Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2022; 6(1): 104-113
Published online February 25, 2022 https://doi.org/10.3807/COPP.2022.6.1.104
Copyright © Optical Society of Korea.
Sung Min Park, Jea-Woo Lee, Sung-Chan Park
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.
We present a method to count the overall length of the zoom system in an initial design stage. In a zoom-lens design using the concept of the group, it has been very hard to precisely estimate the overall length at all zoom positions through the previous paraxial studies. To solve this difficulty, we introduce T_{eq} as a measure of the total track length in an equivalent zoom system, which can be found from the first order parameters obtained by solving the zoom equations. Among many solutions, the parameters that provide the smallest T_{eq} are selected to construct a compact initial zoom system. Also, to obtain an 8× four-group zoom system without moving groups, tunable polymer lenses (TPLs) have been introduced as a variator and a compensator. The final designed zoom lens has a short overall length of 29.99 mm, even over a wide focal-length range of 4‒31 mm, and an f-number of F/3.5 at wide to F/4.5 at tele position, respectively.
Keywords: Aberrations, First orders, Tunable polymer lens, Zoom lens, Zoom position
OCIS codes: (080.2740) Geometric optical design; (120.4570) Optical design of instruments; (220.3620) Lens system design
A zoom system consists of many lens groups, including moving groups such as a variator for zooming and a compensator for fixing the image plane. These moving groups are necessary to achieve different magnifications, but they require additional space for their motion. In addition, since this zoom system must satisfy all first-order requirements and correct aberrations at all zoom positions, it is more likely to consist of many lens elements. However, in the initial design stage of this complex optical zoom system, it is very difficult to precisely estimate the overall length of the zoom optics through a paraxial study.
Much research on initial zoom-system designs using paraxial optics has been reported [1‒6]. These paraxial solutions provide the first orders of each group and zoom locus, but it is difficult to determine whether they provide a suitable solution. To have a reasonable zoom system initially, Park and Shannon proposed zoom-lens design using lens modules, in which each group was replaced with lens modules and then optimized to have a zoom system using the first orders of the modules as design parameters [7‒9]. However, this design method did not provide any information on the overall length needed to achieve a compact zoom system.
To overcome this difficulty, this study proposes a method to count the overall length of the zoom system at the initial design stage. From the fact that the overall length can be estimated indirectly through the air distance and the first orders obtained by solving the zoom equations, we introduce T_{eq} as a measure of the total track length in an equivalent zoom system. When T_{eq} is small, the overall length of a real optical system is expected to be short. Therefore, if the system with the smallest T_{eq} is suggested, we can effectively obtain an optical zoom system having short overall length. This design concept is a key point of this study.
Among many solutions for an 8× four-group zoom system employing two tunable polymer lenses (TPLs) as variator and compensator, the first orders that provide the smallest T_{eq} are selected to construct a compact initial zoom system. This design approach enables us to realize a compact 8× four-group zoom system with short overall length of 29.99 mm over a wide focal length range of 4–31 mm, although all groups are fixed for zooming and compensating.
Each group in a zoom system can be transformed into an equivalent lens group that has the same first-order quantities, such as effective focal length (
Also, another paraxial ray can be traced similarly in the reverse direction of Fig. 1(b); its parameters are marked by double primes as follows:
where
Combining Eqs. (4) and (9), Eqs. (4) and (5), and Eqs. (9) and (10), yields respectively the following three equations:
Substituting Eqs. (1), (4), and (12) into Eq. (3), we obtain Eq. (14). Also, substituting Eqs. (6), (9), and (13) into Eq. (8) yields Eq. (15), and similarly combining Eqs. (14) and (15) leads to Eqs. (16) and (17), as follows:
In addition, inserting Eq. (2) into Eq. (1) yields Eq. (18). Finally, if we substitute Eqs. (12) and (16) into Eq. (18), we obtain Eq. (19), as follows:
If we designate the thickness and the refractive index of the
Consequently, the overall length T_{eq} of an equivalent g-group zoom system within first-order optics can be given by
In Eq. (21), g denotes the number of groups and
In this section, we determine the initial data for an 8× four-group zoom system without moving groups by employing two tunable polymer lenses (TPLs) as a variator and a compensator [10]. As seen in Fig. 2, there are 21 parameters to be determined initially at each zoom position in a four-group zoom system; they should be selected to have the smallest T_{eq}. For convenience, we deal with the zoom system at its wide and tele positions, so there are 42 parameters in total.
