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Abstract
In this paper, the von Karman surface is used in the configuration design of the infrared conformal dome to improve its aerodynamic performance. The principle of differential geometry is used to study the geometric characteristics of the von Karman dome. Additionally, by using ray tracing, the geometric aberrations and wave aberrations of the von Karman dome are analyzed. Further, considering the geometric characteristics and aberration characteristics, an optimization method for the configuration of the von Karman dome is proposed. To prove the effectiveness of the optimization method, the aberrations introduced by the conformal dome after the configuration optimization and the original von Karman dome are compared. The comparison showed that the geometric aberration of the optimized conformal dome is reduced by 43.68%. The optimization method can significantly correct the aberration introduced by the von Karman dome and improve the guidance capability of infrared detection technology.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Infrared detection technology has become an important member of information warfare which recognizes the target by receiving the infrared band information of the target’s radiation or reflection [1]. Infrared detection systems are used to obtain the original target signals. An infrared detection system must offer excellent imaging quality and aerodynamic performance as much as possible to ensure the precise guidance capability of infrared detection technology. The infrared dome is one of the most important components of an infrared detection system, and hence, can also be regarded as the optical window. The aberration caused in the configuration of the infrared dome directly affects the imaging quality of the infrared detection system [2]. Additionally, since the infrared dome is located at the front end of the infrared detection system, its configuration directly affects its air resistance, which determines the aerodynamic performance [3,4]. Hence, the imaging quality and aerodynamic performance are the main factors considered in the configuration design of the infrared dome, which directly affects the precise guidance capability of the infrared detection technology.
With the development of military technology, the infrared domes in future wars will require faster speeds and greater maneuverability to ensure accurate strikes on targets [1]. To adapt to the rapid development of modern weapons, the concept of conformal optics is proposed. According to the concept of conformal optics, the characteristic of the infrared dome is to first consider the aerodynamic requirements instead of the imaging quality in the configuration [5]. This type of infrared dome is called a conformal dome. The configuration of the conformal dome can be designed as a streamlined surface exhibiting good aerodynamic performance. However, the improved and streamlined surface of the conformal dome exhibits new challenges. Numerous dynamic aberrations that rely on the geometric characteristics of the conformal dome are introduced, which leads to the imaging quality degradation of the infrared detection system [6,7]. Hence, clarifying the geometric characteristics and aberration characteristics of the conformal dome is the focus of conformal optics research. Further, the study for aberration characteristics correlation configuration optimization method of the conformal dome to correct the aberration and improve the imaging quality of the infrared system is imperative.
The von Karman curve is a fluid dynamics concept that helps in efficiently reducing the air resistance and the impact of the load. Under the condition of the same aspect ratio, the air resistance of the von Karman surface will be the lowest. Feng et al. proposes a novel waverider generated from axisymmetric supersonic flows past a pointed von Karman ogive, and it possesses higher lift-to-drag ratios, smaller trim drag, and larger internal volume than the conventional waverider [8]. Hence, depending on the design concept of the conformal optics, it is also a better choice to use the von Karman surface as the basis for the conformal dome configuration. However, at present, the conformal dome configurations are dominated by the quadric surfaces, and the research mainly focuses on the ellipsoidal domes or the parabolic domes [6,7,9–13]. There is a lack of research on the application of von Karman surface in conformal dome configuration design. In addition, using the von Karman surface configuration, the geometric characteristics and aberration characteristics of the conformal dome are still unclear. As a result, the application of von Karman surface in conformal optics cannot be realized. Hence, it is essential to apply the von Karman surface to the design of the conformal dome and analyze the geometrical and aberration characteristics of the von Karman dome.
At present, there are currently some correction methods for correcting the aberrations introduced by the conformal dome. Zhang W et al. designed a complete imaging system for the ellipsoidal conformal dome [3]. Dang et al. presented a method to reduce the dynamic aberrations through designing the inner surface of conformal domes [14]. To the best of our knowledge, no study has been done on the conformal dome configuration optimization method for correcting the aberrations introduced by the conformal dome. Moreover, these methods are all aimed at the traditional ellipsoidal conformal dome. There is also a lack of aberration correction method for conformal dome based on von Karman surface. Hence, a study for aberration characteristic correlation configuration optimization of conformal dome based on von Karman surface is essential.
In this paper, the von Karman surface is used to design the configuration of the conformal dome. Firstly, based on the principle of differential geometry, the geometric characteristics of the von Karman conformal dome have been studied. Next, we used ray tracing to analyze the geometric aberrations and the wave aberrations of the von Karman conformal dome, respectively. Among them, Zernike polynomial coefficients are used to characterize the wave aberration. The diameter of all the diffuse spots (GEO) and the root mean square (RMS) of the spot diagram were used to characterize the geometric aberrations quantitatively. Further, according to the geometric characteristics and aberration characteristics of the von Karman dome, the conformal dome configuration based on the von Karman surface was optimized to correct the aberration. Finally, the aberrations introduced by the conformal dome after the configuration optimization and the original von Karman dome have been compared to prove the effectiveness of the optimization method.
