Aberration analysis for freeform surface terms overlay on general decentered and tilted optical surfaces

Tong Yang; Dewen Cheng; Yongtian Wang

doi:10.1364/OE.26.007751

1. Introduction

Freeform optical surface can be defined as the surface without rotationally symmetry. It have more degrees of design freedom for optical design, and can lead to much higher system performance while achieving much more compact configuration. In recent years, the developments in advanced manufacturing technologies have promoted freeform surfaces being successfully used in many areas, such as head-mounted-display [1–3], off-axis reflective systems [4–10], imaging spectrometer [11,12], projection optics [13], corneal imaging system [14], etc. Therefore, the theory of freeform optics design and analysis is very important. Among them, the aberration theory of freeform surfaces helps designers better understanding the nature of freeform systems. In addition it is of great importance in developing analytical methods for the system design, tolerance, and assembly. However, the shape and mathematical expression of freeform surface is very complicated. In addition, freeform surfaces are generally used in decentered and tilted systems. As a result, traditional Seidel or Hopkins aberration theory do not work for freeform imaging systems.

Different from traditional aberration theory, nodal aberration theory (NAT) uses vector-based field position and pupil position coordinate, and it can be used to describe the aberration fields of nonrotationally symmetric systems. The theory is first discovered by Shack [15] and developed by Thompson [16]. Later, Thompson investigated the third- and fifth-order vector aberrations of optical systems without rotational symmetry and analyzed the multi-nodal property [16–19]. The design and alignment of imaging systems based on NAT have also been studied [20–24]. As NAT fully describes the nonsymmetric aberration fields, it can be used in the design and analyses of the nonsymmetric freeform optical surfaces. As Zernike polynomials are both continuous and orthogonal over a unit circle, and they have the same types of wave aberrations as the ones observed in optical tests, Zernike polynomial freeform surface is generally used in the aberration analysis. Schmid et al analyzed the nodal aberration property when Zernike astigmatism figure error is added on the stop surface [25]. Fuerschbach et al presented an analytical theory path to integrate Zernike polynomial surfaces with NAT [26]. With his method, the nodal aberration response generated by the three point mounting error (Zernike trefoil deformation) was analyzed. Then, Fuerschbach et al analyzed the aberration field behavior of Zernike polynomial surface through Fringe Zernike term 17/18 [27]. Yang et al analyzed the multi- and complex nodal aberration behavior when freeform surfaces are used in coaxial systems [28]. Shi et al analyzed the nodal aberration properties when the freeform surface terms are added in off-axis systems [29]. However, the above research works are all limited to the case where the central section of the freeform surfaces are used. For a general optical system with decentered and tilted surfaces, the off-axis section of freeform surfaces are often used. The aberration properties generated by this case need to be developed. In addition, the existing aberration analysis methods are established based on the premise that the footprint of each field is circular. However, in general imaging systems, the actual footprint is not a circle, but approximately an ellipse, especially for the systems using surfaces with large tilt angles. As a result, the aberration properties of freeform terms on general decentered and tilted optical surfaces remains to be discovered.

In this paper, the aberrations generated by freeform surface terms overlay on general decentered and tilted optical surfaces are analyzed in detail. Firstly the framework of integrating freeform surface with NAT are revisited and it is the basis of the following sections. Then, for the case when the off-axis section of freeform surface is used, the aberration equation for the cases using the stop and nonstop surfaces are discussed, and the aberrations generated by Zernike terms up to Z_17/18 are analyzed. To solve the problem of elliptical light footprint for tilted freeform surfaces, the scaled pupil vector is used in the aberration analysis. The mechanism of aberration transformation is discovered, and the aberrations generated by different Zernike terms in this case are calculated. Finally we proposed the aberration equations of freeform surface terms overlay on general decentered and tilted surface. For all the cases no aberration with new pupil-dependence are generated, but only the ones with special and complex field-dependence. The research result given in this paper offers important reference for the optical designers and engineers, and it is of great importance in developing analytical methods for the general freeform system design, tolerance analysis, and system assembly.

2. Basic theory for integrating freeform surface into NAT

In this Section, the basic theory for integrating freeform surface into NAT is revisited. The field coordinate H and pupil coordinate ρ in the aberration terms are expressed in vector form in NAT: $\vec{H} = H e^{i θ}$ and $\vec{ρ} = ρ e^{i ϕ}$ , where θ and ϕ denote the orientations of the two vectors respectively. For the integration of freeform surface into NAT, the freeform surface also need to be expressed in vector form. Zernike polynomial surface is a kind of ϕ-polynomial surface [4]. Besides the traditional expressions using traditional Cartesian coordinates, it can be also written in the polar coordinates form

z = F (ρ, ϕ),

where ρ is the normalized radial distance, ϕ is the azimuth angle. In the optical design field, the Zernike polynomial surface generally has a base conic

z = \frac{c (x^{2} + y^{2})}{1 + \sqrt{1 - (1 + k) c^{2} (x^{2} + y^{2})}} + \sum_{j = 1}^{n} C_{j} Z_{j},

where c is the surface curvature, k is the conic constant, Z_j is the jth Zernike polynomial, C_j is the corresponding coefficient. The Fringe Zernike polynomial set is used in this paper without the loss of generality [30]. In this paper, we mainly focus on the aberration generated by the Fringe Zernike polynomial up to the sixth order in wavefront (Z₁-Z₁₆), which correspond to the sixth order wave aberration. In addition, the Zernike tetrafoil terms (Z₁₇/Z₁₈ terms in Fringe set) are also considered. They are 4th order in mathematics and 8th order in wave aberration. These terms are also important in minimize astigmatism in optical design tasks and will be generated by other Zernike terms (such as Z₉ and Z₁₂/Z₁₃) in Sections 4 and 5. So they are also discussed in this paper. When the first 18 Fringe terms are considered, all the 4th order terms in mathematics are included. Other terms after 18th Fringe term are not considered in this paper, which are corresponding to the 8th, 10th and 12th wave aberrations and are rarely considered in the design process. The first 18 Fringe Zernike terms and their corresponding traditional Seidel aberration types and wavefront order are given in Table 1. Most of the Zernike terms can be divided into pairs of two. A pair of Zernike polynomial terms can be written as C_xZ_x = C_xZ_x(ρ,ϕ) and C_yZ_y = C_yZ_y(ρ,ϕ), with C_x and C_y being the coefficients for the two components. We can express a Zernike pair in vector form

\vec{C} \cdot Z (\vec{ρ})

, where

Z (\vec{ρ}) = Z (ρ, ϕ) = (\begin{matrix} Z_{x} (ρ, ϕ) \\ Z_{y} (ρ, ϕ) \end{matrix})

,

\vec{C} = C e^{i α}

is the vector coefficient of the Zernike pair with

C = \sqrt{C_{x}^{2} + C_{y}^{2}}

being the magnitude and

α = \arctan (C_{y} / C_{x})

being the orientation. For the single term such as defocus (Z₄) and spherical terms (Z₉), it can be written as

C Z (ρ)

.

