Abstract

Our paper mainly separates the specific aberration contributions of third-order astigmatism and third-order coma from the total aberration fields, on the framework of the modified nodal aberration theory (NAT), for the perturbed off-axis telescope. Based on the derived aberration functions, two alignment models for the same off-axis two-mirror telescope are established and compared. Among them, one is based on third-order NAT, the other is based on fifth-order NAT. By comparison, it is found that the calculated perturbations based on fifth-order NAT are more accurate. It illustrates that third-order astigmatism and third-order coma contributed from fifth-order aberrations can’t be neglected in the alignment process. Then the fifth-order NAT is used for the alignment of off-axis three-mirror telescopes. After simulation, it is found that the perturbed off-axis three-mirror telescope can be perfectly aligned as well. To further demonstrate the application of the alignment method based on fifth-order NAT (simplified as NAT method), Monte-Carlo simulations for both off-axis two-mirror telescope and off-axis three-mirror telescope are conducted in the end. Meantime, a comparison between NAT method and sensitivity table method is also conducted. It is proven that the computation accuracy of NAT method is much higher, especially in poor conditions.

© 2016 Optical Society of America

1. Introduction

Compared to coaxial telescopes, off-axis reflective telescopes [1,2] own the better diffractive performance while maintaining the same optical performance. That means the redder, fainter and smaller target can be easily detected by off-axis telescopes. In recent years, some astronomical telescopes with off-axis structure (e.g. New Planetary Telescope, Advanced Technology Solar Telescope and Dark Energy Space Telescope) have been constructed or under construction. All these telescopes can be designed and aligned perfectly on the ground. However, their performance may become degraded when they are on orbit. To maintain the image quality, an active optics system [3,4] is usually equipped with astronomical telescopes. In active optics system, one important issue is how to determine the perturbations (including misalignments and figure errors) based on the measured wavefront errors. To solve it, some alignment algorithms have been presented.

At present, there are several alignment algorithms that have been widely recognized. Most of them can be used for active optical alignment of off-axis telescopes as well. Upton, Rimmele and Hubbard [5] discussed the active optical control of an off-axis Gregorian solar telescope (Advanced Technology Solar Telescope) by sensitivity table method (STM). Lundgren and Wolfe [6] aligned a three-mirror off-axis telescope by reverse optimization (RO). As is known, both STM and RO are numerical algorithms, which have some inherent defects. On one hand, the numerical algorithms are easily restricted by the coupling effect among optical elements. Some perturbations may not be decoupled if they have the same contribution to the aberration field, then the perturbed system can’t be perfectly aligned. On the other hand, the numerical algorithms are linear approximate. Their computing accuracy is restricted by the nonlinearity of aberration coefficients to perturbation parameters. When the perturbation ranges increase, the nonlinearity of aberration coefficients to perturbation parameter will decline, then the accuracy of linear approximation will decline, consequently, the calculated results are not accurate. To cover these shortages, an analytical alignment algorithm is in need. Actually, the analytical alignment algorithm, which is based on nodal aberration theory (NAT), has been developed recently. But this analytical algorithm is only studied for the alignment of coaxial telescopes [7–9]. To align off-axis telescopes, some new work should be done in this paper.

2. Wave aberration expansion in vector form for the perturbed off-axis telescope

The vector form of wave aberration expansion is the theoretic basis of NAT. It was discovered by Shack [10] and developed by Thompson [11–15]. For rotationally symmetric systems, it can be expressed by

W=jpnm(Wklm)j(HH)p(ρρ)n(Hρ)m,k=2p+m,l=2n+m,
where H is the normalized field vector, ρ is the normalized pupil vector, (Wklm)j is the wave aberration coefficient for surface j. Note that the total wave aberration is the sum of the individual surface contribution.

But for off-axis systems (an off-axis system is just regarded as an off-axis section of an on-axis parent system in this paper), their aberration fields can’t be described by Eq. (1). Some modifications must be made to Eq. (1) in order to describe the aberration field of an off-axis section of the on-axis system. As shown in Fig. 1, the decentered pupil is only an off-axis section of the parent pupil. The relationship between them can be described by pupil vector, which is given by

ρ=Bρ+h,
where ρ and ρ are the normalized pupil vectors of the decentered pupil and the parent pupil, h is the pupil decenter vector normalized to the parent pupil, B is the scaling factor. h and B are defined as
h=hxi+hyj,B=rorO,hx=oxrO,hy=oyrO,
where hx and hy are the x-component and y-component of h, ro is the radius of the decentered pupil, rO is the radius of the parent pupil, ox and oy are the coordinates of point o in XOY coordinate system.

 

Fig. 1 Pupil vector transformation between decentered pupil and parent pupil. The black circle represents the pupil of an on-axis system. The red circle represents the pupil of an off-axis section of the on-axis system.

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Then the vector form of wave aberration expansion for the off-axis telescope can be modified as

W=jpnm(Wklm)j(HH)p[(Bρ+h)(Bρ+h)]n[H(Bρ+h)]m.
Equation (4) is similar to Eq. (3) in [16] excluding the scaling factor B. Eq. (3) in [16] is just a special case when B = 1. By introducing the scaling factor, the wave aberration expansion for off-axis telescopes becomes more generalized.

However, Eq. (4) only describes the aberration field for the off-axis telescope without perturbations. If the off-axis telescope is perturbed, the center of the aberration field will displace again. (The center of the aberration field of the off-axis telescope has displaced before perturbations as described in [16,17]) To describe this effect, Buchroder [18] introduced a new vector to NAT. The new vector describes the effective field height after perturbations. It is given by

HAj=Hσj,
where HAj is the normalized field vector for surface j after perturbation, σj is the aberration field decenter vector for surface j. The aberration field decenter vector only depends on the misalignments of optical elements. It determines the center of the aberration field after perturbations.

