### I. Introduction

Reflector antennas are increasingly being used in small- and medium-sized satellites because they are lightweight and operate over a broad bandwidth. Moreover, deployable reflector antennas can increase the storage efficiency of satellites in launch vehicles. Compared to a mesh reflector antenna, an antenna composed of solid material has higher surface accuracy and possesses the advantage of low cross-polarization (cross-pol) [1, 2]. Recently, the cross-pol characteristics of the antenna have been gaining significance in SAR systems, which can be operated in a polarimetric mode [3, 4]. This is because the cross-polar level in an antenna pattern induces cross-talk in polarimetric SAR images [3].

In a perfect parabolic reflector, cross-polarization is generated by an unbalanced feeder whose feed pattern is rotationally asymmetrical [5]. However, the surface error of the reflector may also generate cross-polarized radiation, even in the case of a balanced feeder. In solid-surface deployable antennas, panels are folded during launch and unfolded during deployment in space [6]. If the antenna is imperfectly deployed in space, the panels are misaligned from the ideal shape [7]. Furthermore, in the case of incomplete deployment, the panels are uniformly misaligned. The panels may also misalign non-uniformly, with the error changing smoothly in the angular direction due to malfunctions in the deployment mechanism. In view of these circumstances, it would be interesting to explore the effects of misalignment error on the radiation pattern of a deployable antenna.

The surface error in a reflector antenna leads to a phase error in the aperture plane due to the path-length difference [8]. For panel misalignment, degradations in co-polarization characteristics are calculated using the aperture phase error (APE) method, which calculates the radiated field from the aperture field by accounting for the phase errors caused by the surface error of the reflector [9]. When the panels are uniformly misaligned, the gain decreases, while the side lobe level also increases with an increase in the error. Apart from this, for an asymmetrical distribution of panel misalignment errors, tilting of the main beam is observed.

Cross-pol in far-field radiation is caused by the cross-pol component in the aperture [5]. However, the APE method does not provide sufficient information when it comes to analyzing cross-pol performances because the cross-polar component is not included in the calculation of the aperture field [8, 9]. When panels are misaligned, the direction of the normal vector on the surface rotates in terms of the misalignment angle, in addition to the path-length difference. In the aperture plane, disturbance of the polarization purity arises from the rotation of the normal vector, as well as from the phase difference. Notably, the rotation of the normal vector is a more important factor in cross-pol generation than the phase difference. Therefore, by investigating the rotational effect of the normal vector on the polarization characteristics in the aperture plane, it is possible to estimate the cross-pol performance of radiation in the far-field region. However, although it is convenient to calculate cross-pol radiation using commercial tools based on numerical methods [10], it is difficult to identify the causes of cross-pol generation resulting from panel misalignments. Therefore, to mitigate cross-pol generation from reflector antennas, a physical understanding of the causes leading to cross-pol occurrence in an aperture is required.

In this paper, the effects of panel misalignment errors on the cross-pol performance of a solid surface deployable antenna are analyzed. To separate the investigation of the influence of the panel error from that of the feeding unbalance, the incident field from the feeder is assumed to be a symmetrical Gaussian beam. The cross-pol performances are analyzed using the aperture cross-pol field (AXPF) method, in which the cross-pol radiation field is calculated from the aperture cross-pol field, which is obtained by counting the effects of the normal vector’s rotation as well as the phase error caused by panel misalignment. In other words, the AXPF method provides a physical interpretation of the mechanism of cross-pol occurrence in the aperture. For some typical types of error distributions, including uniform and cosine shapes, the generation of the cross-pol was analyzed and compared to each other by investigating the aperture field.

### II. Cross-Pol Characterization using the Aperture Cross-Pol Field Method

Fig. 1(a) presents the cross-sectional configuration of the reflector of the solid-surface deployable antenna. The reflector consists of a central dish and panels. The diameters of the reflector and the central dish are n ^ ' , is rotated to angle n ^ of the ideal surface. In the spherical coordinates (n ^ ' for arbitrary

