The infrared detection method is widely used in precision guided weapons because of its many advantages. The infrared seeker is the core of a guided weapon and generally consists of a dome, an imaging system, and an electronic cabin^[1]. In order to obtain a high-resolution image of the target in a large field of view (FOV), most of internal imaging systems work in a scanning mode. The traditional dome is hemispherical. When the center of rotation of the imaging system coincides with the center of the sphere, the hemispherical dome introduces constant aberrations during scanning. However, the larger coefficient of air resistance of the hemispherical dome is not conducive to increasing the speed of guided weapons. In order to reduce the coefficient of air resistance of the dome, researchers have proposed a conformal optical technology that uses a streamlined design of the outer surface of the dome^[2]. Conformal domes will introduce dynamic aberrations that vary with look angles (LAs) due to the loss of point symmetry. Therefore, correcting the dynamic aberrations introduced by conformal domes is the core task of conformal optics^[3,4]. Mathematically, the outer surface of the conformal dome can be polynomial, exponential, or spline. Considering aberration correction, dome processing, and testing, the outer surface of the conformal dome is generally a quadric surface, and in most cases it is ellipsoidal. After more than twenty years of development, researchers have proposed a variety of correction methods of dynamic aberrations, including fixed correctors^[5–8], dynamic correctors^[9–11], arch correctors^[12,13], and so on. Applying one or more of the above methods can correct the dynamic aberrations introduced by the conformal dome^[14]. At present, increasing the field of regard (FOR) of the conformal optical system and simplifying the structure of the seeker are some of the problems to be solved in conformal optics.

Inspired by the arch corrector, this Letter proposed a method of correcting dynamic aberrations using the diffractive surface and anamorphic asphere surface. The design results show that this method can better correct the dynamic aberrations introduced by the conformal dome, and it is also conducive to simplifying the structure of the conformal seeker.

A typical conformal optical system includes a conformal dome, an imaging system, and correctors, as shown in Fig. 1.

Parameters describing the conformal optical system are as follows^[15]. The fineness ratio $F$ is the ratio of length $L$ to diameter $D$. FOV is the field of the imaging system. LA is the angle between the axis of the imaging system and the symmetry axis of the conformal dome. FOR is the maximal LA.

Through expanding the wave aberration function in terms of a complete set of Zernike polynomials, we can get the coefficients of Zernike polynomials. Every term of Zernike polynomials has explicit physical meaning, so the coefficients of Zernike polynomials reflect the value of corresponding aberrations^[16,17]. Taking the ellipsoid dome as an example, the amplitudes of Zernike coefficients of the wavefront passing through the dome are shown in Fig. 2. The predominate aberrations in this system are approximately eight waves of astigmatism (Z5) and five waves of defocus (Z9). The astigmatism can be qualitatively understood by visualizing the optical surface as a function of LA: the dome is a rotationally symmetric optical element when the LA is equal to zero, whereas the dome acts more like a cylinder towards the edge of the FOR^[12]. Therefore, astigmatism is the main aberration that needs to be corrected.

The seeker can be divided into two types based on the different gimbals used. One type uses an azimuth-elevation-type gimbal, and the other type uses a Roll-Nod-type gimbal^[18]. The advantage of the azimuth-elevation gimbal is that the control scheme is mature. However, due to the structural limitation, the maximum off-axis angle of the azimuth-elevation gimbal is difficult to exceed 60°, and there is a large tracking blind space. The Roll-Nod-type gimbal uses the Nod frame as the inner frame, and the Roll frame as the outer frame. Because the angle of the Nod frame can reach ±90°, the Roll-Nod-type gimbal has a larger tracking space. Due to the problem of tracking in the zenith zone, the Roll-Nod gimbal was not widely used in the early stage. With the research of control technology, the problem of tracking in the zenith zone has been solved, and a variety of new-generation guided weapons have adopted Roll-Nod gimbals. In general, the seeker using a Roll-Nod gimbal is the development direction.

The fixed corrector is a rotationally symmetric element and has limited correction capabilities of dynamic aberrations. The dynamic corrector will introduce additional motion mechanisms and increases the complexity of the seeker. They are not ideal correction methods. In 1999, Sparrold proposed the arch corrector, which is only capable of Roll-Nod gimbaled missile seekers^[12]. Different from fixed correctors, the arch corrector is an optical element that is bilaterally symmetrical rather than rotationally symmetrical, as shown in Fig. 3^[19].

The arch corrector is mounted on the Roll frame of the Roll-Nod gimbal and rolls with the imaging system. The imaging system is mounted on the Nod frame, and the optical axis swings within the symmetry plane of the arch corrector, as shown in Fig. 4.

Because of broken rotational symmetry, the arch corrector obtains more design freedom and aberration correction capabilities. Through designing the surface shape, the arch corrector can better correct dynamic aberrations introduced by the conformal dome. Because astigmatism aberrations can be corrected well by non-rotationally symmetric surfaces, the arch corrector is one of the potential solutions to correct the dynamic aberrations in the large FOR^[19,20]. The arch corrector has great advantages in structure, but the complex surface shape and large size increase the processing difficulty and limit its application.

