Design and Analysis of a 10× Optical Zoom System for an LWIR Camera
 Author: Ok ChangMin, Park SungChan
 Organization: Ok ChangMin; Park SungChan
 Publish: Journal of the Optical Society of Korea Volume 18, Issue5, p574~581, 25 Oct 2014

ABSTRACT
This paper presents the design and evaluation of the optical zoom system for an LWIR camera. The operating wavelength range of this system is from 7.7 μm to 12.8 μm. Through a paraxial design and optimization process, we have obtained the extended fourgroup innerfocus zoom system with focal lengths of 10 to 100 mm, which consists of the six lenses including four aspheric surfaces and two diffractive surfaces. The diffractive lenses were used to balance the higherorder aberrations, and its diffraction properties were evaluated by scalar diffraction theory. We have calculated the polychromatic integrated diffraction efficiency and the MTF drop generated by background noise. The fnumber of the zoom system is F/1.4 at all positions. Fields of view are given by 51.28°×38.46° at wide field and 5.50°×4.12° at narrow field positions. In conclusion, this design procedure results in a 10× compact zoom lens system useful for an LWIR camera.

KEYWORD
Long wave infrared , Diffractive lens , Zoom lens , Zoom locus

I. INTRODUCTION
A thermal imaging camera has been used to view a target at night for military purposes for the past 40 years. In recent times, civilian demands for the thermal camera have been gradually increasing. Civilian demands are primarily in the security, industry, research and medical fields. The thermal imaging camera measures the radiation of the target, and the measured radiation can be expressed by the distribution of the radiative temperature [1]. It is often necessary for optical systems to have multiple fields of view. A zoom system can have continuous changes of the field of view by moving lens groups along an optical axis.
In this paper, paraxial studies and lens modules are used to obtain the optimum initial design of a 10× extended fourgroup innerfocus zoom system for an LWIR camera. It is integrated into uncooled IR detectors with 384×288 pixels, of which pixel size is 25 µm. The fnumber of the optical zoom system was estimated to be F/1.4 from the Airy disk and MRTD(Minimum Resolvable Temperature Difference) analysis [25].
Through an optimization process, we have obtained an optical zoom system satisfying the requirements for a 10× LWIR camera. In order to balance the wave front aberrations, we located the diffractive surfaces at L3 and L6. Since the diffractive lenses are designed to be very weakly powered and the wavelength to zone period ratio is very small across the entire lens, the scalar predictions of diffraction efficiency are valid to calculate the polychromatic integrated diffraction efficiency. The changes of the MTF(Modulation Transfer Function) due to diffraction efficiency are evaluated from the polychromatic integrated efficiency [6].
II. OPTICAL DESIGN REQUIREMENTS
The main requirements of an LWIR zoom system are listed in Table 1. Since the fnumber has an effect on Airy disk diameter as well as on MRTD, the optical system with uncooled IR detector should have a low fnumber, if possible [7]. Basically spatial resolution of an IR optical system is decided by the specification of IR detector, because target acquisition performance is based on the number of line pairs resolvable across a target critical dimension. Consequently an IR optical system should be designed to have the diffraction limited performance [3].
The Airy disk diameter is given by 2.44·λ·Fnumber. In diffraction limited system, a pixel’s diagonal size of detector should be equal to the Airy disk diameter as follows:
where
d is a pixel width of the detector, λ is an operating wavelength, and F/# is an fnumber of the optical system. For a detector with pixel width of 25 µm, from Eq. (1), the target fnumber of the optical zoom system is given by F/1.4 at central wavelength of 10.2 µm.The MRTD is one of the criteria denoting optical system performances, which characterizes the thermal sensitivity at a given spatial frequency [8], whereas the MTF measures the attenuation in modulation depth without regard for thermal property of an object. This MRTD is given by
where