*D*and*d*, respectively. When the panels are completely deployed, the central dish and the panels are assumed to be ideally parabolic with no errors. The feeder is located at the focal point, which is the origin of the coordinates. If the panels are incompletely deployed, the misaligned panel, denoted by a dotted line in Fig. 1(a), is rotated to the misalignment angle*α*from the ideally aligned panel [9]. The misalignment angle*α*refers to the angle error of an incompletely deployed panel, which is rotated at the rim of the central dish from the ideal shape, as illustrated in Fig. 1(a). The panel misalignment accompanies the path-length difference from the feeder to the aperture plane, as well as the rotation of the normal vector on the reflector surface. In the AXPF method, the aperture field is calculated by counting the effects of the rotation of the normal vector and the phase error resulting from the path length difference. The unit normal vector of the misaligned panel, denoted by*α*from the normal vector*r**,*_{f}*θ**,*_{f}*φ**) of the corresponding rectangular coordinates (*_{f}*x**,*_{f}*y**,*_{f}*z**), the*_{f}*φ**can be expressed, as depicted in Fig. 1(b), as:*_{f}To investigate the effects of panel misalignment on cross-pol generation separately from the feeder unbalance, the feeder is assumed to have a balanced pattern in which the E-plane and H-plane patterns are equal. For a balanced feeder with an

*x-*polarized radiation pattern, the incident wave onto the reflector is expressed as [5]:where n ^ ' · n ^ ' –n ^ ' . For a misaligned panel with a misalignment angle

*C*(*θ**) is the pattern in the E- and H-planes. Furthermore, the reflected wave from the misaligned panel is expressed as*_{f}**E***=2(*_{r}**E***)*_{i}**E***[5], using the rotated unit normal vector*_{i}*α*, the aperture field is given by considering the phase error and the rotated normal vector, which is expressed from the derivation in Appendix A as follows:where

Here,

*δ**indicates the phase error calculated from the path length error at the*_{m}*m*-th panel [8, 9]. In the aperture field expressed in (3),*A**and*_{CO}*A**are the*_{CX}*x*- and*y*-direction components that contribute to the generation of co-polarized and cross-polarized radiation fields, respectively. Meanwhile,*A**refers to the*_{Z}*z*-direction component of the aperture field, which does not contribute to the radiation field. The rotation of the normal vector by angle*α*and phase error*δ**result from the panel misalignment. Moreover, since*_{m}*α*is small positive when the panel is incompletely deployed,*A**exhibits almost unity although*_{CO}*A**is not negligible for the small*_{CX}*α*. While the copol radiation degrades mainly due to the phase error*δ**[9], cross-pol generation is seriously affected by the rotation of the normal vector.*_{m}From (4), it is understood that the cross-pol does not occur in the aperture field if the panels are completely deployed (n ^ ' =n ^ ). For the misaligned panels, the unit normal vector n ^ ' can be decomposed into the normal component in the n ^ ' on the surface of the misaligned panel can therefore be expressed using the unit vectors n ^ and t ^ as:

*α*=0 and*n-*direction and the tangential component in the*t*-direction on the surface of ideally aligned panels, as shown in Fig. 2. The unit normal vectorwhere n ^ ' , respectively. Furthermore, in terms of the geometry in Fig. 1(a), the unit vectors n ^ and t ^ may also be expressed as:

*P**=cos*_{n}*α*and*P**=sin*_{t}*α*are the normal and tangential components ofUsing (2) as an incident n ^ in (5), we can derive the cross-pol component of the aperture field as:

*x*-polarized field from the feeder andFrom (7), it is clear that in the rotated unit normal vector n ^ ' due to the misalignment, the cross-pol field at the aperture is generated from the tangential component

*P**, while the normal component*_{t}*P**is not related to the generation of any cross-pol field. Therefore, to investigate the generation of a cross-pol field, it is necessary to first observe the rotation of the unit normal vector on the surface of the panel that is caused by the misalignment.*_{n}### III. Effects of Panel Misalignment on Cross-Pol Radiation

The generation of cross-pol fields due to panel misalignments in the deployable antenna utilized in this research is investigated by analyzing the aperture fields. The diameters of the antenna and the central dish are 1.5 m (

*D*) and 0.2 m (*d*), respectively. Additionally, the operating frequency is fixed at 9.6 GHz, and the focal length is 0.45 m (*F*). The number of panels is 30. Furthermore, x-polarized balanced feeding is assumed. Additionally,*C*(*θ**) is defined using a Gaussian function when the edge taper is −12 dB.*_{f}### 1. Uniform Panel Misalignment