In this work, we present a four-group zoom system without moving groups by employing TPLs. Thus, we locate the TPLs at the second group for zooming and the fourth group for compensating, as shown in Fig. 3. To obtain a high zoom ratio of 8×, it is desirable to construct the zoom system to be of the retrofocus type at the wide position, and of the telephoto type at the tele position.
The 1^{st} group’s power is always fixed and slightly positive to collect the rays, while the TPL at the 2^{nd} group should have a negative power at the wide position, and a positive power at the tele position. The power of the 3^{rd} group should be positive, to relay the rays to the next group. As opposed to the 2^{nd} group, the TPL at the 4^{th} group should have a positive power at wide position and a negative power at tele position, as in Fig. 3. This large change in power of the TPLs can be obtained by shape-changing the polymeric membrane from plano-concave to plano-convex (or vice versa), as shown in Fig. 4 [11‒14].
From Fig. 3, the paraxial design of this zoom system with an infinite object can be formulated as follows [5, 6, 10]:
where
Figure 4 illustrates the changes in surface sag and central thickness induced by the curvature-changing at different zoom positions. To obtain a physically meaningful 8× four-group zoom system, the TPLs should have reasonable quantities such as refractive index, thickness, aperture size, etc. These parameters should initially be specified to yield a feasible and compact TPL. In Fig. 4,
From the structure of a TPL, while the edge thickness is always fixed for the zooming process, its central thickness is changed by means of a shape-changing polymeric membrane, as shown in Fig. 4 [10, 12‒14]. Assuming that the back surface of the TPL is a rigid plane in this study, the thicknesses at both extreme positions, aperture sizes, and refractive indices of two feasible TPLs are summarized in Table 1.
TABLE 1 Design parameters of the tunable polymer lenses (TPLs) at the 2nd and 4th groups
Parameters | The 2nd group | The 4th group | ||
---|---|---|---|---|
Liquid Lens | Cover Glass | Liquid Lens | Cover Glass | |
Thickness (wide) | ||||
Thickness (tele) | ||||
Semi-aperture | ||||
Refractive Index |
Finally,
These principal-plane distances between adjacent groups at the wide position can be re-expressed in terms of first-order quantities such as the effective focal length (
The curvature of a TPL should be changed to provide the required power at various zoom positions, which consequently causes the principal-point positions to shift. Since the distances between adjacent principal points at the tele position,
(1) Since the 1^{st} and 3^{rd} groups have fixed focal lengths, the first orders of both groups are always fixed at all zoom positions, as follows:
(2) We initially set the focal length of 4 mm at the wide position. To have a zoom ratio of 8×, that at the tele position should be 32 mm. Thus their powers are given by
(3) To avoid changing the air distances between adjacent groups owing to shape change of the polymeric membrane, dummy surfaces are placed at the TPLs of the 2^{nd} and 4^{th} groups, as shown in Fig. 4. Since
(4) Referring to Figs. 2, 4, and 5, when the zoom position is switched from wide to tele, the amount of movement of the principal points of each group, ∆
Here, note that the amount of movement ∆
Among 42 parameters in total, since the distances between the adjacent principal points at the tele position can be expressed using the amount of movement ∆
Here, if the zoom system satisfies Eqs. (22)‒(29), only 10 free parameters remain. It is desirable to select free parameters that can be estimated effectively, such as the focal length of each group and the air distances between groups, rather than the front focal lengths, back focal lengths, and distances between adjacent principal planes.