Consequently, the vector form of wave aberration expansion for the perturbed off-axis telescope can be modified as

W=jpnm(Wklm)j(HAjHAj)p[(Bρ+h)(Bρ+h)]n[HAj(Bρ+h)]m.
If only 3rd-order and 5th-order aberrations are included in Eq. (6), it can be expanded by
W=12jW222j[HAj2(Bρ+h)2]+jW131j[HAj(Bρ+h)][(Bρ+h)(Bρ+h)]+jW040j[(Bρ+h)(Bρ+h)]2+jW240Mj(HAjHAj)[(Bρ+h)(Bρ+h)]2+jW331Mj(HAjHAj)[HAj(Bρ+h)][(Bρ+h)(Bρ+h)]+12jW422j(HAjHAj)[HAj2(Bρ+h)2]+14jW333j[HAj3(Bρ+h)3]+12jW242j[HAj2(Bρ+h)2][(Bρ+h)(Bρ+h)]+jW151j[HAj(Bρ+h)][(Bρ+h)(Bρ+h)]2+jW060j[(Bρ+h)(Bρ+h)]3.
where W240M=W240+12W242,W331M=W331+34W333. In Eq. (7), several terms describing the aberration field of piston, tilt and defocus are omitted. In order to study the specific aberration fields, each aberration in Eq. (7) must be further expanded.

At the same time, wave aberration can also be described as a sum of the weighted Zernike polynomial. The polynomial can be expressed as

W=iCi(Hx,Hy)Zi(ρ,φ).
where Zi is a Fringe Zernike term, Ci is the corresponding Fringe Zernike coefficient, Hx and Hyare the x-component and y-component of H, ρ and φ are the components of ρ. It can be seen that each Fringe Zernike coefficient in Eq. (8) is field dependent. To accurately determine the field dependence of each Fringe Zernike coefficient, all the aberrations in Eq. (7) should be expanded to the form of Eq. (8).

3. Nodal aberration theory for the perturbed off-axis telescope based on 3rd-order aberration expansion

It’s well known that third-order aberrations are dominated in on-axis telescope [19]. And they are independent of each other. In other words, third-order astigmatism only contributes to C5&C6, third-order coma only contributes to C7&C8. Thus the relationship between third-order aberration fields and Fringe Zernike coefficients, which is also called third-order NAT, can be easily determined. But for off-axis telescope, 3rd-order aberrations become complicated as expressed in Eq. (7). Some new aberrations must be derived after transformation. These derived aberrations may also have contributions to the lower-order Fringe Zernike coefficients (C5&C6&C7&C8). So it becomes difficult to determine the relationship between third-order aberration fields and Fringe Zernike coefficients for off-axis telescope.

In this section, we will pay our attention to determine the third-order NAT for the perturbed off-axis system. As expressed in Eq. (7), there are three third-order aberrations (including third-order astigmatism, third-order coma and third-order spherical aberration) altogether. Each aberration will be discussed in detail in the following.

3.1 Third-order astigmatism

As expressed in Eq. (7), third-order astigmatism for the perturbed off-axis telescope is followed by

W=12jW222jHAj2(Bρ+h)2,
According to the vector multiplication of NAT, Eq. (9) can be rewritten as
W=12jW222j[B2(HAj2ρ2)+Δ],
where Δ denotes the contributions to the lowest-order aberration fields (including piston, tilt and defocus), which are not considered in NAT. As expressed in Eq. (10), third-order astigmatism in Eq. (9) only has contributions to C5&C6. Equation (10) should be further expanded by citing Eq. (5) to describe the field dependence. The expanded aberration field can be expressed by a matrix, which is given by
B2[Hx2Hy2HxHy102HxHyHyHx01][W2222A222,x2A222,yB222,x2B222,y2]=2[C5w222C6w222].
where C5w222 and C6w222 denote the 5th and 6th Fringe Zernike coefficients contributed from third-order astigmatism, A222,x and A222,y are the x-component and y-component of A222, B222,x2 and B222,y2 are the x-component and y-component of B2222, A222=jW222jσj, B2222=jW222jσj2. Some analogous parameters used in the following will be defined in the Appendix, Table 15.

3.2 Third-order coma

As expressed in Eq. (7), third-order coma for the perturbed off-axis telescope is followed by

W=jW131j[HAj(Bρ+h)][(Bρ+h)(Bρ+h)],
According to the vector multiplication of NAT, Eq. (12) can be rewritten as
W=jW131j[B3(HAjρ)(ρρ)+B2(HAjhρ2)+Δ],
Equation (13) indicates that third-order coma in Eq. (12) has contributions to C5&C6&C7&C8. Similar to third-order astigmatism, Eq. (13) after expansion can be expressed by two matrices, which are given by
B3[Hx10Hy01][W131A131,xA131,y]=3[C7w131C8w131],
B2[hy00hy][Hy01Hx10][W131A131,xA131,y]=[C5w131C6w131].
where hy is the y-component of h, its x-component is usually regarded as zero because of the bilateral symmetry of off-axis telescopes, A131,x and A131,y are the x-component and y-component of A131, A131=jW131jσj. It can be seen that there is a certain relationship between Eq. (14) and Eq. (15). The relationship can be described by

C5w131=3hyBC8w131C6w131=3hyBC7w131.

3.3 Third-order spherical aberration

As expressed in Eq. (7), third-order spherical aberration for the perturbed off-axis telescope is followed by

W=jW040j[(Bρ+h)(Bρ+h)]2,
According to the vector multiplication of NAT, Eq. (17) can be rewritten as
W=jW040j[B4(ρρ)2+4B3(hρ)(ρρ)+2B2(h2ρ2)+Δ],
Equation (17) indicates that third-order spherical aberration in Eq. (16) has contributions to C5&C6&C7&C8&C9. Similar to subsection 3.1 and 3.2, Eq. (17) after expansion can be expressed by three matrices, which are given by
B4W040=6C9w040,
4B3hyW040=3C8040,
2B2hy2W040=C5w040.
Similar to Eq. (16), there is a certain relationship from Eq. (19) to Eq. (21). The relationship can be described by

C8w040=8hyBC9w040C5w040=12hyBC9w040.

3.4 Third-order NAT for the perturbed off-axis telescope

As described in 3.1, 3.2 and 3.3, we can find that

  • a. The 5th and 6th Fringe Zernike coefficients of the perturbed off-axis telescope contribute from three parts, including third-order astigmatism, third-order coma and third-order spherical aberration.
  • b. The 7th and 8th Fringe Zernike coefficients of the perturbed off-axis telescope contribute from two parts, including third-order coma and third-order spherical aberration.