Table 1. Fringe Zernike polynomials (up to 18th term)

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Next we can analyze the aberration generated by the freeform surface terms. In this section we will make a summary of the simplest case when the light beams use the central on-axis area of non-tilted freeform surface. The “tilt” here can be considered as the surface tilt relative to the light beam of central field. The footprint of each field can be considered as a circle in this case. The Zernike polynomial terms added on the surface can be seen as figure error or deformation. When the freeform surface is located at the stop, the light beams of different fields will use the same area on the surface. So, the contribution of the Zernike polynomial terms to the aberration function can be seen as field-constant [4,27–29]. The aberration equations for different cases are summarized in Table 2 [27,28]. All the aberrations are measured at the system exit pupil in this paper.

Table 2. Aberrations generated by freeform surface terms when the light beams use the central on-axis area of non-tilted surface located at the stop

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For reflective surfaces, n denotes the refractive index of the medium, λ denotes the wavelength of light. In most cases, the reflective surface is in the air and n = 1. For refractive surfaces, n₁ and n₂ are the refractive index before and after the refractive surface. The aberration equations vary with the number of immediate images between the surface and the exit pupil, due to the same or opposite directions of $\vec{ρ}$ at the freeform surface and the exit pupil. From Table 2, we can see that adding Zernike polynomial terms on the stop surface will generate the same type of field-constant aberrations.

When the freeform terms are added on the non-stop surface, the used area on the surface for each field is different. In this way, the beam of the off-axis field displaces from the on-axis field across the surface. A relative beam displacement vector $Δ \vec{h}$ can be used to account for this effect [27]. When the field-of-view (FOV) is not very large, we have

Δ \vec{h} \equiv (\frac{\bar{y}}{y}) \vec{H},

where y is the marginal ray height of the on-axis field on the surface,

\bar{y}

is the chief ray height of the marginal field on the surface. The actual pupil position

{\vec{ρ}}^{'}

in the local surface coordinates

\hat{x} O \hat{y}

equals to

\vec{ρ} + Δ \vec{h}

, as shown in Fig. 1. After replacing

\vec{ρ}

in the equations in Table 2 by

\vec{ρ} + Δ \vec{h}

, the aberration equations for the non-stop freeform surface are summarized in Table 3 [27,28].

Fig. 1 The actual pupil vector on the actual freeform surface when the central section of the nonstop freeform surface is used.

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Table 3. Aberrations generated by freeform surface terms when the light beams use the central on-axis area of non-tilted, non-stop surface

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As $Δ \vec{h} \equiv (\frac{\bar{y}}{y}) \vec{H}$ is involved in the aberration equations, some aberration terms with special field-dependence will be generated in the aberration expansion. These aberrations are consistent with the aberrations generated by non-symmetric configuration. No new aberrations with other pupil-dependence types will be generated. Detailed aberration analyses as well as some nodal points analyses can be founded in [27] and [28].

In the following sections, without the loss of generality, we assume that there are no or an even number of immediate images between the surface and the exit pupil. In addition, we use the general vector coefficient $\vec{M} = M e^{i α}$ to represent the coefficient $- 2 n \vec{C} / λ$ or $(n_{2} - n_{1}) \vec{C} / λ$ in the above aberration equations.

3. Aberration analysis for the case using off-axis section of freeform surface

In the previous Section, the aberrations are derived assuming the light beam use the central part of the freeform surface (the chief ray of the central field intersects the freeform surface at the surface vertex or the origin of the local surface coordinates $\hat{x} O \hat{y}$ ). For actual freeform imaging systems, the light beam may use the off-axis section of the freeform surface. Sometimes the deviation is very large. For the stop surface, the deviation for the center of the actual used area (the light footprint of each field) from the surface vertex can be written as ${\vec{k}}_{1}$ . If normalized to the marginal ray height, the deviation is expressed as

\vec{μ} \equiv \frac{{\vec{k}}_{1}}{y},

where y is the marginal ray height of the central field relative to its chief ray on the freeform surface. In this way, the local pupil coordinate

\vec{ρ}

for each field corresponds to

\vec{ρ} + \vec{μ}

in the actual freeform surface, as shown in Fig. 2(a). Replacing

\vec{ρ}

by

\vec{ρ} + \vec{μ}

for the aberration equations in Table 2, the aberration equation when the off-axis section of freeform terms are used on the stop surface can be written as

W_{s t o p} = \vec{M} \cdot Z (\vec{ρ} + \vec{μ}) .

For the non-stop surface, the light beams of different fields are separated. The deviation for the center of the actual used area for the

\vec{H} = \vec{1}

field from the surface vertex O can be written as

{\vec{k}}_{2}

. In this way the relative beam displacement vector

Δ \vec{h}

(relative to the central field) can be written as

Δ \vec{h} \equiv (\frac{{\vec{k}}_{2} - {\vec{k}}_{1}}{y}) \vec{H} = \vec{λ} \vec{H},

here

{\vec{k}}_{1}

and

{\vec{k}}_{2}

can be calculated by real ray trace data. The local pupil coordinate

\vec{ρ}

for each field corresponds to

\vec{ρ} + \vec{λ} \vec{H} + \vec{μ}

in the actual freeform surface, as shown in Fig. 2(b). Replacing

\vec{ρ}

by

\vec{ρ} + \vec{λ} \vec{H} + \vec{μ}

for the aberration equations in Table 3, the aberration equation when the off-axis section of freeform terms are used on the nonstop surface can be written as Eq. (7). Similar equation can also be found in [31] and [32].

Fig. 2 The actual pupil vector on the actual freeform surface when the off-axis section of the freeform surface is used. (a) The case when the surface is the aperture stop. (b) The case when the surface is away from the stop.

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W_{non s t o p} = \vec{M} \cdot Z (\vec{ρ} + \vec{λ} \vec{H} + \vec{μ}) .

In [33], Ju et al propose their well done work in the computation of astigmatic and trefoil freeform errors and misalignments for two-mirror telescopes. The impact and equations of the decenter of freeform surfaces on the aberrations fields has also been analyzed based on the assumption of circular light footprint. On this basis, the analytic expressions for the aberration fields of two-mirror telescopes in the presence of astigmatic primary mirror figure errors, mount-induced trefoil deformations on both mirrors, and misalignments are presented. Then these effect can be quantitatively separated using the analytical expressions with wavefront measurements at a few field points and pointing errors. These works are of significance for the active optics. In our paper, we focus on the aberrations generated in the general case. When calculating the aberrations of the nonstop surface, a novel vector coefficient $\vec{λ} = (\frac{{\vec{k}}_{2} - {\vec{k}}_{1}}{y})$ calculated by the real ray trace data is used to account for the shift of off-axis fields on the surface. In this way, the shift of $\vec{H} = \vec{1}$ field is not restricted to be along the y-direction, and the general case of non-symmetric systems and can be considered. Furthermore, when analyzing the aberrations generated by the freeform surface terms, the influence of the different number of immediate images between the freeform surface and the exit pupil is taken into account (Tables 2 and 3). This is a key factor in calculating the actual aberrations. Here in Sections 3-5, we assume that there are an even number (including zero) of immediate images between the surface and the exit pupil for the sake of convenience. In addition, besides the trefoil and astigmatism freeform terms, the aberrations generated by the freeform surface terms up to the 18th term in the Fringe Zernike set will be analyzed in details in the following. The analysis method for the off-axis section of freeform surfaces is also successfully integrated with the case when the footprint is not circular (elliptical in this paper) and the nonsymmetrical scaled factor of pupil vector is used. As a result the influence of surface tilt can be involved. For non-tilted surfaces, if the footprint is elliptical, the aberrations can also be calculated for the off-axis section case. With these work, the aberrations generated by freeform surface terms on the general decentered and tilted surface can be analytically derived and calculated. Detailed methods will be discussed in Sections 4 and 5.