First, for the given six free parameters
Next, out of the remaining parameters
To effectively enlarge the magnitude of the power variation at a TPL, the focal lengths of the TPLs at the 2^{nd} and 4^{th} groups are set to be symmetric with opposite signs at wide and tele positions. This configuration reduces two degrees of freedom, as follows:
According to design experience, the closer the 1^{st} group and 2^{nd} group are to each other, the more advantageous they are for a short overall length. When the same rule is applied to the 3^{rd} and 4^{th} groups, their air distances are constrained to be 0.3 mm for a compact zoom system. This eliminates an additional two degrees of freedom:
Finally, only six independent parameters remain; they are
Whenever these six independent parameters are properly given, the remaining 36 design variables are determined by solving the zoom equations (22)‒(29) and satisfying the four conditions of Section 3.3, with the TPL specifications of Table 1.
For the given six parameters
Whenever the focal lengths (
The solutions for these four principal-plane distances at the wide position are initially investigated for
In Fig. 6, note that as
The solutions for the 1^{st} and 4^{th} combinations of
When the focal lengths of all groups, not only
By inputting the focal lengths ranging from
From the finding that solutions suitable for a compact zoom system have positive distances for
The 3^{rd} lens group should have a positive power to relay the rays to the next group and needs many elements to correct all residual aberrations. This situation probably leads the first principal plane to be placed to the right side of the front surface, which yields a positive distance for ∆_{3w}. In the appropriate range of this distance, the constraints of
The other distances
By applying Eq. (21) to a four-group zoom system, the overall length T_{eq} of this equivalent zoom system can be re-expressed as
where
Whenever six independent variables are given, the back (
By inputting the focal lengths ranging from
Table 2 Values of T_{eq} for various
ffl_{1} | −67 | … | −64 | −63 | −62 | … | −58 |
---|---|---|---|---|---|---|---|
air_{2} | |||||||
4 | 100 | … | 100 | 30.267 | 31.751 | … | 37.690 |
4.5 | 100 | … | 100 | 29.481 | 30.966 | … | 36.904 |
… | … | … | … | … | … | … | … |
5.5 | 100 | … | 100 | 27.909 | 29.394 | … | 35.333 |
6 | 100 | … | 100 | 27.124 | 28.608 | … | 34.547 |
6.5 | 100 | … | 100 | 100 | 100 | … | 100 |
… | … | … | … | … | … | … | … |
9.5 | … | … | 100 | 100 | 100 | … | 100 |
10 | … | … | 100 | 100 | 100 | … | 100 |
Among the solutions obtained through this design process, the zoom system with
Table 3 Initial zoom data with the shortest T_{eq} (in mm)
1^{st} group | 2^{nd} group | 3^{rd} group | 4^{th} group | |
---|---|---|---|---|
efl_{w} | 63.0 | -17.5 | 21.0 | 10.0 |
efl_{t} | 17.5 | -10.0 | ||
z_{w} | 4.310 | 17.349 | -14.304 | 8.485 |
z_{t} | 1.635 | 19.287 | -11.783 | 6.71 |
air | 0.3 | 6.0 | 0.3 | 6.033 |
ffl_{w} | -63.0 | 20.175 | -10.337 | -10.0 |
ffl_{t} | -17.5 | 12.521 | ||
bfl_{w} | 61.664 | -18.185 | 35.604 | 7.547 |
bfl_{t} | 14.876 | -10.677 | ||
d_{eq} | 2.025 | 3.675 | 5.27 | 3.521 |
T_{eq} | 27.124 |
To obtain a real lens group equivalent to the zoom data of Table 3, each real lens group should be designed to have the same first-order quantities such as
To have a compact zoom system, the 1^{st} group is designed with two lens elements, and the 2^{nd} and 4^{th} groups each include only one TPL to change the power. As seen in Table 3, to have a higher zoom ratio of 8×, the principal-plane distances (
Thus, we take an initial zoom system with a half-image-size of 1 mm and f-numbers of F/5 at the wide position to F/7 at the tele position, for which reducing the aperture and field size is desirable, to avoid higher-order aberrations. Figure 7 shows an 8× initial real-lens zoom system equivalent to the zoom data of Table 3 within first-order optics. This zoom system has a fixed variator at the 2^{nd} group and compensator at the 4^{th}, in which the former focal lengths are changed from −17.5 mm to 17.5 mm, and the latter from 10 mm to −10 mm.