2. Geometric characteristics of the von Karman conformal dome
2.1 Parametric the von Karman dome and its imaging system
In this paper, to facilitate the study of the geometric characteristics and aberration characteristics of the conformal dome, the conformal dome configuration and imaging system have been parameterized. We describe the von Karman dome configuration based on the common geometric structure expression of the conformal dome [5,15]. The established von Karman conformal dome model exists in a three-dimensional Cartesian coordinate system. The intersection of the meridian and external surfaces of the conformal dome forms a curve that describes the mathematical expression of the dome configuration. This curve is expressed as f(x). The internal surface of the von Karman conformal dome is constructed on the principle of equal thickness, i.e., the distance between the internal surfaces and the external surfaces is equal to d. The curve describes the internal surface. Let us consider the dome apex as the origin of the coordinate axis, the axis of rotational symmetry as the x-axis, and the direction of the dome to be along the positive direction of the x-axis. From Fig. 1, we can establish the dome bottom diameter to be D, the distance from the dome top to the ground to be the dome length, L, and the distance from the top to the bottom of the dome to be the length of the dome. Using the same Fig. 1, we can also define the ratio of the length, L to D to be the aspect ratio F. The ideal lens usually replaces the internal imaging system for making the aberrations of the conformal optical system, which arises from the conformal dome. To high-resolution images of a wide range of targets, the internal imaging system uses a scanning working method [5,15]. The imaging system rotates about a fixed point given by, $O^{\prime}$. Further, the distance between the rotation center and the dome apex x0 characterizes the position of the rotation center. The entrance pupil diameter of the imaging system is given as D0. The angle between the optical axis ray (OAR) and the symmetry axis of the conformal dome is the viewing angle is given as θ. The configuration of the von Karman conformal dome and its imaging system can be described as shown in Fig. 1.
The configuration of the von Karman conformal infrared dome can be expressed as:
where $\phi = \arccos \left( {1 - \frac{{2x}}{L}} \right)$.2.2 Geometric characteristics
The von Karman conformal dome loses its point symmetry in geometry. When the imaging system is at different viewing angles, the parts of the dome that are involved in the imaging are different; this will ultimately introduce dynamic aberrations which vary with the viewing angle. The dynamic aberrations introduced by the conformal dome mainly depend on its geometrical characteristics. Hence, the geometric characteristics of the von Karman conformal dome have been discussed first.
The geometric characteristics of the conformal dome related to the aberration characteristics are mainly divided into two aspects. One is that the meridian and sagittal surfaces of the von Karman conformal dome have different radii of curvature under different viewing angles. This is the main reason for the introduction of astigmatism. This is the main reason for the introduction of astigmatism. The other is the introduction of a coma in the von Karman conformal dome so that the boresight does not coincide with the normal of the dome surface under different viewing angles.
Based on the principle of differential geometry [6,7], the equation of the Dupin indicatrix at any point on the von Karman conformal surface can be obtained. Additionally, the curvature along different directions at each point on the surface can be obtained by the transformation. The definition of the Dupin indicator line is shown in Fig. 2. The von Karman conformal surface is a rotationally symmetric surface that is expressed by a vector function. Let us consider a line segment PN, drawn along the direction of the tangent vector that determines the normal section line through any point P on the von Karman surface. If the starting point of the vector is selected as the origin of the coordinate system, then the von Karman spatial surface, $\overrightarrow r$, can be expressed as:
By calculating three basic values (Ls, Ms, and Ns), the equation of the Dupin indicatrix can be obtained by:
where zx and yα satisfy the relation $\overrightarrow {PN} = {z_x}\overrightarrow {{r_x}} + {y_\alpha }{r_\alpha }$, Ls, Ms, and Ns are the three basic quantities of the second type of von Karman conformal surface, given as follows:Since f(x) > 0, $f^{\prime\prime}(x )={-} \frac{{{R^2}}}{{f{{(x )}^3}}} < 0$, and thus Ls > 0 and Ns > 0. Therefore, the equation of the Dupin indicatrix of the von Karman conformal surface describes an ellipse that can be expressed as follows:
The Dupin indicatrix of the von Karman conformal surface is an ellipse, and hence the curvature has extreme values in the $\overrightarrow {{r_x}}$ and $\overrightarrow {{r_\alpha }}$ directions, and the radius of curvature in the other orientations varies between them. For points on the yox plane, $\overrightarrow {{r_x}}$ and $\overrightarrow {{r_\alpha }}$ are the vectors on the meridian plane and the sagittal surface, respectively, this allows the radius of curvature to have extreme values on the meridian plane and the sagittal surface. The setting ${y_\alpha } = 0$, we get $PN = \sqrt {\frac{1}{{{k_m}}}} = {z_x}\left|{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_x}} \right|= \sqrt {\frac{1}{{{L_s}}}} \left|{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_x}} \right|$. The expression for the radius of curvature, given by Rm, in the meridian plane of the von Karman conformal surface can be obtained as:
Similarly, setting ${z_x} = 0$, we get $PN = \sqrt {\frac{1}{{{k_s}}}} = {y_\theta }|{\overrightarrow {{r_\theta }} } |= \sqrt {\frac{1}{{{N_s}}}} |{\overrightarrow {{r_\theta }} } |$. The expression for the radius of curvature, given by Rs, corresponds to the sagittal surface of the von Karman conformal surface and can be expressed as:
Figure 3 shows a schematic diagram indicating the angle Δθ between the visual axis and the surface normal in a von Karman conformal dome, which can be expressed as:
When analyzing the geometric characteristics of von Karman, the aspect ratio F is a factor that must be considered, which is a necessary limitation for the conformal dome configuration.