This section details the analysis of cross-pol characteristics when the misalignment angle (P t 2 is considered to be a constant for uniform misalignment, with

*α*) is equal for all panels. Since the radiation field is a Fourier integration of the aperture field in (3), the generation of the cross-pol field can be estimated by investigating the behavior of*A**. It has been confirmed that the effect of*_{CX}*δ**on cross-pol generation is negligible for small*_{m}*α*. Drawing on (7),*A**changing along with sin2*_{CX}*φ**. Fig. 3 shows the distribution of*_{f}*A**in the aperture resulting from normal vector rotation when*_{CX}*α*is 1°. It is observed that*A**is negative in the 1st and 3rd quadrants, but positive in the 2nd and 4th quadrants. In addition,*_{CX}*A**is noted to be point symmetric about the center. In addition, the net amount of the cross-pol component in the aperture is found to be zero, since*_{CX}*A**changes two cycles of the sine function in the*_{CX}*φ**-direction.*_{f}The pattern factor of the radiated field in the far-field region can be obtained from the two-dimensional Fourier integral of (3). This integration is performed numerically by dividing the aperture region into trapezoidal micro-regions using a Gaussian quadrature. In radiated electromagnetic fields, the cross-pol field defined by Ludwig [11] is generated from the

*y-*component of the aperture field. Therefore, the distribution of*A**, as shown in Fig. 3, is an important feature in terms of dealing with the cross-pol field.*_{CX}With regard to the E-plane (φ = π 2 or
3 π 2 ), Fig. 3 indicates that the cross-pol field vanishes, since distances from the positive φ = π 4 plane, the radiated cross-pol field exists because

*φ*=0 or*π*) and H-plane (*A**and negative*_{CX}*A**to the observation point are the same, with the corresponding radiated fields canceling each other. In the*_{CX}*A**at the same distances from the observation point are equal.*_{CX}
Fig. 4 depicts the cross-pol patterns in the
φ = π 4 plane for

*α*=1°, 2°, and 3°. It is clear that cross-pol radiation increases along with an increase in the misalignment angle. However, for uniform misalignment error, the peak-to-cross polarization (XPOL) is smaller than −50 dB for a misalignment angle of up to 3°. Moreover, since the distribution of*A**in the aperture is anti-symmetric about the x- and y-axes and the net amount is zero, the cross-pol radiation is very small. Notably, for co-pol radiation, the gain and pattern degrade significantly due to uniform panel misalignment [9].*_{CX}### 2. Non-uniform Panel Misalignments

It would be interesting to investigate the generation of the cross-pol field in cases where the distribution of the panel misalignment error is non-uniform. By investigating aperture cross-pol fields, it is possible to estimate the levels of cross-pol radiation for non-uniform panel misalignments. As a typical model of the non-uniform distribution of misalignment errors, we chose sinusoidal functions. Since misalignment angles resulting from incomplete deployment always have a positive value, we assume the misalignment angle of the

*k*-th panel to be:where

*k*indicates the sequential number of the panel, which increases from the panel at*φ*=0. Meanwhile,*φ**denotes the angle in the*_{k}*φ*-direction at the middle of the*k*-th panel,*α**indicates the maximum error, and*_{m}*φ*_{α}_{m}refers to the angle exhibiting the maximum error. Inside the panel, the misalignment angle is assumed to be uniform.#### 2.1. 1-Cycle Non-uniform Misalignments

For π 4 . Here, φ α m = π 4 .

*N*= 1,*α**varies one cycle for the change of*_{k}*φ**from 0 to 2*_{k}*π*. Fig. 5 depicts the variations in*α**for the panels when*_{k}*φ*_{α}_{m}=0 and*α**is 2° so that the average value is 1°. When*_{m}*φ*_{α}_{m}=0, the misalignment error is maximum at the 1st panel (*φ**=0) and minimum at the 16th panel (*_{k}*φ**=*_{k}*π*). Furthermore, as shown in Fig. 5, the configuration of the error distribution is rotated by*φ*_{α}_{m}in the*φ*-direction forFrom (7), we note that P t 2 = sin 2 α k , with φ α m = π 4 , where the positive quantity in the 2nd quadrant is greater than the negative quantities in the 1st and 3rd quadrants. Fig. 6(c) illustrates the π 4 in the φ = π 4 plane, where it is confirmed that the cross-pol level is higher for
φ α m = π 4 than for 0.

