1. Introduction
With the advantage of eliminating the blockage of feed antennas, offset reflector antennas, such as ellipsoidal and paraboloidal mirrors, are widely used to control the characteristics of beam propagation in free space. Furthermore, the effects of off-axis mirrors on the beam shape and the cross-polarization level were expounded exhaustively [
1]. In the millimeter and submillimeter wavebands, the traditional analysis method, physical optics (PO), becomes more and more inefficient with increasing frequency. In contrast, the theory of Gaussian beam propagation with the inherence of paraxial approximation [
2] is suitable for analyzing electrically large objects. As the application scenarios become complicated, 2-D quasi-optical systems [
3] can no longer meet the requirements, and 3-D systems [
4,
5,
6] receive more and more attention. Compared with 2-D systems, 3-D systems are more flexible and compact, which is more beneficial to being mounted on a spacecraft [
5,
6]. The feeds, reflectors and other components in the 3-D system are not limited to one plane, which is more convenient to set channels with multi-frequency and multi-polarization. This is very beneficial to reduce the space occupied by the system. However, the analysis and design process of 3-D systems is far more complicated than that of 2-D systems. In diffracted Gaussian beam analysis (DGBA), the electric field is expanded into a set of fundamental Gaussian beams with the Gabor expansion. Furthermore, the beam tracing method is used to evaluate the reflected and diffracted fields of each beam from the mirror [
7]. There is a novel Gaussian beam analysis method, in which the field is decomposed into fundamental mode beams by the point matching approach [
8,
9]. In [
10], the Gaussian beam analysis method based on the fundamental mode was generalized to handle 3-D systems based on a three-dimensional diffraction technique. In addition, Gaussian beam mode analysis (GBMA) is another powerful analysis approach based on the multimode Gaussian beam [
2]. Different from Gaussian beam analysis methods mentioned above, the electric field is expanded into the fundamental mode and the higher order mode along the propagation direction due to the orthogonality relationship between Gaussian beam modes. GBMA is applied to assess not only reflector antennas [
11,
12,
13,
14] but also other components, for example, the horn antennas [
15,
16,
17], the phase gratings [
18], and so on. A GBMA method for 2-D multi-reflector systems was expatiated in [
19]. In these works, the beam distortion and the polarization level can be predicted precisely by GBMA compared with PO in the commercial software GRASP10 [
20]. On this basis, some brief explanations were presented in [
21] about the fact that the cross-polarization generated by the preceding mirror can be decreased by adjusting the parameters of the subsequent reflector. Furthermore, a novel design method of 2-D quasi-optical system with low cross-polarization based on the Particle Swarm Optimization (PSO) algorithm was proposed. In the novel design method, the system parameters, such as the distance between two adjacent objects, the incident angle, and so on, can be optimized to control the power of different Gaussian beam modes to reduce the cross-polarization level [
22]. The PSO algorithm is also applied to other antenna designs, for example, the dielectric phase-correcting structure [
23] and the time-delay equalizer metasurface [
24] for the electromagnetic band-gap (EBG) resonator antenna. However, temporarily, GBMA can only be used to analyze and design two-dimensional systems, which cannot meet the more general demands of 3-D system applications.
In this study, we will make further efforts to improve GBMA to deal with off-axis mirrors in 3-D quasi-optical systems. The whole process of algebraic operations and some significant expressions will be presented. The correctness and accuracy of 3-D GBMA will be verified by comparing with PO in GRASP10, taking 3-D dual-reflector systems as examples. Moreover, we will fabricate a 3-D quasi-optical system and give the measured results. Furthermore, the calculation time of 3-D GBMA will be compared with that of PO to illustrate the high efficiency of 3-D GBMA.
