1. Introduction

Optical systems show different wavefronts depending on its alignment state and there have been several studies regarding this subject. Shack and Thompson’s pioneering work has described the field responses of a mis-aligned telescope system using the third-order aberration framework[1]. This theory has recently been revisited by Thompson[2] who added detailed derivations of the vector formulation. The application of this approach has mainly been focused on the secondary mirror alignment problem in two-mirror telescopes such as the Mt. Hopkins telescope and the MMT[3], the NTT[4, 5], the VLT[6], and the VISTA[7].

In parallel with the aforementioned studies on two-mirror telescope alignment problems, there have also been investigations of multi-element cases from more practical perspective : Chapman and Sweeney described a four-mirror microlithographic optics alignment by compensating aberrations in the most effective way[8] and Lee showed a six-element Cassegrain system alignment using wavefront aberrations and non-linear optimisation methods[9]. Such cases show the common problem that multiple elements are mis-aligned simultaneously, thereby coupled non-linear alignment influences could make a significant contribution to the system wavefront error. This problem frequently makes optical alignment procedures challenging and yet, to our knowledge, has not been rigorously studied.

Therefore, this paper aims to provide i) clearer understanding of the coupled non-linear effect onto the resulting system wavefront caused by simultaneously mis-aligned elements and ii) the methodological framework describing such a non-linearity in the practical optical alignment. This framework then, in turn, leads to a CGA model for practical alignment applications to be reported elsewhere[10]. In so doing, we describe the mis-alignment and the wavefront error by mathematical complex quantities. The possibility of adopting this complex approach for alignment study was briefly discussed in early works[1, 2], but has not been explored in depth during the subsequent years.

Starting with the theory on the complex alignment parameters and the relation of the alignment state with the optical wavefront in Sec. 2, we explain the methods used for describing this relation in more detail in Sec. 3. We then discuss the phase and amplitude modulation of the alignment-influenced wavefront in Sec. 4 and present its model using complex alignment parameters in Sec. 5. The model’s physical implications and practical aspects, together with an estimate of the modelling accuracy, are discussed in Sec. 6 and we summarise our findings in Sec. 7. The optical prescriptions of an example system used here and the Zernike coefficients of some of the influence functions shown in the paper are given in the appendix.

2. Wavefront perturbation and complex alignment parameter

Although the alignment tolerance is system-specific, it is generally true that the tolerance range is far smaller than the size of an optical system. Therefore, the alignment effect can be regarded as the perturbation given to the alignment parameters and this leads us to the pertubation formalism in describing the alignment influence.

In a system with M optical elements, including a source, we can define the alignment state of each element, describing the position and orientation with respect to a reference entity, as

‘i’ is the element index from 0 for the source to M for the last element. The linear position parameters, (x _i,y _i) and z _i, are called the lateral and axial parameters respectively. The angular orientation parameters, (θ _i,ϕ _i,ω _i), are rotations about the x, y, z-axis respectively. ω _i may be ignored in an axially symmetric element. The lateral and angular parameters are again called the directional parameters as they are associated with a particular axis on the pupil plane.

Assuming the optical wavefront(Φ) is a mathematically well-defined function of u _i and (ξ,η), the pupil coordinates in the Cartesian coordinate system, it can be expanded into a series by perturbing u _i around its nominal as,

where δ u _i = (δx _i,δy _i,δz _i,δθ _i,δϕ _i,δω _i) and the derivatives of order m occurring in successive terms define the alignment influence functions λ_i,j,... ^(m) for each element or combination of elements, i, j, . . ., for respective perturbation orders. These functions contain the characteristic wavefront variation at each order over the pupil plane. Therefore, analysing it by the complex Zernike polynomials enables us to quantify the aberrations associated with a particular alignment perturbation of order m.

