Quasi-optical design of a millimeter wave imaging system

Abstract. We report a complete design and simulation of a quasi-optical millimeter wave imaging system using ZEMAX and FEKO software, respectively. A Fresnel lens and a horn antenna are combined in this design. Compared to spherical and aspherical lenses, a Fresnel lens can be fabricated much easier at millimeter wavelengths. For focusing millimeter wave radiation, a Fresnel lens can be used to reduce the thickness of the focusing element and to lighten its weight from 25 to 4.5 kg. A horn antenna with a Gaussian profile and corrugated walls is designed for feeding this system at a central frequency of 94 GHz. The symmetrical radiation pattern of the designed corrugated Gaussian horn in E- and H-orthogonal planes, its wide bandwidth, and low side lobe levels make it a good candidate for feeding a W-band millimeter wave imaging system. The designed quasi-optical imaging system is light-weighted, has high resolution, and can be used in detecting hidden objects within a distance of 5 m with a 30-mm resolution in W-band at a central frequency of 94 GHz.

Millimeter waves cover a wavelength range of 1 to 10 mm, which corresponds to a frequency range of 30 to 300 GHz. 7 Regarding the atmospheric transmission, there are two optimal frequency bands [i.e., Q (35 GHz) and W (94 GHz) bands] for use in millimeter wave imaging. Frequency selection is based on a simple rule: selecting a lower frequency for greater penetration depth and optics with larger aperture, and/or selecting a higher frequency for better resolution. Compared to the Q-band, the W-band has better image quality and spatial resolution, which is controlled by the diffraction limit. The distance of the millimeter wave imaging system from the target is usually several meters. 8 The history of millimeter wave technology dates back to 1890, but the first important activity in this field was conducted in 1930. 9 Ditchfield and England 10 introduced the first millimeter wave imaging system in the UK. 6 Since then, the technology has continued with rapid advances in recent years. 11 Dielectric lenses or reflectors can be used for imaging systems. 12 Of course, each of the two methods of refraction and reflection has advantages and disadvantages. Reflective systems are light in weight, but due to their smaller number of components and interconnection, they have narrower field of view than equivalent refractive systems, and the installation of mirrors in these systems is even more difficult. 13 At frequencies above 30 GHz, lens antennas can be considered as great alternative to reflectors since they are not affected by aperture blockage, exhibit great manufacturing tolerance, and use inexpensive plastic materials. However, when a short focal length is accompanied by a large diameter, the thickness and weight of a lens are increased. These limitations can be avoided by using the diffractive equivalent of the lens, i.e., Fresnel zoned lens. 14,15 Fresnel zoned lenses are good options as focusing elements in the millimeter wave band. The lens material should have low values of loss, mass density, and dielectric constant (ε_r ¼ 2 to 4). 16 Millimeter wave band lenses are usually made of materials such as high-density polyethylene (HDPE), silicon, polystyrene, Rexolite, and/or Teflon. HDPE is a cost-effective material and can be easily machined using computer-aided cutting. 17,18 A uniform dielectric lens with two surfaces is equivalent to two reflectors because each surface is equivalent to a degree of freedom. By forming both surfaces, the designers can design a lens that corrects aberrations. 19 The performance of highly accurate optical systems that use spherical optics is limited by aberrations. Using aspherical optics, geometric aberrations can be reduced or removed. New manufacturing methods allow producing high-precision aspherical surfaces. 20 Since quasioptical imaging systems have to reduce the blurring effect for obtaining an acceptable sharpness, an aspherical lens can be used. 8 Zhou et al. 2 developed a 43-cm-diameter HDPE aspherical transition system at 89 GHz that was capable of imaging objects at a distance of 3.5 m with a resolution of 28 mm. Chen et al. 21 reported a similar system with 35-mm resolution at a distance of 3.5 m 2 . In 2011, an HDPE transition system was developed with a diameter of 50 cm at 94 GHz using a one-dimensional array of receivers. 22 A focal array imaging system was also made in 2011 using an acrylic lens with a diameter of 20 cm and a resolution of 2 cm at a distance of 1 m in the frequency range of 75 to 95 GHz. 23 In the millimeter wave band, where large lenses are required because of the diffraction limit, the system becomes very heavy. 8 When thinner, lighter, and easy-to-manufacture systems are required to focus the incoming radiation, Fresnel lenses are preferred over conventional refractive lenses. Fresnel lenses use diffraction as a method of collecting electromagnetic waves at the focal point. In this type of lens, the stepped discrete pattern, which first proposed by Rayleigh, 24 can realize phase correction. According to this theory, different methods have been proposed and the desired phase correction has been achieved. 18,24,25 A 600-GHz Fresnel lens was designed for active and passive imaging by Chen et al. 1 Design and manufacturing of Fresnel zone plate lenses with opaque and transparent zones at millimeter wavelengths was reported by León et al. 26 . Moreover, the use of dielectric Fresnel lenses for imaging was investigated in the frequency range of 75 to 110 GHz in 2017. 27 In passive systems, the target radiation is measured by the systems, whereas in active systems, a source is used to illuminate the target. An important application of passive imaging in millimeter wave region is the detection of weapons hidden under clothing. 28 In short-range stand-off passive millimeter wave imaging, the main purpose is to reduce the costs and complexity of the system. 29 Here, the optical configuration is arranged so that the optical aberration is reduced and the system can be employed in active or passive mode by proper setup.
Wide range applications of electromagnetic spectrum have made it necessary to use methods of designing and developing antennas for lowering the interference that may occur between the wanted signals and undesired ones. Many methods have been presented during recent years and a number of them have received attention and have been employed in real-world applications. These methods can be divided into two broad categories: side lobe reduction and gain reduction in specific narrow directions, which is, in reality, null placement in the radiation patterns. Side lobes can cause noise in phase contrast image and some problems in hidden objects detection. The effect of this noise is considerable on the surface measurement and consequently on the uncertainty in measurement. This noise can be suppressed by a few enhancement techniques and many methods are available for side lobe reduction. Wavelet transform is an example of side lobe reduction methods that has been proposed for demodulation using the Fourier transform. [30][31][32] In quasi-optical systems, the free space energies are focused by the lens to a horn antenna, which transmits it to the detector as shown in Fig. 1. There are different types of horn antennas with advantages and disadvantages that have to be taken into account when designing a quasioptical system. 33 In millimeter wave region, high-performance applications, such as security imaging, radar, and radio astronomy, corrugated horn antennas, are commonly used for feeding due to their high ray symmetry, relatively low side lobe levels, and low cross-polarization. 34 The process of designing a quasi-optical system involves the design of the primary optics (selection of focal element) and the secondary optics (selection of horn profile and wall surface features) according to the system requirements.
In this study, the primary optics is designed using an aspherical lens that has a better spot diagram and corrected aberrations as compared to spherical lenses. A grooved Fresnel lens is then simulated in the ZEMAX software to replace with the aspherical lens due to the heaviness of the aspherical lens. A corrugated Gaussian profile horn is also designed at a central frequency of 94 GHz as the secondary optics and simulated using the electromagnetic software FEKO. This horn can be used with a lens (spherical, aspherical, or Fresnel lens) with an F-number of 1.2, to meet the requirements of the system. This system is capable of forming an image by mechanically scanning the object plane or using an array of detectors in the image plane with a spatial resolution of 30 mm at a distance of 5 m.