So the 5th and 6th Fringe Zernike coefficients contributed from third-order astigmatism can be described as

{C5w222=C5C5w131C5w040C6w222=C6C6w131.
And the 7th and 8th Fringe Zernike coefficients contributed from third-order coma can be described as
{C7w131=C7C8w131=C8C8w040.
For simplicity, the notation hyB in Eq. (16) and Eq. (22) is replaced by A. In this case, Eq. (23) and Eq. (24) can be expressed by
{C5w222=C5+3AC8+12AC924A2C9C6w222=C63AC7,
{C7w131=C7C8w131=C88AC9.
Then the Fringe Zernike coefficients in Eq. (11) and Eq. (14) can be replaced by Eq. (25) and Eq. (26). The equations after replacement are expressed by
B2[Hx2Hy2HxHy102HxHyHyHx01][W2222A222,x2A222,yB222,x2B222,y2]=2[C5+3AC8+12AC924A2C9C63AC7],
B3[Hx10Hy01][W131A131,xA131,y]=3[C7C88AC9].
Then the third-order NAT for the perturbed off-axis telescope is completed.

4. An alignment example for the off-axis two-mirror telescope based on third-order NAT

For the alignment of off-axis two-mirror telescopes, three are three steps altogether based on NAT. At first, the linear perturbation vectors (A222,x&A222,y&A131,x&A131,y) should be determined. Based on Eq. (27) and Eq. (28), it can be seen that they can be easily determined if the Fringe Zernike coefficients can be measured by wavefront sensing. Next, the aberration field decenter vectors of secondary mirror (SM) (σSM,xsph&σSM,xasph&σSM,ysph&σSM,yasph) should be determined. According to Eq. (3-38) and Eq. (3-43) in [19], the linear perturbation vectors (A222,x&A222,y&A131,x&A131,y) for two-mirror telescopes are expressed by

{A222x=W222,SMsphσSM,xsph+W222,SMasphσSM,xasphA222y=W222,SMsphσSM,ysph+W222,SMasphσSM,yasphA131x=W131,SMsphσSM,xsph+W131,SMasphσSM,xasphA131y=W131,SMsphσSM,ysph+W131,SMasphσSM,yasph,
where W222,SMsph&W222,SMasph&W131,SMsph&W131,SMasph are the wave aberration coefficients of SM, which are constant, sph and asph denote the spherical and aspherical contributions, respectively. And A222,x&A222,y&A131,x&A131,y have been determined above. So the aberration field decenter vectors of SM can be directly determined based on Eq. (29). Lastly, the perturbations of SM (XDESM&YDESM&ADESM&BDESM) should be determined. Referred to Eq. (3-34)-Eq. (3-37) in [19], they are expressed by
{XDESM=u¯PMd1σSM,xasphYDESM=u¯PMd1σSM,yasphADESM=u¯PM(1+cSMd1)σSM,ysphcSMYDESMBDESM=u¯PM(1+cSMd1)σSM,xsph+cSMYDESM,
where u¯PM is the paraxial chief ray incident angle at PM, d1 is the thickness of primary mirror (PM), cSM is the curvature of SM, they are all constant. And the aberration field decenter vectors of SM (σSM,xsph&σSM,xasph&σSM,ysph&σSM,yasph) have been determined above. So the perturbations of SM (XDESM&YDESM&ADESM&BDESM) can be determined based on Eq. (30).

In this section, an off-axis two-mirror telescope is selected to simulate the performance of the alignment algorithm only based on third-order NAT. As shown in Fig. 2, the off-axis telescope is the main optical system of New Solar Telescope (NST). It has a 1600mm aperture stop (located at primary mirror) with a 0.03° × 0.03° field of view (FOV). The optical parameters of NST are listed in Table 1. The wave aberration coefficients of SM for the parent system of NST are listed in Table 2.

 

Fig. 2 Layout of the main optical system of New Solar Telescope (NST). It can be seen that the offset of aperture stop is 1840mm in the positive direction of Y axis. Then the defined parameter A in Eq. (25) and Eq. (26) can be determined. A=1840/800.

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Tables Icon

Table 1. Optical Parameters of NST

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Table 2. Wave Aberration Coefficients of SM for NST

During simulation, four FOVs ((−0.03°, −0.03°), (−0.03°, 0.03°), (0.03°, −0.03°), (0.03°, 0.03°)) are selected for Eq. (27) and Eq. (28). For each FOV, there are 5 Fringe Zernike coefficients (from 5-th term to 9-th term) that are needed to be measured in Zemax. To fit these coefficients, a 128 × 128 grid of rays is traced for each measurement. Based on these measured Fringe Zernike coefficients, the alignment process can be finished based on Eq. (27)-Eq. (30) through the dynamic data linking (DDL) between Zemax and Matlab. Here we introduce some misalignments to NST and calculate them based on these equations. The introduced misalignments are listed in Table 3. The calculated misalignments and the relative errors between them are listed in Table 4.

Tables Icon

Table 3. Introduced Misalignments of SM for NST

Tables Icon

Table 4. Calculated Misalignments of SM for NST Based on Third-order NAT and Relative Errors between Calculated and Introduced Misalignments

Compared Table 3 and Table 4, we can find that the calculated angular misalignments (ADESM&BDESM) are very close to the introduced. It is proven that the idea of optical alignment for the off-axis two-mirror telescopes used here is reasonable. But the calculated lateral misalignments (XDESM&YDESM) are not very accurate. That may be because some additional aberration contributions to the low-order aberration fields that are not considered, resulting that the computation accuracy is unacceptable in the alignment process.

Just as described at the beginning of section 3, third-order aberrations is dominated in on-axis telescope. For off-axis telescope, the dominance is unknown. Considering the alignment example, a conclusion can be made that the dominance of third-order aberrations is reduced. The contributions of fifth-order aberrations to the aberration fields can’t be neglected. That means fifth-order NAT should also be included in the alignment algorithm to accurately determine all the misalignments.

5. Nodal aberration theory for the perturbed off-axis telescope based on fifth-order aberration expansion

As concluded in section 4, the contributions from fifth-order aberrations have to be considered for the alignment of the perturbed off-axis telescopes. There are seven fifth-order aberrations altogether in Eq. (7). These aberrations all have influences on the lower-order aberration coefficients. Similar to section 3, fifth-order aberrations should be expanded one by one to determine the specific contributions.