*A**is positive in the 1st and 3rd quadrants, while it is negative in the 2nd and 4th quadrants, regardless of whether the misalignments are uniform or not. Fig. 6(a) shows the*_{CX}*A**distribution for 1-cycle misalignment with*_{CX}*φ*_{α}_{m}=0. Since*α**’s in Fig. 5 for*_{k}*φ*_{α}_{m}=0, the*A**about the*_{CX}*x*-axis is anti-symmetric. Furthermore, since the net amount of*A**in the aperture vanishes, the cross-pol radiation is expected to be small. When*_{CX}*φ*_{α}_{m}changes from zero, the anti-symmetry of the distribution is not sustained, while the net amount of*A**does not vanish. In other words, cross-pol radiation increases when*_{CX}*φ*_{α}_{m}changes its value from zero. Fig. 6(b) presents the*A**distribution for*_{CX}*A**variations for*_{CX}*φ*_{α}_{m}=0 and*φ*-direction for comparison. Meanwhile, Fig. 7 shows the cross-pol patterns in theSince the level of cross-pol radiation depends on the net amount of

*A**in the aperture, the average aperture cross-pol for a balanced feeder pattern may be expressed as:*_{CX}Here,

*F**refers to the integration of the cross-pol component divided by an incident field from the feeder. From*_{A}*F**, we can easily estimate the level of cross-pol radiation for various configurations of the misalignment error. Since*_{A}*F**= 0 for the uniform misalignment error, the cross-pol level is very low. For a 1-cycle misalignment error with a cosine variation,*_{A}*F**= 0 when*_{A}*φ*_{α}_{m}=0, although*F**≠ 0 and the cross-pol level increases when*_{A}*φ*_{α}_{m}≠0.Moreover, for 1-cycle misalignments, a closed-form expression of

*F**can be obtained from*_{A}*α*in (8). A smooth change in the misalignment angle is assumed and*φ**is used instead of*_{f}*φ**. By implementing Taylor’s series,*_{k}*P*_{t}^{2}is expanded into two terms, expressed as follows:Substituting (10) in (9),

*F**for a 1-cycle misalignment error with a maximum misalignment angle of*_{A}*α**at*_{m}*φ*_{α}_{m}is expressed as:
Fig. 8(a) shows the calculated φ α m = π 4 ( 2 n - 1 ) , φ α m = n π 4 ,

*F**with respect to the change in*_{A}*φ*_{α}_{m}when*α**is 2° and 4°. It is observed that*_{m}*F**changes along with*_{A}*φ*_{α}_{m}, reaching its maximum value when*n*=1, 2, 3, 4 and minimum when*n*=1, 2, 3, 4. Fig. 8(b) shows the change in the XPOL of the far-field radiation. When*α**is 2°, the cross-pol level in the far-field radiation is low, while the change in*_{m}*φ*_{α}_{m}is small. However, when*α**is 4°, it is clearly observable that*_{m}*φ*_{α}_{m}'s for maximum and minimum of XPOL coincide with those of*F**. Effectively, the analysis of the cross-pol component in the aperture indicates that it is possible to minimize the level of cross-pol radiation by adjusting the angular position of maximum misalignment in a 1-cycle non-uniform error.*_{A}
Fig. 9 depicts the cross-pol patterns in the
φ = π 4 plane for

*α**= 2°, 4°, and 6°. The XPOL is −68 dB when*_{m}*α**is 2°, while it increased to −45 dB when*_{m}*α**is increased to 6°. Notably, the XPOL increased by more than about 7 dB compared to the uniform misalignment error. This indicates that the cross-pol radiation for the 1-cycle misalignment error is higher since the average of the cross-pol components in the aperture does not vanish.*_{m}In Fig. 10, cross-pol patterns calculated by the AXPF method are compared with the results of physical optics approximation using the commercial GRASP program for one-cycle non-uniform errors. In the case of the unbalanced Gaussian feeder, in which the ratio of the standard deviations of