Aspherical Lens
The spatial resolution in imaging applications is diffraction limited. The angle is limited by the diffraction corresponding to the central zone of the diffraction pattern, which comprises 84% of the irradiance distribution. According to Rayleigh's criterion, the optical system resolution is given in Eq. (1), E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 1 1 6 ; 6 0 0 where λ is the central frequency, R is the imaging distance, D is the diameter of the input aperture, and s is the resolution required at R. 13 According to Rayleigh's criterion, for achieving a resolution of 30 mm at 94 GHz and a distance of 5 m, the aperture diameter should be about 65 cm. The designed optical system is a transmission dielectric aspherical lens, which is simulated by the ZEMAX software (Fig. 2); this system consists of the object plane (left), lens and image plane (right). Moreover, the colored lines show different positions of the object in the field of view of the designed lens. HDPE lens dielectric material with a refractive index of 1.5147 is suitable for imaging (at 94 GHz, tan δ ¼ 0.0003 and ε ¼ 2.2). HDPE has not been listed in ZEMAX™ Glass catalog, so it was entered in the frequency range of 45 to 145 GHz using "appropriate coefficient data" and Conrady's formula according to Eq. (2), 1 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 1 1 6 ; 4 2 6 Conrady's formula is a well-known equation for changing the refractive index of a material in the wavelength region and uses three pairs of wavelength-refractive index data to create continuous capability. 1 The formula of the aspherical lens surface is given as follows: 2 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 1 1 6 ; 3 3 7 where c is the curvature, r is the radial coordinate, and k is the conic constant. 2 Table 1 shows the optimized values of aspherical surface constants of the simulated lens according to ZEMAX. The diameter (D) and thickness (d) of the lens are 650 and 71 mm, respectively. Using these parameters and mass density, which were entered in ZEMAX, this lens weighs 25 kg.   Figure 3 shows the spot diagram of the designed system for seven different positions in a field of view (0 deg, 0.12 deg, 0.17 deg, 2.5 deg, −2.5 deg, 5 deg, and −5 deg) that covers an area of 50 × 50 cm 2 . In this system, the aberrations were corrected using the optimization program in ZEMAX, and the root mean square size of the spot at the end of the field of view was approximately 3 mm. The image quality curves of the spot diagram, focused energy, and MTF (MTF describes the transfer of modulation from the object to the image as a function of spatial frequency and commonly used to specify lens performance, and as an optimization and tolerancing target during lens design) as well as their analysis indicate that the designed system has a high image quality.