On the other hand, an on-axis system, like the one we used here(Table 1 in the appendix), has an axially symmetric circular field-of-view(FOV) and optical surfaces. As a result, its alignment influence is also axially symmetric and the influences of δx and δy are the same with each other so are the influences of δϕ and δθ. This means that these parameters can be modeled as complex quantities as expressed in Eq. 3.

Because of the symmetry, ω _i can be ignored and δz is modeled as a complex number without the imaginary part. Therefore its phase g is either 0 or π while z_a and z_c have phase from 0 to 2π, forming a circle on the complex plane.

Then Eq. 2 can be described in a new way by those two complex quantities and, for this, we investigated the detailed physical relation between them using the methods described in the next section.

3. Simulation and analysis approaches

We defined a grid by considering a discrete set of N values of the alignment perturbation for each element within the range given in Table 2, such that we have finer sampling of each perturbation close to the nominal value. The k-th point, x_k, was determined by x_k = cos(π(k- 0.5)/N) within [-1,1] and then normalised to the actual range[11]. For optical components, the range of perturbation was set to 10 times the nominal build tolerance, considered sufficient for the analysis. For the source, the alignment perturbation can be defined for only its angular alignment parameters with respect to the stop as it is plane parallel and its range was defined as large as the system FOV.

Table 2. The perturbation grid

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At each grid point, we evaluated Φ of a ray, passing through a particular pupil coordinate (ξ,η) with respect to the chief ray, at the exit pupil of the system 500.63mm behind the nominal focus. This was repeated over the entire range of pupil plane coordinates, yielding Φ as a function of ξ and η. After sampling Φ at all perturbation grid points, a set of sampled wavefronts was created and we denote this set by {Φ}.

The dependency of {Φ} on the alignment perturbations can be computed by approximation using some form of polynomials. We used the Chebyshev polynomial for this approximation because it is not only able to give the smallest deviation from the true function, but also easily computed as described by Press et al.[11]

We then extracted the subset of wavefronts corresponding to a particular pupil coordinate (ξ,η). Decomposing this subset into Chebyshev polynomials yields a set of coefficients which are then converted into the coefficients of Eq. 2 using the method given in [11]. By performing the calculations to the entire {Φ}, i.e. over the entire pupil coordinates, one can obtain the complete coefficient for each term in Eq. 2, i.e. λ ^(m).

Without the dependency on δ u _i, λ ^(m) is now just a function of two pupil coordinates. This contains the characteristic wavefront perturbation at a particular order m and therefore one can identify the associated aberration components by decomposing λ ^(m) into an orthogonal basis such as the Zernike series. This is mathematically equivalent to solving λ ^(m) = Z x, where Z and x are the Zernike matrix and the coefficient vector respectively.

4. Computed alignment influence functions and perturbation characteristics

4.1. The first-order influence function : λ⁽¹⁾

λ ⁽¹⁾ corresponds to the coefficient of the first term in Eq. 2, which linearly depends on δ u _i. It is plotted for different alignment elements in rows (1–4) and for different alignment parameters in columns (A-B) in Fig. 1. Two pupil coordinates, ξ and η, are normalised to unity and the unit of the z-axis is ‘μm/100μm’ for (δx, δy, δz), ‘μm/0.1°’ for (δθ, δϕ) of optical components, and ‘μm/0.8°’ for (δθ, δϕ) of the source.

 

Fig. 1. λ⁽¹⁾ as made by ray-tracing and Chebyshev decomposition.

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Tilt and comatic patterns are generated by the directional alignment perturbations (column A,B,D,E), whereas axial perturbations show circularly symmetric influence patterns (column C). A symmetry is seen from the fact that the lateral perturbations show the same influence power but in the directions orthogonal to each other. The two angular perturbations also behave in the same way except that they are larger in magnitude than the lateral ones.

