This paper presents a design method for improving the low relative illumination and large distortion due to widening the field of a system. A telecentric optical system in image space was suggested to increase the relative illumination. Through the analyses of the thirdorder aberrations affected by introducing aspherical surfaces, we have proposed a method to determine analytically what surface should be aspheric to correct each aberration effectively. By utilizing this method to design a wide field lens, a telecentric wide field lens with fnumber of F/2.0 was obtained. Even though the field angle is 120 degrees, it has a very low distortion less than 2% and high relative illumination more than 73.7%. In conclusion, this analytic method for selecting aspherical surfaces is expected to serve as a useful way to find design solutions.
I. INTRODUCTION
In modern optical instruments such as CCTV and black box cameras, a wide field system is widely used. Such a system is required to cover an extremely wide field, normally more than 120 degrees, without additional instruments. Because of its inherent wide angle, however, the large incidence angles of rays into an image plane significantly reduces the illumination around the margin field and induces large aberrations, especially great distortion being proportional to the cube of field size
[1]
. Many wide field lenses subtending the field angle of 120 degrees have been reported, but their distortions are from −40% to −50%
[2

4]
. These distortions are so large that the images are significantly distorted, to a level that needs software to correct distorted images.
To overcome these problems, in this paper we propose a design method for improving the low relative illumination and large distortion due to widening the field of a system. A telecentric optical system in image space is suggested to reduce the incidence angles of rays into an image, which results in increasing the relative illumination
[5

8]
.
All the thirdorder aberrations can be corrected by introducing aspherical surfaces. Through the analyses of the thirdorder aberrations affected by aspherical parameters
[9

11]
, this study proposes a method to determine analytically the aspheric surface that is most effective to correct each aberration. From the aberration analyses, aspherization of rear surface of the 1st lens is confirmed to be most appropriate for distortion correction.
By utilizing this design method, a telecentric wide field lens with field angle of 120 degrees has been obtained. In addition, it has a very small distortion less than −2% at the margin field.
II. DESIGN OF A TELECENTRIC SYSTEM
In this study, to overcome low illumination due to wide field, a telecentric optical system in image space is suggested to improve the relative illumination. The starting wide field lens is selected from the patented lens composed of two groups, as shown in
Fig. 1
[2]
. This lens can realize the wide field, but has great distortion of −48.53% at the margin field. The goal of our study is to design an optical system having small distortion less than ±2% and higher relative illumination at all fields, by using the proposed design concepts.
Patented lens for a wide field camera (US Patent 8,917,460 B2).
In
Fig. 1
, the configuration of a negative first group and positive second group gives a retrofocus system so that it is suitable for a wide field lens. Our telecentric system can be realized by adding a lens element before the image plane, as shown in
Fig. 2
.
Layout of a telecentric optical system obtained by locating a positive element (ϕ 3).
By denoting the distances between adjacent principal planes as
z_{i}
(
i
= 1, 2, 3), the focal length (
efl
= 1/
ϕ_{t}
), imagery condition of an axial ray (
z
_{3}
), and telecentric condition are given
[4
,
5]
, as follows:
Solving Eqs. (1)~(3) simultaneously results in an expression for the unknown parameters of
Figure 2
illustrates the telecentric system found by locating the element designed from Eqs. (4)~(6).
III. PROJECTION METHOD IN A WIDE FIELD LENS
The full field angle of the system is aimed to 120°, therefore the gnomonic projection method is generally used to define the image size
[6
,
12

13]
. The paraxial image height (
) and distortion (%) are respectively given by
If the real image height (
) and full field angle (2
θ
) subtending an image sensor are given, the focal length can be determined to make a system have targeted distortion. In other words, if the distortion is allowed to ±2%, the paraxial image height is calculated from Eq. (8) at a given sensor size, and inserting it into Eq. (7) gives the range of focal lengths.
Table 1
lists the ranges of focal lengths for several image sensors, which can correct distortion within ±2% at field angle of 120 degrees.
Ranges of EFLs for several image sensors to correct distortion within ±2% at field angle of 120 degree
Ranges of EFLs for several image sensors to correct distortion within ±2% at field angle of 120 degree
IV. ANALYSES OF THE THIRDORDER ABERRATIONS OF A LENS SYSTEM
The thirdorder aberration coefficients for spherical aberration (
S_{I}
), coma (
S_{II}
), astigmatism (
S_{III}
), Petzval blur (
S_{IV}
), and distortion (
S_{V}
) of a system are expressed, as
[9