Fresnel Lens
A zone plate is a tool for image formation whose mechanism is not refraction, but rather diffraction in the aperture rings. The interference of the diffracted radiation generates the image. The spherical wave-front can be modified by using materials with different permittivities [ Fig. 4(a)] or phase correction areas [ Fig. 4(b)]. 28,35   When the wave-front passes through the lens, the lens imposes a phase difference on the wave. The result is nearly a spherical wave-front that converges on the focal point of the lens. 36 By applying simple rules during the design process, a Fresnel lens antenna can achieve high efficiency with low side lobe levels. In a grooved Fresnel zone plate lens, for having acceptable complexity, phase transfer steps can be used when the phase change reaches the half phase, quarter phase, and p phase. Therefore, each time the phase change reaches 180 deg, 90 deg, and 2π∕p, the Fresnel lens compensates the phase. The parameter p corresponds to the number of compensations made during a 360-deg phase change. Since this compensation is not complete, the lens will have limited efficiency. [37][38][39] In a Fresnel lens, each groove forms a prism, so the Fresnel lens is made up of a set of prisms. The grooves near the center of the Fresnel lens are almost flat and shallow, and the grooves near the outside points have deep and sharp angles. [40][41][42] All diffraction areas contribute to focusing the light at the right point. For the design of the diffraction lens, the transition points should be calculated for each area, which is a correct multiple of the wavelength relative to the focal point of the lens (Fig. 5 36 ).
In a lens with the focal length f, which operates at wavelength λ (for 2π phase, the change is equal to 1λ of the optical path difference), the radius of the area p, where the phase change is equal to 2π, can be obtained as follows using Pythagorean theorem: 36 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 1 1 6 ; 3 6 7 r 2 p ¼ 2fpλ: The phase difference caused by the lens is the difference between the input wave at any point and the phase delay due to passing through the lens at distance r from the axis, which is given as follows: 36 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 1 1 6 ; 2 9 8 ΔφðrÞ ¼ 2πðn − 1Þr 2 ∕2Rλ: The optical path difference at different points of the wave-front can be written as follows: 36 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 1 1 6 ; 2 5 3 OPD ¼ ΔφðrÞλ∕2π: ZEMAX calculates the location of the transition points for a symmetric lens phase profile (even powers of radius) by minimizing the aberrations. The phase function in this program is as follows: 36 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 7 ; 1 1 6 ; 1 8 5 The Fresnel lens substrate that is designed by ZEMAX is a flat disk. Its profile is made of radial flat surfaces whose endpoints are defined by SAG expression. 36 The two-dimensional design of the Fresnel lens is illustrated in Fig. 6. Its input aperture has a radius of 330 mm, and there are 16 grooves in this Fresnel lens. The central thickness of the designed Fresnel lens is 12.7 mm. Additionally, the spherical surface radius of this lens is 778.9 mm and its conical coefficient is equal to −1.115. For designing this type of lens in ZEMAX, the non-sequential component of Fresnel was used and its 3D design is shown in Fig. 7. This lens weighs 4.5 kg, which is considerably lower compared to the corresponding spherical and aspherical lenses. This design can be used in a low-weight millimeter wave imaging system with a resolution of 30 mm at a distance of 5 m. Figure 8 shows the optimized Fresnel lens spot diagram. The intensity profile of the image spot contains a central lobe and several side lobes with gradually decreasing intensity containing 84% received energy of the target. The RMS of the lens spot's radius is 5.626 mm. To optimize the lens, the image plane is set as the global coordinate reference, and the default merit function is used with RMS of the spot's radius of all configurations as the convergence criteria.