The linear relation makes it obvious that the coma aberration due to one alignment perturbation can be enhanced or canceled by each other, when they are associated with the same axis. This happens in any combination of two elements or groups of elements. The linear terms are independent of couplings between parameters. As a result, the comatic perturbation due to one alignment element remains constant against the alignment perturbation of another. For example, the optical components’ perturbation is the same over the entire range of the alignment perturbation of the source, i.e. the FOV, known as the constant field coma[1, 2, 4]. In this particular case, we can find the point on the FOV where the field-driven coma compensates the alignment-driven coma[1, 2, 4]. This point forms at a different location on the FOV depending on the alignment state of a system.

Another interesting characteristic of the alignment perturbations is shown in Fig. 2. Based on the symmetry among the alignment parameters, we set the lateral position errors of M1 to be δx = r(t) cost, δy = r(t) sint, where r(t) = 100 (1 - t/2π) μm, and sampled the wavefronts while varying t from 0 to 2π. The amplitude at a normalised pupil coordinate (-1,0), as marked by yellow circles in (B), is plotted against t in (A) and the top-view of the comatic component of the wavefront is shown in (B). The normalised amplitude exactly follows the input perturbation amplitude with 2π period(A). The physical interpretion of this is that these two alignment parameters are actually components of a single complex parameter which controls the magnitude and phase of λ ⁽¹⁾ simultaneously as we predicted in Sec. 2, which also happens for the angular parameters in the same way.

 

Fig. 2. (A) Amplitude at (ξ,η) = (-1,0) as a function of t, (B) Top-view of the wavefront.

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4.2. The second-order influence function : λ⁽²⁾

λ ⁽²⁾ corresponds to the coefficient of the second-order term in Eq. 2, which quadratically depends on δ u _i, and there are two possible cases : one is the coupling between two alignment parameters associated with the same alignment element and the other is the coupling between two alignment parameters but associated with different alignment elements. We discuss only M1 for the first case and then include M2 for the second. Note that in any cases the unit of λ ⁽²⁾ is ‘μm/(100μm)²’ for two lateral alignment parameters, ‘μm/(0.1°)²’ for two angular parameters, or ‘μm/(100μm) (0.1°)’ for one lateral and angular parameters. ξ and η are normalised to unity.

In Fig. 3(A), λ ⁽²⁾ of δx ² shows an astigmatic pattern. The influence of δz ² in (B) is symmetric as expected. Strong astigmatic influence is seen in (C) which shows the influences of δθ ². λ ⁽²⁾ of δθδϕ in (D) has a factor of ~ 2 more power than λ ⁽²⁾ of δθ ²(C). As shown in (E), the coupling of δϕ with δz generates comatic influence which is in sharp contrast to the other cases, showing that the comatic perturbation can also be generated from the second-order term. The main difference is that δϕδz includes only one directional parameter whereas terms like δϕδθ have two such parameters. Coupling between the lateral and angular parameters shows an astigmatic pattern (F).

 

Fig. 3. A selection of λ ⁽²⁾ for couplings between two parameters associated with M1.

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In the case of coupling between two elements(Fig. 4), the influences behave in a similar way to those in Fig. 3, except that the amplitudes depend on the elements involved in the coupling. For example, as M1 has stronger influence power than M2, the coupled influence of δϕ ₁ δx ₂ is few times stronger than that of δϕ ₂ δx ₁ (B,C). δϕ ₁ δz ₂ shows comatic influence again (E).

 

Fig. 4. A selection of λ ⁽²⁾ for couplings between one parameter of M1 and another of M2.

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As seen from previous two cases, the second-order perturbations implicitly depend on couplings between parameters and therefore the corrections are more complicated than those required for the first-order effects. Writing the second-order perturbation term for two alignment elements, we have

Fixing either δ u _i or δ u _j, Eq. 4 becomes a quadratic function for any combination of two alignment elements. In this case, it is interesting that the coupling term reduces to the first-order for δ u _i if we fix δ u _j. This means that the linear influence function of δ u _j appears differently depending on δ u _i. For instance, if we set the i-th and the j-th elements as the source and M1 of the system respectively and define the influences of M1 with respect to the source, M1 would develop the astigmatic influence in the first-order term as a function of the source’s alignment state. The magnitude of this influence is linearly proportional to the coupled influence λ_ij ⁽²⁾ in Eq. 4, known as the linear field-astigmatism[1, 2]. In this case, since Eq. 4 is a quadratic function, it embraces two solutions of the source’s alignment state given M1’s alignment state. These solutions correspond to the points on the FOV where astigmatism vanishses, known as the binodal points[1, 2].