In the initial design of Fig. 7, the aperture and image size should be extended, to satisfy the specifications for a zoom camera. The aperture is increased to F/3.5 at wide and F/4.5 at tele position. The half-image-size should be 1.8 mm for a 1/6-inch CMOS image sensor with 2048 × 1536 pixels. To improve the entire performance of an extended aperture and field system, we reduce the aberrations of the starting zoom lens by using aspheric surfaces.
Figure 8 illustrates the layout of the final designed zoom system, which consists of 10 elements including two TPLs and six aspheric lenses. The maximum diameter of the 1^{st} lens group is 13.59 mm. The distortions are balanced to less than 4.6%, and the ratio of relative illumination (RI) is more than 70% at the corner field when measured at all zoom positions, as in Fig. 8. The modulation transfer function (MTF) at 180 lp/mm is more than 25% over all fields at both extreme positions, from Fig. 9. Thus, all aberrations are significantly reduced.
The variation of the chief ray angle of incidence (AOI) from wide to tele position is less than 7.11 degrees. Because this variation is small, a stable image quality can be achieved for zooming. The overall length of this zoom lens, even at a large zoom ratio of 8×, is less than 29.99 mm. This is 5.5 mm shorter than the previous design, even for the same focal-length range of 4–31 mm [10]. Thus this design concept, based on the analysis for the overall length T_{eq} of an equivalent zoom system, greatly reduces the effort required to obtain a compact zoom lens system. This is an advantage of the study.
To have a compact zoom system with a high zoom ratio of 8×, it is most effective to construct an optical system with a short overall length. In a zoom system consisting of several groups, however, it is very difficult to precisely estimate the overall length through a paraxial study. To solve this difficulty, this study has proposed a method to count the overall length of the zoom system in an initial design stage. From the fact that the overall length can be estimated indirectly through paraxial ray tracing, we introduce T_{eq} as a measure of the total track length in an equivalent zoom system, which can be found from the first orders obtained by solving the zoom equations.
Among many solutions, the first-order quantities that provide the smallest T_{eq} are selected to construct an initial zoom system with a high zoom ration of 8×. Also, to have a compact 8× four-group zoom system with no moving groups, TPLs have been suggested for variator and compensator.
Utilizing the design approaches proposed in this study, we have achieved a compact 8× four-group zoom system with a short overall length of 29.99 mm. This zoom lens had a wide focal-length range of 4–31 mm, and f-numbers of 3.5 at wide to 4.5 at tele position. Through this study, we illustrated the design process of a zoom system considering the overall length by introducing T_{eq}, along with a numerical, paraxial design approach.
The proposed design approaches are expected to be useful in estimating the overall length of any optical zoom system, regardless of whether there is a moving group or not. Even if a TPL were replaced by a moving group, we could estimate the overall length by inserting the first order quantities of the moving group into Eq. (34) for T_{eq}. Thus, we have been investigating an effective way to discover the solutions with the shortest overall length and applying this method to a zoom system with moving groups.
National Research Foundation of Korea (NRF); Korea government (MSIT, No. 2019R1F1A1060065).
Curr. Opt. Photon. 2022; 6(1): 104-113
Published online February 25, 2022 https://doi.org/10.3807/COPP.2022.6.1.104
Copyright © Optical Society of Korea.
Sung Min Park, Jea-Woo Lee, Sung-Chan Park
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.
We present a method to count the overall length of the zoom system in an initial design stage. In a zoom-lens design using the concept of the group, it has been very hard to precisely estimate the overall length at all zoom positions through the previous paraxial studies. To solve this difficulty, we introduce T_{eq} as a measure of the total track length in an equivalent zoom system, which can be found from the first order parameters obtained by solving the zoom equations. Among many solutions, the parameters that provide the smallest T_{eq} are selected to construct a compact initial zoom system. Also, to obtain an 8× four-group zoom system without moving groups, tunable polymer lenses (TPLs) have been introduced as a variator and a compensator. The final designed zoom lens has a short overall length of 29.99 mm, even over a wide focal-length range of 4‒31 mm, and an f-number of F/3.5 at wide to F/4.5 at tele position, respectively.
