## Abstract

A conformal dome optical system was established and aberration characteristics of the dome were investigated using Zernike aberration theory. The conformal dome was designed with gradient index element. The designing method was introduced and the optimizing results were analyzed in detail. The results show that the Zernike aberrations produced by the conformal dome decreased dramatically. Also, a complete conformal optical system was designed to further illustrate the aberration correction effect of gradient index elements. The results show that the utilization of gradient index optical elements not only improves the imaging quality, but also simplifies the structure of the conformal optical system.

© 2014 Optical Society of America

## 1. Introduction

The missile domes of today leave much to be desired in the realm of aerodynamic performance. Traditional optical domes have been sections of concentric spheres or hemispheres. The domes in the shape of hemispheres have axially symmetric aberrations that are invariant to the field of regard (FOR), leading to the optical seekers possess good imaging quality, but produce huge air resistance. Therefore, these domes might affect the speed and working distance of the missiles. A conformal dome is one in which the optical dome conforms in some sense to the needs of the host platform. For example, the dome on the front of an infrared-guided missile would have a pointed shape if it were to conform to the best (lowest drag) aerodynamic shape. With the advantages of the shape, a conformal dome can make great contribution to the military defense and attack. However, the conformal dome might sacrifice the imaging quality of its optical system. A dome that is not hemispherical will have large amounts of non-symmetric aberrations which also varying with FOR (as shown in Fig. 1). The main mission of conformal optical design is to correct these aberrations [1,2].

To reduce the varying aberrations, new types of correctors were designed into the conformal optical systems. Sparrold *et al.* designed a set of counter-rotating phase plates and simulated its aberration correction effect on the conformal optical system [3]. Zhang *et al.* tried to utilize diffractive optical elements to eliminate the varying aberrations produced by the conformal domes [4]. Li *et al.* reduced the varying aberrations to a permissible level by combining the static optical elements and dynamic optical elements to work together [5]. Although the methods used for past aberration correction have improved imaging quality to some extent, additional optical devices were added to the conformal optical system. Consequently, its weight was increased and the structure also became complicated.

In this study, we designed the conformal dome by using gradient index (GRIN) optical elements. Since the refractive index of GRIN optical elements varied continuously, the optical path difference from the incident rays could be minimized after travelling through the dome, and the imaging quality of the conformal optical system was therefore improved. Moreover, the optical seekers possessed the merits of light weight and simple structure since no additional optical devices were introduced. Taking an ellipsoidal dome for example, the objectives of this study were to (1) establish an ellipsoidal dome and analyze its aberration characteristics at different FOR, (2) design an ellipsoidal dome with GRIN optical elements and examine the aberration correction effect, and (3) design a complete cooled conformal dome optical system and evaluate the imaging quality of its system.

## 2. Establishing a conformal dome and analyzing aberration characteristics

#### 2.1 Establishing a conformal dome

The parameters of nose radius $r$ and conic constant $k$ are used to describe the structure of conformal domes. The equations used to calculate the nose radius and conic constant can be expressed as [6].

where $L$ is the length of the dome, $D$ is the diameter of the dome, and $F$ is the fineness ratio. The ellipsoidal dome was established after calculating radius $r$ and conic constant $k$. Each parameter of the dome is shown in Fig. 2.#### 2.2 Analyzing aberration characteristics of the conformal dome

Since conformal domes lose rotational symmetry compared with spherical domes at nonzero FOR (as shown in Fig. 1), traditional Seidel aberration theory could not evaluate exactly the imaging quality of conformal optical systems. In this study, Zernike aberration theory was used to evaluate the imaging quality of conformal optical systems by decomposing the incident wavefront at the exit pupil. As in a unit circle, Zernike polynomials are orthogonal, linear, and independent. Owing to the fact that each Zernike polynomials have explicit physical meaning, the coefficients of Zernike polynomials directly reflect the aberrations influence on the imaging quality. Thus Zernike polynomials are used as an effective tool to study wavefront aberrations and to describe the wavefront distortion [7]. The primary Zernike polynomials and their corresponding aberrations are listed in Table 1 [8].