11]
where
In these equations,
u_{j}
and
y_{j}
(
j
= 1, 2, ···,
l
) are the convertgence angle and height of the ray from the axial object point, also
and
(
j
= 1, 2, ···,
l
) are the convergence angle and height of the chief ray from the offaxial object point.
P_{j}
(
j
= 1, 2, ···,
l
) is the Petzval curvature of the jth surface, also
c_{j}
and
cn_{j}
are the curvature and refractive index of the jth surface.
The conic constant (
k
) and fourthorder coefficient (A) of an aspheric surface have an effect on the thirdorder aberrations, meanwhile the higherorder coefficients (B, C, D, ···) are useful in correcting the higherorder aberrations. The equation for the aspheric surface is given as
Introducing the conic constant (
k_{j}
) and fourthorder coefficient (
A_{j}
) into the jth surface generates the additional thirdorder aberrations, as follows
[9

11]
:
where
Therefore the conditions that all the thirdorder aberrations except Petzval blur are corrected, by combining Eqs. (9)~(13) with Eqs. (15)~(18), are expressed as follows:
V. DESIGN FOR AN INITIAL WIDE FIELD LENS USING THE THIRDORDER ABERRATION ANALYSES
In this research, unlike general methods correcting the aberrations using aspheric surfaces, we suggest approaches to analytically determine what surface is most effective to correct each thirdorder aberration. Before determining these corrections, Petzval curvature should be firstly removed from the spherical lens, not the aspheric.
 5.1. Petzval Field Curvature Correction
In Section Ⅱ, adding a positive element in front of an image gave a telecentric system, but in parallel it made the system have a much more negative Petzval sum. Since the selection of optical glasses is limited, it is better to change the curvature than the refractive index so that the Petzval sum can be easily corrected.
Because the first group is located before the stop, even if its parameters are changed, the telecentric system is still effective. To have zero Petzval sum, the radius of curvature of the jth surface is required to have the value given by Eq. (24).
where
j
= 1, 2, ···, 6.
In Eq. (24), the denominator denotes the Petzval sum except the Petzval curvature of the jth surface. Consequently, the radius of curvature of each surface can be calculated to have zero Petzval sum, and
Table 2
lists the calculated radius of curvature of each surface to make the Petzval sum be zero.
Calculated radius of curvature of each surface to correct the Petzval sum
Calculated radius of curvature of each surface to correct the Petzval sum
To have zero Petzval sum in
Table 2
, changing the radius of the 4th surface (
r
_{4}
) is most effective, because its rate of change is the smallest. But if
r
_{4}
is chosen, it should be reduced to 2.516 mm, which is too small to have a necessary aperture size. Changing the radius of the 2nd surface (
r
_{2}
) gives a similar result. So, two radii of
r
_{2}
and
r
_{4}
are selected to correct the Petzval sum. The additional Petzval curvature, being induced from adding a lens element, can be canceled out by changing two radii of curvature of minus lenses. After allocating the additional Petzval curvature to two surfaces so that they are proportional to the rates of change of
Table 2
, two radii of
r
_{2}
and
r
_{4}
are changed to correct each additional Petzval curvature.
Table 3
shows two radii of curvature correcting the Petzval sum. Also this system is still satisfying the telecentric condition. During this design process, the focal length has been changed to 1.294 mm, which can realize the system having distortion of ±2%, if a 1/4inch image sensor is selected from
Table 1
.
Radii of curvature to correct the Petzval sum
Radii of curvature to correct the Petzval sum
 5.2. Spherical Aberration Correction Using Conic Constant
Among the thirdorder aberrations, we firstly correct spherical aberration using an aspherical surface. By introducing an aspheric to the jth surface, the condition that spherical aberration is corrected is expressed in terms of a conic constant (
k_{j}
) and fourthorder coefficient (
A_{j}
), as
[9