A is a proportional constant depending on the detector,D * is the normalized detectivity of which the average value is known as 1.44×10^{8} cmHz^{1/2}W^{1} [9], and MTF(𝜉_{t}) is the modulation transfer function of the optical system at target frequency of 𝜉_{t}. For a given set of parameters, the MRTD can be calculated from the Eq. (2) [8]. The relationships between the fnumber and MRTD are summarized in Table 2.Since the diffraction MTF depends on the fnumber of an optical system, a large fnumber results in a large value of MRTD, which reduces the resolution for resolvable temperature difference. Meanwhile, a small fnumber gives a good value of MRTD, however, the large aperture system is limited by the cost and the detector. From the exact evaluation for the relationship between fnumber and MRTD, optimum fnumber of this system is confirmed to be 1.4. The MRTD at F/1.4 is so small that it may give enough resolution.
III. INITIAL DESIGN FOR AN LWIR ZOOM SYSTEM
Figure 1 illustrates an extended fourgroup innerfocus zoom system that has the same total track length at all zoom positions. The zoom system is composed of fixed three groups, the second group for zooming, and the fourth group for focusing.
The powers of the groups are denoted by
k _{1},k _{2},k _{3},k _{4}, andk _{5}, respectively. The distances between the adjacent principal planes are represented byz_{iW} andz_{iN} (i=1, 2, 3, 4) at wide and narrow field positions, respectively. The first, the third, and the fifth groups are always fixed. While the second group moves to have a longer focal length, the fourth group should move to keep the image position stationary. For the zoom system having zoom ratio of 10×, it is desired that the powers of all groups are positive, except the negative second group for zooming.We can analytically derive the set of zoom equations for an extended fourgroup innerfocus zoom system with infinite object by using the Gaussian brackets. The initial design of this zoom system can be formulated as follows [1012]:
where
K is the optical power at a zoom position. Equations (3) ~ (4) can be rearranged as follows:Therefore, Eqs. (5) and (6) can be expressed in a matrix form at wide and narrow field positions as follows:
where
a_{ij} is given as a function ofa_{ij} (K _{W} ,K _{N} ,k _{5},z _{1}_{W} ,z _{2}_{W} ,z _{3}_{W} ,z _{4}_{W} ,z _{1}_{N} ,z _{2}_{N} ,z _{3}_{N} ,z _{4}_{N} ,bfd ) withK _{W} andK _{N} being the optical powers at wide and narrow field positions, respectively. If the GaussJordan elimination method is applied, Eq. (7) is replaced bywhere
b_{ij} is given as a function ofa_{ij} . In order to derive the unknown quantities ofk _{1},k _{2},k _{3}, andk _{4}, four equations are obtained from Eq. (8):Inserting the conditions of Eqs. (9) ~ (11) into the first row equation of Eq. (8) results in an expression for the unknown parameter
k _{1}:where
A_{i} (i =1,2,3),B_{i} (i =1,2,3),A ,B ,C ,D andE are given as a function ofb_{ij} . Solving Eq. (12) for a givenk _{5}, we can getk _{1}. As a result,k _{2},k _{3}, andk _{4} are determined from Eqs. (9) ~ (11).For the initial design of a 10× extended fourgroup innerfocus system, we input the proper distances between principal points and the targeted total powers at wide and narrow positions. These starting values of z_{1}, z_{2}, z_{3}, z_{4}, and
bfd are empirically selected to satisfy the basic requirements by using paraxial studies, as shown in Table 3.By inputting these starting values of Table 3 into Eqs. (5) and (6), we can obtain the proper values for the power of each group from Eqs. (9) ~ (12). Figures 2 illustrates the four solution sets of Eq. (12) for various
k _{5}.In the solutions of case 1 and case 2, the powers of the second group (
k _{2}) are positive. Since the second group is the variator, these positive powers of this group are not valid. In the solutions of case 3, the powers of the third group are negative, so they are also not valid to an extended fourgroup innerfocus zoom system. Also, the powers of the second group are so small that the displacement amount of this group is to be larger, not compact in the high zoom ratio system.Finally, the solutions of case 4 satisfy the requirements for power distribution available to our zoom system having zoom ratio of 10×. Among the solutions of case 4, we took the powers of
k _{1},k _{2},k _{3},k _{4}, andk _{5}, maintaining the most stable zooming locus. Table 4 lists the paraxial design data for the powers obtained from above process.We have set up a zoom lens system with five thinlens modules, for which the power of each group and the zoom locus inputs are taken from Table 4 and Table 3. The lens modules do not reflect higherorder aberrations, so reducing the aperture and the field size of the system is desirable [13, 14]. Thus, we took a zoom system with a half image size of 3 mm and fnumber of F/3 at all positions. The air distances between modules were constrained to be longer than 3.0 mm for the mounting space.
