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Curr. Opt. Photon. 2023; 7(3): 273-282

Published online June 25, 2023 https://doi.org/10.3807/COPP.2023.7.3.273

Copyright © Optical Society of Korea.

Achromatic and Athermal Design of a Mobile-phone Camera Lens by Redistributing Optical First-order Quantities

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: March 26, 2023; Revised: May 15, 2023; Accepted: May 16, 2023

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 a new method for redistributing effectively the first orders of each lens element to achromatize and athermalize an optical system, by introducing a novel method for adjusting the slope of an achromatic and athermal line. This line is specified by connecting the housing, equivalent single lens, and aberration-corrected point on a glass map composed of available plastic and glass materials for molding. Thus, if a specific lens is replaced with the material characterized by the chromatic and thermal powers of an aberration-corrected point, we obtain an achromatic and athermal system. First, we identify two materials that yield the minimum and maximum slopes of the line from a housing coordinate, which specifies the slope range of the line spanning the available materials on a glass map. Next, redistributing the optical first orders (optical powers and paraxial ray heights) of lens elements by moving the achromatic and athermal line into the available slope range of materials yields a good achromatic and athermal design. Applying this concept to design a mobile-phone camera lens, we efficiently obtain an achromatic and athermal system with cost-effective material selection, over the specified temperature and waveband ranges.

Keywords: Aberrations, Achromatization, Athermalization, First order, Glass map

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

An optical system generally suffers from chromatic aberration, owing to wide changes in wavelength. In addition, variations in the ambient temperature induce changes in the curvature radius, refractive index, and lens thickness. The thermal defocus caused by such changes significantly degrades the image quality. Therefore, an optical system, including refractive elements and housing, should be designed for stable performance over the specified waveband and temperature ranges.

The aspherical lenses of a mobile-phone camera mainly employ plastic materials that are easy to inject. However, since these plastics are sensitive to temperature, the thermal defocus due to temperature changes should be corrected at the optical-design stage.

Numerous graphical methods have been reported to correct the chromatic aberration and thermal defocus that cause an optical system to deteriorate in the visible and infrared wavebands [110]. Among them, to correct these errors, matching the aberration-corrected point to a specific glass and redistributing the optical powers of lens elements has been presented [10]. These methods make it difficult, though, to select a material on a glass map when the aberration-corrected point deviates greatly from the available material distributions [9, 10].

To solve the above problem, this paper presents a new graphical method by introducing the achromatic and athermal line that satisfies the achromatic and athermal conditions on a glass map. By changing the slope of this line to the available material boundary, the first orders are redistributed, which yields an optical system that simultaneously reduces chromatic aberration and thermal defocus. Even if a current achromatic and athermal line is far from the available range of materials, we can easily adjust the first orders of elements by altering the line’s slope so that the aberration-corrected point moves into the range of available materials.

This approach results in an achromatic and athermal design even for an optical system made of plastic and glass materials, along with a cost-effective material selection for glass molding. An effective solution with small chromatic aberration and thermal defocus is obtained by using this method to design a mobile-phone camera lens with an f-number of 1.8 and a 16-megapixel image sensor.

2.1. Achromatic and Athermal Conditions

The chromatic power ωi and thermal power γi of the element material Mi have a significant effect on the changes in optical power owing to the fluctuations in the wavelength and temperature, and they are expressed as follows [3, 4, 9, 10]:

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 center wavelength, αi is the coefficient of thermal expansion (CTE) of the ith lens material, and T is the temperature.

The longitudinal chromatic aberration comes from the changes (Δfbch') in the back focal length (BFL) with wavelength, and is expressed as Eq. (3). The thermal defocus ∆z′ is evaluated as the difference between the change (Δfbth') in the BFL with the temperature and the change ΔHb') in the flange back length (FBL) with the temperature, as follows [9, 10]:

Δfb'-ch=1ϕT2 i=1kωi'ϕi'

Δz'=Δfb'-thΔHb'=1ϕT2 i=1kγi'ϕi'αhLΔT

where ϕT is the total power and k is the total number of elements.

In the above, the primed parameters denote that they are weighted by the ratio of the paraxial ray heights; That is, the weighted element optical power, weighted chromatic power, and weighted thermal power are ϕi′ = (hi /h1)ϕi, ωi′ = (hi /h1)ωi, and γi′ = (hi /h1)γi respectively. These imply that the air spacings between elements are included in Eqs. (3) and (4), which can handle the lens system more practically.

2.2. Graphical Expression for an Achromatic and Athermal Line on a Glass Map

In this study, an equivalent single lens Le is used to simplify an optical system with an arbitrary number of elements into a doublet system [8]. Thus an optical system with k elements can be recomposed into a doublet system composed of the specific jth 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 power ϕT, achromatic (Δfbch'), and athermal (∆z′ = 0) conditions are respectively given by [911]

ϕT= i=1kϕi'=ϕj'+ϕe'

Δfb'-ch=1ϕT2ωj'ϕj'+ωe'ϕe'=0

Δz'=1ϕT2 i=1kγi'ϕi'αhLΔT=0

where ϕe'= i=1kϕi 'ϕj',ωe'= i=1 k ω i' ϕ i ' ω j' ϕ j'/ ϕ e ',andγe'= i=1 k γ i' ϕ i ' γ j' ϕ j'/ ϕ e '.

In Eq. (7) it can be assumed that the FBL (L) is approximately the same as the BFL, i.e. L ≅ (hk /h1)/ϕT. While the achromatic condition of Eq. (6) and the athermal condition of Eq. (7) in this doublet system are divided by the ratio of the paraxial ray height (hj /h1), we can easily identify the specific lens location without weighting on a glass map [10]. Finally, dividing Eqs. (5), (6), and (7) by the total power ϕT results in expressions for the achromatic and athermal conditions in a doublet system, as follows:

pj+pe=1

ωjpj+ωe''pe=0

γjpj+γe''pe=αh''

where ωe″ = (h1/hj)ωe′, γe″ = (h1/hj)γe′, and αh″ = (hk /hj)αh. The two parameters pj and pe are the ratios of optical powers of the specific lens and equivalent single lens with respect to the total power.