Keywords: Aberrations, First orders, Tunable polymer lens, Zoom lens, Zoom position
A zoom system consists of many lens groups, including moving groups such as a variator for zooming and a compensator for fixing the image plane. These moving groups are necessary to achieve different magnifications, but they require additional space for their motion. In addition, since this zoom system must satisfy all first-order requirements and correct aberrations at all zoom positions, it is more likely to consist of many lens elements. However, in the initial design stage of this complex optical zoom system, it is very difficult to precisely estimate the overall length of the zoom optics through a paraxial study.
Much research on initial zoom-system designs using paraxial optics has been reported [1‒6]. These paraxial solutions provide the first orders of each group and zoom locus, but it is difficult to determine whether they provide a suitable solution. To have a reasonable zoom system initially, Park and Shannon proposed zoom-lens design using lens modules, in which each group was replaced with lens modules and then optimized to have a zoom system using the first orders of the modules as design parameters [7‒9]. However, this design method did not provide any information on the overall length needed to achieve a compact zoom system.
To overcome this difficulty, this study proposes a method to count the overall length of the zoom system at the initial design stage. From the fact that the overall length can be estimated indirectly through the air distance and the first orders obtained by solving the zoom equations, we introduce T_{eq} as a measure of the total track length in an equivalent zoom system. When T_{eq} is small, the overall length of a real optical system is expected to be short. Therefore, if the system with the smallest T_{eq} is suggested, we can effectively obtain an optical zoom system having short overall length. This design concept is a key point of this study.
Among many solutions for an 8× four-group zoom system employing two tunable polymer lenses (TPLs) as variator and compensator, the first orders that provide the smallest T_{eq} are selected to construct a compact initial zoom system. This design approach enables us to realize a compact 8× four-group zoom system with short overall length of 29.99 mm over a wide focal length range of 4–31 mm, although all groups are fixed for zooming and compensating.
Each group in a zoom system can be transformed into an equivalent lens group that has the same first-order quantities, such as effective focal length (
Also, another paraxial ray can be traced similarly in the reverse direction of Fig. 1(b); its parameters are marked by double primes as follows:
where
Combining Eqs. (4) and (9), Eqs. (4) and (5), and Eqs. (9) and (10), yields respectively the following three equations:
Substituting Eqs. (1), (4), and (12) into Eq. (3), we obtain Eq. (14). Also, substituting Eqs. (6), (9), and (13) into Eq. (8) yields Eq. (15), and similarly combining Eqs. (14) and (15) leads to Eqs. (16) and (17), as follows:
In addition, inserting Eq. (2) into Eq. (1) yields Eq. (18). Finally, if we substitute Eqs. (12) and (16) into Eq. (18), we obtain Eq. (19), as follows:
If we designate the thickness and the refractive index of the
Consequently, the overall length T_{eq} of an equivalent g-group zoom system within first-order optics can be given by
In Eq. (21), g denotes the number of groups and
In this section, we determine the initial data for an 8× four-group zoom system without moving groups by employing two tunable polymer lenses (TPLs) as a variator and a compensator [10]. As seen in Fig. 2, there are 21 parameters to be determined initially at each zoom position in a four-group zoom system; they should be selected to have the smallest T_{eq}. For convenience, we deal with the zoom system at its wide and tele positions, so there are 42 parameters in total.
In this work, we present a four-group zoom system without moving groups by employing TPLs. Thus, we locate the TPLs at the second group for zooming and the fourth group for compensating, as shown in Fig. 3. To obtain a high zoom ratio of 8×, it is desirable to construct the zoom system to be of the retrofocus type at the wide position, and of the telephoto type at the tele position.
The 1^{st} group’s power is always fixed and slightly positive to collect the rays, while the TPL at the 2^{nd} group should have a negative power at the wide position, and a positive power at the tele position. The power of the 3^{rd} group should be positive, to relay the rays to the next group. As opposed to the 2^{nd} group, the TPL at the 4^{th} group should have a positive power at wide position and a negative power at tele position, as in Fig. 3. This large change in power of the TPLs can be obtained by shape-changing the polymeric membrane from plano-concave to plano-convex (or vice versa), as shown in Fig. 4 [11‒14].