5.1 Fifth-order astigmatism

As expressed in Eq. (7), fifth-order astigmatism for the perturbed off-axis telescope is followed by

W=12jW422j(HAjHAj)[HAj2(Bρ+h)2],
According to the vector multiplication of NAT, Eq. (31) can be rewritten as
W=12jW422j[B2(HAjHAj)(HAj2ρ2)+Δ],
Equation (31) is similar to Eq. (10). To describe the field dependence, Eq. (32) should be further expanded. The expanded matrix is followed by
B2[Hx4Hy44Hx34Hy32HxHy(Hx2+Hy2)6Hx2Hy2Hy32Hx36HxHy2][W422A422,xA422,y]=2[C5w422C6w422],
where A422,x and A422,y are the x-component and y-component of A422, A422=jW422jσj, and the higher-order terms with respect to field aberration decenter vectors are omitted because of their few contributions to the total aberration field.

5.2 Field-cubed coma

As expressed in Eq. (7), field-cubed coma for the perturbed off-axis telescope is followed by

W=jW331Mj(HAjHAj)[HAj(Bρ+h)][(Bρ+h)(Bρ+h)],
According to the vector multiplication of NAT, Eq. (34) can be rewritten as
W=jW331Mj[(HAjHAj)(HAjρ)(ρρ)+(HAjHAj)(HAjhρ2)+Δ],
Equation (35) is similar to Eq. (13). To describe the field dependence, Eq. (35) should be further expanded. The expanded matrices are followed by
B3[Hx3+HxHy23Hx2Hy22HxHyHx2Hy+Hy32HxHyHx23Hy2][W331MA331M,xA331M,y]=3[C7w331MC8w331M],
B2[hy00hy][Hx2Hy+Hy32HxHyHx23Hy2Hx3+HxHy23Hx2Hy22HxHy][W331MA331M,xA331M,y]=[C5w331MC6w331M],
where A331M,x and A331M,y are the x-component and y-component of A331M, A331M=jW331Mjσj, and the higher-order terms with respect to field aberration decenter vectors are also omitted. Similar to Eq. (16), C5&C6 contributed from field-cubed coma can be described by

C5w331M=3hyBC8w331MC6w331M=3hyBC7w331M.

5.3 Elliptical coma/trefoil

As expressed in Eq. (7), elliptical coma for the perturbed off-axis telescope is followed by

W=14jW333jHAj3(Bρ+h)3,
According to the vector multiplication of NAT, Eq. (39) can be rewritten as
W=14jW333j[B3(HAj3ρ3)+3B2(HAj3hρ2)+Δ],
Equation (40) indicates that elliptical coma in Eq. (39) has contributions to C5&C6&C10&C11. Similar to Eq. (38), C5&C6 contributed from elliptical coma can be described by C10&C11. It is given by

C5w333=3hyBC11w333C6w333=3hyBC10w333.

5.4 A component of oblique spherical aberration

As expressed in Eq. (7), the component of oblique spherical aberration for the perturbed off-axis telescope is followed by

W=jW240Mj(HAjHAj)[(Bρ+h)(Bρ+h)]2,
According to the vector multiplication of NAT, Eq. (42) can be rewritten as
W=jW240Mj[B4(HAjHAj)(ρρ)2+4B3(HAjHAj)(hρ)(ρρ)+2B2(HAjHAj)(h2ρ2)+Δ],
Equation (43) is similar to Eq. (18). The specific contributions to C5&C8 can be described by

C8w240M=8hyBC9w240MC5w240M=12hyBC9w240M.

5.5 Oblique spherical aberration

As expressed in Eq. (7), oblique spherical aberration for the perturbed off-axis telescope is followed by

W=12jW242j[HAj2(Bρ+h)2][(Bρ+h)(Bρ+h)],
According to the vector multiplication of NAT, Eq. (45) can be rewritten as
W=12jW242j[B4(HAj2ρ2)(ρρ)+B3(hHAj2ρ3)+3B3(HAj2hρ)(ρρ)+3B2(hh)(HAj2ρ2)+Δ],
Equation (46) indicates that oblique spherical aberration in Eq. (45) has contributions to C5&C6& C7&C8&C10&C11&C12&C13. Similar to subsection 5.2-5.4, C5&C6&C7&C8&C10&C11 contributed from oblique spherical aberration can be described by

{C10w242=4hyBC13w242C11w242=4hyBC12w242C7w242=4hyBC13w242C8w242=4hyBC12w242C5w242=12hy2B2C12w242C6w242=4hy2B2C13w242.

5.6 Fifth-order field-linear coma

As expressed in Eq. (7), fifth-order field-linear coma for the perturbed off-axis telescope is followed by

W=jW151j[HAj(Bρ+h)][(Bρ+h)(Bρ+h)]2,
According to the vector multiplication of NAT, Eq. (48) can be rewritten as
W=jW151j[B5(HAjρ)(ρρ)2+2B4(HAjhρ2)(ρρ)+B3(HAjh2ρ3)+3B4(HAjh)(ρρ)(ρρ)+6B3(hh)(HAjρ)(ρρ)+3B3(h2HAjρ)(ρρ)+2B2(hh)(HAjhρ2)+2B2(HAjh)(h2ρ2)+Δ],
Equation (49) indicates that fifth-order field-linear coma in Eq. (48) has contributions to C5&C6& C7&C8&C9&C10&C11&C12&C13&C14&C15. Similar to subsection 5.2-5.5, C5&C6& C7&C8&C9&C10&C11&C12&C13 contributed from fifth-order field-linear coma can be described by

{C12w151=5hyBC15w151C13w151=5hyBC14w151C10w151=10hy2B2C14w151C11w151=10hy2B2C15w151C9w151=5hyBC15w151C7w151=10hy2B2C14w151C8w151=30hy2B2C15w151C5w151=40hy2B2C15w151C6w151=20hy2B2C14w151.