When an equivalent single lens is given, the point designated as Lc(ωc, γc) in Fig. 1(a) 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). Thus, substituting Lj(ωj, γj) into Lc(ωc, γc) and inserting Eqs. (9) and (10) into Eq. (8) results in expressions for the achromatic and athermal conditions in a doublet system, as follows [10]:

Figure 1.Achromatic and athermal conditions on a glass map: (a) Causes of chromatic aberration and thermal defocus, (b)–(d) various combinations of an achromatic and athermal line according to the signs of the optical powers of pj and pe.

pjpe=1ωe''ωcωe''ωc

pjpe=1γe''γcγe''+αh''γcαh''

where ωc = −ωe″(pe /pj), and γc = −(γe″ + αh″)(pe /pj) − αh″. Rearranging Eqs. (11) and (12) yields expressions for pj and pe,

pj=ωe''ωe''ωc=γe''+αh''γe''γcγe''γcωe''ωc=γe''+αh''ωe''

pe=ωcωe''ωc=γc+αh''γe''γcγe''γcωe''ωc=γc+αh''ωc

Each term of these equations is characterized as the optical material properties and first orders of a specific lens and an equivalent lens. If the optical material properties and the first orders do not satisfy the achromatic and athermal conditions, they also cannot satisfy Eqs. (13) and (14). To have an achromatic and athermal system, a specific lens should be located at the achromatic and athermal point Lc(ωc, γc) by changing the lens material. At this time, while the material properties of Lj are changed into those of Lc, the first orders should be maintained to obtain an achromatic and athermal system. Therefore, if a specific lens Lj is replaced with Lc as shown in Fig. 1(a), we obtain the following relationship:

Slope of line(MhLc):γc+αh''ωc=(MhLe):γe''+αh''ωe''

Note that this relationship of Eq. (15) verifies Eqs. (13) and (14). For an achromatic and athermal system, from Eq. (15) and Fig. 1(a) the slope of the line connecting points Mh(0, −αh″) and Le(ωe″, γe″) should be equal to that of the line connecting points Mh(0, −αh″) and Lc(ωc, γc). Therefore, If the specific lens Lj is replaced with the lens Lc characterized by two powers of ωc and γc, then the lens Lc lies on the line connecting the housing Mh and an equivalent lens Le as depicted in Fig. 1(a). If a specific lens is not replaced with Lc(ωc, γc), the line connecting the points of Mh(0, −αh″) and Lj(ωj, γj) has a different slope, so the system does not meet the achromatic and athermal conditions.

Since the sum of normalized powers for a specific lens and an equivalent single lens is unity, i.e. pj + pe = 1, either pj or pe must have a positive value in an imaging system. Figure 1(b) shows the case of positive pj and negative pe. For the sum of two power ratios to be 1.0, pj should have an absolute value greater than pe. Here, to satisfy the condition ωc = −ωe″(pe /pj), ωc and ωe″ should have the same sign. Also, since | pe /pj| is less than 1.0, ωe″ should have an absolute value greater than ωc. Consequently, the line slope between Mh and Le is the same as that between Mh and Lc, in which the latter line exists within the line between Mh and Le, as shown in Fig. 1(b).

Similarly, Fig. 1(c) shows the case of negative pj and positive pe. Since pe have an absolute value greater than pj, ωc should have an absolute value greater than ωe″. Finally, Fig. 1(d) shows the case of positive pj and positive pe. To satisfy the condition ωc = −ωe″(pe /pj), ωc and ωe″ should have different signs. This situation means that the line connecting Mh and Le has the same slope as the line connecting Mh and Lc, but these two line segments exist on opposite sides with respect to Mh. Thus the signs of pj and pe determine the positions of Lc(ωc, γc) and Le(ωe, γe) from Mh(0, −αh″) with the same slope, so they should be considered when reconfiguring an optical system by redistributing the first orders of the lens elements.

2.3. A Method for Graphically Controlling the Slope of a Line on a Glass Map

To have an achromatic and athermal design, we propose the graphical method of adjusting the slope of the line representing achromatic and athermal conditions, which we refer to as the achromatic and athermal line, or briefly, line of MhLcLe. This achromatic and athermal line satisfies the conditions of Eq. (15) on the glass map of Fig. 2.

Figure 2.Slope-control method of shifting an achromatic and athermal line (MhLcLe) into available material distributions.

We first evaluate the optical properties and first orders of an initial optical system, which yields the coordinates of Mh(0, −αh″), Le(ωe″, γe″), Lj(ωj, γj), and Lc(ωc, γc). Next, as can be observed in Fig. 2, we determine the slope range of the line connecting all available materials from housing Mh on a glass map. Subsequently, redistributing the optical first orders of elements by moving the line connecting the housing Mh and equivalent single lens Le into the available slope range specified by the line of MhLj1 and the line of MhLj2, yields a good achromatic and athermal design. Here the first orders include the optical power and paraxial ray height of each element. Although the line of MhLe is far from the available range of materials, we can easily adjust the paraxial ray heights and optical powers of elements by controlling the line slope so that the aberration-corrected point Lc moves into the range of available materials. Thus, this approach enables us to effectively obtain an achromatic and athermal design. This is a key point of this study.

We first identify the distribution of available materials on a glass map, and then mark two extreme glasses that create the minimum and maximum slopes Lj1(ωj1, γj1) and Lj2(ωj2, γj2), and then evaluate the slope range of the line spanning from MhLj1 to MhLj2, as Fig. 2. To have an achromatic and athermal design, the slope of the line between Mh and Le should be within the following two extreme slope limits, namely the available slope ranges as

γj1+αh''ωj1γe''+αh''ωe''γj2+αh''ωj2

γe"+αh''ωe''= i=1kγi'ϕi'γj'ϕj'+hkh1αhϕe' i=1kωi'ϕi'ωj'ϕj'

Equation (17) denotes that the optical powers and paraxial ray heights of elements constituting an equivalent lens are parameters used to change the slope of the line between Mh and Le. Thus, while changing the slope of that line to meet the conditions of Eq. (16), the line between Mh and Lc also leads to the available material boundary, which yields a reasonable solution for the glass of a specific lens. Here, we can easily adjust the slope of the line by checking the relative locations of Le and Lc from Mh, owing to the refracting-power ratios pj and pe.

3.1. Design of a 16-megapixel Mobile-phone Camera Lens

A mobile-phone camera lens taken from the patented lens [12] operating in the visible range from −20 ℃ to +60 ℃ is presented as a design example using the proposed method. However, to work this patent lens as an optical system for a mobile-phone camera with a low f-number of 1.8 and a 16-megapixel image sensor, we first optimize that lens to have a modulation transfer function (MTF) of more than 25% over all fields at 250 cycles/mm and room temperature [13]. In this design process, we retain the optical power configuration without considering achromatic and athermal conditions. Figure 3 shows the layout for this initial camera lens, and Table 1 lists the targeted specifications to be designed by utilizing the method proposed in this study.

Table 1 Target specification of a mobile-phone camera lens

ParametersTarget Value
SensorType1/3 inch, 16-megapixel (4,656 × 3,496)
Pixel Size (μm)1.0 × 1.0
f-number1.8
Image Height (mm)±2.92
Field of View (deg)±35–37
Effective Focal Length (mm)3.87–4.17
Back Focal Length (including filter) (mm)More than 0.6
Total Track Length (mm)Less than 5.5
Distortion (%)Less than 2
Wavelengths435.8 nm (g)–656.3 nm (C)
Temperature Range (℃)−20~+60
Housing MaterialAL6061 (αh = 23.4 × 10−6/℃)
MTF at 250 cycles/mmMore than 30% at 0.0–0.7 F
More than 25% at 0.8–1.0 F


Figure 3.Layout of an initial mobile-phone camera lens.