From Fig. 3, the paraxial design of this zoom system with an infinite object can be formulated as follows [5, 6, 10]:
where
Figure 4 illustrates the changes in surface sag and central thickness induced by the curvature-changing at different zoom positions. To obtain a physically meaningful 8× four-group zoom system, the TPLs should have reasonable quantities such as refractive index, thickness, aperture size, etc. These parameters should initially be specified to yield a feasible and compact TPL. In Fig. 4,
From the structure of a TPL, while the edge thickness is always fixed for the zooming process, its central thickness is changed by means of a shape-changing polymeric membrane, as shown in Fig. 4 [10, 12‒14]. Assuming that the back surface of the TPL is a rigid plane in this study, the thicknesses at both extreme positions, aperture sizes, and refractive indices of two feasible TPLs are summarized in Table 1.
TABLE 1. Design parameters of the tunable polymer lenses (TPLs) at the 2nd and 4th groups.
Parameters | The 2nd group | The 4th group | ||
---|---|---|---|---|
Liquid Lens | Cover Glass | Liquid Lens | Cover Glass | |
Thickness (wide) | ||||
Thickness (tele) | ||||
Semi-aperture | ||||
Refractive Index |
Finally,
These principal-plane distances between adjacent groups at the wide position can be re-expressed in terms of first-order quantities such as the effective focal length (
The curvature of a TPL should be changed to provide the required power at various zoom positions, which consequently causes the principal-point positions to shift. Since the distances between adjacent principal points at the tele position,
(1) Since the 1^{st} and 3^{rd} groups have fixed focal lengths, the first orders of both groups are always fixed at all zoom positions, as follows:
(2) We initially set the focal length of 4 mm at the wide position. To have a zoom ratio of 8×, that at the tele position should be 32 mm. Thus their powers are given by
(3) To avoid changing the air distances between adjacent groups owing to shape change of the polymeric membrane, dummy surfaces are placed at the TPLs of the 2^{nd} and 4^{th} groups, as shown in Fig. 4. Since
(4) Referring to Figs. 2, 4, and 5, when the zoom position is switched from wide to tele, the amount of movement of the principal points of each group, ∆
Here, note that the amount of movement ∆
Among 42 parameters in total, since the distances between the adjacent principal points at the tele position can be expressed using the amount of movement ∆
Here, if the zoom system satisfies Eqs. (22)‒(29), only 10 free parameters remain. It is desirable to select free parameters that can be estimated effectively, such as the focal length of each group and the air distances between groups, rather than the front focal lengths, back focal lengths, and distances between adjacent principal planes.
First, for the given six free parameters
Next, out of the remaining parameters
To effectively enlarge the magnitude of the power variation at a TPL, the focal lengths of the TPLs at the 2^{nd} and 4^{th} groups are set to be symmetric with opposite signs at wide and tele positions. This configuration reduces two degrees of freedom, as follows:
According to design experience, the closer the 1^{st} group and 2^{nd} group are to each other, the more advantageous they are for a short overall length. When the same rule is applied to the 3^{rd} and 4^{th} groups, their air distances are constrained to be 0.3 mm for a compact zoom system. This eliminates an additional two degrees of freedom:
Finally, only six independent parameters remain; they are
Whenever these six independent parameters are properly given, the remaining 36 design variables are determined by solving the zoom equations (22)‒(29) and satisfying the four conditions of Section 3.3, with the TPL specifications of Table 1.