In a similar way as shown in the λ ⁽¹⁾ case, the variation in the phase and amplitude of the wavefront is observed in the second-order case but with different characteristics as shown in Fig. 5. We used the same parameterisation of t and r(t) as used in Sec. 4.1, but for δϕ and δθ of M1. We plotted the astigmatic component of the sampled wavefronts. The amplitude curve exactly follows r ²(t)cos(2t) with the period halved to π. This can be interpreted as the phase and amplitude modulations by the complex alignment parameters.

 

Fig. 5. (A) Amplitude at (ξ,η) = (-1,0) with π period, (B) Top-view of the wavefront.

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5. Alignment-influenced wavefront model with complex alignment parameters

5.1. Wavefront perturbation function

From Fig. 2 and 5, we see that the alignment perturbation and its order determine the resulting wavefront shape by regulating the phase and amplitude of the alignment influence functions. This leads to the alignment-influenced wavefront model as follows.

We analyse λ ^(m) in more detail by expressing it as the Zernike polynomials in polar coordinates on the pupil plane (ρ,φ) instead of rectangular coordinates (ξ,η).

where δ _l0 = 0 if l ≠ 0 and δ _l0 = 1 if l = 0 and _mA_k,l, $R_k^l$, and ϴ^l are the coefficient, radial, and angular functions of the Zernike polynomials[12]. Now, we introduce Δ^l_m to represent the alignment perturbation in the form of a product of all participating complex perturbations. Its general expression is given below although it appears differently depending on m and l.

where p₁(i), p₂(i), p₃(i) are the number of complex perturbations of the i-th element for z_a, z_b, and z_c, respectively.

The wavefront perturbation model must be in the form of Eq. 2 where the alignment perturbations are multiplied by the alignment influence functions. As Δ^l_m needs to modify the phase of λ ^(m), the multiplication must occur within the ‘Real{}’ operator in Eq. 5. Omitting the normalisation constant for convenience, we get the wavefront perturbation function at order m,

where the integers, $n={\displaystyle {\sum }_{\text{i}=0}^{\text{M}}\{p_1(\text{i})+p_3(\text{i})\}},q={\displaystyle {\sum }_{\text{i}=0}^{\text{M}}{p}_2(\text{i})},$, and m = n + q, can be either zero of positive.

For any alignment element, z_a and z_c affect the phase of ϴ^l_m in Eq. 7 and generate non-circularly symmetric perturbations. z_b modulates only the amplitude of the perturbation in either ‘+’ or ‘-’ direction. For a given n, there are a set of admissible ls which in turn determine a set of admissible ks. For example, in λ ⁽¹⁾ of δx, n = 1 and l = 1 result in k = 1,3,5, ⋯ corresponding to R ¹ _k of comatic and tilt components in the Zernikes.

Physically, l represents the total number of z_a and/or z_c modulating the wavefront phase in the direction as set by Eq. 3. Therefore when l < n, in most of the higher-order cases, some of these parameters become complex conjugate by which the effective number of parameters changing the phase in clockwise direction becomes l. In this case there will be several possible conjugate pairs, all of which must be taken into account in calculating δΦ^(m).

5.2. Examples

(A). The first-order perturbation associated with z_a of a single alignment element

The only possible angular order is l = 1 and Δ¹ ₁ = z_a = a e ^-if , resulting in

The fact that only l = 1 is admissible implies that the influence functions associated with the first-order lateral perturbation have tilted and comatic patterns which is consistent with the calculations in Sec. 4.1.