The MTF at 250 cycles/mm at room temperature is good, but the MTFs from −20 ℃ to +60 ℃ are very unstable, as presented in Figs. 4(a) and 4(b) respectively.

Figure 4.Optical performance analysis of an initial mobile-phone camera lens: (a) Modulation transfer function (MTF) at room temperature, (b) MTFs from −20 ℃ to +60 ℃, (c) chromatic aberration, and (d) thermal defocus of an initial mobile-phone camera lens.

For this system the chromatic aberration between the C- and g-lines is evaluated as +40.24 µm, which is too large to get a good image, as shown in Fig. 4(c). Moreover, the BFL of this system is thermally unstable from −20 ℃ to +60 ℃, as illustrated in Fig. 4(d). The thermal defocus expressed as ∆z′ = BFL − FBL ranges from −31.28 µm to +28.99 µm at the extreme temperatures, which leaves the system far out of focus. These significant thermal defocuses lead to an unstable MTF at the two extremal temperatures, as depicted in Fig. 4(b).

3.2. Achromatic and Athermal Design by Redistributing the Optical First-order Quantities

In this study we apply the achromatic and athermal method outlined in Section 2 to a mobile-phone camera lens design through the analysis of the optical properties and the first orders. Since the thermal powers of most plastic materials are very negative, it is extremely hard to configure a plastic-only optical system corrected for thermal defocus. The glasses for molding supplied by Schott are used to construct a glass map for an achromatic and athermal design [14].

Next, we classify several cases according to the selection of a specific lens Lj and an equivalent single lens Le to find the most suitable combination for the achromatic and athermal design. As can be observed in Fig. 3, it is desirable to select the specific lens in consideration of the diameter and aspherical shape of that lens. This mobile-camera lens consists of five elements, so five doublet systems can be created. However, the shapes and diameters of the fourth and fifth lenses are complicated and large, so it is difficult to mold them into aspherical lenses made of molding glass. Hence we select three available lens candidates used for a specific lens Lj; They are Lj = L1, L2, and L3 respectively. Figure 5 illustrates the three cases and their aberration-corrected points Lc(ωc, γc) for each specific lens Lj ( j = 1, 2, 3), used to recompose the optical system as a doublet with a corresponding equivalent single lens. Table 2 lists the material properties and the first orders of an initial mobile-phone camera lens.

Table 2 Optical material properties and first orders of an initial mobile-phone camera lens

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−3/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1A551427.7−260.10.1581.089
2OKP4HT69.9−243.9−0.1110.806
3A551427.7−260.10.2420.804
4OKP4HT69.9−243.90.1970.437
5OKP4HT69.9−243.9−0.4160.206


Figure 5.Glass maps of an initial mobile-phone camera lens: (a) case 1 (Lj = L1), (b) case 2 (Lj = L2), and (c) case 3 (Lj = L3).

In the glass map of Fig. 5, case 1 is the doublet system composed of the specific lens Lj = L1 and the equivalent single lens Le comprising the remaining elements L2, L3, L4 and L5. Similarly, case 2 is Lj = L2 and Le = L1 + L3 + L4 + L5, and case 3 is Lj = L3 and Le = L1 + L2 + L4 + L5. In that figure, the slope of the line connecting Mh and Le is equal to that of the line connecting Mh and Lc. However, the line connecting Mh and Lj composed of a material not on the achromatic and athermal line shows a large difference from the line of MhLc, which may result in large chromatic aberration and thermal defocus. Since Figs. 5(a) and 5(c) feature the case of positive pj and positive pe, this situation means that the slope of the line connecting Mh and Le is equal to that of the line connecting Mh and Lc, but these two lines exist on the opposite sides of Mh. Meanwhile, while redistributing the first orders of an equivalent lens, pj and pe should be maintained at positive values. In two cases, note that only when the line of MhLe is turned clockwise can the line of MhLc reach the available material range. Figure 5(b) shows the case of negative pj and positive pe. Since pe has an absolute value greater than pj, ωc should have an absolute value greater than ωe″, which reveals that the line of MhLe is within the line of MhLc. In this case, even if the slope is adjusted through optical power redistribution, it is difficult for the line of MhLc to exist within the range of available material. Thus, cases 1 and 3 have an advantage over case 2. Among them, as shown in Fig. 5, case 1 may change the slope of the line relatively less than case 3, so case 1 is more useful than case 3. In addition, looking at the lens layout in Fig. 3, it is judged that applying molding materials to the first lens is advantageous for cost reduction and aspherical molding, rather than to the third one as in case 3. For this reason, by selecting L1 in case 1 as a specific lens and adjusting the slope of the line, an achromatic and athermal design can be realized by redistributing the optical first-order properties of elements.

Two steps can be considered to design a lens corrected for thermal defocus and chromatic aberration.

The first step is selecting the slope range of the line by considering available material distributions. Among them, the material NKZFS2, marked as Lj2 (28.42, 4.96) on a glass map of Fig. 6(a), has a positive maximum slope. Similarly, NPK52A, marked as Lj1(18.79, −25.50), has a negative maximum slope. The slope range of the lines from the housing Mh(0, −4.421) to the available materials of Lj1 and Lj2 are −1.122 to 0.330 (×10−3/℃) in an initial mobile-phone camera lens, as depicted in Fig. 6(a). Unfortunately, the line connecting Mh and Le(or Lc) has a slope of −40.486 (×10−3/℃), which is outside the slope range spanning the available materials.

Figure 6.Achromatic and athermal design process using the slope-control method for a mobile-phone camera lens: (a) optical first-order redistribution process by adjusting the slope of an achromatic and athermal line, (b) material-change process for a specific lens.

The second step is redistributing the optical first-order quantities of an equivalent single lens so that the aberration-corrected point Lc moves into an available material region by changing the slope of the line of MhLe [see Fig. 6(b)]. While adjusting the slope of an achromatic and athermal line, pj and pe must have positive values.

This design approach yields an achromatic and athermal system, for which the material properties and the first orders of elements are listed in Table 3. The slope range of the line from the housing Mh(0, −4.343) to the available materials of Lj1 and Lj2 is −1.126 to 0.327 (×10−3/℃) in a redesigned camera lens. Fortunately, the newly altered line connecting Mh and Le(or Lc) has a slope of 0.275(×10−3/℃), which exists within the slope range specified by the available materials. In the redesigned system of Fig. 6(b), the line of MhLc exists within the available materials for molding. Table 4 illustrates the available material candidates used for a specific lens Lj = L1 and their prices relative to that of NBK7. We replace the plastic of A5514 used as a specific lens (Lj = L1) with NFK5, which is a cost-effective crown glass, rather than other glasses, as shown in Table 4. Here, as with general mobile-phone camera lenses, aspherical surfaces are applied to all lens surfaces. The first lens is made of molding glass and has an aspherical shape that can be fabricated sufficiently. While the glass for a specific lens is replaced, the optical first orders of that element remain unchanged in this process. This design approach outlined in Fig. 6 yields an achromatic and athermal system, for which the optical powers of the front group (1st to 3rd lenses) and the rear group (4th and 5th lenses) are kept positive and negative respectively for the optimization process. This layout is typically a telephoto system.