In the realm of conformal optical design, the system with a large aperture and a wide FOR has difficulty in correcting aberrations. To examine the aberration correction method presented in this paper and to make it find its wide applications, the conformal optical system with a large aperture and wide FOR was designed (as shown in Fig. 3). The parameters of the system are listed in Table 2. A perfect lens was placed behind the ellipsoidal dome to investigate the varying aberration characteristics of the dome. The image of the dome was therefore observed at the focal plane of the perfect lens. Since the lens was introduced for no aberrations, all the aberrations were produced by the dome. By representing the aberrations with corresponding Zernike coefficients, the aberration characteristics of the dome at different FOR were obtained as shown in Fig. 4.

As shown in Fig. 4, the dominant aberrations produced by the ellipsoidal dome are focus, astigmatism and coma. These aberrations are large and also vary with FOR. Thus, effective methods should be applied to correcting these aberrations. Although the focus term has large magnitude, it is perhaps less significant as it does not represent a true aberration insofar as the image can be restored by a proper selection of image location. Astigmatism (since Z6 Y astigmatism equals to 0 within all the FOR, here, astigmatism represents Z5 X astigmatism), however, is a true image distorting aberration, and its large magnitude suggests that it is indeed the performance limiting characteristic of the system. The shape of the astigmatism curve is representative of the dome subsection presented to the imaging system (as shown in Fig. 4). At 0° FOR, the optical system images through, and along the optical axis of, a rotationally symmetric dome element with surfaces that are approximately spherical. The incident plane wave tends to a concentric spherical wave after traversing the section of the dome. Therefore, the astigmatism was 0. As the sensor is gimbaled to larger angles with respect to the dome axis, the dome subsection element appears increasingly cylindrical. Each section of the dome could be regarded as the combination of a spherical element and a cylindrical element.

To facilitate the understanding of the varying trend of astigmatism curve, supposing that a spherical element produces positive astigmatism and a cylindrical element generates negative astigmatism. When the FOR varies from 0° to 13°, the positive astigmatism produced by the spherical element is larger than the absolute value of the negative astigmatism produced by the cylindrical element, the total value of astigmatism being positive. As the sensor is gimbaled from 13° to 40°, the positive astigmatism generated by the spherical element is less than the absolute value of the negative astigmatism generated by the cylindrical element, so the total astigmatism being negative. Beyond 40°, the dome appears to have a more constant, cylindrical shape, resulting in the large magnitude and relatively constant nature of the astigmatism curve at these angles. Also, coma (since Z7 X coma equals to zero within all the FOR, here, coma represents Z8 Y coma) varies greatly, which was slightly less than astigmatism and should be corrected either. Spherical aberration was small and nearly constant compared with astigmatism and coma. Therefore, our conclusion about spherical aberration can be drawn as specific correctors are not needed and by the compensation of the rear imaging optical system, this kind of fixed aberrations could be corrected.

## 3. Designing a conformal dome with GRIN optical element

GRIN optical elements are similar in index and dispersion to homogeneous optical elements, however, the index also varies with position in the elements. GRIN optical elements can change the optical path by varying index. When incident beam of light travelled through GRIN optical element, the beam actually traversed several regions with different indices. On the surface of each region especially, the travelling direction of beam has changed. As a result, the optical path of beam has varied much greater after travelling through the GRIN optical elements than homogeneous medium elements. The researches have shown that GRIN optical elements could replace homogeneous medium elements for improving imaging quality [9,10].

Generally speaking, there are three types of index gradients having been used: axial gradient, in which the index varies parallel to the optical axis, radial gradient, in which the index varies perpendicular to the optical axis, and spherical gradient, in which the index is symmetric about a point. The radial GRIN optical elements are mostly used in correcting chromatic aberrations and manufacturing optical fibers. Spherical GRIN optical elements are often used in microwave region of the spectrum. Especially, the axial gradient optical elements can provide aberrations and are therefore used to improve imaging quality of optical system. It has been shown that axial GRIN optical elements both theoretically and experimentally to be capable of replacing aspheric surfaces [11], which indicating that they may produce similar aberrations to aspheric surfaces in any surface shapes. As the ellipsoidal dome is actually an aspheric optical element, the ellipsoidal dome and axial GRIN optical element may produce equal aberrations opposite being in sign. As a result, the total aberrations produced by the ellipsoidal dome with a reasonable GRIN property may close to zero. In addition, both the aberrations introduced by conformal domes and the optical path difference provided by axial GRIN are rotationally symmetric about the dome axis. Thus, it is very suitable for controlling the aberrations of conformal dome by using axial GRIN optical elements. The refractive index of axial GRIN optical elements can be expressed as [12]