Using Eqs. (6) and (7), we can analyze the aberration types and properties when the freeform terms are added onto non-stop surfaces and off-axis sections of the freeform surfaces are used. Assuming kth Fringe Zernike polynomial terms ${\vec{M}}_{k} \cdot Z_{k} (\vec{ρ})$ are added, the generated aberrations are given in Table 4. Among the Fringe Zernike set, terms 1-4 (piston, tilt and defocus) are actually related to the position of the reference sphere or do not affect the image quality, and they are generally not applied in fabrication, so these terms will not be discussed in this section. In addition, in some Zernike terms, there are extra terms which are used to minimize the RMS wavefront error, such as the tilt term in Z₇ and Z₈, the defocus and piston terms in Z₉, the astigmatism term in Z₁₂ and Z₁₃, the third-order spherical term, defocus and piston terms in Z₁₆, etc. These extra terms can be easily compensated by adding corresponding Zernike terms, so they are not included in the discussed below. The Zernike terms with extra terms removed can be called “adjusted terms” [27].

Table 4. Aberrations generated by freeform surface terms when the light beams use the off-axis area of non-tilted, non-stop surface

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The results of Table 4 are the generated aberrations when Fringe Zernike polynomial terms ${\vec{M}}_{k} \cdot Z_{k} (\vec{ρ})$ (or $M_{k} Z_{k} (ρ)$ ) are added. So the ${\vec{M}}_{k}$ or $M_{k}$ contained in the aberration terms indicate that they are generated by the kth Fringe Zernike polynomial term. In fact, the aberrations generated by freeform surface terms when the light beams use the off-axis area of non-tilted stop surface are also included in Table 4. These aberrations are the ones which do not include $\vec{λ}$ (the power of $\vec{λ}$ , the conjugate of $\vec{λ}$ , the magnitude of $\vec{λ}$ ). The aberrations generated by specific Zernike polynomial terms can be extracted according to the subscript of $\vec{M}$ . It can be seen that most of aberration terms with traditional field dependence can be generated by the Zernike spherical aberration terms. This is why that aspherical surfaces are important in the traditional optical design. In addition, Table 5 demonstrates how the aberration terms given in Table 4 are linked to the existing concepts of NAT. These concept are summarized from [16–19]. The aberrations given in the NAT column and the corresponding aberration terms from Table 4 have the same field and pupil dependences. (Note that Terms (35), (43) and (85) which are generated by the Zernike tetrafoil terms have no related NAT concept by now, as these concepts will be discovered when higher-order optical aberrations of optical systems without rotational symmetry are considered.) With this table, we can clearly see that when using the off-axis section of freeform surface, no new aberration types are generated, but only the ones with special and complex field-dependence, which are also consistent with the aberrations generated by non-symmetric configuration. This demonstrates that off-axis sections of freeform surface are also very useful in the non-symmetric systems design.

Table 5. The link of the generated aberration terms listed in Table 4 to the existing concept of NAT

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4. Aberration analysis of freeform terms on tilted surface

In conventional non-symmetric systems, as tilted surfaces are included, the projection of the light beam on the surface, or the actual footprint of each field on the tilted surface is generally not circular, but approximately an ellipse. A two-mirror example system is as shown in Fig. 3. The light coming from the circular aperture stop is reflected by two tilted mirrors and the light footprints on two mirrors are ellipses approximately. As the existing aberration theories given in Section 2 and 3 are based on the premise of circular footprint, they are not applicable for the case of elliptical footprint using tilted surfaces.

Fig. 3 A two-mirror example system.

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To deal with this problem, the scaled pupil vector is introduced in the aberration analysis. Here, we consider the conventional case when the optical system is symmetric about the yOz plane. The central part of the freeform surface is used and only the on-axis central field is considered. This is the same case when the tilted surface is the stop surface. The ratio between the semi-axes of the elliptical footprint in x and y direction is defined as s. Here, the semi-axis in y direction is taken as the normalization radius of the freeform surface. Then the normalized semi-axis in x direction is s and it is taken as the scale factor. In this way, the ray with pupil coordinate $\vec{ρ} = (ρ_{x}, ρ_{y})$ in the exit pupil plane has a scaled pupil coordinate ${\vec{ρ}}_{s} = (s ρ_{x}, ρ_{y})$ on the tilted surface, as shown in Fig. 4. When kth Fringe Zernike polynomial terms ${\vec{M}}_{k} \cdot Z_{k} (\vec{ρ})$ are added onto the tilted surface, the generated aberrations is

W_{k} = {\vec{M}}_{k} \cdot Z_{k} ({\vec{ρ}}_{s}) = {\vec{M}}_{k} \cdot Z_{k} ([s ρ_{x}, ρ_{y}]) .

As we focus on the conventional case when the optical system is symmetric about the yOz plane, here only the Zernike terms which are symmetric about the yOz plane are considered. In addition, the adjust Zernike terms with wavefront aberration balance terms removed are used. Therefore, the generated aberration for the kth term can be written as

W_{k} = M_{k} Z_{k}^{a d j} (s ρ_{x}, ρ_{y}) .

Expand the aberration equation and arrange the equation into XY polynomials, we can get

W_{k} = M_{k} Z_{k}^{a d j} (s ρ_{x}, ρ_{y}) = \sum c_{k, i} g_{i} (ρ_{x}, ρ_{y}) = C_{k}^{T} G,

where

g_{i} (ρ_{x}, ρ_{y})

are the ith xy monomial, c_k_,_i is the corresponding coefficient, G is the vector containing the used xy terms, C_k is the coefficient vector. Then the XY polynomials can be rearranged into Zernike polynomials

W_{k} = \sum M_{k, j}^{r e a r r} Z_{j}^{a d j} (ρ_{x}, ρ_{y}) = M_{k}^{r e a r r}^{T} Z^{a d j} = \sum c_{k, i} g_{i} (ρ_{x}, ρ_{y}) = C_{k}^{T} G,

where

Z_{j}^{a d j} (ρ_{x}, ρ_{y})

are the jth Zernike term,

M_{k, j}^{r e a r r}

is the corresponding coefficient, Z^adj is the vector containing the used adjusted Zernike polynomial terms,

M_{k}^{r e a r r}

is the coefficient vector.

Fig. 4 The elliptical footprint and the scaled pupil vector.