Table 3 Optical material properties and first orders of a camera lens after changing the slope of an achromatic and athermal line

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−3/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1A551427.7−260.10.2311.135
2OKP4HT69.9−243.9−0.1440.763
3A551427.7−260.10.2400.484
4OKP4HT69.9−243.90.1860.487
5OKP4HT69.9−243.9−0.3890.211


Table 4 Glass codes and prices relative to that of NBK7 of available materials for molding

No.MaterialsGlass CodeRelative Price
1NFK5489.4621.5
2SF57855.1492.6
3NKZFS8725.2212.9
4SF57HTultra855.1493.6
5NLASF46B911.1983.8
6NKZFS5658.2543.9
7NLAF33790.2834.4
8NPK52A499.5327.0
9NKZFS4617.2878.8
10NKZFS2561.3529.3


Figure 7 presents the achromatic and athermal mobile-phone camera lens designed using the above processes, which meets the targeted specifications of Table 1. Figures 8(a) and 8(b) depict the MTFs at room temperature and various temperatures. The MTF of the lens designed using the achromatic and athermal process is much more stable than that of the initial lens, over the specified temperature range and waveband. Furthermore, the MTF at a maximum frequency of 250 cycles/mm is greater than 20% over all fields and the entire temperature range. The chromatic aberration between the C- and g-lines is reduced to +6.43 µm, as illustrated in Fig. 8(c). Moreover, as indicated in Fig. 8(d), the thermal defocus of this lens is significantly reduced. The thermal defocus from −20 ℃ to +60 ℃ ranges from −6.46 µm to +5.64 µm at the two extreme temperatures. Thus both chromatic aberration and thermal defocus are significantly reduced, to be within the focusing area. The final designed lens has an f-number of 1.8, a focal length of 4.07 mm, and stable chromatic and thermal focusing. In conclusion, the designed lens is achromatic in visible light and passively athermalized from −20 ℃ to +60 ℃, while incorporating cost-effective material selection for glass molding.

Figure 7.Layout of the final mobile-phone camera lens.

Figure 8.Optical performance analysis of the final mobile-phone camera lens: (a) Modulation transfer function (MTF) at room temperature, (b) MTFs from −20 ℃ to +60 ℃, (c) chromatic aberration, and (d) thermal defocus of the final mobile-phone camera lens.

We have proposed a method of adjusting the slope of an achromatic and athermal line for the reasonable selection of optical materials on a glass map, to correct chromatic aberration and thermal defocus by altering the optical properties and first orders of lens elements. If the line connecting Mh and Le (or Lc) is out of the slope range spanning the available materials, unfortunately this situation may result in the occurrence of large chromatic aberration and thermal defocus.

To solve this problem, we have proposed a method to effectively alter the slope of an achromatic and athermal line by considering available material distributions. We identify the materials, marked as Lj1 and Lj2, that yield the minimum and maximum slopes of the lines from a housing Mh, which specify the slope range of the line spanning the available materials on a glass map. Next, redistributing the optical first orders of elements by moving the achromatic and athermal line into the available slope range yields a good achromatic and athermal design, which is a key point of this study.

By using this concept to design a mobile-phone camera lens, a reasonable solution with small chromatic aberration and thermal defocus was obtained. The maximum chromatic aberration was +6.43 µm, and the thermal defocus from −20 ℃ to +60 ℃ ranged from −6.46 µm to +5.64 µm at the two extreme temperatures. Both errors were found to be so small that this system was within the focusing area. In conclusion, this proposed design approach is expected to provide a useful means of determining the glasses for an achromatic and athermal designs.

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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  11. W. J. Smith, “Optical-design technique,” in Handbook of Optics, W. G. Driscoll and W. Vaughan, Eds. (McGraw-Hill, USA, 1978), Chapter 2.
  12. R. I. Mercado, “Camera lens system with five lens components,” U.S. Patent 20160313537A1 (2016).
  13. OmniVision OV16A10 16MP product brief, OmniVision Technologies Inc., California, USA (2020).
  14. SCHOTT Optical Glass Overview Excel Table, Schott AG, Mainz, Germany (2022).

Article

Research Paper

Curr. Opt. Photon. 2023; 7(3): 273-282

Published online June 25, 2023 https://doi.org/10.3807/COPP.2023.7.3.273

Copyright © Optical Society of Korea.

Achromatic and Athermal Design of a Mobile-phone Camera Lens by Redistributing Optical First-order Quantities

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: March 26, 2023; Revised: May 15, 2023; Accepted: May 16, 2023

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 a new method for redistributing effectively the first orders of each lens element to achromatize and athermalize an optical system, by introducing a novel method for adjusting the slope of an achromatic and athermal line. This line is specified by connecting the housing, equivalent single lens, and aberration-corrected point on a glass map composed of available plastic and glass materials for molding. Thus, if a specific lens is replaced with the material characterized by the chromatic and thermal powers of an aberration-corrected point, we obtain an achromatic and athermal system. First, we identify two materials that yield the minimum and maximum slopes of the line from a housing coordinate, which specifies the slope range of the line spanning the available materials on a glass map. Next, redistributing the optical first orders (optical powers and paraxial ray heights) of lens elements by moving the achromatic and athermal line into the available slope range of materials yields a good achromatic and athermal design. Applying this concept to design a mobile-phone camera lens, we efficiently obtain an achromatic and athermal system with cost-effective material selection, over the specified temperature and waveband ranges.

Keywords: Aberrations, Achromatization, Athermalization, First order, Glass map

I. INTRODUCTION

An optical system generally suffers from chromatic aberration, owing to wide changes in wavelength. In addition, variations in the ambient temperature induce changes in the curvature radius, refractive index, and lens thickness. The thermal defocus caused by such changes significantly degrades the image quality. Therefore, an optical system, including refractive elements and housing, should be designed for stable performance over the specified waveband and temperature ranges.

The aspherical lenses of a mobile-phone camera mainly employ plastic materials that are easy to inject. However, since these plastics are sensitive to temperature, the thermal defocus due to temperature changes should be corrected at the optical-design stage.