2. Proposed System Design
The analysis of a single off-axis reflector antenna and the transformation of co- and cross-polarization in a 2-D quasi-optical system were discussed in detail [
19]. In brief, we choose an individual Hermite–Gaussian mode, for example, a
Gaussian beam mode as the incident field to an offset mirror, the cross-polarization will emerge, and the co-polarization will be distorted as a result of the generated other order modes. The significant expressions of the input and output fields are given below, and the geometry of an off-axis ellipsoidal mirror with the incident and reflected beams in the local coordinate systems is plotted in
Figure 1.
Here,
is the co-polarization of the incident field linearly polarized in the
direction, and
and
are the co- and cross-polarization of the reflected field, respectively.
represents the amplitude of
order Hermite–Gaussian mode. The exponential term in Equation (1) is the phase of
order Hermite–Gaussian mode. In Equations (2) and (3),
and
are coefficients of co-polar modes and cx-polar modes, respectively. The last two exponential terms,
and
, are additional phases to ensure the phase matching and phase reversal of the incident and reflected beams on the mirror according to the reflection theorem. Furthermore, the subscript
means that the reflected
mode is generated by the incident
mode. The expressions of
,
, and
were given in [
19].
is the wavenumber.
is the distance between the beam waist
and the center of the mirror.
and
are the radius of curvature and the phase slippage, respectively, which are functions of
. Furthermore, the relationship between parameters of the input and output Gaussian beams can be figured out by the ABCD matrix techniques [
25,
26]. From Equations (2) and (3), we can see that the reflected co-polarization will no longer be a single
mode, which consists of a set of
modes close to the
mode. Additionally, the cross-polarization is composed of a set of
modes.
Different from 2-D quasi-optical systems in [
19], the optical axes of beams are not on the same plane in 3-D systems. Referring to the analysis of a single offset reflector, we set up two coordinate systems of the output beam from the first mirror
and the input beam to the second mirror
, as shown in
Figure 2. It can be seen from the figure that there is a rotation angle
between two planes of optical axes of
and
. Obviously, it becomes a 2-D system when
equals
or
. In 3-D systems, some appropriate mode conversions and polarization conversions are essential to ensure that the reflected beams from
can be used as the incident beams to
. Then, GBMA could be applied to analyze 3-D multi-reflector systems.
Figure 3 presents the view of
and
coordinate systems (shown in
Figure 2) from the
-axis direction. From
Figure 2 and
Figure 3, two coordinates are connected by:
where
and
are the distances from the beam waist
to
and to
, respectively. Therefore,
is the distance between the two mirrors. Apparently, the modes of reflected waves from
are in the
coordinate system. These modes must be converted into a new set of modes in the
system to fit
. Now, substitute Equation (4) into Equation (2):
In the process of replacing
with
, the functions
and
in Equation (2) can be written as
and
, and then we can obtain Equation (5). The expression of cross-polarization is similar to this and not given here. The amplitude
of the
Gaussian beam mode needs to be further discussed, which is a function of
,
, and
.
where
represents the beam radius and
is the
order Hermite polynomial.
where
is the beam waist radius of the output wave from
, which is also the beam waist radius of the input wave to
. Then, we obtain
in the
coordinate system.
Combining Equation (4), Equation (6), Equation (7), and Equation (8), we could obtain the formula of
(
).
Then, according to the expressions of Hermite polynomials, we could transform the amplitude
of a single
mode into the amplitude sum of a new set of modes in the
system. Here, we give three simple results of the mode conversion. By the way, these three modes are generated by one reflection of the fundamental mode [
11].
By substituting Equation (10) into Equation (5), the formula for the co-polarization in the
x3y3z3 coordinate system will be obtained. We need to pay attention to the fact that the phase
in Equation (5) is the phase of the
mode. However, there is not only the
mode but also other modes after the mode conversion. Therefore, to achieve modular computing, we apply the same way in Equation (10) in [
11] to approximate the phase
so that it can be paired with modes other than the (
m,
n) mode. Only when the sum of
x and
directional orders of the new modes is not equal to that of the original mode, that is,
, the phase
needs to be replaced by the phase
.