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The relationship between Z^adj and G can be written as

Z^{a d j} = T G,

where T is the transform matrix. When considering up to 4th order, we have

\begin{array}{l} G = [\begin{matrix} 1 & ρ_{x} & ρ_{y} & ρ_{x}^{2} & ρ_{x} ρ_{y} & ρ_{y}^{2} & ρ_{x}^{3} & ρ_{x}^{2} ρ_{y} & ρ_{x} ρ_{y}^{2} & ... \end{matrix} \\ \begin{matrix} ... & ρ_{y}^{3} & ρ_{x}^{4} & ρ_{x}^{3} ρ_{y} & ρ_{x}^{2} ρ_{y}^{2} & ρ_{x} ρ_{y}^{3} & ρ_{y}^{4} \end{matrix}]^{T}, \end{array}

\begin{array}{l} Z^{a d j} = [\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} 1 & ρ_{x} \end{matrix} & ρ_{y} & 2 (ρ_{x}^{2} + ρ_{y}^{2}) \end{matrix} & ρ_{x}^{2} - ρ_{y}^{2} & 2 ρ_{x} ρ_{y} \end{matrix} & 3 ρ_{x} (ρ_{x}^{2} + ρ_{y}^{2}) & ... \end{matrix} \\ \begin{matrix} ... & 3 ρ_{y} (ρ_{x}^{2} + ρ_{y}^{2}) & 6 {(ρ_{x}^{2} + ρ_{y}^{2})}^{2} & ρ_{x} (ρ_{x}^{2} - 3 ρ_{y}^{2}) & ρ_{y} (3 ρ_{x}^{2} - ρ_{y}^{2}) & ... \end{matrix} \\ \begin{matrix} ... & 4 (ρ_{x}^{4} - ρ_{y}^{4}) & 8 (ρ_{x}^{2} + ρ_{y}^{2}) ρ_{x} ρ_{y} & ρ_{x}^{4} + ρ_{y}^{4} - 6 ρ_{x}^{2} ρ_{y}^{2} & 4 ρ_{x} ρ_{y} (ρ_{x}^{2} - ρ_{y}^{2})]^{T} \end{matrix}, \end{array}

T = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 12 & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & - 3 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & - 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 8 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & - 6 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & - 4 & 0 \end{matrix}] .

Based on Eqs. (11) and (12), we have

M_{k}^{r e a r r} = {(T^{T})}^{- 1} C_{k} .

Using Eq. (16), we can analyze the generated aberrations when the freeform surface term are added on the central area of tilted stop surface, which are summarized in Table 6.

Table 6. Aberrations generated by freeform surface terms when the light beams use the central on-axis area of tilted stop surface

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From the analysis we can find that when freeform terms are added on the tilted stop surface, besides the aberrations corresponding to the employed Zernike term, some other aberrations will also be generated. However, no new aberration types are generated. In addition, the method presented in this Section also opens a pathway to analyze the aberrations generated by freeform surface terms other than Zernike polynomial surface terms.

We use the two-mirror system shown in Fig. 3 to quantitatively validate the aberrations generated by the added Zernike polynomial terms. The entrance pupil diameter is 10mm. Only the 0° field is used here to show the aberration property. The primary mirror (M1) and secondary mirror (M2) are both freeform surfaces. Both M1 and M2 have a 45° tilt relative to the incident light beam. The footprints of the beam on two mirrors are ellipses approximately. For the 45° tilted M2 the scale factor s approximately equals to 1/cos45° = 1/√2. We optimized the system so that the top, bottom, left and right marginal rays intersect M2 at local surface coordinates (0mm, 5√2mm) and (0mm, −5√2mm), (−5mm, 0mm) and (5mm, 0mm) respectively. The normalization radius of M2 is 5√2mm. Then we added different Zernike terms (the coefficients all equal to 0.001) onto M2 to observe the generated types and amount of aberrations in CODE V and compare them with theoretical values, as shown in Fig. 5. From this figure we can see that actual generated aberration types and value coincide with theory approximately.

Fig. 5 The comparison of the theoretical and actual values when different Zernike terms (the coefficients all equal to 0.001) are added onto M2

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5. Aberration of freeform terms on general decentered and tilted surface

In the above sections, the aberration equations using freeform terms for different cases are discussed. Based on these discussions, it is easy to obtain the aberration equation when the freeform terms are added on general decentered and tilted surfaces. When the freeform terms are added onto the stop surface, the generated aberration can be written as

W_{s t o p} = \vec{M} \cdot Z ({\vec{ρ}}_{s} + \vec{μ}) .

When the freeform terms are added onto the nonstop surface, the generated aberration is

W_{non s t o p} = \vec{M} \cdot Z ({\vec{ρ}}_{s} + \vec{λ} {\vec{H}}_{s_{H}} + \vec{μ}) .

where

{\vec{ρ}}_{s}

,

\vec{λ}

and

\vec{μ}

have the same meanings as they have in Section 3 and 4; The scaled field vector

{\vec{H}}_{s_{H}}

is used for tilted surface and elliptical projection, which is similar to

{\vec{ρ}}_{s}

. Note that the scale factor

s_{H}

in field vector generally does not equal to s, which have to be calculated respectively using the footprints data. Using the above equations, we can get the aberration expansions. It can be easily inferred that when freeform surface terms are added onto general decentered and tilted surfaces, no new aberration types are generated, but only the ones with special and complex field-dependence. These aberrations are consistent with the aberration types generated by non-symmetric configuration. It proves that freeform surfaces are very useful in the general non-symmetric systems design. For the conventional case when the optical system is symmetric about the yOz plane, only the Zernike terms which are symmetric about the yOz plane are considered. The aberration generated by the kth term can be written as

W_{k} = M_{k} Z_{k}^{a d j} (s ρ_{x} + s_{H} λ_{y} H_{x}, ρ_{y} + λ_{y} H_{y} + μ_{y}) .

Using the method given in Section 4, we can analyze the aberrations, which are summarized in Table 7. Besides the direct derivation using Eqs. (17)-(19), these results can also be calculated based on the specific aberration expansions given in Section 3 combining the analyzing method given in Section 4.

Table 7. Aberrations generated by different freeform surface terms when the off-axis section of tilted surface is used

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We use another two-mirror example system which has the same optical layout and system specification to the one (Fig. 3) given in Section 4 to quantitatively validate the aberrations generated by different freeform surface terms when the off-axis section of tilted surface is used. The difference is that the used area of M2 is the off-axis section of the freeform surface. The deviation for the center of the actual used area from the surface vertex is 5√2/2 mm. Both M1 and M2 have a 45° tilt relative to the incident light beam. We also optimized the system so that the top, bottom, left and right marginal rays intersect M2 at local surface coordinates (0mm, 5√2mm) and (0mm, −5√2mm), (−5mm, 0mm) and (5mm, 0mm) respectively. The normalization radius of M2 is 5√2mm. Then we added different Zernike terms (the coefficients all equal to 0.00015) onto M2 to observe the types and amount of generated aberrations of 0° field in CODE V and compare them with theoretical values, as shown in Fig. 6. From this figure we can see that actual generated aberration types and value coincide with theory approximately. In this paper, we focus on the simple and common case of the elliptical footprint for the tilted surface. The error between the theoretical and actual values for both two examples is partially due to the imperfect elliptical footprint or other possible footprint shapes due to intermediate aberrations. In addition, a significant error source comes from the approximation of the aberration equations given in Tables 2 and 3 [28]. If the aberration equations are to be accurate, the incident and outgoing rays on the surface have to be always along the local z direction of the surface. However, this condition cannot be satisfied generally. Then, extra optical paths will be added to (or subtracted from) the ideal incident and outgoing paths. The total extra optical path will induce an extra aberration. It can be classified into different aberration types according to the pupil dependence and the error will be different when different freeform terms are added. However, the impact of imperfect elliptical or other types footprint shapes as well as the mechanism of the error generation and analysis is very complicated, and this is a research we hope to be done in the future.