Numerous graphical methods have been reported to correct the chromatic aberration and thermal defocus that cause an optical system to deteriorate in the visible and infrared wavebands [110]. Among them, to correct these errors, matching the aberration-corrected point to a specific glass and redistributing the optical powers of lens elements has been presented [10]. These methods make it difficult, though, to select a material on a glass map when the aberration-corrected point deviates greatly from the available material distributions [9, 10].

To solve the above problem, this paper presents a new graphical method by introducing the achromatic and athermal line that satisfies the achromatic and athermal conditions on a glass map. By changing the slope of this line to the available material boundary, the first orders are redistributed, which yields an optical system that simultaneously reduces chromatic aberration and thermal defocus. Even if a current achromatic and athermal line is far from the available range of materials, we can easily adjust the first orders of elements by altering the line’s slope so that the aberration-corrected point moves into the range of available materials.

This approach results in an achromatic and athermal design even for an optical system made of plastic and glass materials, along with a cost-effective material selection for glass molding. An effective solution with small chromatic aberration and thermal defocus is obtained by using this method to design a mobile-phone camera lens with an f-number of 1.8 and a 16-megapixel image sensor.

II. ACHROMATIC AND ATHERMAL DESIGN METHOD

2.1. Achromatic and Athermal Conditions

The chromatic power ωi and thermal power γi of the element material Mi have a significant effect on the changes in optical power owing to the fluctuations in the wavelength and temperature, and they are expressed as follows [3, 4, 9, 10]:

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 center wavelength, αi is the coefficient of thermal expansion (CTE) of the ith lens material, and T is the temperature.

The longitudinal chromatic aberration comes from the changes (Δfbch') in the back focal length (BFL) with wavelength, and is expressed as Eq. (3). The thermal defocus ∆z′ is evaluated as the difference between the change (Δfbth') in the BFL with the temperature and the change ΔHb') in the flange back length (FBL) with the temperature, as follows [9, 10]:

Δfb'-ch=1ϕT2 i=1kωi'ϕi'

Δz'=Δfb'-thΔHb'=1ϕT2 i=1kγi'ϕi'αhLΔT

where ϕT is the total power and k is the total number of elements.

In the above, the primed parameters denote that they are weighted by the ratio of the paraxial ray heights; That is, the weighted element optical power, weighted chromatic power, and weighted thermal power are ϕi′ = (hi /h1)ϕi, ωi′ = (hi /h1)ωi, and γi′ = (hi /h1)γi respectively. These imply that the air spacings between elements are included in Eqs. (3) and (4), which can handle the lens system more practically.

2.2. Graphical Expression for an Achromatic and Athermal Line on a Glass Map

In this study, an equivalent single lens Le is used to simplify an optical system with an arbitrary number of elements into a doublet system [8]. Thus an optical system with k elements can be recomposed into a doublet system composed of the specific jth 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 power ϕT, achromatic (Δfbch'), and athermal (∆z′ = 0) conditions are respectively given by [911]

ϕT= i=1kϕi'=ϕj'+ϕe'

Δfb'-ch=1ϕT2ωj'ϕj'+ωe'ϕe'=0

Δz'=1ϕT2 i=1kγi'ϕi'αhLΔT=0

where ϕe'= i=1kϕi 'ϕj',ωe'= i=1 k ω i' ϕ i ' ω j' ϕ j'/ ϕ e ',andγe'= i=1 k γ i' ϕ i ' γ j' ϕ j'/ ϕ e '.

In Eq. (7) it can be assumed that the FBL (L) is approximately the same as the BFL, i.e. L ≅ (hk /h1)/ϕT. While the achromatic condition of Eq. (6) and the athermal condition of Eq. (7) in this doublet system are divided by the ratio of the paraxial ray height (hj /h1), we can easily identify the specific lens location without weighting on a glass map [10]. Finally, dividing Eqs. (5), (6), and (7) by the total power ϕT results in expressions for the achromatic and athermal conditions in a doublet system, as follows:

pj+pe=1

ωjpj+ωe''pe=0

γjpj+γe''pe=αh''

where ωe″ = (h1/hj)ωe′, γe″ = (h1/hj)γe′, and αh″ = (hk /hj)αh. The two parameters pj and pe are the ratios of optical powers of the specific lens and equivalent single lens with respect to the total power.

When an equivalent single lens is given, the point designated as Lc(ωc, γc) in Fig. 1(a) 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). Thus, substituting Lj(ωj, γj) into Lc(ωc, γc) and inserting Eqs. (9) and (10) into Eq. (8) results in expressions for the achromatic and athermal conditions in a doublet system, as follows [10]:

Figure 1. Achromatic and athermal conditions on a glass map: (a) Causes of chromatic aberration and thermal defocus, (b)–(d) various combinations of an achromatic and athermal line according to the signs of the optical powers of pj and pe.

pjpe=1ωe''ωcωe''ωc

pjpe=1γe''γcγe''+αh''γcαh''

where ωc = −ωe″(pe /pj), and γc = −(γe″ + αh″)(pe /pj) − αh″. Rearranging Eqs. (11) and (12) yields expressions for pj and pe,

pj=ωe''ωe''ωc=γe''+αh''γe''γcγe''γcωe''ωc=γe''+αh''ωe''

pe=ωcωe''ωc=γc+αh''γe''γcγe''γcωe''ωc=γc+αh''ωc

Each term of these equations is characterized as the optical material properties and first orders of a specific lens and an equivalent lens. If the optical material properties and the first orders do not satisfy the achromatic and athermal conditions, they also cannot satisfy Eqs. (13) and (14). To have an achromatic and athermal system, a specific lens should be located at the achromatic and athermal point Lc(ωc, γc) by changing the lens material. At this time, while the material properties of Lj are changed into those of Lc, the first orders should be maintained to obtain an achromatic and athermal system. Therefore, if a specific lens Lj is replaced with Lc as shown in Fig. 1(a), we obtain the following relationship:

Slope of line(MhLc):γc+αh''ωc=(MhLe):γe''+αh''ωe''

Note that this relationship of Eq. (15) verifies Eqs. (13) and (14). For an achromatic and athermal system, from Eq. (15) and Fig. 1(a) the slope of the line connecting points Mh(0, −αh″) and Le(ωe″, γe″) should be equal to that of the line connecting points Mh(0, −αh″) and Lc(ωc, γc). Therefore, If the specific lens Lj is replaced with the lens Lc characterized by two powers of ωc and γc, then the lens Lc lies on the line connecting the housing Mh and an equivalent lens Le as depicted in Fig. 1(a). If a specific lens is not replaced with Lc(ωc, γc), the line connecting the points of Mh(0, −αh″) and Lj(ωj, γj) has a different slope, so the system does not meet the achromatic and athermal conditions.

