To reduce the degradation of radiation characteristics, both transmission coefficient of dielectrics and inducted current of metal should be cut down. In this study, we mainly paid attention to the effects of transmission coefficient, and tried to reduce the phase difference in transmission aperture.

The phase of transmission coefficient *T* is the key factor that causes the distortion of radiation pattern. In the original aperture, the phase of the wave is a constant, namely the phase difference is 0, and the radiation pattern has no distortion without the radome. In contrast, with the radome, the phase of *T* in the transmission aperture is not a constant, but varies with the incidence angle *γ*. A different *γ* leads to a different IPD, thus induces a phase difference of *T* in the transmission aperture, and finally causes the distortion of radiation pattern.

In antenna-radome systems, radome is an indispensible part and we cannot remove it, but we could change the phase distribution in the original aperture. If phase difference in the original aperture has an inverse trend to that in the transmission aperture. The wave propagates through the radome. Thus, the phase change will turn out to be a constant in the transmission aperture. Finally, an ideal radiation pattern can be obtained.

The reflector antenna is usually composed of a main reflector, a sub-reflector, and a feed. There are three ways to change the original aperture: shaping the main reflector, moving the sub-reflector and moving the feed, as shown in Fig. 3.

### 3.1 Shaping the Main Reflector

A small displacement of one point in the main reflector along the axial direction *d*
_{mz} leads to the change of wave transmission distance from the sub-reflector to the original aperture. Figure 3 shows (the red line indicates the shaped reflector, moved feed or sub-reflector). The phase shift at the corresponding point in the original aperture is obtained as follows by the geometric relations [25]

$$ \eta_{\text{m}} = \frac{2\pi }{\lambda }d_{\text{mz}} \left( {1 + \cos \xi } \right), $$

(4)

where *ξ* is the flare angle from the sub-reflector to the main reflector.

However, it is not practical to shape the main reflector for an installed reflector antenna.

### 3.2 Moving the Feed

Moving the feed along the axial direction *d*
_{fz} (offset focus) leads to the change of transmission distance from the feed to the sub-reflector as shown in Fig. 3. The phase shift in the original aperture is calculated by [25]

$$ \eta_{\text{f}} = - \frac{2\pi }{\lambda }d_{\text{fz}} \cos \xi^{\prime}, $$

(5)

where *ξ*′ is the flare angle from the feed to the sub-reflector. In the center of the reflector, the flare angle *ξ*′ minimizes to 0 and the phase shift is −2π*d*
_{fz}/*λ*. In the edge of the reflector, the flare angle *ξ*′ maximizes to \( \xi^{\prime}_{\hbox{max} } \) and the phase shift is \( \eta_{\text{f}} = - 2\pi d_{\text{fz}} \cos \xi^{\prime}_{\hbox{max} } /\lambda \). Thus, the phase difference in the original aperture is

$$ \Delta \eta_{\text{f}} = - \frac{2\pi }{\lambda }d_{\text{fz}} \left( {1 - \cos \xi^{\prime}_{\hbox{max} } } \right). $$

(6)

### 3.3 Moving the Sub-Reflector

Similarly, the change of the wave transmission distance by moving the sub-reflector along the axial direction *d*
_{sz} includes two parts: one is from the feed to the sub-reflector and the other is from the sub-reflector to the main reflector. Therefore, the phase shift in the aperture is also compensated by two parts,

$$ \eta_{\text{s}} = \frac{2\pi }{\lambda }d_{\text{sz}} \left( {\cos \xi + \cos \xi^{\prime}} \right). $$

(7)

In the center of the reflector, both flare angles *ξ*′ and *ξ* minimize to 0 and the phase shift is 4π*d*
_{sz}/*λ*. In the edge of the reflector, flare angles *ξ*′ and *ξ* maximize to \( \xi^{\prime}_{\hbox{max} } \) and *ξ*
_{max} respectively, and the phase shift is \( \eta_{\text{s}} = 2\pi d_{\text{sz}} \left( {\cos \xi_{\hbox{max} } + \cos \xi^{\prime}_{\hbox{max} } } \right)/\lambda \). Thus, the phase difference in the original aperture is

$$ \Delta \eta_{\text{s}} = \frac{2\pi }{\lambda }d_{\text{sz}} \left( {2 - \cos \xi_{\hbox{max} } - \cos \xi^{\prime}_{\hbox{max} } } \right). $$