11]
For
A_{j}
= 0, Equation (25) is reduced to
Introducing an aspheric to the 7th surface at stop has advantage in that any aberration is not changed, except for spherical aberration. For spherical aberration to be corrected, from Eq, (26), the conic constant of the 7th surface should be
k
_{7}
= −14.964. This lens system is still satisfying the zero Petzval sum and telecentric condition.
 5.3. Correction of Spherical Aberration and Distortion
By aspherization of the 7th and other surfaces, we can simultaneously correct spherical aberration and distortion that is the most troubling aberration in a wide field system. From Eqs. (15), (18), (20), and (23), the following equations must be satisfied for these aberrations to be corrected:
Assuming that two fourthorder coefficients(
A
_{7}
,
A_{j}
) are zero and solving the above two equations, the solutions for
A
_{7}
and
k_{j}
are given by
Table 4
shows the conic constants of
k
_{7}
and
k_{j}
that correct spherical aberration and distortion simultaneously. The new conic constant of
k
_{7}
, being used to correct spherical aberration, should be selected to be not far away from the initial value of
k
_{7}
= −14.964. Because the conic constant (
k
_{2}
) of the 2nd surface is smallest, aspherization for the second surface is most desirable to have distortion be removed. By selecting
k
_{2}
= −1.057 and
k
_{7}
= −15.163, we can obtain the optical system corrected for distortion and spherical aberration.
Conic constants to correct spherical aberration and distortion
Conic constants to correct spherical aberration and distortion
 5.4. Correction of Spherical Aberration, Distortion, and Astigmatism
The 2nd and 7th surfaces are aspherized to correct distortion and spherical aberration. By introducing an aspheric into another surface, the conditions for correction of astigmatism and the above two aberrations are given by
Assuming that three fourthorder coefficients (
A
_{2}
,
A
_{7}
,
A_{j}
) are zero and solving three simultaneous equations Eqs. (31)~(33), the parameters of
k
_{2}
,
k
_{7}
, and
k_{j}
can be calculated for all surfaces that may be aspherized.
Table 5
illustrates the conic constants of
k
_{2}
,
k
_{7}
, and
k_{j}
that correct distortion, spherical aberration, and astigmatism simultaneously.
The conic constant
k
_{2}
, being used for distortion correction in the previous section, has changed astigmatism from positive to negative value. Because astigmatism is overcorrected, it is desirable to select the surface that gives the
k
_{2}
less than −1.057 of Table 4 with small
k
_{7}
and
k_{j}
for corrections of spherical aberration and astigmatism. From
Table 5
, selection of
k
_{2}
= −1.017,
k
_{7}
= −8.135,
k
_{12}
= −23.103 gives a good solution which removes distortion, spherical aberration, and astigmatism.
Conic constants to correct distortion, spherical aberration, and astigmatism simultaneously
Conic constants to correct distortion, spherical aberration, and astigmatism simultaneously
 5.5. Correction of All ThirdOrder Aberrations
Among the thirdorder aberrations, the Petzval sum was already corrected using two negative surfaces of the front group, and three aberrations other than coma were also removed by aspherization of three surfaces. By introducing another aspherical surface, the conditions correcting all the thirdorder aberrations are given by
Through the same design process of Section 5.4, solving four simultaneous equations of Eqs. (34)~(37) provides the values of
k
_{2}
,
k
_{7}
,
k
_{12}
, and
k_{j}
for all surfaces that may be aspherized.
Table 6
illustrates the conic constants of
k
_{2}
,
k
_{7}
,
k
_{12}
, and
k_{j}
that correct the thirdorder aberrations.
Conic constants to correct distortion, spherical aberration, astigmatism, and coma simultaneously
Conic constants to correct distortion, spherical aberration, astigmatism, and coma simultaneously
In the aberrations correction of Section 5.4, the best three surfaces are independently selected to correct three aberrations so that three aspheric lenses are required. In this study, we hope to use just three aspherical lenses without an additional aspherical lens. Therefore, the 1st, 8th, and 13th surfaces can be aspherized to correct coma. Since the 8th surface is near to a stop, the height of a chief ray is low, and that of an axial marginal ray is high. Therefore, aspherization of the 8th surface is very useful in removing coma aberration. From
Table 6
, the design parameters of
k
_{2}
= −1.018,
k
_{7}
= 31.229,
k
_{12}
= −21.988, and
k
_{8}