Design and Simulation of Secondary Optics
For the design of the secondary optics, one should specify the horn profile and its wall type (flat or corrugated). It is very important to match the feed horn beam width to the system F-number (F#) to optimize the performance of the system. The usual values of the optimum feed taper (a measure of the ratio of the power reduction at the opening edge to the power on the axis) range from −10 to −13 dB (Fig. 9). If θ −10 dB ≫ θ 0 , spillover loss will occur and some power will be lost. If θ −10 dB ≪ θ 0 , the amplitude taper loss and phase error loss result in deviations from amplitude and constant phase in the opening field. 43 The performance of the horn antenna with a linear opening wall is improved by profiling the wall. 43 Horn profile can be sinusoidal, tangential, exponential, hyperbolic, polynomial, etc. 44 The Gaussian profiled horn antenna provides a smooth transition from the waveguide to the flare, which improves the matching between the antenna and free space. This improved matching ensures a better radiation pattern and more bandwidth and increases the total efficiency of the system. 45 Horn antennas with Gaussian profile can be used to optimize various parameters, such as beam diffraction, bandwidth, side lobe levels, transverse polarization, orientation, gain, and efficiency. In Gaussian horn antennas, the main features of the radiation pattern of conical horn  antenna are preserved. Furthermore, side lobe levels and cross-polarization surface are significantly reduced. 45,46 The next step after selecting the wall profile of the horn is choosing the feature of the wall (flat or corrugated). Flat-wall horn antennas have some problems that can be resolved by wall corrugation. These problems include uneven beam widths, uneven phase centers in two orthogonal planes, more side lobes on E-plane than H-plane, and diffraction from E-plane walls that will cause back lobes. 19 First, we designed an optimized Gaussian profile horn with a flat wall in FEKO software for feeding the desired optical system. The structure of this horn is shown in Fig. 10. Fig. 9 Spillover loss of the combined lens and horn antenna. θ 0 is the opening angle of the lens, θ −10 dB is the angle for −10-dB illumination taper, F is the focal length, and D is the lens diameter. The optimized parameters of the horn are listed in Table 2, where Dg is the waveguide diameter, Lg is the waveguide length, Df is the final diameter of the Gaussian profile, and Lf is the profile length. Figure 11 shows the radiation patterns of the flat-wall Gaussian horn at 94 GHz in two orthogonal planes.
Then, we designed a Gaussian profile horn with a corrugated wall and reduced side lobes. Side lobes allow receiving radiation from unwanted directions. Moreover, the symmetry of patterns means less aberrations, so more focused energy with lower reflection can be transferred from the aperture to the detector. The horn is well designed to match the primary optics.

Designing Corrugated Gaussian Feeding Horn
There are quite a number of parameters to define the design of corrugated horns; the main ones are choice of input radius, output radius, and length of the horn and corrugated-surface profiles. Aside from these quantities, there are several other parameters to consider such the center frequency and the lowest and highest operating frequencies. 47 The structure of this horn was designed according to the instructions in Ref. 47, and the obtained parameters were optimized in FEKO. The waveguide must have a certain minimum cross section, relative to the central wavelength to be applicable. If the wavelength compared to the cross section of the waveguide is too long (frequency is too low), the electromagnetic fields cannot propagate in it. The lowest frequency at which a waveguide will function is where the cross section is large enough to fit one complete central wavelength. In this design, the input radius (the waveguide radius) is 1.275 mm. It is a standard waveguide in 94-GHz central frequency. To calculate the output radius, we must typically consider that a corrugated horn is designed to have a taper of between −12 to −18 dB at an angle corresponding to the edge of the first optics as mentioned in Sec. 2.3. Also, it is related to center frequency and input radius. 47 First, we estimate the output radius for a −13-dB taper at an illumination angle of the first optics with diameter of 65 cm. It was about three times of the central wavelength.
Also, the length of the horn is set by the application, but about 5λ to 10λ is usually required. It will influence on the side lobes. Although some applications may need a horn 20λ to 30λ long. 47 In this design, the length was optimized to be as compact as possible to reduce cost and weight and was considered about 7λ. After estimation of all of these parameters, they were entered and optimized in FEKO software and are precisely stated in the following sections.
This horn comprises a circular waveguide with a transition from a flat linear section to a corrugated linear section, which is used as a mode converter (Fig. 12). The depth of the grooves in the conical convertor section starts with an initial value of approximately half the wavelength at the maximum frequency and ends with one quarter of the wavelength at the central frequency.
The mode converter is a phasing section that feeds the Gaussian profile section. The depths of all slots after the converter are equal. These sections are described below.