Since the converted modes and the original modes are essentially the same fields, the condition of phase matching between the new mode
and the mode
still needs to be met on
. Therefore, the additional phase term in Equation (5),
, needs to be replaced by
, when
is not equal to
. Finally, a new set of modes and coefficients in the
coordinate system will be obtained. After the mode conversion, the co-polarization can be written in the form of a set of
modes:
Here, is the coefficient of the mode, which is a function of the mode order and the rotation angle , and can be obtained from the above analysis. For example, the coefficients of new modes converted from , , and modes can be obtained from Equation (10). is the amplitude term of the mode. and are additional phase terms.
Up to now, the mode conversion was fully described. In the following part, we will give details about the polarization conversion. According to the previous analyses, the co-polarization
and the cross-polarization
reflected from
are linearly polarized in the
and
directions, respectively. Similarly, we should convert these two orthogonal polarizations into linear polarizations in the
system after the mode conversion. First, the original co- and cross-polarization need to be decomposed into two components in the
and
directions.
can be decomposed into
and
, which are linearly polarized in the
and
directions, respectively. Furthermore,
can be decomposed into
and
, which are linearly polarized in the
and
directions, respectively. Second, the decomposed co- and cross-polarization in the same direction, for example, the
or
direction, need to be accumulated to obtain two new polarizations linearly polarized in the
and
directions. Whether these components are added or subtracted depends on the rotation angle
. We have modelled and simulated various systems, including the systems with two, three or more reflectors, the feeds with or without cross-polarization, the feeds containing one, two or more Gaussian beam modes, and so on. Through the calculation and analysis of these systems, we finally obtained the following equation, which can correctly calculate the converted co- and cross-polarization.
Here, the subscripts and mean the co-polarization of the incident beam to and the cross-polarization of the reflected beam from .
Ultimately, we will obtain the new co- and cross-polarization in the coordinate system by substituting and into Equation (12). can be calculated by Equation (11) and can be obtained in a similar way. It should be noted that and are the fields after the mode conversion. Then, and can be used as the incident fields to the next mirror .
Finally, 3-D multi-reflector quasi-optical systems can be analyzed by 3-D GBMA. With 3-D GBMA, the proportion of different Gaussian beam modes in the system can be calculated accurately, which can be used to analyze the performance of the system. At the same time, the amplitude and phase field distribution at any position in the system can be evaluated quickly, which directly shows the electrical properties. Therefore, 3-D GBMA can be considered to be very useful for system analysis and design.
3. Results and Discussion
First, the rationality of the approximate calculation of the phase mentioned above will be analyzed. There are three modes,
,
, and
, reflected from
with the fundamental Gaussian beam mode incident to
. These three modes need to be converted to match the next mirror
. According to the previous analysis of the mode conversion and Equation (10), only the mode
will be converted to other modes that meet the condition
. In
Figure 4, we show the curves of the coefficients of the new modes, varying with
in the example of
mode. The coefficients of new modes can be calculated by Equation (10). We can find that coefficients of other modes remain low through the process of
varying, except
and
modes, which means that the energy of other modes is kept at a low level, and its influence on
and
modes is small. In addition, coefficients of
and
modes that need to be approximated for phase remain at a low level, so the impact of the approximation is relatively low. In addition, the energy of
and
modes reaches a high level with
approaching
,
, etc., which means that the effect of approximation is great in these cases.
The further accuracy analysis of 3-D GBMA will be given below, verified by the simulated and measured results of 3-D double ellipsoidal reflector systems. To make the whole process easier to understand, we select a fundamental mode Gaussian beam feed without cross-polarization and two identical mirrors to compose the systems with different
. The details of the feed and reflector are given in
Table 1. It is worth mentioning that the proposed 3-D GBMA is programmed in MATLAB. The commercial electromagnetic simulation software GRASP based on PO is used as a contrast. We can model and calculate different systems with the parameters of the feed and reflector in
Table 1 and different rotation angles
. Then, we present near-field amplitude and phase distributions on the plane of the beam waist
(shown in
Figure 2) in 3-D systems with different
in
Figure 5 and
Figure 6.