(B). The second-order perturbation associated with z_c of a single alignment element

The second-order perturbation of z_c has two angular orders of l = 0 and l = 2. When l = 0, the angular term of ${\Delta }_2^0$ vanishes so that ${\Delta }_2^0=z_c{z}_c^{*}+z_c^{*}{z}_c=2c^2$ where we wrote the sum of z_c z*_c and z*_c z_c to underscore two possible combinations at l = 0. ${\Delta }_2^2$ has the angular order l = 2 and therefore ${\Delta }_2^2$ = z_c z_c = c ² e ^-2ih. Inserting these into Eq. 7 yields

As seen in Sec. 4.2, axially symmetric (l = 0) and astigmatic (l = 2) terms appear in Eq. 10 and the power of the cross-coupled term δϕδθ is exactly twice of the amplitude of δϕ ² - δθ ². If δϕ and δθ are associated with the source, Eq. 10 can be interpreted as a combination of two components known as field curvature and astigmatism and λ ⁽²⁾ for this case is shown in Fig. 6.

 

Fig. 6. λ ⁽²⁾ of the source (unit : μm/(0.8°)²)

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(C). The second-order perturbation associated with z_a of M1 and z_c of M2

Similar to (B), we have two possible angular orders of l = 0 and l = 2 and the perturbation terms for two cases are

The two terms in the first line of the above have physically different meanings : z _a,1 z*_c,2 indicates anti-clockwise phase modulation by M2 whereas z*_a,1 z _c,2 means anti-clockwise phase modulation by M1, resulting in two different coefficients, ₂ A _k,0 and ₂ Ã _k,0, at l = 0 (Eq. 12).

Eq. 10 and 12 reproduce part of the astigmatism due to the lateral and angular perturbations from a single alignment element as well as from pairs of elements, which retains the same form but with different coefficients for the different alignment elements. The particular case of the source and a single optical component has been investigated in a different way elsewhere [1, 2].

In a similar way, it is possible to derive the aberration components for the higher-orders and to identify their behaviour and contribution to the overall wavefront perturbation. For example, one can expect trefoil pattern in the third-order perturbation of z_a and/or z_c at l = 3. For z_c of the source, we have Δ¹ ₃ = c ³ ₀ e ^-ih₀ and Δ³ ₃ = c ³ ₀ e ^-3ih₀ and the perturbation function becomes

As shown in Fig. 7, λ ⁽³⁾ clearly shows trefoil pattern at l = 3 and λ ⁽³⁾ of δθ ² ₀ δϕ ₀ shows three times larger amplitude (9.051×10^-3 μm/(0.8°)³) than λ ⁽³⁾ of δϕ ³ ₀ (3.017×10^-3 μm/(0.8°)³) as predicted by Eq. 13. Using the same parameterisation used in Sec. 4.1, we found that the trefoil amplitude follows the cube of r(t) with 2π/3 period (C). (D) shows the rotation of the wavefront on the pupil plane, confirming the presence of the phase and amplitude modulation as observed in Sec. 4.1 and 4.2.

 

Fig. 7. (A), (B) λ ⁽³⁾ of each term associated with cos 3φ in Eq. 13 (unit : μm/(0.8°)³), (C) Amplitude at (ξ,η) = (-1,0) with 2π/3 period, (D) Top-view of the wavefront.

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6. Discussions

As we have seen so far, the resultant wavefront is created by complicated interactions among the elements involved in the system. The same aberration pattern can be generated by all elements but with different magnitudes and these can enhance or cancel each other. This can provide many ways of aberration compensation, which makes it hard to align multi-element systems due to the presence of local minima in optimisation-based methods. Therefore, when multiple elements are mis-aligned, quantifying the coupled multi-element effect should be an essential prerequisite leading to the proper use of the linear or quadratic alignment influence matrices in practical alignment applications.