5.7 Fifth-order spherical aberration

As expressed in Eq. (7), fifth-order spherical aberration for the perturbed off-axis telescope is followed by

W=jW060j[(Bρ+h)(Bρ+h)]3,
According to the vector multiplication of NAT, Eq. (51) can be rewritten as
W=jW060j[B6(ρρ)3+6B5(ρρ)2(ρh)+6B4(ρρ)(ρ2h2)+2B3(ρ3h3)+9B4(ρρ)2(hh)+18B3(ρρ)(ρh)(hh)+6B2(ρ2h2)(hh)+Δ],
Equation (52) indicates that fifth-order spherical aberration in Eq. (51) has contributions to C5&C6&C7&C8&C9&C10&C11&C12&C13&C14&C15&C16. Similar to subsection 5.2-5.6, C5&C6&C7&C8&C9&C10&C11&C12&C13&C14&C15 contributed from fifth-order spherical aberration can be described by

{C5w060=20hy4B4C16w060C8w060=120hy3B3C16w060C9w060=30hy2B2C16w060C11w060=40hy3B3C16w060C12w060=30hy2B2C16w060C15w060=12hyBC16w060.

5.8 Fifth-order NAT for the perturbed off-axis telescope

As described in subsection 3.1-3.4 and 5.1-5.7, we can find that the lower-order Fringe Zernike coefficients contribute from multiple aberrations. Among them, the 5th and 6th Fringe Zernike coefficients contributed from third-order astigmatism and fifth-order astigmatism can be expressed by

{C5w222+C5w422=C5sumC5w060C5w151C5w242C5w040+w240MC5w131+w331MC5w333C6w222+C6w422=C6sumC6w151C6w242C6w131+w331MC6w333.
And the 7th and 8th Fringe Zernike coefficients contributed from third-order coma and field-cubed coma can be expressed by
{C7w131+C7w331M=C7sumC7w151C7w242 C8w131+C8w331M=C8sumC8w060C8w151C8w242C8w040+w240M .
All the parameters in Eq. (54) and Eq. (55) have been stated in the Appendix Table 15. They have been derived in subsection 3.1-3.4 and 5.1-5.7. On the premise that hy/Bis simplified by A, Eq. (54) and Eq. (55) can be rewritten as
{C5w222+C5w422=C5+40A2C15+3AC812A2C93AC11+12A2C12+360A4C16480A3C16C6w222+C6w422=C63AC720A2C14+3AC10+12A2C13
{C7w131+C7w331M=C74AC13+10A2C14C8w131+C8w331M=C88AC9+4AC12+30A2C15+120A2C16240A3C16
Then the fifth-order NAT can be described as
B2[Hx2Hy22HxHyHxHyHyHx1001Hx4Hy42HxHy(Hx2+Hy2)4Hx36Hx2Hy2Hy34Hy32Hx36HxHy2]T[W2222A222,x2A222,yB222,x2B222,y2W422A422,xA422,y]=2[C5w222+C5w422C6w222+C6w422],
B3[HxHy1001Hx3+HxHy2Hx2Hy+Hy33Hx2Hy22HxHy2HxHyHx23Hy2]T[W131A131,xA131,yW331MA331M,xA331M,y]=3[C7w131+C7w331MC8w131+C8w331M],
where {C5w222+C5w422C6w222+C6w422 and {C7w131+C7w331MC8w131+C8w331Mare expressed as Eq. (56) and Eq. (57).

In Eq. (56) and Eq. (57), we can see that the weights of higher-order Fringe Zernike coefficients, which correspond to fifth-order aberrations, are much larger than the weights of lower-order Fringe Zernike coefficients. So fifth-order aberrations must be considered even if their contributions are relatively small. At the same time, we can also see that the 16th Fringe Zernike coefficient remains unchanged for different FOVs and different perturbed states. That means fifth-order NAT is enough for the alignment of the off-axis two-mirror telescope. It will be verified in next section.

6. An alignment example for the off-axis two-mirror telescope based on fifth-order NAT

In this section, the same off-axis two-mirror telescope as section 4 is used for simulation. And the introduced misalignments remain unchanged. But the alignment algorithm here is based on fifth-order NAT rather than third-order NAT. The determination of linear perturbation vectors are based on Eq. (58) and Eq. (59) rather than Eq. (27) and Eq. (28). In Eq. (58) and Eq. (59), the needed FOVs are the same as that in section 4. But the measured Fringe Zernike coefficients increase to 12 terms (from 5-th term to 16-th term) rather than 5 terms. And the rest equations (Eq. (29) and Eq. (30)) still remain unchanged. After simulation, the calculated misalignments and relative errors are listed in Table 5.

Tables Icon

Table 5. Calculated Misalignments of SM for NST Based on Fifth-order NAT and Relative Errors between Calculated and Introduced Misalignments

Compared Table 3 and Table 5, we can find that the calculated results are very accurate for each misalignment. It is proven that the alignment algorithm based on fifth-order NAT is successful.

However, the perturbed mirror figure errors are not discussed above. Actually, they should also be considered in active optical alignment, especially the astigmatic figure errors of primary mirror (PM). In on-axis telescope, they can be determined as described in [20]. But for off-axis telescope, some modifications should be made to determine them, which can be expressed by

B222,Fig2=B2(B2222B222,Mis2)
where B222,Fig2=2(nn)[C5,FigPMC6,FigPM], nn=2, C5,FigPM and C6,FigPM denote the astigmatic figure errors on PM, B is the scaling factor presented in the pupil vector transformation, B2222can be determined in Eq. (58), B222,Mis2 can be determined in the equation B222,Mis2=W222,SMsph(σSMsph)2+W222,SMasph(σSMasph)2

To verify Eq. (60), some astigmatic figure errors on PM as listed in Table 6 are introduced on the basis of Table 3. After simulation, the calculated perturbations (including misalignments of SM and astigmatic figure errors of PM) and their relative errors are listed in Table 7.

Tables Icon

Table 6. Introduced Astigmatic Figure Errors on PM

Tables Icon

Table 7. Calculated Perturbations Based on Fifth-order NAT and Relative Errors between Calculated and Introduced Perturbations

In Table 7, it can be seen that the calculated astigmatic figure errors of PM are very accurate. At the same time, the calculated misalignments of SM are comparatively accurate. The alignment algorithm based on fifth-order NAT is validated again.

To further demonstrate the computation accuracy of NAT method (the alignment method based on fifth-order NAT) for off-axis two-mirror telescopes, four Monte-Carlo simulations that correspond to four different cases, as shown in Table 8, will be performed. In Case 1, Case 2 and Case 3, the perturbation ranges increase step by step, but without any measurement error. In Case 4, the perturbation ranges are the same as Case 3, but with 1% measurement error for each Fringe Zernike coefficient.