Since the sum of normalized powers for a specific lens and an equivalent single lens is unity, i.e. pj + pe = 1, either pj or pe must have a positive value in an imaging system. Figure 1(b) shows the case of positive pj and negative pe. For the sum of two power ratios to be 1.0, pj should have an absolute value greater than pe. Here, to satisfy the condition ωc = −ωe″(pe /pj), ωc and ωe″ should have the same sign. Also, since | pe /pj| is less than 1.0, ωe″ should have an absolute value greater than ωc. Consequently, the line slope between Mh and Le is the same as that between Mh and Lc, in which the latter line exists within the line between Mh and Le, as shown in Fig. 1(b).

Similarly, Fig. 1(c) shows the case of negative pj and positive pe. Since pe have an absolute value greater than pj, ωc should have an absolute value greater than ωe″. Finally, Fig. 1(d) shows the case of positive pj and positive pe. To satisfy the condition ωc = −ωe″(pe /pj), ωc and ωe″ should have different signs. This situation means that the line connecting Mh and Le has the same slope as the line connecting Mh and Lc, but these two line segments exist on opposite sides with respect to Mh. Thus the signs of pj and pe determine the positions of Lc(ωc, γc) and Le(ωe, γe) from Mh(0, −αh″) with the same slope, so they should be considered when reconfiguring an optical system by redistributing the first orders of the lens elements.

2.3. A Method for Graphically Controlling the Slope of a Line on a Glass Map

To have an achromatic and athermal design, we propose the graphical method of adjusting the slope of the line representing achromatic and athermal conditions, which we refer to as the achromatic and athermal line, or briefly, line of MhLcLe. This achromatic and athermal line satisfies the conditions of Eq. (15) on the glass map of Fig. 2.

Figure 2. Slope-control method of shifting an achromatic and athermal line (MhLcLe) into available material distributions.

We first evaluate the optical properties and first orders of an initial optical system, which yields the coordinates of Mh(0, −αh″), Le(ωe″, γe″), Lj(ωj, γj), and Lc(ωc, γc). Next, as can be observed in Fig. 2, we determine the slope range of the line connecting all available materials from housing Mh on a glass map. Subsequently, redistributing the optical first orders of elements by moving the line connecting the housing Mh and equivalent single lens Le into the available slope range specified by the line of MhLj1 and the line of MhLj2, yields a good achromatic and athermal design. Here the first orders include the optical power and paraxial ray height of each element. Although the line of MhLe is far from the available range of materials, we can easily adjust the paraxial ray heights and optical powers of elements by controlling the line slope so that the aberration-corrected point Lc moves into the range of available materials. Thus, this approach enables us to effectively obtain an achromatic and athermal design. This is a key point of this study.

We first identify the distribution of available materials on a glass map, and then mark two extreme glasses that create the minimum and maximum slopes Lj1(ωj1, γj1) and Lj2(ωj2, γj2), and then evaluate the slope range of the line spanning from MhLj1 to MhLj2, as Fig. 2. To have an achromatic and athermal design, the slope of the line between Mh and Le should be within the following two extreme slope limits, namely the available slope ranges as

γj1+αh''ωj1γe''+αh''ωe''γj2+αh''ωj2

γe"+αh''ωe''= i=1kγi'ϕi'γj'ϕj'+hkh1αhϕe' i=1kωi'ϕi'ωj'ϕj'

Equation (17) denotes that the optical powers and paraxial ray heights of elements constituting an equivalent lens are parameters used to change the slope of the line between Mh and Le. Thus, while changing the slope of that line to meet the conditions of Eq. (16), the line between Mh and Lc also leads to the available material boundary, which yields a reasonable solution for the glass of a specific lens. Here, we can easily adjust the slope of the line by checking the relative locations of Le and Lc from Mh, owing to the refracting-power ratios pj and pe.

III. EXAMPLE OF ACHROMATIC AND ATHERMAL DESIGN

3.1. Design of a 16-megapixel Mobile-phone Camera Lens

A mobile-phone camera lens taken from the patented lens [12] operating in the visible range from −20 ℃ to +60 ℃ is presented as a design example using the proposed method. However, to work this patent lens as an optical system for a mobile-phone camera with a low f-number of 1.8 and a 16-megapixel image sensor, we first optimize that lens to have a modulation transfer function (MTF) of more than 25% over all fields at 250 cycles/mm and room temperature [13]. In this design process, we retain the optical power configuration without considering achromatic and athermal conditions. Figure 3 shows the layout for this initial camera lens, and Table 1 lists the targeted specifications to be designed by utilizing the method proposed in this study.

Table 1 . Target specification of a mobile-phone camera lens.

ParametersTarget Value
SensorType1/3 inch, 16-megapixel (4,656 × 3,496)
Pixel Size (μm)1.0 × 1.0
f-number1.8
Image Height (mm)±2.92
Field of View (deg)±35–37
Effective Focal Length (mm)3.87–4.17
Back Focal Length (including filter) (mm)More than 0.6
Total Track Length (mm)Less than 5.5
Distortion (%)Less than 2
Wavelengths435.8 nm (g)–656.3 nm (C)
Temperature Range (℃)−20~+60
Housing MaterialAL6061 (αh = 23.4 × 10−6/℃)
MTF at 250 cycles/mmMore than 30% at 0.0–0.7 F
More than 25% at 0.8–1.0 F


Figure 3. Layout of an initial mobile-phone camera lens.

The MTF at 250 cycles/mm at room temperature is good, but the MTFs from −20 ℃ to +60 ℃ are very unstable, as presented in Figs. 4(a) and 4(b) respectively.

Figure 4. Optical performance analysis of an initial mobile-phone camera lens: (a) Modulation transfer function (MTF) at room temperature, (b) MTFs from −20 ℃ to +60 ℃, (c) chromatic aberration, and (d) thermal defocus of an initial mobile-phone camera lens.

For this system the chromatic aberration between the C- and g-lines is evaluated as +40.24 µm, which is too large to get a good image, as shown in Fig. 4(c). Moreover, the BFL of this system is thermally unstable from −20 ℃ to +60 ℃, as illustrated in Fig. 4(d). The thermal defocus expressed as ∆z′ = BFL − FBL ranges from −31.28 µm to +28.99 µm at the extreme temperatures, which leaves the system far out of focus. These significant thermal defocuses lead to an unstable MTF at the two extremal temperatures, as depicted in Fig. 4(b).

3.2. Achromatic and Athermal Design by Redistributing the Optical First-order Quantities

In this study we apply the achromatic and athermal method outlined in Section 2 to a mobile-phone camera lens design through the analysis of the optical properties and the first orders. Since the thermal powers of most plastic materials are very negative, it is extremely hard to configure a plastic-only optical system corrected for thermal defocus. The glasses for molding supplied by Schott are used to construct a glass map for an achromatic and athermal design [14].