As shown in Sec. 4, there is the influence symmetry among the alignment parameters due to the axial symmetry of surfaces and the FOV. This symmetry would certainly break for non-symmetric elements, for example off-axis and mechanically deformed surfaces. The complex parameter of such elements would appear differently with curves other than a circle on the complex plane. Rotation parameters about the optical axes of elements, which have been ignored here, would play another important role.

Throughout the calculations, we have witnessed that both the field- and alignment-driven aberrations show the same characteristics. This enabled us to describe the field as part of the alignment state of the source and therefore to model the alignment and field effect in terms of a single perturbation parameter Δ^l_m in Eq. 6. This modeling approach leads to Eq. 7 accounting for both the field- and alignment-driven influences on the wavefront.

Using this framework, we can describe the interdependencies among alignment parameters and/or elements in a more consistent way. For example, as we briefly noted in Sec. 4.2, the optical elements’ alignment influences can be described as a function of the source’s alignment state, i.e. field. For fixed δ u ₀, λ⁽¹⁾ of M1 becomes,

where the subscript (1,0) means the coupling between M1 and the source. This implies that λ ⁽¹⁾ of M1 can show higher-order influence patterns for different alignment states of the source as well as the intrinsic comatic influence. To see this effect, we plotted λ ⁽¹⁾ of δϕ ₁ sampled at δθ ₀ = 0.8° and δθ ₀ = 0.0° in Fig. 8(A) and (B) respectively, where (A) shows a slightly distorted shape compared to (B). As two angular parameters are considered, Eq. 14 indicates the presence of an astigmatic pattern due to λ ⁽²⁾ _1,0 in λ ⁽¹⁾, which is shown in (C) by subtracting (B) from (A). This additional effect corresponds to the terms after the second in Eq. 14 and can be reconstructed by summing up those terms as shown in (D). Note that negligible difference, dominated by numerical noise, between (C) and (D) is plotted in (E).

 

Fig. 8. λ ⁽¹⁾ of δϕ ₁ sampled by fixing (A) δθ ₀ = 0.8° and (B) δθ ₀ = 0.0°, (C) Difference of (A) from (B), (D) Difference computed from summing up the coupled influences of δϕ ₁ and δθ ^j ₀ from j = 1 up to 6, (E) Difference of (D) from (C) (unit : μm/0.1°)

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Considering the above and the fact that λ ^(m) can be described by the Zernike aberration coefficients, it is natural that the linear(or even higher-order) alignment sensitivity matrix of the Zernike coefficients for optical elements appears differently at different field position. This, in the presented framework, is a result from defining elements’ influences with respect to the alignment state of a particular element. That element is the source in the case of Fig. 8.

Finally, we simplified the model by including only the first three terms of Eq. 2. We sampled the wavefront by ray-tracing at 10,000 alignment states which were randomly generated within the range in Table 3. We then compared the result with the simplified model to estimate the modeling accuracy. The results are shown in Fig. 9.

 

Fig. 9. Modeling accuracy estimation

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We have two sample wavefronts as made by ray-tracing and the model in Fig. 9(A) and (B), respectively. The model deviation is plotted in (C). In (D), the root-mean-square(RMS) of the model deviation is plotted against the RMS wavefront aberrations. The mean value of the deviation remains of order 10^-3 μm and decreases with reduction of the wavefront aberration. In (E), the deviations within 3σ are plotted over the FOV, showing that the distribution appears rotationally symmetric. This naturally results from the fact that the FOV is rotationally symmetric and the terms excluded from the model apparently depend on the source’s alignment state as discussed in Fig. 8. This should be taken into account when one attempts to use a set of alignment sensitivity matrices sampled at different field positions in optical alignment.