*C**and*_{H}*C**is*_{E}*v*, the cross-pol patterns of the two methods show good agreement. The agreement between the results from the two methods in the unbalanced case implies that our analysis is valid for calculating cross-pol. In the balanced case, however, the level of cross-pol radiation is very low compared to that in the unbalanced case. Moreover, the patterns do not exhibit good agreement. The results of deploying the AXPF method clearly indicate tilting of the radiation field due to non-uniform panel misalignment. Apart from this, the AXPF analysis also provides adequate results with regard to the level of cross-pol radiation resulting from panel misalignments. Additionally, it demonstrates the physical implications for the generation of cross-pol components from panel misalignments.#### 2.2. 2-Cycle non-uniform misalignments

For φ α m = π 4 , the error distribution is rotated, as shown in Fig. 11.

*N*= 2,*α**varies across two cycles with regard to the change in*_{k}*φ**from 0 to 2*_{k}*π*, as shown in Fig. 11. When*φ*_{α}_{m}= 0, the misalignment error reaches its maximum value at the 1st and 16th panels, and a minimum value at the 8th and 23rd panels. The configuration of the misalignment error is symmetric about the x- and y-axes. Meanwhile, the radiation patterns are symmetric, but vary in tandem with different angles of the observation planes. ForThe distribution of φ α m = π 4 , the negative quantities in the 1st and 3rd quadrants prevail over the positive ones in the 2nd and 4th quadrants, as shown in Fig. 12(b). The noticeable net amount of cross-pol in the aperture is the cause of the high levels of cross-pol radiation for
φ α m = π 4 . Fig. 12(c) illustrates the change in φ α m = π 4 .

*A**in the aperture on the occurrence of a 2-cycle non-uniform misalignment is elaborated in Fig. 12(a) and 12(b), where*_{CX}*α**is 2°. These distributions are observed to be point symmetric about the center. For*_{m}*φ*_{α}_{m}=0, the net amount of*A**is zero, while the cross-pol radiation is very small. However, for*_{CX}*A**in the*_{CX}*φ**-direction. It is clear that the average of*_{f}*A**is zero for*_{CX}*φ*_{α}_{m}=0, but it does not completely vanish in the case ofFor the 2-cycle panel misalignment error (

*N*= 2), we obtained the expression of the factor of the average cross-pol in the aperture*F**using (8) and (10).*_{A}
Fig. 13(a) illustrates changes in φ α m = π 4 ( 2 n - 1 ) ,

*F**with*_{A}*φ*_{α}_{m}for*N*= 2, with*F**reaching its maximum value when*_{A}*n*= 1, 2, 3, 4. The variation in*F**shows a similar feature as that in the case of*_{A}*N*= 1, although the magnitude of*F**is larger for*_{A}*N*= 2. The change in XPOL in the far field is shown in Fig. 13(b), which confirms that the*φ*_{α}_{m}generating the maximum and minimum XPOLs coincides with the generation of the maximum and minimum*F**in the aperture.*_{A}
Fig. 14 shows the cross-pol patterns in the
φ = π 4 plane for

*α**= 2°, 4°, and 6°. When a 2-cycle misalignment error occurs, the cross-pol radiation pattern splits into two since these patterns are symmetric, and the peak value is lowered. Compared to the 1-cycle error,*_{m}*F**for*_{A}*N*= 2 is larger, but XPOL is lower by about 2 dB.### IV. Conclusion

This study first assumed the shape of the panel misalignment error that can occur in a deployable reflector antenna and then analyzed the generation mechanism of the cross-pol radiation caused by the panel misalignment error. The aperture field was calculated by considering the phase error of the signal and the rotation of the normal vector on the reflective surface resulting from the misalignment of the panel. The rotation of the normal vector caused by misalignment is an essential factor in the analysis of cross-pol performance. Moreover, the net amount of the cross-pol component in the aperture was found to be closely related to the level of cross-pol radiation. For non-uniform misalignment, the cross-pol radiation was higher, and the maximum cross-pol level was generated when
φ α m = π 4 ( 2 n - 1 ) ,

*n*= 1, 2,3, …. It was also noted that the XPOL for the 2-cycle misalignment is higher by a maximum of 7 dB than that in the uniform misalignment, with*α**being 3°. Overall, the present method of analysis provides a design rule for deployable antennas, along with instructions for reducing the level of cross-pol radiation by adjusting the features of panel misalignment.*_{m}