In Table 3 and Table 4 of the previous paraxial study, the thickness of each group and the air spaces are not presented. They can be derived by specifying the design variables of the lens modules, such as the effective focal length (
FL_{M} ), the front (FF_{M} ) and the back focal lengths (BF_{M} ), and the air spaces between the modules. Figure 3 shows the initial zoom design composed of five thick lens modules obtained from this process. The focal lengths range from 10.0 to 100.0 mm. Table 5 shows the firstorder properties and the zoom loci at each position, and Table 6 lists the design data for each module. In Table 5,d_{ij} (j =1,2,3,4) are the air spaces between the modules at thei th zoom position.A real lens group is generally composed of several lens elements, which should be equivalent to the lens module given in Table 6. In this paper, an optimization design method is used to design a real lens group equivalent to the module of each group. We choose an appropriate structure for each real lens group and scale it up or down so that the focal length of each real lens group is the same as that of the lens module. By using the optimization method of CodeV, we matched the firstorder quantities of the lens module to those of a real lens. For the conversion process, the design variables of the real lens are the radius of each surface, the thickness, and the refractive index. Therefore, the three constraints given in Table 6 can be satisfied by specifying the lens design variables. After a few iterations, the real lenses of the groups are obtained.
If a zoom system equivalent to the lens module zoom system is to be achieved, the air spaces (
d_{ji} ) between groups should be set according to the zoom loci of Table 5 at each position. This procedure results in a zoom system equivalent to the lens module zoom system within paraxial optics. From the evaluation of this zoom system, the agreement of the firstorder quantities between the two systems is good.IV. OPTIMIZED DESIGN FOR A ZOOM SYSTEM USING DIFFRACTIVE LENSES
In the initial design of the zoom system, we reduced the aperture and the field size so that they were too small. If current specifications for an LWIR zoom camera are to be met, the aperture and the field size should be increased. The fnumbers are extended to
F /1.4 at all positions. The half image size should be 6.0 mm for an uncooled IR detector. In an extended aperture and field system, however, the aberrations that were not corrected in the previous design become significant. In order to improve the overall performance of the zoom system, we balance the aberrations of the starting data. In this process, the firstorder layouts are fixed, and the residual aberrations are corrected using aspheric surfaces and diffractive lenses. Finally, a zoom system having good performance was obtained, and its layout is shown in Fig. 4. Table 7 lists the design data.This system consists of the six lenses including two diffractive surfaces. The diffractive surfaces are used to balance the wavefront aberrations. They are located in the front surface of L3 and the back surface of L6. The hybrid lenses such as L3 and L6 have refractive and diffractive properties at the same time. Therefore the hybrid lens differs from a conventional refractive lens in that it can produce many images due to diffraction orders. These diffraction images serve to lower the contrast of the desired image.
The rotationally symmetric diffractive lens has the surfacerelief of ring type. The number of rings (
N ) on the diffractive surface is given by Eq. (13):where
D is the diameter of diffractive surface andf _{0} is the focal length at design wavelength of λ_{0}. For a diffractive lens having short focal length, as shown in Eq. (13), it can be seen thatN is to be large. Therefore, we need to budget the small power on the diffractive lens. Table 8 gives the specification of the designed hybrid lens in the zoom system. Table 9 lists the radius of each zone, given by .Figure 5 compares two point spread functions by L3, Fig. 5(a) is obtained from refracted image only and Fig. 5(b) from refracted plus diffracted images. Other diffraction orders, not design order, generate the background noises such as flares. They result in degrading the contrast at around zero frequency. However, the width of the central peak is much more narrow than that of the refracted image, which eventually improves the image quality at high frequency.