For the given six parameters
Whenever the focal lengths (
The solutions for these four principal-plane distances at the wide position are initially investigated for
In Fig. 6, note that as
The solutions for the 1^{st} and 4^{th} combinations of
When the focal lengths of all groups, not only
By inputting the focal lengths ranging from
From the finding that solutions suitable for a compact zoom system have positive distances for
The 3^{rd} lens group should have a positive power to relay the rays to the next group and needs many elements to correct all residual aberrations. This situation probably leads the first principal plane to be placed to the right side of the front surface, which yields a positive distance for ∆_{3w}. In the appropriate range of this distance, the constraints of
The other distances
By applying Eq. (21) to a four-group zoom system, the overall length T_{eq} of this equivalent zoom system can be re-expressed as
where
Whenever six independent variables are given, the back (
By inputting the focal lengths ranging from
Table 2 . Values of T_{eq} for various
ffl_{1} | −67 | … | −64 | −63 | −62 | … | −58 |
---|---|---|---|---|---|---|---|
air_{2} | |||||||
4 | 100 | … | 100 | 30.267 | 31.751 | … | 37.690 |
4.5 | 100 | … | 100 | 29.481 | 30.966 | … | 36.904 |
… | … | … | … | … | … | … | … |
5.5 | 100 | … | 100 | 27.909 | 29.394 | … | 35.333 |
6 | 100 | … | 100 | 27.124 | 28.608 | … | 34.547 |
6.5 | 100 | … | 100 | 100 | 100 | … | 100 |
… | … | … | … | … | … | … | … |
9.5 | … | … | 100 | 100 | 100 | … | 100 |
10 | … | … | 100 | 100 | 100 | … | 100 |
Among the solutions obtained through this design process, the zoom system with
Table 3 . Initial zoom data with the shortest T_{eq} (in mm).
1^{st} group | 2^{nd} group | 3^{rd} group | 4^{th} group | |
---|---|---|---|---|
efl_{w} | 63.0 | -17.5 | 21.0 | 10.0 |
efl_{t} | 17.5 | -10.0 | ||
z_{w} | 4.310 | 17.349 | -14.304 | 8.485 |
z_{t} | 1.635 | 19.287 | -11.783 | 6.71 |
air | 0.3 | 6.0 | 0.3 | 6.033 |
ffl_{w} | -63.0 | 20.175 | -10.337 | -10.0 |
ffl_{t} | -17.5 | 12.521 | ||
bfl_{w} | 61.664 | -18.185 | 35.604 | 7.547 |
bfl_{t} | 14.876 | -10.677 | ||
d_{eq} | 2.025 | 3.675 | 5.27 | 3.521 |
T_{eq} | 27.124 |
To obtain a real lens group equivalent to the zoom data of Table 3, each real lens group should be designed to have the same first-order quantities such as
To have a compact zoom system, the 1^{st} group is designed with two lens elements, and the 2^{nd} and 4^{th} groups each include only one TPL to change the power. As seen in Table 3, to have a higher zoom ratio of 8×, the principal-plane distances (
Thus, we take an initial zoom system with a half-image-size of 1 mm and f-numbers of F/5 at the wide position to F/7 at the tele position, for which reducing the aperture and field size is desirable, to avoid higher-order aberrations. Figure 7 shows an 8× initial real-lens zoom system equivalent to the zoom data of Table 3 within first-order optics. This zoom system has a fixed variator at the 2^{nd} group and compensator at the 4^{th}, in which the former focal lengths are changed from −17.5 mm to 17.5 mm, and the latter from 10 mm to −10 mm.
In the initial design of Fig. 7, the aperture and image size should be extended, to satisfy the specifications for a zoom camera. The aperture is increased to F/3.5 at wide and F/4.5 at tele position. The half-image-size should be 1.8 mm for a 1/6-inch CMOS image sensor with 2048 × 1536 pixels. To improve the entire performance of an extended aperture and field system, we reduce the aberrations of the starting zoom lens by using aspheric surfaces.
Figure 8 illustrates the layout of the final designed zoom system, which consists of 10 elements including two TPLs and six aspheric lenses. The maximum diameter of the 1^{st} lens group is 13.59 mm. The distortions are balanced to less than 4.6%, and the ratio of relative illumination (RI) is more than 70% at the corner field when measured at all zoom positions, as in Fig. 8. The modulation transfer function (MTF) at 180 lp/mm is more than 25% over all fields at both extreme positions, from Fig. 9. Thus, all aberrations are significantly reduced.