where z is the coordinate in the direction of dome axis, $n\left(z\right)$ is the index, ${n}_{0}$ is the base index, ${n}_{1},\dots {n}_{11}$ are the coefficients.The optimizing of the ellipsoidal dome with axial GRIN material consists of three steps. In the first step, the base index should be determined reasonably, since it is the basis for optimizing the optical system. Secondly, the number of coefficients used to optimize the system should be determined, which is important for both optimizing and manufacturing the GRIN elements. The last step is a process of accurate optimizing. In this step, the base index and the coefficients of GRIN function should be adjusted. Also, different weights should be distributed to different FOR, which ensures that each FOR could possess similar performance. The specific process of designing a conformal optical system with axial GRIN elements is described as follows.

In the process of designing the ellipsoidal dome with axial GRIN optical element, the index range of GRIN element should be carefully considered, because properly and efficiently bounding the GRIN index function between the minimum and maximum material limits was one key to successful lens optimization and putting into practice, same as the conclusion drawn by Flynn *et al.* [13]. As a phenomenon found in optimizing our optical system, the lower base index we choose, the better optimizing result will occur. (Since the ellipsoidal dome possesses the fineness ratio of 1, a smaller index range delta *n* can satisfy the demand for the aberration correction of this dome. However, for the dome with the fineness ratio larger than 1, the GRIN profile with a smaller delta *n* may be not enough. In this case, GRIN profile with higher base index and larger delta *n* may provide better performance.) In the range of mid-wave infrared spectrum, the refractive indices of MgF_{2} and CaF_{2} seem to be smaller than most infrared materials. More importantly, due to their physical properties, both of them can be used for manufacturing optical domes. The previous studies suggested that the index of MgF_{2} was 1.35 at the wavelength of 4μm, while the index of CaF_{2} was 1.41 [14]. According to the refractive index mixture rule [15], any two homogeneous materials with known refractive indices can be modeled as a GRIN material. (Although this rule is not strictly true for all materials, it is a useful approximation for modeling the material properties.) In this manner, the GRIN material could be constrained to real material properties and the gradient can change smoothly from one known composition to a second known composition. The refractive index of GRIN material is represented as

_{2}/CaF

_{2}GRIN material could be modeled with refractive index ranges from 1.35 to 1.41.

While optimizing the dome, the base index ${n}_{0}$ and the coefficients ${n}_{1},\dots {n}_{11}$ are used as the variable parameters for optimizing. We find that if the base index and the coefficients are all set to be variable, the index soon converges into 1. However, the medium whose refractive index of 1 is air, not the solid medium. Therefore, the base index should set to be frozen and only the coefficients of axial GRIN are set to be the variable parameters in the process of optimization. We tried to set the base index to be 1.41 and the coefficients to be variable parameters for optimizing at first. Based on this, the GRIN optical material after optimizing may be suitable for the mid-wave infrared (MWIR) optical system.

While optimizing the coefficients of GRIN, the number of coefficients should also be considered. With fewer number of coefficients selected, the amount of variable parameters is not enough, leading to a bad aberration correction effect. With more number of coefficients selected, the computing efficiency becomes lower and the error function slowly converges. Moreover, the GRIN optical elements would become more difficult in manufacturing. Figure 5 shows the RMS ray aberration results of the conformal optical system with different number of coefficients used to be the variable parameters for an optimizing purpose. From Fig. 5 we can see that, the aberration correction results are not sensitive to high order terms of index distribution. The aberration correction result of using 4 expansion terms for axial index distribution is much better than using 1, 2, and 3 expansion terms. When we use 4 to 11 expansion terms for optimizing separately, little difference between the results is presented. Taking the computing efficiency and manufacturing cost into account, the first 4 terms of coefficients are adopted for optimizing.