The geometric characteristics of the von Karman dome are shown in Fig. 4. In terms of the difference between Rm and Rs, we can observe that the meridian radius of the curvature is larger than the sagittal radius of the curvature. Moreover, it can be observed that the difference in the radius of curvature of the small-angle viewing angle changes rapidly. Additionally, larger values of F correspond to a larger range of the radius of curvature.
Fig. 4. Geometric characteristics of the von Karman dome, (a) shows the difference between Rm and Rs,(b) shows the plot of Δθ as a function of θ.
On the other hand, we can observe that as θ increases, Δθ has the largest extreme value. Higher values of F correspond to a greater angle between the boresight and the surface normal. Therefore, at a small viewing angle, θ δ 10°, the change in Δθ is drastic.
3. Aberration characteristics of the von Karman conformal dome
The evaluation methods of the aberration introduced by the conformal dome evaluation mainly include geometric aberration and wave aberration. Simplicity and intuitiveness are the benefits of using geometric aberrations in these two evaluation methods. Moreover, the analysis of wave aberration can introduce the specific type of aberration. In order to clarify the aberration characteristics of the von Karman dome more comprehensively, the geometric aberrations and wave aberrations of the von Karman dome have been analyzed separately.
When analyzing the aberration of von Karman, the aspect ratio F is a factor that must be considered, which is a necessary limitation for the conformal dome configuration. The model of von Karman conformal domes has been established in optical design software CODE V, and the Zernike aberrations under different F have been researched. Since the optical dome is not a complete optical system, we join a perfect lens in the optical design software CODE V. The ideal lens usually replaces the internal imaging system for making the aberrations of the conformal optical system only arises from the von Karman conformal dome.
3.1 Geometric aberration of the von Karman conformal dome
A point map on the imaging surface is obtained by tracing the light rays passing through the dome to reach the imaging surface. This point map can characterize the geometric aberrations introduced by the dome. Additionally, the RMS and GEO radius of the dot map are used as evaluation indicators for geometric aberrations. In this section, we study the geometric characteristics of the von Karman dome.
Figure 5 displays the results of the geometric aberration introduced by the von Karman conformal dome for different values of F. Larger values of F correspond to larger geometric aberrations introduced by the von Karman conformal dome. In particular, when θ < 12.5°, the geometric aberration is relatively large and also changes drastically. This indicates that the image quality in the small-angle market has deteriorated drastically.
3.2 Wave aberration of the von Karman conformal dome
In terms of the wave aberration analysis, the traditional Seidel aberration theory is inapplicable for resolving wave aberrations for the von Karman conformal dome. In optics, the Zernike polynomials are widely used for the description, detection, and aberration analysis of freeform surfaces [16]. By decomposing the wave aberrations of the imaging system from different viewing angles into infinite Zernike bases, the Zernike coefficients thus obtained can directly reflect the magnitude of the corresponding aberrations. This assists the quantitative evaluation of the aberrations introduced by the von Karman conformal dome. Some related studies prove that the aberrations introduced by the conformal dome mainly include astigmatism, coma, and spherical aberration corresponding to the Z5, Z8, and Z9 terms in the fringe Zernike polynomial, respectively [17,18]. Further, there is also a big change in the defocus, which corresponds to the Z4 in the fringe Zernike polynomial [17,18].
The Zernike polynomial in the polar coordinate system is expressed as
According to the aforementioned formula, an infinite number of Zernike polynomials can be obtained in turn. The coefficients discussed in this paper are shown in Table 1.