Table 3. The modeling accuracy for three cases

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As shown in Table 3, the coupled effect between elements makes a substantial contribution to the model. Without the coupled effect in C2 and C3, the accuracy drops by a few orders of magnitude. This can highly affect the quality of the alignment state estimation by alignment methods relying on the linear sensitivity matrix and wavefront data obtained under the presence of measurement, control, and surface deformation uncertainty. Note that the source is generally less influential than the optical elements in terms of power per unit perturbation although its perturbation range is several times larger and therefore the model accuracy is strongly affected by ignoring the coupling effect of the source(C3). In summary, the interdependecies among elements make a considerable contribution to the alignment-influenced wavefront.

7. Conclusion

We described the phase and amplitude modulation observed in the alignment-influenced optical wavefront with special attention to the multi-element alignment problem. This modulation results from the interaction between the alignment perturbation and the influence function, both of which constitute the presented alignment-influenced wavefront model. This wavefront model clearly shows how the alignment state influences the optical wavefront and enables us to explain the coupled interaction between the alignment parameters. This naturally describes the field-dependency of the alignment sensitivity matrix without invoking the field separately. Also the model not only reproduces the formulae for the alignment-influenced low-order aberrations derived by past researchers[1, 2, 4], but generalises those for higher-order aberrations. Furthermore it shows the importance of the interdependencies among the optical elements, which substantially contribute to the resultant wavefront, particularly when multiple elements are mis-aligned simultaneously. We therefore conclude that our approach and the generalised model using the complex alignment parameters are not necessarily limited in use to the single-element alignment problem in a two-element system, but applicable to the multiple-element alignment cases in various systems. Also, in principle, it should be possible to extend our approach to the off-axis or segmented system alignment problems to obtain similar models as we derived here.

From a more practical perspective, the alignment-influenced wavefront model, particularly its interdependency part, can serve as the physical foundation for the alignment-guidance model, which could provide reliable alignment state estimations for the multi-element misalignment rectification. We will describe the investigation of this approach in a later paper[10] where various real-world factors, such as measurement and control uncertainties as well as surface deformations, are to be accounted for in order to examine the model’s practical feasibility.

Appendix : The optical prescription and Zernike coefficients

Table 1. The optical prescription of the example system

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Table 4. Zernike coefficients of some of the presented alignment influence functions

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References and links

1. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” in Optical alignment, R. M. Shagam and W. C. Sweatt, eds., Proc. SPIE 251,146–153 (1980)

2. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22,1389–1401 (2005). [CrossRef]  

3. B. McLeod, “Collimation of Fast Wide-Field Telescopes,” PASP 108,217–219 (1996). [CrossRef]  

4. R. N. Wilson and B. Delabre, “Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT,” PASP 109,53–60 (1997). [CrossRef]  

5. R. N. Wilson, Reflecting Telescope Optics Vol.I 2nd ed. (Springer-Verlag, Berlin, 2004).

6. L. Noethe and S. Guisard, “Analytic expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” A&A Supp. 144,157–167 (2000).

7. W. Sutherland, Alignment and Number of Wavefront Sensors for VISTA, VIS-TRE-ATC-00112-0012 (Technical report, Astronomy Technology Center, UK, 2001).

8. H. N. Chapman and D.W. Sweeney, “Rigorous method for compensation selection and alignment of microlithographic optical systems,” in Emerging Lithographic Technology, Y. Vladimirsky, eds., Proc. SPIE3331,102–113 (1998). [CrossRef]  

9. H. Lee, “Amon-Ra system alignment,” in Novel space optical instrument for deep space earth albedo monitoring, pp250–289 (Master thesis, Yonsei University, 2005).

10. H. Lee, G. B. Dalton, I. A. J. Tosh, and S. Kim are preparing a manuscript to be called “Computer-guided alignment II : Optical alignment via deliberate mis-alignment”.

11. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C 2nd ed., (Cambridge University Press, Cambridge, 1999).

12. M. Born and E. Wolf, Principles of Optics 7th ed, (Cambridge University Press, Cambridge, 2004).

13. K. B. Doyle, V. L. Genberg, and G. J. Michels, Integrated Optomechanical Analysis, (SPIE Press, Washington, 2002). [CrossRef]