= −2.524 give a good solution for correction of distortion, spherical aberration, astigmatism, and coma simultaneously.
Figure 3
shows a telecentric optical system of which all the thirdorder aberrations are corrected, and
Fig. 4
illustrates the ray aberrations of this system.
A telecentric optical system of which all the thirdorder aberrations are corrected.
Ray aberrations of a telecentric optical system of which all the thirdorder aberrations are corrected: (a) Longitudinal spherical aberration, (b) astigmatic field curves, and (c) distortion.
VI. COMPLETE WIDE FIELD LENS DESIGN
A telecentric system of
Fig. 3
has been designed to have the focal length of 1.294 mm, which is proper to realize the system having distortion of ±2%, if a 1/4inch image sensor is used, as shown in Table 1. In the initial design, however, we corrected the thirdorder aberrations useful in reduced aperture and field so that practicable aperture and image size were small. If current specifications for a wide angle camera are to be met, the aperture and field size should be increased. The aperture is extended to F/2.0. The full field size is increased to 4.4 mm for a 1/4inch CCD, which corresponds to 120 degree at focal length of 1.294 mm. In an extended aperture and field system, however, there are higherorder aberrations that are not corrected in the previous design, as shown in
Fig. 4
.
In order to improve the overall performance of the lens system with an extended aperture and field, we balance the aberrations of the starting lens given in
Fig. 3
by using the higherorder aspheric coefficients not used in correction of the thirdorder aberrations.
Finally, a wide field lens having good performance is obtained. The layout of the system is shown in
Fig. 5
.
Table 7
lists the specifications of this lens. Compared to
Fig. 4
, all residual aberrations are significantly reduced, as shown in
Fig. 6
. Especially distortion is dramatically reduced to 2%, from 48.53% of the patented lens. Even though this system covers an extremely wide field angle of 120 degrees, distortion is so small that the distorted image is nearly not seen.
Layout of an aberrationbalanced wide field lens.
Specifications for a wide field camera lens
Specifications for a wide field camera lens
Ray aberrations of an aberrationbalanced wide field lens: (a) Longitudinal spherical aberration, (b) astigmatic field curves, and (c) distortion.
Figure 7
shows the modulation transfer function (MTF) characteristics of the system. The MTF at 200
lp/mm
is more than 31% over all fields. Also, the ratio of relative illuminations is more than 73.7% at all fields. The chief ray’s angle of incidence (AOI) into the image plane is just +3.3° at margin field. It is so small that this lens is near to a telecentric system, as targeted in the starting design.
MTF characteristics of an aberrationbalanced wide field lens.
Even though it is extremely wide angle, the total track of the lens is just 23.56 mm, and large aperture with F/2.0 is realized. Consequently, this system has enough performance to fulfill the requirements of a modern wide field camera.
VII. CONCLUSION
By use of the thirdorder aberration analyses and telecentric design, in this paper large distortion and low relative illumination due to widening the field have been solved. To correct all the thirdorder aberrations without changing configuration, aspheric lenses are most effective so that they are used to correct aberrations.
In a design using general software like CodeV, the designer usually determines the surfaces to be aspherized from empirical judgments. Through the analyses of the thirdorder aberrations affected by aspherical parameters, however, this study proposed the method to determine analytically what surface should be aspheric to correct each aberration effectively. By utilizing this method to design a wide field lens, an initial telecentric system was obtained, of which all the thirdorder aberrations were removed. To improve the performance of the starting lens in the extended aperture and field, three aspheric lenses are used to balance the residual aberrations.
A wide field lens with a total track of 23.56 mm, whose aperture was F/2.0, was obtained. In addition, even though this lens subtends the field angle of 120 degree, it has a very low distortion less than −2%. The optical system developed in this work performs reasonably as a compact camera system with wide field angle. In conclusion, this analytic method for selecting aspherical surfaces is expected to serve as a useful way to find design solutions.
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