For a diffractive optical system that utilizes spectrally broadband illumination, the wavelength dependence of diffraction efficiency needs to be considered when evaluating imaging performance. To determine the polychromatic transfer function for a diffractive lens, it is first to define a polychromatic integrated efficiency for the wavelength band ranging from λ_{min} to λ_{max}. Denoting the integrated efficiency at wavelength λ by
η_{int} (λ), the polychromatic integrated efficiency is given by [6]In Table 8, the diffractive components are very weakly powered, to the 4 % of the hybrid component’s powers. In addition to that, the wavelength to zone period ratio is very small across the entire lens, as shown in Table 9. In these situations, it is known that scalar predictions of diffraction efficiency are valid to replace the
η_{int} (λ) of Eq. (14) [6, 15]. The scalar diffraction theory value for the diffraction efficiency in order m for wavelength λ is given bywhere λ_{0} is the design wavelength for the diffractive lens. The expression for
η_{scalr} (λ) given by Eq. (15) cannot be explicitly integrated, but an approximate expression for can be found by expandingη_{scalr} (λ) in a power series and integrating term by term. The resulting approximate expression for the first order diffraction ofm = 1 is given byAs a result, the polychromatic integrated efficiency of an LWIR zoom system is 93.39 % and background noise reduces the diffraction efficiency by 6.61 % for λ_{min} = 7.7 µm, λ_{max} = 12.8 µm, and λ = 10.2 µm. These losses in diffraction efficiency may reduce the contrast of image. Figure 6 shows the polychromatic modulation transfer functions calculated by using the Eq. (16). The polychromatic MTF is scaled by and drops down at zerofrequency. The resulting curves show the behavior predicted by point spread functions of Fig. 5(b).
V. CONCLUSION
We have designed and evaluated the optical zoom system for an LWIR camera with an uncooled IR detector. The fields of view are 51.28° × 38.46° at wide and 5.50° × 4.12° at narrow field positions. The optimum fnumber of the zoom system was determined to be F/1.4 from Airy disk and MRTD analysis.
Through paraxial design and optimization method, we have obtained an extended fourgroup innerfocus zoom system having a focal length of 10 to 100 mm. Balancing the wavefront aberrations by use of diffractive surfaces, we improved the performance of the zoom system further. To evaluate the polychromatic transfer function for a weakly powered diffractive lens, the scalar diffraction theory was effectively used.
As a result, the polychromatic integrated efficiency is 93.39% and background noise reduces the diffraction efficiency by 6.61%. This loss in diffraction efficiency slightly reduced the contrast of image at zerofrequency, and scaled down the polychromatic MTF by . However, the designed optical zoom system has enough image qualities for an LWIR camera, as shown in Fig. 6. Consequently, the zoom system developed in this work performs reasonably as an LWIR zoom camera.

[TABLE 1.] Optical requirements for a LWIR zoom system

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[TABLE 2.] Fnumber versus MRTD

[FIG. 1.] Layout of an extended fourgroup innerfocus zoom system.

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[TABLE 3.] The starting parameters for zoom loci (in mm)

[FIG. 2.] Solution sets of the fourthorder equation of Eq. (12) for various k5.

[TABLE 4.] Paraxial design data for the powers (in mm1)

[FIG 3.] Design data for the lens modules in the optimized lens module zoom system (in mm)

[TABLE 5.] Firstorder properties and zooming loci of the zoom system consisting of five thick lens modules (in mm)

[TABLE 6.] Design data for the lens modules in the optimized lens module zoom system (in mm)

[FIG. 4.] Layout of an optimized zoom system

[TABLE 7.] Design data of an optimized zoom system (in mm)

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[TABLE 8.] Specifications of the hybrid lenses (in mm)

[TABLE 9.] The radius of each zone rn (in mm)

[FIG. 5.] Point spread functions by L3 on image plane. (a) refracted image only and (b) refracted plus diffracted image.

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[FIG. 6.] Polychromatic MTF calculated by the polychromatic integrated diffraction efficiency. (a) Wide field position, (b) Narrow field position.