Next, we classify several cases according to the selection of a specific lens Lj and an equivalent single lens Le to find the most suitable combination for the achromatic and athermal design. As can be observed in Fig. 3, it is desirable to select the specific lens in consideration of the diameter and aspherical shape of that lens. This mobile-camera lens consists of five elements, so five doublet systems can be created. However, the shapes and diameters of the fourth and fifth lenses are complicated and large, so it is difficult to mold them into aspherical lenses made of molding glass. Hence we select three available lens candidates used for a specific lens Lj; They are Lj = L1, L2, and L3 respectively. Figure 5 illustrates the three cases and their aberration-corrected points Lc(ωc, γc) for each specific lens Lj ( j = 1, 2, 3), used to recompose the optical system as a doublet with a corresponding equivalent single lens. Table 2 lists the material properties and the first orders of an initial mobile-phone camera lens.

Table 2 . Optical material properties and first orders of an initial mobile-phone camera lens.

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−3/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1A551427.7−260.10.1581.089
2OKP4HT69.9−243.9−0.1110.806
3A551427.7−260.10.2420.804
4OKP4HT69.9−243.90.1970.437
5OKP4HT69.9−243.9−0.4160.206


Figure 5. Glass maps of an initial mobile-phone camera lens: (a) case 1 (Lj = L1), (b) case 2 (Lj = L2), and (c) case 3 (Lj = L3).

In the glass map of Fig. 5, case 1 is the doublet system composed of the specific lens Lj = L1 and the equivalent single lens Le comprising the remaining elements L2, L3, L4 and L5. Similarly, case 2 is Lj = L2 and Le = L1 + L3 + L4 + L5, and case 3 is Lj = L3 and Le = L1 + L2 + L4 + L5. In that figure, the slope of the line connecting Mh and Le is equal to that of the line connecting Mh and Lc. However, the line connecting Mh and Lj composed of a material not on the achromatic and athermal line shows a large difference from the line of MhLc, which may result in large chromatic aberration and thermal defocus. Since Figs. 5(a) and 5(c) feature the case of positive pj and positive pe, this situation means that the slope of the line connecting Mh and Le is equal to that of the line connecting Mh and Lc, but these two lines exist on the opposite sides of Mh. Meanwhile, while redistributing the first orders of an equivalent lens, pj and pe should be maintained at positive values. In two cases, note that only when the line of MhLe is turned clockwise can the line of MhLc reach the available material range. Figure 5(b) shows the case of negative pj and positive pe. Since pe has an absolute value greater than pj, ωc should have an absolute value greater than ωe″, which reveals that the line of MhLe is within the line of MhLc. In this case, even if the slope is adjusted through optical power redistribution, it is difficult for the line of MhLc to exist within the range of available material. Thus, cases 1 and 3 have an advantage over case 2. Among them, as shown in Fig. 5, case 1 may change the slope of the line relatively less than case 3, so case 1 is more useful than case 3. In addition, looking at the lens layout in Fig. 3, it is judged that applying molding materials to the first lens is advantageous for cost reduction and aspherical molding, rather than to the third one as in case 3. For this reason, by selecting L1 in case 1 as a specific lens and adjusting the slope of the line, an achromatic and athermal design can be realized by redistributing the optical first-order properties of elements.

Two steps can be considered to design a lens corrected for thermal defocus and chromatic aberration.

The first step is selecting the slope range of the line by considering available material distributions. Among them, the material NKZFS2, marked as Lj2 (28.42, 4.96) on a glass map of Fig. 6(a), has a positive maximum slope. Similarly, NPK52A, marked as Lj1(18.79, −25.50), has a negative maximum slope. The slope range of the lines from the housing Mh(0, −4.421) to the available materials of Lj1 and Lj2 are −1.122 to 0.330 (×10−3/℃) in an initial mobile-phone camera lens, as depicted in Fig. 6(a). Unfortunately, the line connecting Mh and Le(or Lc) has a slope of −40.486 (×10−3/℃), which is outside the slope range spanning the available materials.

Figure 6. Achromatic and athermal design process using the slope-control method for a mobile-phone camera lens: (a) optical first-order redistribution process by adjusting the slope of an achromatic and athermal line, (b) material-change process for a specific lens.

The second step is redistributing the optical first-order quantities of an equivalent single lens so that the aberration-corrected point Lc moves into an available material region by changing the slope of the line of MhLe [see Fig. 6(b)]. While adjusting the slope of an achromatic and athermal line, pj and pe must have positive values.

This design approach yields an achromatic and athermal system, for which the material properties and the first orders of elements are listed in Table 3. The slope range of the line from the housing Mh(0, −4.343) to the available materials of Lj1 and Lj2 is −1.126 to 0.327 (×10−3/℃) in a redesigned camera lens. Fortunately, the newly altered line connecting Mh and Le(or Lc) has a slope of 0.275(×10−3/℃), which exists within the slope range specified by the available materials. In the redesigned system of Fig. 6(b), the line of MhLc exists within the available materials for molding. Table 4 illustrates the available material candidates used for a specific lens Lj = L1 and their prices relative to that of NBK7. We replace the plastic of A5514 used as a specific lens (Lj = L1) with NFK5, which is a cost-effective crown glass, rather than other glasses, as shown in Table 4. Here, as with general mobile-phone camera lenses, aspherical surfaces are applied to all lens surfaces. The first lens is made of molding glass and has an aspherical shape that can be fabricated sufficiently. While the glass for a specific lens is replaced, the optical first orders of that element remain unchanged in this process. This design approach outlined in Fig. 6 yields an achromatic and athermal system, for which the optical powers of the front group (1st to 3rd lenses) and the rear group (4th and 5th lenses) are kept positive and negative respectively for the optimization process. This layout is typically a telephoto system.

Table 3 . Optical material properties and first orders of a camera lens after changing the slope of an achromatic and athermal line.

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−3/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1A551427.7−260.10.2311.135
2OKP4HT69.9−243.9−0.1440.763
3A551427.7−260.10.2400.484
4OKP4HT69.9−243.90.1860.487
5OKP4HT69.9−243.9−0.3890.211


Table 4 . Glass codes and prices relative to that of NBK7 of available materials for molding.

No.MaterialsGlass CodeRelative Price
1NFK5489.4621.5
2SF57855.1492.6
3NKZFS8725.2212.9
4SF57HTultra855.1493.6
5NLASF46B911.1983.8
6NKZFS5658.2543.9
7NLAF33790.2834.4
8NPK52A499.5327.0
9NKZFS4617.2878.8
10NKZFS2561.3529.3


Figure 7 presents the achromatic and athermal mobile-phone camera lens designed using the above processes, which meets the targeted specifications of Table 1. Figures 8(a) and 8(b) depict the MTFs at room temperature and various temperatures. The MTF of the lens designed using the achromatic and athermal process is much more stable than that of the initial lens, over the specified temperature range and waveband. Furthermore, the MTF at a maximum frequency of 250 cycles/mm is greater than 20% over all fields and the entire temperature range. The chromatic aberration between the C- and g-lines is reduced to +6.43 µm, as illustrated in Fig. 8(c). Moreover, as indicated in Fig. 8(d), the thermal defocus of this lens is significantly reduced. The thermal defocus from −20 ℃ to +60 ℃ ranges from −6.46 µm to +5.64 µm at the two extreme temperatures. Thus both chromatic aberration and thermal defocus are significantly reduced, to be within the focusing area. The final designed lens has an f-number of 1.8, a focal length of 4.07 mm, and stable chromatic and thermal focusing. In conclusion, the designed lens is achromatic in visible light and passively athermalized from −20 ℃ to +60 ℃, while incorporating cost-effective material selection for glass molding.