A major advantage of the arch corrector is that it can produce sufficient astigmatism, which is the main aberration introduced by a conformal dome. Reducing the size of the correction element while retaining sufficient design freedom is the target of this Letter. Inspired by the arch corrector, the design layout of this Letter is shown in Fig. 5. For the seeker with a Roll-Nod gimbal, the arch corrector is replaced with a diffractive surface superimposed on the inner surface of the conformal dome. As a rotationally symmetric surface has limited ability to correct astigmatism, anamorphic asphere surfaces are used in the imaging system in this Letter.

Diffractive elements are a type of optical elements based on the principle of optical diffraction, which can change the phase and amplitude of the wave front by etching two or several level relief structures on the substrate. A diffractive lens is mathematically equivalent to a thin lens with an infinite refractive index and has similar aberration characteristics. In order to simplify the structure of the system, the diffractive surface is superimposed on the inner surface of the conformal dome in this Letter.

The phase function of the rotationally symmetric diffraction surface can be described as where $A_i$ is the coefficient of the diffraction phase.

Some researchers have used diffractive elements in correction systems for conformal optics and have verified the feasibility of correcting dynamic aberrations. By analyzing the Zernike aberrations of the wavefront, the phase coefficients and the phase orders of diffractive optical elements used to correct primary Zernike aberrations can be obtained^[21].

The anamorphic asphere surface is bilateral symmetry in both $x$ and $y$ directions but not necessarily with rotational symmetry. The surface form without the additional aspheric terms is sometimes referred to as a bi-conic surface. The equation is where $z_{\mathrm{sag}}$ is the sag of the surface parallel to the $z$ axis; $C_x$, $C_y$ are the curvatures in $x$ and $y$ directions, respectively; $k_x$, $k_y$ are the conic coefficients in $x$ and $y$ directions, respectively; $A$, $B$, $C$, $D$ are aspheric coefficients from the 4th to 10th order; $A_p$, $B_p$, $C_p$, $D_p$ are deviations of axisymmetric components from the 4th to 10th order.

For a thin pencil beam, the position of focal lines in the meridian plane and the sagittal plane can be solved by Young’s formulas. The formulas are where $I$ is the angle of incidence; $I^{\prime }$ is the angle of refraction; $t$ and $t^{\prime }$ are the object distance and image distance in the meridian plane, respectively; $s$ and $s^{\prime }$ are the object distance and image distance in the sagittal plane, respectively; $r_t$ and $r_s$ are the curvature radius in the meridian plane and the sagittal plane, respectively.

For different LAs of the conformal dome, the parameters of the diffractive surface and the anamorphic asphere surface can be solved successively by the method of ray tracing. A conformal optical system is designed in this Letter based on the design ideas described above. The parameters are shown in Table 1.

The initial inner and outer surfaces of the conformal dome are ellipsoids of equal thickness, and they are gradually expanded into asphere surfaces with ${\mathrm{10}}^{\mathrm{th}}$-order terms during the design process. The diffractive surface is superimposed on the inner surface, and the phase equation of the diffractive surface is^[22]

The optimization result is $A_1=0.036$ and $A_2=-2.0892\times {\mathrm{10}}^{-5}$. The phase curve of the diffractive surface after optimizing is shown in Fig. 6.

For the diffractive element, the diffraction efficiency is a key parameter. The diffraction efficiency can be calculated using the following formula:

For a center wavelength ${\lambda }_0=4\mathrm{\mu }\mathrm{m}$, diffraction order $m=-1$, and the diffraction efficiency at $3.7–4.8\mathrm{\mu }\mathrm{m}$ is over 92%.

The optimized conformal optical system at different LAs is shown in Fig. 7, and the designed structure of the conformal seeker is shown in Fig. 8. The mirror in the folded optical path used a ${\mathrm{10}}^{\mathrm{th}}$-order anamorphic asphere surface to correct astigmatism. The exit pupil of the conformal optical system coincides with the cooled stop of the detector by using secondary imaging, and the system reaches 100% cooled efficiency. The conformal optical system can be divided into three parts, as shown in Fig. 7(a). The first part contains four lenses and one mirror, and they are mounted on the Nod frame. The second part is a prism with three reflective surfaces and is mounted on the Roll frame. The remaining elements are the third part, and they are fixed relative to the missile. The biggest advantage of this optical system is that it has fewer moving optical elements and has high system reliability.

Parameters of the conformal optical system after optimization are shown in Table 2. The first four lenses used aspheric surfaces, and the following mirror used an anamorphic aspheric surface in the conformal system.

The spot diagrams of the conformal system at different LAs are shown in Fig. 9. The size of spot diagrams in the FOV at different LAs is less than 30 μm. The modulation transfer function (MTF) results of the conformal optical system are shown in Fig. 10, and all of the MTFs in the FOV at 16 lp/mm are higher than 0.5. In summary, the conformal optical system designed using this method achieves better design results without introducing additional optical components.

In conclusion, a correction method of dynamic aberrations based on the diffraction surface and anamorphic aspheric surface is described. This method is derived from the arch corrector and can only be used on the Roll-Nod gimbal but has a more compact structure. The design results show this method can obtain good imaging performance when designing the conformal optical system. This design method has great advantages in improving system reliability.