the second group lens
Case | Rank | Material | Material Selection Index (MSI) |
---|---|---|---|
Case 4 | 1 | PSF68 | 0.8464 |
2 | NZK7 | 0.8589 | |
3 | SF11 | 0.8617 | |
4 | NZK7A | 0.8623 | |
5 | NKZFS2 | 0.8647 | |
Case 5 | ⁝ | ||
4 | NPK51 | 0.6335 | |
5 | NPK52A | 0.6342 | |
6 | PPK53 | 0.7139 | |
7 | NPSK53A | 0.7402 |
TABLE 11 Specifications of the final designed telephoto optical system for SWIR
Parameters | Target Values |
---|---|
Sensor Type/Format/Pixel Pitch | InGaAs/1280×1024/10 µm |
Wavelengths (µm) | 0.9–1.7 (SWIR) |
Effective Focal Length (mm) | 600.0 |
F-number | 5.0 |
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 Material | AL6061 (CTE = 23.4 × 10^{−6}/℃) |