How to constrain the GRIN function to the index range of MgF_{2}/CaF_{2} material is vital for our design. As a phenomenon found in optimizing our optical dome with axial GRIN element, the index gradually decreased from the vertex to the bottom of the dome. We set the base index of the GRIN function to be 1.41 in CODE V, it means that the top of the dome would possess the maximum index of 1.41, and the other parts of the dome would possess the index less than it after optimizing. By adjusting the GRIN coefficients appropriately while optimizing the optical system according to the optimizing results and change trend of index, the index will get to the stable solution and be bounded from 1.35 to 1.41. To achieve similar imaging quality within all the FOR, the residual aberrations of the ellipsoidal dome at different FOR should be about the same. Therefore, different weights should be appropriately distributed to each FOR. Because the error function had a few minimal values in the process of optimizing, the coefficients of GRIN should be adjusted according to the values and types of the residual aberrations. Repeating the above optimizing process, the least minimal error function is achieved and an excellent imaging quality is also obtained.

The refractive index distribution of the conformal dome after optimizing is shown in Fig. 6. Along the dome axis, the refractive index gradually declines from the top to the bottom of the dome. The top of the dome possesses the maximum index of 1.410, while the minimum index is 1.356. To illustrate the reason of the refractive index distribution (as shown in Fig. 6), the traversing of the rays at 0° FOR is made for example. The traversing of the rays at other FOR is nearly the same.

As shown in Fig. 7, the incident paraxial rays A and B reach the dome at different heights. If the optical system could image perfectly, rays A and B will converge at the focal point and possess the same optical path. From Fig. 7, we see that the geometrical path of ray B traverses from plane C to the focal point is obviously longer than that of ray A. To ensure the two rays possessing the same optical path, the refractive index of the section of dome, in which ray A traverses, must be larger than that of ray B does. Since the area of the dome which ray A travels through is closer to the top of the dome than ray B does, the refractive index of the ellipsoidal dome should be gradually declined from the top to the bottom along the dome axis. This index variation trend makes the rays, at the same FOR and different heights, satisfy the equal optical path theory, which leading to a much smaller aberrations produced by the ellipsoidal dome. As a result, the imaging quality of the conformal optical system is improved.

The residual aberrations of the ellipsoidal dome after optimizing are shown in Fig. 8. Compared with the Zernike aberrations of the ellipsoidal dome before optimization (as shown in Fig. 4), the aberrations decrease dramatically. The peak to valley (P-V) value of astigmatism is lessened from 19.5 to 0.71, and the P-V value of coma is reduced from 4.2 to 0.79.

## 4. Designing a complete conformal optical system

To further illustrate GRIN optical element is an effective tool in designing conformal optical system, the perfect lens was replaced with a catadioptric optical system. A catadioptric system was designed with diffraction limited imaging quality and the same optical parameters as the perfect lens, such as entrance pupil diameter and focal length, to replace the perfect lens for imaging. The distance between the pivot point of catadioptric system and the vertex of the dome is 134mm.

As above mentioned, the residual constant aberrations, left by the ellipsoidal dome with GRIN property, can be corrected by designing the rear imaging optical system. Therefore, the catadioptric system was re-optimized to compensate the residual constant aberrations. The final structures of the conformal optical system after optimizing are shown in Fig. 9. The catadioptric system consists of four refractive lenses. The last element of the catadioptric system is a plane optical window which is made of Ge. It was to be the protective window of the cooled detector. With secondary imaging, the exit pupil of the conformal optical system coincides with the cooled stop of the detector. The conformal optical system reaches 100% cooled efficiency.

The Root Mean Square (RMS) wave aberrations of the conformal optical system at each FOR are shown in Fig. 10, in which the optical system produces small wave aberrations within all the FOR. The maximum wave aberration is only 0.041 waves. The modulation transfer function (MTF) results of the conformal optical system are shown in Fig. 11. The system reaches the diffractive limited imaging quality within all the FOR. The conformal optical system achieves excellent imaging quality which proves the GRIN optical elements are efficient in correcting conformal dome aberrations, compared to the conformal aberration correction method using fixed correctors for the optical system with the entrance pupil less than 20mm. The fixed corrector was made up of three aspherical surfaces and the MTF of the optical system was 0.5 at 16lp/mm of marginal FOR [16]. The conformal optical system designed with GRIN elements introduced no additional optical elements and achieved better performance.