Figure 7. Layout of the final mobile-phone camera lens.

Figure 8. Optical performance analysis of the final mobile-phone camera lens: (a) Modulation transfer function (MTF) at room temperature, (b) MTFs from −20 ℃ to +60 ℃, (c) chromatic aberration, and (d) thermal defocus of the final mobile-phone camera lens.

IV. CONCLUSION

We have proposed a method of adjusting the slope of an achromatic and athermal line for the reasonable selection of optical materials on a glass map, to correct chromatic aberration and thermal defocus by altering the optical properties and first orders of lens elements. If the line connecting Mh and Le (or Lc) is out of the slope range spanning the available materials, unfortunately this situation may result in the occurrence of large chromatic aberration and thermal defocus.

To solve this problem, we have proposed a method to effectively alter the slope of an achromatic and athermal line by considering available material distributions. We identify the materials, marked as Lj1 and Lj2, that yield the minimum and maximum slopes of the lines from a housing Mh, which specify the slope range of the line spanning the available materials on a glass map. Next, redistributing the optical first orders of elements by moving the achromatic and athermal line into the available slope range yields a good achromatic and athermal design, which is a key point of this study.

By using this concept to design a mobile-phone camera lens, a reasonable solution with small chromatic aberration and thermal defocus was obtained. The maximum chromatic aberration was +6.43 µm, and the thermal defocus from −20 ℃ to +60 ℃ ranged from −6.46 µm to +5.64 µm at the two extreme temperatures. Both errors were found to be so small that this system was within the focusing area. In conclusion, this proposed design approach is expected to provide a useful means of determining the glasses for an achromatic and athermal designs.

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.

FUNDING

Research fund of Dankook University in 2023.

Fig 1.

Figure 1.Achromatic and athermal conditions on a glass map: (a) Causes of chromatic aberration and thermal defocus, (b)–(d) various combinations of an achromatic and athermal line according to the signs of the optical powers of pj and pe.
Current Optics and Photonics 2023; 7: 273-282https://doi.org/10.3807/COPP.2023.7.3.273

Fig 2.

Figure 2.Slope-control method of shifting an achromatic and athermal line (MhLcLe) into available material distributions.
Current Optics and Photonics 2023; 7: 273-282https://doi.org/10.3807/COPP.2023.7.3.273

Fig 3.

Figure 3.Layout of an initial mobile-phone camera lens.
Current Optics and Photonics 2023; 7: 273-282https://doi.org/10.3807/COPP.2023.7.3.273

Fig 4.

Figure 4.Optical performance analysis of an initial mobile-phone camera lens: (a) Modulation transfer function (MTF) at room temperature, (b) MTFs from −20 ℃ to +60 ℃, (c) chromatic aberration, and (d) thermal defocus of an initial mobile-phone camera lens.
Current Optics and Photonics 2023; 7: 273-282https://doi.org/10.3807/COPP.2023.7.3.273

Fig 5.

Figure 5.Glass maps of an initial mobile-phone camera lens: (a) case 1 (Lj = L1), (b) case 2 (Lj = L2), and (c) case 3 (Lj = L3).
Current Optics and Photonics 2023; 7: 273-282https://doi.org/10.3807/COPP.2023.7.3.273

Fig 6.

Figure 6.Achromatic and athermal design process using the slope-control method for a mobile-phone camera lens: (a) optical first-order redistribution process by adjusting the slope of an achromatic and athermal line, (b) material-change process for a specific lens.
Current Optics and Photonics 2023; 7: 273-282https://doi.org/10.3807/COPP.2023.7.3.273

Fig 7.

Figure 7.Layout of the final mobile-phone camera lens.
Current Optics and Photonics 2023; 7: 273-282https://doi.org/10.3807/COPP.2023.7.3.273

Fig 8.

Figure 8.Optical performance analysis of the final mobile-phone camera lens: (a) Modulation transfer function (MTF) at room temperature, (b) MTFs from −20 ℃ to +60 ℃, (c) chromatic aberration, and (d) thermal defocus of the final mobile-phone camera lens.
Current Optics and Photonics 2023; 7: 273-282https://doi.org/10.3807/COPP.2023.7.3.273

Table 1 Target specification of a mobile-phone camera lens

ParametersTarget Value
SensorType1/3 inch, 16-megapixel (4,656 × 3,496)
Pixel Size (μm)1.0 × 1.0
f-number1.8
Image Height (mm)±2.92
Field of View (deg)±35–37
Effective Focal Length (mm)3.87–4.17
Back Focal Length (including filter) (mm)More than 0.6
Total Track Length (mm)Less than 5.5
Distortion (%)Less than 2
Wavelengths435.8 nm (g)–656.3 nm (C)
Temperature Range (℃)−20~+60
Housing MaterialAL6061 (αh = 23.4 × 10−6/℃)
MTF at 250 cycles/mmMore than 30% at 0.0–0.7 F
More than 25% at 0.8–1.0 F

Table 2 Optical material properties and first orders of an initial mobile-phone camera lens

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−3/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1A551427.7−260.10.1581.089
2OKP4HT69.9−243.9−0.1110.806
3A551427.7−260.10.2420.804
4OKP4HT69.9−243.90.1970.437
5OKP4HT69.9−243.9−0.4160.206

Table 3 Optical material properties and first orders of a camera lens after changing the slope of an achromatic and athermal line

ElementMaterialChromatic Power (×10−3)Thermal Power (×10−3/℃)Optical Power (mm−1)Paraxial Ray Height (mm)
1A551427.7−260.10.2311.135
2OKP4HT69.9−243.9−0.1440.763
3A551427.7−260.10.2400.484
4OKP4HT69.9−243.90.1860.487
5OKP4HT69.9−243.9−0.3890.211

Table 4 Glass codes and prices relative to that of NBK7 of available materials for molding

No.MaterialsGlass CodeRelative Price
1NFK5489.4621.5
2SF57855.1492.6
3NKZFS8725.2212.9
4SF57HTultra855.1493.6
5NLASF46B911.1983.8
6NKZFS5658.2543.9
7NLAF33790.2834.4
8NPK52A499.5327.0
9NKZFS4617.2878.8
10NKZFS2561.3529.3

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