Fig. 6 The comparison of the theoretical and actual values when off-axis sections of different Zernike terms (the coefficients all equal to 0.00015) are added onto M2 in the second example.

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6. Conclusion

In this paper, the aberrations generated by freeform surface terms overlay on general decentered and tilted optical surfaces are analyzed. The pathway of integrating freeform surface with NAT are revisited. Then, for the case when the off-axis section of freeform surface is used, the aberration equation for using the stop and nonstop surfaces are discussed, and the aberrations generated by Zernike terms up to Z_17/18 are analyzed in detail. The scaled vector is used in the aberration analysis to deal with the elliptical light footprint. The mechanism of aberration transformation is discovered, and the aberrations generated by different Zernike terms in this case are calculated. Finally we proposed the aberration equations of freeform terms added onto general decentered and tilted freeform surface. The research result given in this paper offers important reference for the optical designers and engineers, and it is of great importance in developing analytical methods for the general freeform system design, tolerance, and assembly tasks.

Funding

The National High Technology Research and Development Program of China (2015AA016301); National Key Research and Development Program of China (2016YFB1001502); National Natural Science Foundation of China (NSFC) (61727808).

Acknowledgments

We thank Synopsys for the educational license of CODE V.

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Term	Zernike polynomial	Corresponding aberration type	Term	Zernike polynomial	Corresponding aberration type
1	1	Piston (constant) 0th order in wavefront	10	ρ³cos(3ϕ)	Trefoil, Primary (x-axis) 6th order in wavefront
2	ρcosϕ	Distortion-Tilt (x-axis) 2nd order in wavefront	11	ρ³sin(3ϕ)	Trefoil, Primary (y-axis) 6th order in wavefront
3	ρsinϕ	Distortion-Tilt (y-axis) 2nd order in wavefront	12	(4ρ⁴−3ρ²) cos(2ϕ)	Astigmatism, Secondary (axis at 0° or 90°) 6th order in wavefront
4	2ρ²−1	Defocus 2nd order in wavefront	13	(4ρ⁴−3ρ²) sin(2ϕ)	Astigmatism, Secondary (axis at ± 45°) 6th order in wavefront
5	ρ²cos(2ϕ)	Astigmatism, Primary (axis at 0° or 90°) 4th order in wavefront	14	(10ρ⁵−12ρ³ + 3ρ) cos(ϕ)	Coma, Secondary (x-axis) 6th order in wavefront
6	ρ²sin(2ϕ)	Astigmatism, Primary (axis at ± 45°) 4th order in wavefront	15	(10ρ⁵−12ρ³ + 3ρ) sin(ϕ)	Coma, Secondary (y-axis) 6th order in wavefront
7	(3ρ³−2ρ) cosϕ	Coma, Primary (x-axis) 4th order in wavefront	16	20ρ⁶−30ρ⁴ + 12ρ²−1	Spherical aberration, Secondary 6th order in wavefront
8	(3ρ³−2ρ) sinϕ	Coma, Primary (y-axis) 4th order in wavefront	17	ρ⁴cos(4ϕ)	Tetrafoil, Primary (x-axis) 8th order in wavefront
9	6ρ⁴−6ρ² + 1	Spherical aberration, Primary 4th order in wavefront	18	ρ⁴sin(4ϕ)	Tetrafoil, Primary (y-axis) 8th order in wavefront

Surface type	Even number of immediate images	Odd number of immediate images
Reflective surface	$W = - \frac{2 n}{λ} \vec{C} \cdot Z (\vec{ρ})$	$W = - \frac{2 n}{λ} \vec{C} \cdot Z (- \vec{ρ})$
Refractive surface	$W = \frac{(n_{2} - n_{1})}{λ} \vec{C} \cdot Z (\vec{ρ})$	$W = \frac{(n_{2} - n_{1})}{λ} \vec{C} \cdot Z (- \vec{ρ})$

Surface type	Even number of immediate images	Odd number of immediate images
Reflective surface	$W = - \frac{2 n}{λ} \vec{C} \cdot Z (\vec{ρ} + Δ \vec{h})$	$W = - \frac{2 n}{λ} \vec{C} \cdot Z (- \vec{ρ} + Δ \vec{h})$
Refractive surface	$W = \frac{(n_{2} - n_{1})}{λ} \vec{C} \cdot Z (\vec{ρ} + Δ \vec{h})$	$W = \frac{(n_{2} - n_{1})}{λ} \vec{C} \cdot Z (- \vec{ρ} + Δ \vec{h})$