The variation of the chief ray angle of incidence (AOI) from wide to tele position is less than 7.11 degrees. Because this variation is small, a stable image quality can be achieved for zooming. The overall length of this zoom lens, even at a large zoom ratio of 8×, is less than 29.99 mm. This is 5.5 mm shorter than the previous design, even for the same focal-length range of 4–31 mm [10]. Thus this design concept, based on the analysis for the overall length T_{eq} of an equivalent zoom system, greatly reduces the effort required to obtain a compact zoom lens system. This is an advantage of the study.
To have a compact zoom system with a high zoom ratio of 8×, it is most effective to construct an optical system with a short overall length. In a zoom system consisting of several groups, however, it is very difficult to precisely estimate the overall length through a paraxial study. To solve this difficulty, this study has proposed a method to count the overall length of the zoom system in an initial design stage. From the fact that the overall length can be estimated indirectly through paraxial ray tracing, we introduce T_{eq} as a measure of the total track length in an equivalent zoom system, which can be found from the first orders obtained by solving the zoom equations.
Among many solutions, the first-order quantities that provide the smallest T_{eq} are selected to construct an initial zoom system with a high zoom ration of 8×. Also, to have a compact 8× four-group zoom system with no moving groups, TPLs have been suggested for variator and compensator.
Utilizing the design approaches proposed in this study, we have achieved a compact 8× four-group zoom system with a short overall length of 29.99 mm. This zoom lens had a wide focal-length range of 4–31 mm, and f-numbers of 3.5 at wide to 4.5 at tele position. Through this study, we illustrated the design process of a zoom system considering the overall length by introducing T_{eq}, along with a numerical, paraxial design approach.
The proposed design approaches are expected to be useful in estimating the overall length of any optical zoom system, regardless of whether there is a moving group or not. Even if a TPL were replaced by a moving group, we could estimate the overall length by inserting the first order quantities of the moving group into Eq. (34) for T_{eq}. Thus, we have been investigating an effective way to discover the solutions with the shortest overall length and applying this method to a zoom system with moving groups.
National Research Foundation of Korea (NRF); Korea government (MSIT, No. 2019R1F1A1060065).
TABLE 1 Design parameters of the tunable polymer lenses (TPLs) at the 2nd and 4th groups
Parameters | The 2nd group | The 4th group | ||
---|---|---|---|---|
Liquid Lens | Cover Glass | Liquid Lens | Cover Glass | |
Thickness (wide) | ||||
Thickness (tele) | ||||
Semi-aperture | ||||
Refractive Index |
Table 2 Values of T_{eq} for various
ffl_{1} | −67 | … | −64 | −63 | −62 | … | −58 |
---|---|---|---|---|---|---|---|
air_{2} | |||||||
4 | 100 | … | 100 | 30.267 | 31.751 | … | 37.690 |
4.5 | 100 | … | 100 | 29.481 | 30.966 | … | 36.904 |
… | … | … | … | … | … | … | … |
5.5 | 100 | … | 100 | 27.909 | 29.394 | … | 35.333 |
6 | 100 | … | 100 | 27.124 | 28.608 | … | 34.547 |
6.5 | 100 | … | 100 | 100 | 100 | … | 100 |
… | … | … | … | … | … | … | … |
9.5 | … | … | 100 | 100 | 100 | … | 100 |
10 | … | … | 100 | 100 | 100 | … | 100 |
Table 3 Initial zoom data with the shortest T_{eq} (in mm)
1^{st} group | 2^{nd} group | 3^{rd} group | 4^{th} group | |
---|---|---|---|---|
efl_{w} | 63.0 | -17.5 | 21.0 | 10.0 |
efl_{t} | 17.5 | -10.0 | ||
z_{w} | 4.310 | 17.349 | -14.304 | 8.485 |
z_{t} | 1.635 | 19.287 | -11.783 | 6.71 |
air | 0.3 | 6.0 | 0.3 | 6.033 |
ffl_{w} | -63.0 | 20.175 | -10.337 | -10.0 |
ffl_{t} | -17.5 | 12.521 | ||
bfl_{w} | 61.664 | -18.185 | 35.604 | 7.547 |
bfl_{t} | 14.876 | -10.677 | ||
d_{eq} | 2.025 | 3.675 | 5.27 | 3.521 |
T_{eq} | 27.124 |