Tables Icon

Table 8. Four Different Cases Considered in Monte-Carlo Simulations

In each case, 150 pairs of random perturbations following a standard uniform distribution are generated. Then there are 600 perturbed states altogether in the simulation. For each perturbed state, one set of perturbations will be calculated based on NAT method. Then we use the calculated results to recover the perturbed system. Here the average RMS wavefront errors (WFE) before and after alignment, calculated with 7 × 7 equally spaced field point in 0.03° × 0.03°, are compared in each perturbed state for each case, as shown in Fig. 3. Meantime, the root mean square deviation (RMSD) of every perturbation before and after alignment is calculated for each case. The RMSD can be expressed by

RMSDi=1150n=1150[Xi(n)xi(n)]2
where Xi(n) denotes the introduced perturbation, xi(n) denotes the calculated perturbation. The computation results for different cases are presented in Table 9.

 

Fig. 3 Average RMS WFE before and after alignment for different cases based on NAT method. (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4. Note that the pink spot represents the RMS WFE before alignment. The blue spot represents the RMS WFE after first alignment. The red spot represents the RMS WFE after second alignment. It is the same as the RMA WFE of the nominal design (0.062 waves).

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Tables Icon

Table 9. Root Mean Square Deviations (RMSDs) between Introduced and Computed Perturbations

From Fig. 3(a)-Fig. 3(c), it can be seen that the perturbed system can always be aligned perfectly (the average RMS WFE after alignment is 0.062 waves) based on the fifth-order NAT if the average RMS WFE before alignment is not more than 1 waves. From Fig. 3(d), it can be seen that the perturbed system can also be aligned to the nominal even if there are 1% measurement errors in the wavefront sensing. It is demonstrated that NAT method presented here is also successful in practical application. From Table 9, it can be seen that the RMSDs of all the perturbations are less than 0.01, which demonstrates that the computation accuracy is very high for every perturbation parameter in each case. So NAT method is applicable for active optical alignment of off-axis two-mirror telescopes.

7. An alignment example for the off-axis three-mirror telescope based on fifth-order NAT

Compared to off-axis two-mirror telescopes, off-axis three telescopes own more misalignment parameters. As a result, the aberration fields after misalignments may become more complicated. And the lower-order Fringe Zernike coefficients may be dominated by higher-order aberrations. In other words, fifth-order NAT may become inapplicable for the alignment of off-axis three-mirror telescopes. This question will be considered and settled in this section.

For an off-axis three-mirror telescope, its alignment process is similar to the alignment of off-axis two-mirror telescopes. Firstly, the determination of linear perturbation vectors (A222,x&A222,y&A131,x&A131,y) can also be obtained by solving Eq. (58) and Eq. (59). Then, the aberration field decenter vectors should be determined. To determine them, two more equations describing boresight errors (defined as the height of optical axis ray on TM and Image Plane) should be combined with the solved linear perturbation vectors. For off-axis three-mirror telescopes, they are expressed by

{HTM,x=CAσSM,xsphHTM,y=CAσSM,ysphHIMAGE,x=CBσSM,xsph+CCσTM,xsphHIMAGE,y=CBσSM,ysph+CCσTM,ysphA222x=W222,SMsphσSM,xsph+W222,SMasphσSM,xasph+W222,TMsphσTM,xsph+W222,TMasphσTM,xasphA222y=W222,SMsphσSM,ysph+W222,SMasphσSM,yasph+W222,TMsphσTM,ysph+W222,TMasphσTM,yasphA131x=W131,SMsphσSM,xsph+W131,SMasphσSM,xasph+W131,TMsphσTM,xsph+W131,TMasphσTM,xasphA131y=W131,SMsphσSM,ysph+W131,SMasphσSM,yasph+W131,TMsphσTM,ysph+W131,TMasphσTM,yasph
where HTM,x&HTM,y&HIMAGE,x&HIMAGE,y are the components of boresight errors that can be directly measured, CA&CB&CC are constants that are related to optical system parameters. W222,SMsph&W222,SMasph&W131,SMsph&W131,SMasph are the wave aberration coefficients of SM, W222,TMsph&W222,TMasph&W131,TMsph&W131,TMasph are the wave aberration coefficients of TM, both of them are constant for an optical system. σSM,xsph&σSM,xasph&σSM,ysph&σSM,yasph are the aberration field decenter vectors of SM, σSM,xsph&σSM,xasph&σSM,ysph&σSM,yasph are the aberration field decenter vectors of TM. So the aberration field decenter vectors can be determined based on Eq. (62). Lastly, the misalignments of SM and TM need to be determined. On the premise that the aberration field decenter vectors of SM and TM are known, the misalignments can be easily determined by
{XDESM=u¯PMd1σSM,xasphYDESM=u¯PMd1σSM,yasphADESM=u¯PM(1+cSMd1)σSM,ysphcSMYDESMBDESM=u¯PM(1+cSMd1)σSM,xsph+cSMYDESMXDETM=[d2+d1(2cSMd21)]u¯PMσTM,xasph+2d2(BDESM+cSMXDESM)YDETM=[d2+d1(2cSMd21)]u¯PMσTM,yasph+2d2(ADESM+cSMXDESM)ADETM=[cTM(d2d1)+2cSM(cTMd1d2+d1)+1]u¯PMσTM,ysph+2(1+cTMd2)(cSMYDESM+ADESM)cTMYDETMBDETM=[cTM(d2d1)+2cSM(cTMd1d2+d1)+1]u¯PMσTM,xsph2(1+cTMd2)(cSMXDESMBDESM)+cTMXDETM
Where XDESM&YDESM&ADESM&BDESM and XDETM&YDETM&ADETM&BDETM are the misalignments of SM and TM, u¯PM is the paraxial chief ray incident angle at PM, d1 and d2 are the thicknesses of PM and SM, cSM and cTM are the curvatures of SM and TM, they are all constant. And the values of aberration field decenter vectors have been determined above. So the misalignments of SM and TM can be calculated based on Eq. (63).