Aberration type	Aberrations with traditional field dependence	Aberrations with special field dependence
Fifth-order spherical aberration (W₀₆₀)	$(1) 20 M_{16} {(\vec{ρ} \cdot \vec{ρ})}^{3}$	None
Oblique spherical aberration (W₂₄₂)	$(2) 120 M_{16} ({\vec{λ}}^{2} {\vec{H}}^{2} \cdot {\vec{ρ}}^{2}) (\vec{ρ} \cdot \vec{ρ})$	$\begin{array}{l} (23) 120 M_{16} ({\vec{μ}}^{2} \cdot {\vec{ρ}}^{2}) (\vec{ρ} \cdot \vec{ρ}) \\ (24) 240 M_{16} (\vec{λ} \vec{μ} \vec{H} \cdot {\vec{ρ}}^{2}) (\vec{ρ} \cdot \vec{ρ}) \\ (25) 20 ({\vec{M}}_{14 / 15} \vec{λ} \vec{H} \cdot {\vec{ρ}}^{2}) (\vec{ρ} \cdot \vec{ρ}) \\ (26) 20 ({\vec{M}}_{14 / 15} \vec{μ} \cdot {\vec{ρ}}^{2}) (\vec{ρ} \cdot \vec{ρ}) \\ (27) 4 ({\vec{M}}_{12 / 13} \cdot {\vec{ρ}}^{2}) (\vec{ρ} \cdot \vec{ρ}) \end{array}$
Focal surface for medial oblique spherical (W_240M)	$(3) 180 M_{16} {\| \vec{λ} \|}^{2} (\vec{H} \cdot \vec{H}) {(\vec{ρ} \cdot \vec{ρ})}^{2}$	$\begin{array}{l} (28) 360 M_{16} (\vec{λ} \vec{H} \cdot \vec{μ}) (^{\vec{ρ} \cdot \vec{ρ}) 2} \\ (29) 30 ({\vec{M}}_{14 / 15} {\vec{λ}}^{*} \cdot \vec{H}) (^{\vec{ρ} \cdot \vec{ρ}) 2} \end{array}$
Fifth order aperture coma (W₁₅₁)	$(4) 120 M_{16} (\vec{λ} \vec{H} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2}$	$\begin{array}{l} (30) 120 M_{16} (\vec{μ} \cdot \vec{ρ}) (^{\vec{ρ} \cdot \vec{ρ}) 2} \\ (31) 10 ({\vec{M}}_{14 / 15} \cdot \vec{ρ}) (^{\vec{ρ} \cdot \vec{ρ}) 2} \end{array}$
Elliptical coma (trefoil, W₃₃₃)	$(5) 40 M_{16} ({\vec{λ}}^{3} {\vec{H}}^{3} \cdot {\vec{ρ}}^{3})$	$\begin{array}{l} (32) 120 M_{16} ({\vec{λ}}^{2} \vec{μ} {\vec{H}}^{2} \cdot {\vec{ρ}}^{3}) \\ (33) 120 M_{16} (\vec{λ} {\vec{μ}}^{2} \vec{H} \cdot {\vec{ρ}}^{3}) \\ (34) 40 M_{16} ({\vec{μ}}^{3} \cdot {\vec{ρ}}^{3}) \\ (35) 4 {\vec{M}}_{17 / 18} {\vec{λ}}^{} {\vec{H}}^{} \cdot {\vec{ρ}}^{3} \\ (36) 4 {\vec{M}}_{17 / 18} {\vec{μ}}^{*} \cdot {\vec{ρ}}^{3} \\ (37) 10 ({\vec{M}}_{14 / 15} {\vec{λ}}^{2} {\vec{H}}^{2} \cdot {\vec{ρ}}^{3}) \\ (38) 20 ({\vec{M}}_{14 / 15} \vec{λ} \vec{μ} \vec{H} \cdot {\vec{ρ}}^{3}) \\ (39) 10 ({\vec{M}}_{14 / 15} {\vec{μ}}^{2} \cdot {\vec{ρ}}^{3}) \\ (40) 4 ({\vec{M}}_{12 / 13} \vec{λ} \vec{H} \cdot {\vec{ρ}}^{3}) \\ (41) 4 ({\vec{M}}_{12 / 13} \vec{μ} \cdot {\vec{ρ}}^{3}) \\ (42) {\vec{M}}_{10 / 11} \cdot {\vec{ρ}}^{3} \end{array}$
Tetrafoil (W₄₄₄)	None	$(43) {\vec{M}}_{17 / 18} \cdot {\vec{ρ}}^{4}$
Third-order spherical aberration (W₀₄₀)	$\begin{array}{l} (6) 180 M_{16} {\| \vec{μ} \|}^{2} (^{\vec{ρ} \cdot \vec{ρ}) 2} \\ (7) 30 ({\vec{M}}_{14 / 15} \cdot \vec{μ}) (^{\vec{ρ} \cdot \vec{ρ}) 2} \\ (8) 6 M_{9} (^{\vec{ρ} \cdot \vec{ρ}) 2} \end{array}$	None
Third-order aperture coma (W₁₃₁ + W_331M)	$\begin{array}{l} (9) 360 M_{16} {\| \vec{λ} \|}^{2} (\vec{H} \cdot \vec{H}) (\vec{λ} \vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (10) 720 M_{16} {\| \vec{μ} \|}^{2} (\vec{λ} \vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (11) 60 ({\vec{M}}_{14 / 15}^{} \vec{λ} \vec{μ} \vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (12) 60 ({\vec{M}}_{14 / 15} {\vec{μ}}^{} \vec{λ} \vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (13) 24 M_{9} (\vec{λ} \vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \end{array}$	$\begin{array}{l} (44) 720 M_{16} (\vec{λ} \vec{H} \cdot \vec{μ}) (\vec{λ} \vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (45) 360 M_{16} {\| \vec{λ} \|}^{2} (\vec{H} \cdot \vec{H}) (\vec{μ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (46) 360 M_{16} {\| \vec{μ} \|}^{2} (\vec{μ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (47) 360 M_{16} ({\vec{μ}}^{2} {\vec{λ}}^{} {\vec{H}}^{} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (48) 30 {\| \vec{λ} \|}^{2} (\vec{H} \cdot \vec{H}) ({\vec{M}}_{14 / 15} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (49) 60 (\vec{λ} \vec{H} \cdot {\vec{M}}_{14 / 15}) (\vec{λ} \vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (50) 60 {\| \vec{μ} \|}^{2} ({\vec{M}}_{14 / 15} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (51) 60 ({\vec{M}}_{14 / 15} \vec{μ} {\vec{λ}}^{} {\vec{H}}^{} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (52) 30 ({\vec{M}}_{14 / 15}^{} {\vec{μ}}^{2} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (53) 12 ({\vec{M}}_{12 / 13} {\vec{λ}}^{} {\vec{H}}^{} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (54) 12 ({\vec{M}}_{12 / 13} {\vec{μ}}^{} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (55) 24 M_{9} (\vec{μ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ (56) 3 ({\vec{M}}_{7 / 8} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \end{array}$
Third-order plus fifth-order astigmatism (W₂₂₂ + W₄₂₂)	$\begin{array}{l} (14) 120 M_{16} {\| \vec{λ} \|}^{2} (\vec{H} \cdot \vec{H}) ({\vec{λ}}^{2} {\vec{H}}^{2} \cdot {\vec{ρ}}^{2}) \\ (15) 360 M_{16} {\| \vec{μ} \|}^{2} ({\vec{λ}}^{2} {\vec{H}}^{2} \cdot {\vec{ρ}}^{2}) \\ (16) 30 {\vec{M}}_{14 / 15}^{} ({\vec{λ}}^{2} \vec{μ} {\vec{H}}^{2} \cdot {\vec{ρ}}^{2}) \\ (17) 30 {\vec{M}}_{14 / 15} ({\vec{λ}}^{2} {\vec{μ}}^{} {\vec{H}}^{2} \cdot {\vec{ρ}}^{2}) \\ (18) 12 M_{9} ({\vec{λ}}^{2} {\vec{H}}^{2} \cdot {\vec{ρ}}^{2}) \end{array}$	$\begin{array}{l} (57) 240 M_{16} {\| \vec{λ} \|}^{2} (\vec{H} \cdot \vec{H}) (\vec{λ} \vec{μ} \vec{H} \cdot {\vec{ρ}}^{2}) \\ (58) 360 M_{16} {\| \vec{λ} \|}^{2} (\vec{H} \cdot \vec{H}) ({\vec{μ}}^{2} \cdot {\vec{ρ}}^{2}) \\ (59) 240 M_{16} {\| \vec{μ} \|}^{2} (\vec{λ} \vec{μ} \vec{H} \cdot {\vec{ρ}}^{2}) \\ (60) 120 M_{16} {\| \vec{μ} \|}^{2} ({\vec{μ}}^{2} \cdot {\vec{ρ}}^{2}) \\ (61) 240 M_{16} (\vec{λ} \vec{H} \cdot \vec{μ}) ({\vec{λ}}^{2} {\vec{H}}^{2} \cdot {\vec{ρ}}^{2}) \\ (62) 120 M_{16} ({\vec{μ}}^{3} {\vec{λ}}^{} {\vec{H}}^{} \cdot {\vec{ρ}}^{2}) \\ (63) 120 M_{16} (\vec{λ} \vec{μ} \vec{H} \cdot {\vec{ρ}}^{2}) \\ (64) 20 (\vec{λ} \vec{H} \cdot {\vec{M}}_{14 / 15}) ({\vec{λ}}^{2} {\vec{H}}^{2} \cdot {\vec{ρ}}^{2}) \\ (65) 20 {\| \vec{λ} \|}^{2} (\vec{H} \cdot \vec{H}) ({\vec{M}}_{14 / 15} \vec{λ} \vec{H} \cdot {\vec{ρ}}^{2}) \\ (66) 30 {\vec{M}}_{14 / 15}^{} (\vec{λ} {\vec{μ}}^{2} \vec{H} \cdot {\vec{ρ}}^{2}) \\ (67) 30 {\vec{M}}_{14 / 15} ({\vec{λ}}^{} {\vec{μ}}^{2} {\vec{H}}^{} \cdot {\vec{ρ}}^{2}) \\ (68) 10 {\vec{M}}_{14 / 15}^{} ({\vec{μ}}^{3} \cdot {\vec{ρ}}^{2}) \\ (69) 60 {\| \vec{λ} \|}^{2} (\vec{H} \cdot \vec{H}) ({\vec{M}}_{14 / 15} \vec{μ} \cdot {\vec{ρ}}^{2}) \\ (70) 60 {\| \vec{μ} \|}^{2} ({\vec{M}}_{14 / 15} \vec{λ} \vec{H} \cdot {\vec{ρ}}^{2}) \\ (71) 30 {\| \vec{μ} \|}^{2} ({\vec{M}}_{14 / 15} \vec{μ} \cdot {\vec{ρ}}^{2}) \\ (72) 12 {\| \vec{λ} \|}^{2} {\| \vec{H} \|}^{2} {\vec{M}}_{12 / 13} \cdot {\vec{ρ}}^{2} \\ (73) 8 {\vec{M}}_{12 / 13} \vec{λ} {\vec{μ}}^{} \vec{H} \cdot {\vec{ρ}}^{2} \\ (74) 8 {\vec{M}}_{12 / 13} {\vec{λ}}^{} \vec{μ} {\vec{H}}^{} \cdot {\vec{ρ}}^{2} \\ (75) 12 {\| \vec{μ} \|}^{2} {\vec{M}}_{12 / 13} \cdot {\vec{ρ}}^{2} \\ (76) 3 {\vec{M}}_{10 / 11} ({\vec{μ}}^{} \cdot {\vec{ρ}}^{2}) \\ (77) 4 ({\vec{M}}_{12 / 13} \vec{μ} {\vec{λ}}^{} {\vec{H}}^{} \cdot {\vec{ρ}}^{2}) \\ (78) 4 ({\vec{M}}_{12 / 13} {\vec{μ}}^{} \vec{λ} \vec{H} \cdot {\vec{ρ}}^{2}) \\ (79) 3 {\vec{M}}_{10 / 11} ({\vec{λ}}^{} {\vec{H}}^{} \cdot {\vec{ρ}}^{2}) \\ (80) 24 M_{9} (\vec{λ} \vec{μ} \vec{H} \cdot {\vec{ρ}}^{2}) \\ (81) 12 M_{9} ({\vec{μ}}^{2} \cdot {\vec{ρ}}^{2}) \\ (82) 6 {\vec{M}}_{17 / 18} (^{{\vec{μ}}^{}) 2} \cdot {\vec{ρ}}^{2} \\ (83) 3 {\vec{M}}_{7 / 8} (\vec{λ} \vec{H} \cdot {\vec{ρ}}^{2}) \\ (84) 3 {\vec{M}}_{7 / 8} (\vec{μ} \cdot {\vec{ρ}}^{2}) \\ (85) 6 {\vec{M}}_{17 / 18} (^{{\vec{λ}}^{} {\vec{H}}^{}) 2} \cdot {\vec{ρ}}^{2} \\ (86) {\vec{M}}_{5 / 6} \cdot {\vec{ρ}}^{2} \\ (87) 12 {\vec{M}}_{17 / 18} {\vec{λ}}^{} {\vec{μ}}^{} {\vec{H}}^{*} \cdot {\vec{ρ}}^{2} \end{array}$
Third-order plus fifth-order medial focal surface (W_220M + W_420M)	$\begin{array}{l} (19) 180 M_{16} {\| \vec{λ} \|}^{4} (^{\vec{H} \cdot \vec{H}) 2} (\vec{ρ} \cdot \vec{ρ}) \\ (20) 720 M_{16} {\| \vec{λ} \|}^{2} {\| \vec{μ} \|}^{2} (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{ρ}) \\ (21) 120 {\| \vec{λ} \|}^{2} ({\vec{M}}_{14 / 15} \cdot \vec{μ}) (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{ρ}) \\ (22) 24 M_{9} {\| \vec{λ} \|}^{2} (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{ρ}) \end{array}$	$\begin{array}{l} (88) 720 M_{16} {\| \vec{λ} \|}^{2} (\vec{H} \cdot \vec{H}) (\vec{λ} \vec{H} \cdot \vec{μ}) (\vec{ρ} \cdot \vec{ρ}) \\ (89) 360 M_{16} ({\vec{λ}}^{2} {\vec{H}}^{2} \cdot {\vec{μ}}^{2}) (\vec{ρ} \cdot \vec{ρ}) \\ (90) 720 M_{16} {\| \vec{μ} \|}^{2} (\vec{λ} \vec{H} \cdot \vec{μ}) (\vec{ρ} \cdot \vec{ρ}) \\ (91) 60 {\| \vec{λ} \|}^{2} ({\vec{M}}_{14 / 15} \cdot \vec{λ} \vec{H}) (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{ρ}) \\ (92) 60 ({\vec{M}}_{14 / 15} \cdot {\vec{λ}}^{2} {\vec{μ}}^{*} {\vec{H}}^{2}) (\vec{ρ} \cdot \vec{ρ}) \\ (93) 120 {\| \vec{μ} \|}^{2} ({\vec{M}}_{14 / 15} \cdot \vec{λ} \vec{H}) (\vec{ρ} \cdot \vec{ρ}) \\ (94) 60 ({\vec{M}}_{14 / 15} \vec{λ} \vec{H} \cdot {\vec{μ}}^{2}) (\vec{ρ} \cdot \vec{ρ}) \\ (95) 12 ({\vec{M}}_{12 / 13} \cdot {\vec{λ}}^{2} {\vec{H}}^{2}) (\vec{ρ} \cdot \vec{ρ}) \\ (96) 24 ({\vec{M}}_{12 / 13} \cdot \vec{λ} \vec{μ} \vec{H}) (\vec{ρ} \cdot \vec{ρ}) \\ (97) 48 M_{9} (\vec{λ} \vec{H} \cdot \vec{μ}) (\vec{ρ} \cdot \vec{ρ}) \\ (98) 6 ({\vec{M}}_{7 / 8} \cdot \vec{λ} \vec{H}) (\vec{ρ} \cdot \vec{ρ}) \end{array}$

