2.2. Coupling Effect and Compensation Relationship of Net Aberration Fields Induced by Different Perturbation Parameters
According to NAT, when an optical system is perturbed, new aberrations will not be produced but the field dependence of aberrations (such as the linear dependence of the field, and the quadratic dependence of the field) will be changed [
23,
33]. However, different perturbation parameters often induce the same type of net aberration fields, that is, there are coupling effects among different perturbation parameters. They either compensate and offset each other to make the image quality better or accumulate and superimpose each other to make the image quality worse. This section aims to determine the compensation correction strategy for the off-axis TMA telescope described in the previous section by analyzing the coupling effect and compensation relationship between the net aberration fields induced by perturbation parameters. The wave aberration function of perturbed pupil-offset off-axis optical systems with designed tilts and decenters can be expressed as [
11]
where
,
,
is the optical surface number;
represents the aberration coefficient for a particular aberration type;
represents the net change of
induced by axial misalignments of optical surfaces;
denotes the aberration field decenter vector induced by lateral misalignments of optical surfaces;
denotes the aberration field decenter vector introduced in the design process of optical systems. Both
and
can be obtained by ray tracing using the method proposed in [
34].
and
represent the normalized effective field vectors before and after the aberration field center shift, respectively;
and
represent the normalized pupil vectors of the off-axis system and its on-axis parent system, respectively;
represents the normalized position change vector of the pupil center of the off-axis system relative its on-axis parent system;
and
denote the pupil radius of the off-axis system and its on-axis parent system, respectively.
In our previous work [
11], based on Equation (1), we derived the functions of third-order net astigmatic fringe Zernike coefficients (C5/C6), third-order net coma fringe Zernike coefficients (C7/C8), and third-order net spherical aberration fringe Zernike coefficient (C9) induced by axial and lateral misalignments. To facilitate the description of the proposed compensation correction principle, their field dependence is further clearly expressed in this section, as shown in the following equations.
In Equations (2)–(4), the subscripts
,
, and
represent the field-constant dependence, field-linear dependence, and field-quadratic dependence of different aberration types, respectively; the subscripts
and
denote the x-component and y-components of a vector, respectively,
,
, and
. In addition, because the aperture of PM in astronomical telescopes is usually relatively large, it is prone to figure errors due to factors such as gravity, support stress, and thermal stress. In this paper, third-order astigmatic figure errors, third-order coma figure errors, and third-order spherical aberration figure errors of PM are specifically considered. When PM is used as the stop surface, the beams emitted from different FOVs entering the optical system have the same footprint on PM and completely cover PM. At that time, it can be considered that the aberration contribution of PM to different FOVs is the same, that is, the figure errors of PM will only introduce field-constant aberrations [
22,
26]. In consequence, we can add the field-constant aberrations induced by PM figure errors to the field-constant aberrations induced by misalignments. According to the relationship between optical path difference and figure error, Equations (2)–(4) can be modified as
where
and
, respectively, represent the refractive index of the space where the incident light and the reflected light are located (it could be noted that
,
for odd reflections and
,
for even reflections);
and
represent field-constant astigmatism induced by PM astigmatic figure errors;
and
represent field-constant coma induced by PM coma figure errors;
represents the field-constant spherical aberration induced by PM spherical aberration figure errors.
In principle, only the same type of aberration fields can compensate for each other [
33]. It can be seen from Equations (5)–(7) that misalignments and PM figure errors of the off-axis optical systems simultaneously introduce field-constant astigmatism, field-constant coma, and field-constant spherical aberration; this indicates that there are coupling effects between the aberration fields induced by misalignments and aberration fields induced by PM figure errors in the off-axis optical systems, and the possibility of mutual compensation exists. However, aberration compensation in the true sense includes not only the aberration fields induced by the target perturbation parameters (referring to PM figure errors) that need to be compensated for but also other types of aberration fields induced by compensation perturbation parameters (referring to misalignments), which need to be compensated at the same time. For instance, while using the field-constant aberrations induced by misalignments to compensate for the field-constant aberrations induced by PM figure errors, it can be seen from Equations (5)–(7) that misalignments will inevitably introduce additional field-linear aberrations and field-quadratic aberrations; these two types of aberration fields cannot be ignored in wide-field off-axis telescopes. To solve this problem, compensation perturbation parameters can be decomposed into different components, so that the different components can compensate for each other, thereby hopefully compensating for the extra-induced aberration fields. In fact, it is possible and relatively easy to decompose compensation perturbation parameters into different compensators according to different mirrors, that is, compensation between misalignments of different mirrors. This is mainly because misalignments of different mirrors will introduce the same type of aberration fields, and no additional type of aberration fields will be introduced after compensation is done between different mirrors. Taking the off-axis three-mirror telescope as an example, when PM is used as the coordinate reference, the compensation parameters can be decomposed into two components, one of which is the misalignments of SM and the other is the misalignments of TM. The misalignments of SM will introduce field-constant aberrations, field-linear aberrations, and field-quadratic aberrations, and so does the misalignments of TM. Among them, field-constant aberrations induced by the misalignments of SM and TM can be used to compensate for field-constant aberrations induced by PM figure errors, while the additional field-linear aberrations and field-quadratic aberrations induced by the misalignments of SM and TM can compensate each other. Based on this analysis, we propose the following mechanism of aberration field compensation for an off-axis three-mirror telescope.
Taking SM to compensate misalignments of TM and figure errors of PM as an example, we can adjust SM to introduce appropriate field-constant aberrations (astigmatism, coma, spherical aberration), field-linear aberrations (astigmatism, coma) and field-quadratic aberrations (astigmatism). Among them, the introduced field-constant aberrations can be used to compensate for the field-constant aberrations caused by TM misalignments of PM figure errors. Meanwhile, the introduced field-linear aberrations and field-quadratic aberrations can be used to compensate for the field-linear aberrations and field-quadratic aberrations caused by TM misalignments.
In the following two sections, based on the above aberration field compensation mechanism, the optical compensation of misaligned TM and deformed PM by adjusting SM will be discussed; moreover, the optical compensation of misaligned SM and deformed PM by adjusting TM will also be discussed.