In this section, an off-axis Cook-TMA telescope [21,22] is selected to simulate the computation accuracy of NAT method presented above. As shown in Fig. 4, it’s a 600mm F/10 telescope with a 2.75° × 0.3° FOV and a −0.3° field offset. The wave aberration coefficients of SM and TM for the three-mirror parent telescope are listed in Table 10, including the values of CA&CB&CC at the same time.

 

Fig. 4 Layout of an off-axis Cook-TMA telescope. It can be seen that the offset of aperture stop is −460mm in the negative direction of Y axis. Then the defined parameter A in Eq. (56) and Eq. (57) can be determined. A=460/300.

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Tables Icon

Table 10. Wave Aberration Coefficients of SM and TM for Three-mirror Parent Telescope and Values of CA&CB&CC

In this simulation, six FOVs ((−1.375°,-0.175°), (0°, −0.175°), (1.375°,-0.175°), (−1.375°,-0.425°), (0°, −0.425°), (1.375°,-0.425°)) are selected for Eq. (58) and Eq. (59). For each FOV, there are 12 Fringe Zernike coefficients that need to be obtained in Zemax. To validate the application of fifth-order NAT for aligning off-axis three-mirror telescopes, we introduce some misalignments to the designed system and calculate them based on the algorithm. The introduced misalignments of SM and TM are listed in Table 11. The calculated misalignments and their relative errors are listed in Table 12.

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Table 11. Introduced Misalignments of SM and TM

Tables Icon

Table 12. Calculated Misalignments of SM and TM and Their Relative Errors

Compared Table 11 and Table 12, it can be found that all the misalignments can be accurately solved. It is demonstrated that the alignment algorithm based on fifth-order NAT for off-axis three-mirror telescopes is successful as well.

Similar to section 6, four Monte-Carlo simulations for different cases will also be performed in this section to demonstrate the computation accuracy of NAT method used for the alignment of off-axis three-mirror telescopes, as shown in Table 13. Besides, the sensitivity table method (STM) is compared with NAT method by aligning the same off-axis three-mirror telescope for the same cases. Here the RMS WFE is averaged at 45 (15 × 3) FOVs. After calculation, the average RMS WFE based on NAT method for each case is shown in Fig. 5, the average RMS WFE based on STM for each case is shown in Fig. 6. Meantime, the RMSDs between the introduced misalignments and the calculated misalignments based on Eq. (61) for the two alignment methods are also calculated, as shown in Table 14.

Tables Icon

Table 13. Four Different Cases Considered in Monte-Carlo Simulations

 

Fig. 5 Average RMS WFE before and after alignment for different cases based on NAT method. (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4. Note that the pink spot represents the RMS WFE before alignment. The blue spot represents the RMS WFE after alignment. The red spot represents the RMS WFE in nominal design (0.072 waves).

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Fig. 6 Average RMS WFE before and after alignment for different cases based on sensitivity table method (SMT). (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4. Note that the pink spot represents the RMS WFE before alignment. The blue spot represents the RMS WFE after alignment. The red spot represents the RMS WFE in nominal design (0.072 waves).

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Tables Icon

Table 14. RMSDs between Introduced and Calculated Misalignments Based on NAT and STM for Different Cases

Compared Fig. 5 with Fig. 6, we can see that the perturbed systems in any case can always be aligned to the nominal design based on NAT method. But for STM, the perturbed systems can only be aligned in small perturbation ranges and without any measurement errors (Case 1 and Case 2). If the perturbation ranges are larger (Case 3) or the measured Fringe Zernike coefficients are not very accurate (Case 4), the alignment process based on STM becomes unsuccessful at some perturbed states. For Case 3, that may be because the residual error after linear approximation in STM is also dominating. But this part of contributions has been omitted in large perturbation ranges. For Case 4, that may be because the linear system of equations in STM is ill-conditioned. As a result, the calculated results are easily influenced by measurement errors.

In Table 14, we can see that the computation accuracy of all the misalignments based on NAT method and STM are both high in Case 1 and Case 2. In Case 3 and Case 4, the computation accuracy of all the misalignments based on NAT method is also acceptable. But the computation accuracy of some misalignments based on STM is unacceptable, especially the misalignments of TM. That’s because SM is more sensitive to the aberration field than TM. When the perturbation range of TM is the same as that of SM, the aberration contribution of TM is easily drowned by that of SM. Hence computation accuracy of the misalignments of TM is relatively low.

Therefore, NAT method is also applicable for active optical alignment of off-axis three-mirror telescopes. And the alignment result based on NAT method is better than the alignment result based on STM. In active optical alignment, NAT method is a better choice.

8. Conclusion

In this paper, the vector-form wave aberration expansion for the off-axis telescope is firstly presented by pupil transformation of the on-axis telescope. Then both third-order NAT and fifth-order NAT of the perturbed off-axis telescope are derived based on the modified wave aberration expansion. In the end, an off-axis two-mirror telescope and an off-axis three-mirror telescope are selected to verify whether third-order NAT and fifth-order NAT are applicable for active optical alignment or not.

Compared to on-axis telescopes, there exist some derived aberrations in wave aberration expansion for off-axis telescopes. In third-order NAT, it can be seen that both third-order spherical aberration and third-order coma have contributions to C5&C6, which is only contributed from third-order astigmatism in on-axis system. This is the main difference between off-axis telescopes and on-axis telescopes. In order to determine the single contribution from third-order astigmatism, the contributions from third-order spherical aberration and third-order coma should be subtracted from the total astigmatic field. It is same to third-order comatic field. If fifth-order aberrations are considered, more derived terms should be subtracted to determine the fifth-order NAT.

According to Eq. (56) and Eq. (57), it can be found that the weights of higher-order Fringe Zernike coefficients are much larger than the weights of lower-order Fringe Zernike coefficients. That means some third-order aberrations derived from fifth-order aberrations have great influence on the low-order aberration field. So fifth-order aberrations must be considered even if their contributions are relatively small.

As expected and simulated, the alignment result based on fifth-order NAT is much more accurate than that only based on third-order NAT. Specially, the maximum calculated error is more than 16% in the alignment of off-axis two-mirror telescope based on third-order NAT. But it is less than 1% if the alignment process is based on fifth-order NAT. So the computation accuracy of fifth-order NAT used for the alignment of off-axis two-mirror telescopes is very high. For off-axis three-mirror telescope, only one parameter’s calculated error is more than 1%. The calculated errors of other parameters are all less than 1%. So NAT method is also applicable to the alignment of off-axis three-mirror telescopes.