Mode converter
The Gaussian horn antenna has to be fed by a purely HE 11 mode. Therefore, in the horn neck region, an impedance converter was used to accommodate flat single-mode waveguide TE 11 with the corrugated waveguide. 48 This mode converter usually starts with a single-mode circular flat propagator waveguide TE 11 and terminates in the opening diameter required for the phasing section, which feeds the Gaussian profile of the horn antenna. The parameters proposed for the corrugated conical horn (Fig. 13) are listed in Table 3. In Table 3, 2a is the waveguide diameter, Lg is the waveguide length, and Lt is the transition region length; Sc1 and ScN are the depths of the first and last grooves in the converter, respectively; Ws is the groove width, Wr is the ridge width, Lc is the axial length in the converter section, and Nc is the number of grooves in the converter section.

Phasing section
Due to the compression of the corrugated horn antenna in the first section, the directivity is low and the phase centers on Eand H-planes do not match. Therefore, a phasing section is used to overlap the phase centers in the desired frequency band. This section also improves the combination of TE 11 and TM 11 modes. The phasing section is shown in Fig. 14 and its design parameters are given in Table 4.    In Table 4, Dc is the phasing section diameter, Sp is the groove depth in the phasing section, Lp is the length of the phasing section, and Np is the number of grooves in the phasing section; Ws and Wr are the groove and ridge widths, respectively.

Gaussian section parameters
The values of the design parameters in the Gaussian section (Fig. 15) are given in Table 5.
In Table 5, Dgp is the final diameter of the Gaussian profile, Sgp is the Gaussian profile depth, Lgp is the Gaussian profile length, and Ngp is the number of grooves in the Gaussian profile section; Ws and Wr are the groove and ridge widths, respectively.

Simulation results of Gaussian corrugated horn
The most important parameters in our design are the symmetry of the radiation pattern and low side lobe level. Figure 16 shows the 3D radiation pattern of the designed horn, which has a gain of about 20 dBi. Moreover, the designed horn has a very symmetrical pattern in both orthogonal planes and the side lobes are reduced to a large extent compared to other conventional horns in this wavelength range. Figure 17 shows the radiation pattern of the corrugated horn at the desired design frequency in Eand H-planes. At the optimal frequency band, the side lobes levels are below −38.5 dB, which is a very good result.
The pattern symmetry is very good and the horn −10-dB beamwidth is about 40 deg, which makes it appropriate for quasi-optical transition systems.

Conclusion
In this study, we designed and simulated a millimeter wave imaging system, which included a lens and a feedhorn. With a lens aperture diameter of 65 cm, the system was capable of imaging an object with a resolution of 30 mm in a 5-m imaging range at a central frequency of 94 GHz. The aspherical lens, which was designed for the millimeter wave camera, yielded a high-quality system with well-corrected aberrations. The image quality curves also showed  the high quality of the designed system. In addition, to reduce the weight and volume of the imaging system, a Fresnel lens was designed and simulated in this wavelength band. An optimized corrugated Gaussian horn was designed with a gain of about 20 dB and a 10-dB beamwidth of 40 deg in Eand H-planes to obtain proper axially symmetric patterns. According to the analytical and simulation results, it was concluded that the combination of a Fresnel lens and Gaussian corrugated horn is very advantageous and this complete quasi-optical system can be used in high-resolution millimeter wave imaging to detect hidden objects in security imaging applications.