NAT concept	Corresponding aberration terms in Table 4	NAT concept	Corresponding aberration terms in Table 4
$W_{060} {(\vec{ρ} \cdot \vec{ρ})}^{3}$	(1)	$- (\vec{H} \cdot \vec{H}) ({\vec{A}}_{331 M} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ})$	(45)(48)
$\frac{1}{2} W_{242} ({\vec{H}}^{2} \cdot {\vec{ρ}}^{2}) (\vec{ρ} \cdot \vec{ρ})$	(2)	$({\vec{B}}_{331 M}^{2} {\vec{H}}^{*} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ})$	(47)(51)(53)
$- (\vec{H} {\vec{A}}_{242} \cdot {\vec{ρ}}^{2}) (\vec{ρ} \cdot \vec{ρ})$	(24)(25)	$\begin{array}{l} - ({\vec{C}}_{331 M} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ - ({\vec{A}}_{131} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \end{array}$	(46)(50)(52)(54)(55)(56)
$\frac{1}{2} ({\vec{B}}_{242}^{2} \cdot {\vec{ρ}}^{2}) (\vec{ρ} \cdot \vec{ρ})$	(23)(26)(27)	$W_{420 M} {(\vec{H} \cdot \vec{H})}^{2} (\vec{ρ} \cdot \vec{ρ})$	(19)
$W_{240 M} (\vec{H} \cdot \vec{H}) {(\vec{ρ} \cdot \vec{ρ})}^{2}$	(3)	$\begin{array}{l} W_{220 M} (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{ρ}) \\ + 4 B_{420 M} (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{ρ}) \end{array}$	(20)(21)(22)
$- 2 (\vec{H} \cdot {\vec{A}}_{240 M}) {(\vec{ρ} \cdot \vec{ρ})}^{2}$	(28)(29)	$- 4 (\vec{H} \cdot \vec{H}) (\vec{H} \cdot {\vec{A}}_{420 M}) (\vec{ρ} \cdot \vec{ρ})$	(88)(91)
$W_{151} (\vec{H} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2}$	(4)	$2 ({\vec{H}}^{2} \cdot {\vec{B}}_{420 M}^{2}) (\vec{ρ} \cdot \vec{ρ})$	(89)(92)(95)
$- ({\vec{A}}_{151} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2}$	(30)(31)	$\begin{array}{l} - 4 (\vec{H} \cdot {\vec{C}}_{420 M}) (\vec{ρ} \cdot \vec{ρ}) \\ - 2 (\vec{H} \cdot {\vec{A}}_{220 M}) (\vec{ρ} \cdot \vec{ρ}) \end{array}$	(90)(93)(94)(96)(97)(98)
$\frac{1}{4} W_{333} ({\vec{H}}^{3} \cdot {\vec{ρ}}^{3})$	(5)	$\frac{1}{2} W_{422} (\vec{H} \cdot \vec{H}) ({\vec{H}}^{2} \cdot {\vec{ρ}}^{2})$	(14)
$- \frac{3}{4} {\vec{H}}^{2} {\vec{A}}_{333} \cdot {\vec{ρ}}^{3}$	(32)(37)	$\begin{array}{l} \frac{1}{2} W_{222} ({\vec{H}}^{2} \cdot {\vec{ρ}}^{2}) \\ + \frac{3}{2} B_{422} ({\vec{H}}^{2} \cdot {\vec{ρ}}^{2}) \end{array}$	(15)(16)(17)(18)
$\frac{3}{4} \vec{H} {\vec{B}}_{333}^{2} \cdot {\vec{ρ}}^{3}$	(33)(38)(40)	$- (\vec{H} \cdot \vec{H}) (\vec{H} {\vec{A}}_{422} \cdot {\vec{ρ}}^{2})$	(57)(65)
$- \frac{1}{4} {\vec{C}}_{333}^{3} \cdot {\vec{ρ}}^{3}$	(34)(36)(39)(41) (42)	$\frac{3}{2} (\vec{H} \cdot \vec{H}) ({\vec{B}}_{422}^{2} \cdot {\vec{ρ}}^{2})$	(58)(69)(72)
$W_{040} {(\vec{ρ} \cdot \vec{ρ})}^{2} + B_{240 M} {(\vec{ρ} \cdot \vec{ρ})}^{2}$	(6)(7)(8)	$- (\vec{H} \cdot {\vec{A}}_{422}) ({\vec{H}}^{2} \cdot {\vec{ρ}}^{2})$	(61)(64)
$W_{331 M} (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ})$	(9)	$- \frac{1}{2} ({\vec{C}}_{422}^{3} {\vec{H}}^{*} \cdot {\vec{ρ}}^{2})$	(67)(74)(77)(79)(87)
$\begin{array}{l} W_{131} (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ + 2 B_{331 M} (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \end{array}$	(10)(11)(12)(13)	$- \vec{H} {\vec{A}}_{222} \cdot {\vec{ρ}}^{2} - \frac{3}{2} \vec{H} {\vec{C}}_{422} \cdot {\vec{ρ}}^{2}$	(59)(66)(70)(73)(78)(80)(83)
$- 2 (\vec{H} \cdot {\vec{A}}_{331 M}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ})$	(44)(49)	$\frac{1}{2} {\vec{B}}_{222}^{2} \cdot {\vec{ρ}}^{2} + \frac{1}{2} {\vec{D}}_{422}^{2} \cdot {\vec{ρ}}^{2}$	(60)(68)(71)(75)(76)(81)(82)(84)(86)