These amplitude results of 3-D GBMA are in good agreement with those of PO within the range of twice the beam radius. The polarization direction of the feed is chosen to define the co- and cross-polarization direction of the system. The mutual conversion between the co- and cross-polarization is caused by the change of
. It is not hard to see that cross-polarization is higher than co-polarization with
. The maximum difference between the two results is less than 2.6 dB, which occurs on the edge of the sampling plane with the rotation angle
equal to
. Here, the results of two special 3-D systems with
and
are no longer given, which have been presented in [
19]. As
increases from 0 to
, the difference is increasing. Then, the difference drops until
equals
. From the phase results, we can obtain the same conclusions. The maximum difference is less than 24.7° on the edge of the sampling plane with
. The reason is that the approximate calculation of phase is used in the mode conversion.
Following, we present detailed information about the fabrication and measurement of a system with
to verify the accuracy of 3-D GBMA further. The prototypes of the corrugated horn and the fabricated double ellipsoidal reflector systems are illustrated in
Figure 7. The parameters of the corrugated horn and manufactured reflectors are also consistent with those given in
Table 1. The Root Mean Square (RMS) surface tolerance of mirrors is less than 5 microns.
We applied near field scanning to measure the output electric field from this system.
Figure 8 presents the scanning devices, including the probe, scanning frame and absorbing baffle. The scanning plane is chosen at a distance of 200 mm from the beam waist
(shown in
Figure 2). The dimension of the measured plane is 212 mm by 212 mm, which covers a range of two times the beam radius. The number of sampling points is 143*143, and the sampling interval is 1.49 mm, slightly less than
at 183 GHz. Furthermore, the far field results are calculated by performing Fast Fourier Transform (FFT) on the near field data. The far field radiation patterns on E- and H-planes are plotted in
Figure 9, including the simulated results of 3-D GBMA and PO and the measured results. The near field results on the scanning plane are given in
Figure 10 and
Figure 11. It should be noted that the radiation field from the corrugated horn contains not only the fundamental Gaussian beam, but also some higher order modes with very low energy. The purity of the fundamental Gaussian beam mode in the horn feed is about 99.8%, and there is a cross-polarization with a peak amplitude of about −30 dB. In the simulations, two single fundamental Gaussian beam modes with peaks of 0 dB and −30 dB are used to approximate the co- and cross-polarization in the horn, respectively, which caused a small difference between the simulated and measured results. Furthermore, the ripples of measured results are caused by the background noise, the reflection, and diffraction of various metal surfaces. However, we can still see that the differences between these three results are acceptable. Moreover, we can find that the amplitudes and phases simulated by 3-D GBMA and measured are very similar through observing the near field results. Finally, we can conclude that both the amplitude and phase of the co- and cross-polarization are simulated by 3-D GBMA accurately, and 3-D GBMA is correct and precise compared with PO and the measured results. Now, the calculation efficiency of 3-D GBMA will be discussed and compared with PO in GRASP10. We program 3-D GBMA in MATLAB. It should be noted that the calculation speed is not optimized, such as parallel computing. The execution time of 3-D GBMA and PO in the 3-D dual-reflector system with
is given in
Table 2. In the first reflection, 3-D GBMA takes 0.08s to compute the modes, and PO (single-core) takes 0.52 s to obtain the currents on
. Then, the consuming time of the mode and polarization conversions is 0.13 s and 0.06 s, respectively. In the second reflection, 3-D GBMA and PO (single-core) take 0.17 s and 15.92 s. Finally, two methods cost 0.08 s and 3.71 s to calculate the field distributions. It is worth noting that the time of 3-D GBMA in
Table 2 is executed in the single-core, which can be further reduced with the multi-core parallel computing. In addition, the execution time of PO in the multi-core operation is also given as a reference. We can conclude that 3-D GBMA is efficient and accurate compared with PO in GRASP10.