Table 1. Zernike Aberration Polynomial
Figure 6 displays the results of the wave aberration introduced by the von Karman conformal dome for different values of F. We can conclude that among the aberrations introduced by von Karman, the most dramatic change occurs in the defocus, followed by astigmatism and coma. Additionally, the larger values of F correspond to larger values of wave aberration introduced by the von Karman conformal dome. Among the wave aberration, F has a larger impact on the defocus and astigmatism introduced by the dome. Moreover, all types of aberration changes are extremely drastic if a small angle of view is considered.
4. Configuration optimization of a conformal dome based on the von Karman surface
In this paper, we studied the geometric characteristics and aberration characteristics of the von Karman conformal dome. The results obtained from the foregoing studies show that in a small viewing angle, the aberration changes drastically, and the introduced aberration of the von Karman conformal dome is particularly large. This observation is consistent with the geometric characteristics of the von Karman dome. According to the geometric characteristics and aberration characteristics of the von Karman dome, this paper proposes a conformal dome configuration optimization method based on the von Karman surface. This optimization method replaces a part of the von Karman surface that participates in the imaging with a small angle of view with an ellipsoidal surface. This is because the aberration introduced by the ellipsoidal surface is easier to correct, and this can be corroborated by many related correction studies [6,12]. Figure 7 displays the optimization method diagram.
The replacement position depends on the values of θ and x0. The two surfaces are linked as per the principle of the equal tangent slope. The internal surface of the positive flow hood changes with the replacement of the external surface. $\gamma$ is the angle between the tangent and the x-axis.
The mathematical expression describing the basic configuration of the dome von Karman surface is expressed as fo(x), which can be described as:
where $\phi = \arccos \left( {1 - \frac{{2x}}{L}} \right)$.In the coordinate system, the OAR function is expressed as:
When ${F_{\textrm{OAR}}} = {f_o}$, the available replacement position is (xs, fo(xs)). The tangent slope of the replacement position can be given as:
The mathematical expression describing the replace surface is defined as fs(x), which can be described as:
where R is the radius of curvature of the vertex (also called the base circle radius) and k is the conic constant. To calculate the value of R and k, we subsequently differentiated as:Hence we get the expressions of R and k to be:
In this paper, the conformal dome based on the von Karman surface with F = 1.7 has been optimized for configuration. To verify the effectiveness of the optimization method, a comparison of the geometric aberrations introduced by the conformal dome after the configuration design and the original von Karman dome is made. The geometric aberration is used to evaluate the aberration introduced by the conformal dome to get a more intuitive and comprehensive comparison result. The optical system parameters used during comparison, as mentioned in Table 2.
Table 2. Optical System Parameters
Figure 8 displays the RMS and GEO comparison spot diagrams of the von Karman conformal dome after configuration design. The comparison results show that the RMS of the optimized von Karman conformal dome spot diagram is reduced by 24.94%, and the GEO of the spot diagram is reduced by 43.68%. The optimization method proposed in this paper can effectively correct the aberration introduced by the von Karman dom
5. Summary
In this paper, the von Karman surface has been used to design the configuration of the conformal dome. Firstly, the geometric characteristics and aberration characteristics of the von Karman dome have been obtained. Specifically, in terms of geometric characteristics, we can conclude that if the viewing angle is small, the changes in the radius of ${R_m} - {R_s}$ and $\Delta \theta$ are relatively sharp. In terms of aberration characteristics, the general change rule of aberration introduced by the von Karman dome is similar to its geometric characteristics. When the viewing angle is small, the geometric aberration and wave aberration introduced by the von Karman conformal dome changes drastically, and the range of cut changes is large. From an in-depth analysis, we can observe that among the aberrations introduced by the von Karman curved dome, the defocus aberration is the largest, followed by the coma and the astigmatism, and the spherical aberration is observed to be the smallest. The clear von Karman geometric characteristics and aberration characteristics seem to be promising as a basis for the configuration design of the conformal dome.
More importantly, based on the von Karman dome's geometric characteristics and aberration characteristics, an optimization method for the conformal dome configuration based on the von Karman surface has been proposed. We observed that the RMS of the optimized conformal dome spot diagram is reduced by 24.94%, and the GEO of the spot diagram is reduced by 43.68%, compared with the Von Karman dome. The configuration optimization method can correct the aberration introduced by the von Karman conformal dome and improve the imaging quality of the infrared system. The correction method for optimized configuration can be combined with other methods to correct further the conformal dome's aberration based on the von Karman surface. The results of this paper are promising to apply the von Karman surface to the design of conformal dome and reduce the difficulty of subsequent corrector and optical system design. Furthermore, the method is promising to advance the development of conformal optics.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the finding of this study are available within this article.
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