Aberration analysis for freeform surface terms overlay on general decentered and tilted optical surfaces

Abstract

1. Introduction

2. Basic theory for integrating freeform surface into NAT

3. Aberration analysis for the case using off-axis section of freeform surface

4. Aberration analysis of freeform terms on tilted surface

5. Aberration of freeform terms on general decentered and tilted surface

6. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (6)

Tables (7)

Equations (19)

Optics Express

Zernike term added on surface	Generated Zernike aberrations
$M_{4} Z_{4}^{a d j}$	$(s^{2} - 1) M_{4} Z_{5}$
$M_{5} Z_{5}$	$(0.5 s^{2} + 0.5) M_{5} Z_{5}$
$M_{8} Z_{8}^{a d j}$	${\begin{array}{l} (0.25 s^{2} + 0.75) M_{8} Z_{8}^{a d j} \\ (0.75 s^{2} - 0.75) M_{8} Z_{11} \end{array}$
$M_{9} Z_{9}^{a d j}$	${\begin{array}{l} (0.375 s^{4} + 0.25 s^{2} + 0.375) M_{9} Z_{9}^{a d j} \\ (0.75 s^{4} - 0.75) M_{9} Z_{12}^{a d j} \\ (0.75 s^{4} - 1.5 s^{2} + 0.75) M_{9} Z_{17} \end{array}$
$M_{11} Z_{11}$	${\begin{array}{l} (0.25 s^{2} - 0.25) M_{11} Z_{8}^{a d j} \\ (0.75 s^{2} + 0.25) M_{11} Z_{11} \end{array}$
$M_{12} Z_{12}^{a d j}$	${\begin{array}{l} (0.25 s^{4} - 0.25) M_{12} Z_{9}^{a d j} \\ (0.5 s^{4} + 0.5) M_{12} Z_{12}^{a d j} \\ (0.5 s^{4} - 0.5) M_{12} Z_{17} \end{array}$
$M_{17} Z_{17}$	${\begin{array}{l} (0.0625 s^{4} - 0.125 s^{2} + 0.0625) M_{17} Z_{9}^{a d j} \\ (0.125 s^{4} - 0.125) M_{17} Z_{12}^{a d j} \\ (0.125 s^{4} + 0.75 s^{2} + 0.125) M_{17} Z_{17} \end{array}$