(8)

At the same point of the sub-reflector, \( \xi^{\prime}_{\hbox{max} } \) is less than *ξ*, and their relationship is \( M = \tan \left( {\xi_{\hbox{max} } /2} \right)/\tan \left( {\xi^{\prime}_{\hbox{max} } /2} \right) \), where *M* is the amplification factor of the double reflector antenna, tan (*ξ*/2) = *D*/4*f*, *D* is the diameter of the main reflector, and *f* is the focal length.

In the aperture S, from the center to the edge, the two flare angles increase from 0 to \( \xi^{\prime}_{\hbox{max} } \) and *ξ*
_{max}. If *d*
_{fz} is negative, namely the feed moves to the −Z direction, the distribution of phase difference in the original aperture is bigger in the center and smaller at the edge. If *d*
_{sz} is positive, namely the sub-reflector moves to the +*Z* direction, the distribution of the phase difference is similar to that with a negative *d*
_{fz}.

For a hemisphere radome in common use, the maximum *γ*
_{max} locates at the edge of the aperture, and the minimum *γ*
_{min} is 0 in the center of the aperture. Thus, the phase difference in the transmission aperture is

$$ \Delta \eta_{\text{T}} = \eta_{\hbox{max} } - \eta_{\hbox{min} } = \frac{2\pi }{\lambda }d\left( {1 - \cos \gamma_{\hbox{max} } } \right). $$

(9)

The distribution of \( \Delta \eta_{\text{T}} \) is smaller in the center and bigger at the edge, which is just inverse to the distribution of phase difference caused by negative *d*
_{fz} or positive *d*
_{sz}. Hence, the two-phase difference can offset each other and realize the compensation of the distortion of radiation pattern. This means \( \Delta \eta_{\text{T}} = \Delta \eta_{\text{f}} \) or \( \Delta \eta_{\text{T}} = \Delta \eta_{\text{s}} \).

Then, the value of the offset focus of the feed is

$$ d_{\text{fz}} = \frac{{\Delta \eta_{\text{s}} \lambda }}{{2\pi \left( {1 - \cos \xi^{\prime}_{\hbox{max} } } \right)}}. $$

(10)

The value of the offset focus of the sub-reflector is

$$ d_{\text{sz}} = \frac{{\Delta \eta_{\text{T}} \lambda }}{{2\pi \left( {2 - \cos \xi_{\hbox{max} } - \cos \xi^{\prime}_{\hbox{max} } } \right)}}. $$

(11)

From the above analysis, it is known that the offset focus or *T* leads to a phase difference in the transmission aperture and result in the distortion of radiation pattern as well as the degradation of radiation characteristics. However, if they work together, the degradation will be mitigated significantly.

Flare angle *ξ*′ is less than *ξ* at the same point in the sub-reflector. To compensate the same Δ*η*
_{T}, the value of *d*
_{fz} is larger than that of *d*
_{sz}. Thus, the compensation by moving the sub-reflector is more appropriate than by moving the feed in practice. For most reflector antenna in engineering, both the sub-reflector and feed are fixed by bolts, if we adjust the bolts, the position of the sub-reflector or feed could be changed in a small range.

Once *d*
_{sz} is determined, the phase shift *η*
_{a} of each point in the aperture can be obtained by Eq. (9). Then, substitute *η*
_{a} into Eq. (3), and the far field of an antenna with a radome can be expressed by

$$ \varvec{E}(\theta ,\phi ) = \iint_{S} {T\left( {\rho ,\phi } \right)f\left( {\rho ,\phi } \right)} \cdot \exp j\left( {\varphi \left( {\rho ,\phi } \right) + \eta_{\text{a}} } \right)\rho {\text{d}}\rho {\text{d}}\phi .{\kern 1pt} $$

(12)

By numerical integration, the calculation of Eq. (12) is implemented [22].