## 5. Conclusions

To realize the design demands of simple structure and excellent imaging quality in the conformal seeker optical system, a conformal dome was designed with GRIN optical elements. Taking an ellipsoidal dome as an example, the aberration characteristics were investigated well. By designing the ellipsoidal dome with GRIN element, the Zernike aberrations introduced by the dome can decrease dramatically. Also, a complete cooled conformal optical system was designed to further the examination of the aberration correction effect on adopting GRIN optical elements. Compared with the traditional aberration correction method using fixed correctors, the method presented in this paper have not only improved the imaging quality, but also simplified the structure of the conformal optical system. The design method of correcting conformal aberrations presented in this paper cannot only be applicable to the seekers with ellipsoidal domes, but also be suitable for the seekers with other kinds of conformal domes.

Although GRIN material was used to design the conformal optical system for a single wavelength in this paper, our previous work showed that GRIN material was also an effective tool for designing the conformal optical system at the full MWIR band. In recent years, a large number of researches have been conducted for the infrared GRIN material. However, few of them reached the stage that can be commercialized. The goal of this paper is to provide optical designers to explore the benefits of GRIN materials in conformal optical systems. This, in turn, may motivate the development of infrared GRIN material.

## Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant NO. 61275020) and Aeronautical Science Foundation of China (Grant No. 20130177003). The authors are grateful to the reviewers and editors for their helpful and invaluable comments.

## References and links

**1. **P. A. Trotta, “Precision conformal optics technology program,” Proc. SPIE **4375**, 96–107 (2001). [CrossRef]

**2. **B. G. Crowther, D. B. McKenney, and J. P. Mills, “Aberrations of optical domes,” Proc. SPIE **3482**, 48–61 (1998). [CrossRef]

**3. **S. W. Sparrold, J. P. Mills, D. J. Knapp, K. S. Ellis, T. A. Mitchell, and P. K. Manhart, “Conformal dome correction with counter-rotating phase plates,” Opt. Eng. **39**(7), 1822–1829 (2000). [CrossRef]

**4. **W. Zhang, B. J. Zuo, S. Q. Chen, H. S. Xiao, and Z. G. Fan, “Design of fixed correctors used in conformal optical system based on diffractive optical elements,” Appl. Opt. **52**(3), 461–466 (2013). [CrossRef] [PubMed]

**5. **Y. Li, L. Li, Y. F. Huang, and J. G. Liu, “Conformal optical design with combination of static and dynamic aberration corrections,” Chinese Physics B **18**(2), 565–570 (2009). [CrossRef]

**6. **S. W. Sparrold, “Arch corrector for conformal optical systems,” Proc. SPIE **3705**, 189–200 (1999). [CrossRef]

**7. **C. E. Leroux, A. Tzschachmann, and J. C. Dainty, “Pupil matching of Zernike aberrations,” Opt. Express **18**(21), 21567–21572 (2010). [CrossRef] [PubMed]

**8. **Y. M. Liu, J. Ma, H. P. Ma, and X. Z. Jiang, “Zernike aberration characteristics of precision conformal optical windows,” Proc. SPIE **7544**, 1–7 (2010). [CrossRef]

**9. **D. T. Moore, “Gradient-index optics: a review,” Appl. Opt. **19**(7), 1035–1038 (1980). [CrossRef] [PubMed]

**10. **G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express **16**(15), 11540–11547 (2008). [PubMed]

**11. **D. T. Moore, “Design of singlets with continuously varying indices of refractions,” J. Opt. Soc. Am. **61**(7), 886–894 (1971). [CrossRef]

**12. **CODE V Reference Manual, ORA (2004).

**13. **R. A. Flynn, E. Fleet, C. Kretzer, and G. Beadie, “Practical Design of a Layered Polymer GRIN lens,” *CLEO: Applications and Technology* (OSA, 2012).

**14. **M. Wakaki, K. Kudo, and T. Shibuya, *Physical Properties and Data of Optical Materials* (CRC Press, 2007).

**15. **P. McCarthy and D. T. Moore, “Optical design with gradient-index elements constrained to real material properties,” Optical Fabrication and Testing, OSA (2012).

**16. **W. Fu and B. G. Chen, “Design of conformal optical system based on fixed aspheric corrector,” Infrared Technology **32**, 408–410 (2010).