What’s more, Monte-Carlo simulations for both off-axis two-mirror telescope and off-axis three-mirror telescope are conducted in this paper. And a comparison between NAT method and sensitivity table method is also conducted in the end. It is demonstrated that NAT method is applicable for larger perturbation ranges. It is also demonstrated that NAT method is applicable even considering the measurement error. It is a better choice in active optical alignment.

Therefore, the work in this paper facilitates the active optical alignment of the perturbed off-axis telescope, either two-mirror or three-mirror. It is meaningful for the development of active optics in astronomical telescopes.

Appendix

Tables Icon

Table 15. Parameter Definitions and Vector Identities

Funding

National Natural Science Foundation of China (NSFC) (61205143).

References and links

1. H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998). [CrossRef]  

2. J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999). [CrossRef]  

3. R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987). [CrossRef]  

4. M. Liang, V. Krabbendam, C. F. Claver, S. Chandrasekharan, and B. Xin, “Active Optics in Large Synoptic Survey Telescope,” Proc. SPIE Astronomical Telescopes + Instrumentation. International Society for Optics and Photonics, 84444Q–84444Q–13 (2012). [CrossRef]  

5. R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006). [CrossRef]  

6. M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991). [CrossRef]  

7. R. Tessieres, “Analysis for alignment of optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).

8. T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment of two-mirror astronomical telescopes; the astigmatic component,” Proc. SPIE 7017, 701711 (2008).

9. Z. Gu, C. Yan, and Y. Wang, “Alignment of a three-mirror anastigmatic telescope using nodal aberration theory,” Opt. Express 23(19), 25182–25201 (2015). [CrossRef]   [PubMed]  

10. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980). [CrossRef]  

11. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).

12. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005). [CrossRef]   [PubMed]  

13. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009). [CrossRef]   [PubMed]  

14. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010). [CrossRef]   [PubMed]  

15. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821–836 (2011). [CrossRef]   [PubMed]  

16. J. Wang, B. Guo, Q. Sun, and Z. Lu, “Third-order aberration fields of pupil decentered optical systems,” Opt. Express 20(11), 11652–11658 (2012). [CrossRef]   [PubMed]  

17. H. Hu, J. Liu, and Z. Fan, “Interaction of pupil offset and fifth-order nodal aberration field properties in rotationally symmetric telescopes,” Opt. Express 21(15), 17986–17998 (2013). [CrossRef]   [PubMed]  

18. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).

19. T. Schmid, “Misalignment Induced Nodal Aberration Fields and Their Use in the Alignment of Astronomical Telescopes,” Ph.D. dissertation (University of Central Florida Orlando, Florida, 2010).

20. T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010). [CrossRef]   [PubMed]  

21. M. L. Lampton, M. J. Sholl, and M. E. Levi, “Off-axis telescopes for dark energy investigations,” Proc. SPIE Astronomical Telescopes + Instrumentation. International Society for Optics and Photonics, 77311G–77311G (2010). [CrossRef]  

22. L. G. Cook, “Three-mirror anastigmat used off-axis in aperture and field,” Proc. SPIE 183, 207–211 (1979). [CrossRef]  

References

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  1. H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
    [Crossref]
  2. J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
    [Crossref]
  3. R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987).
    [Crossref]
  4. M. Liang, V. Krabbendam, C. F. Claver, S. Chandrasekharan, and B. Xin, “Active Optics in Large Synoptic Survey Telescope,” Proc. SPIE Astronomical Telescopes + Instrumentation. International Society for Optics and Photonics, 84444Q–84444Q–13 (2012).
    [Crossref]
  5. R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006).
    [Crossref]
  6. M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
    [Crossref]
  7. R. Tessieres, “Analysis for alignment of optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).
  8. T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment of two-mirror astronomical telescopes; the astigmatic component,” Proc. SPIE 7017, 701711 (2008).
  9. Z. Gu, C. Yan, and Y. Wang, “Alignment of a three-mirror anastigmatic telescope using nodal aberration theory,” Opt. Express 23(19), 25182–25201 (2015).
    [Crossref] [PubMed]
  10. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
    [Crossref]
  11. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).
  12. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005).
    [Crossref] [PubMed]
  13. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009).
    [Crossref] [PubMed]
  14. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010).
    [Crossref] [PubMed]
  15. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821–836 (2011).
    [Crossref] [PubMed]
  16. J. Wang, B. Guo, Q. Sun, and Z. Lu, “Third-order aberration fields of pupil decentered optical systems,” Opt. Express 20(11), 11652–11658 (2012).
    [Crossref] [PubMed]
  17. H. Hu, J. Liu, and Z. Fan, “Interaction of pupil offset and fifth-order nodal aberration field properties in rotationally symmetric telescopes,” Opt. Express 21(15), 17986–17998 (2013).
    [Crossref] [PubMed]
  18. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).
  19. T. Schmid, “Misalignment Induced Nodal Aberration Fields and Their Use in the Alignment of Astronomical Telescopes,” Ph.D. dissertation (University of Central Florida Orlando, Florida, 2010).
  20. T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
    [Crossref] [PubMed]
  21. M. L. Lampton, M. J. Sholl, and M. E. Levi, “Off-axis telescopes for dark energy investigations,” Proc. SPIE Astronomical Telescopes + Instrumentation. International Society for Optics and Photonics, 77311G–77311G (2010).
    [Crossref]
  22. L. G. Cook, “Three-mirror anastigmat used off-axis in aperture and field,” Proc. SPIE 183, 207–211 (1979).
    [Crossref]

2015 (1)

2013 (1)

2012 (1)

2011 (1)

2010 (2)

2009 (1)

2008 (1)

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment of two-mirror astronomical telescopes; the astigmatic component,” Proc. SPIE 7017, 701711 (2008).

2006 (1)

R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006).
[Crossref]

2005 (1)

1999 (1)

J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
[Crossref]

1998 (1)

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

1991 (1)

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
[Crossref]

1987 (1)

R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

1980 (1)

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

1979 (1)

L. G. Cook, “Three-mirror anastigmat used off-axis in aperture and field,” Proc. SPIE 183, 207–211 (1979).
[Crossref]

Cook, L. G.

L. G. Cook, “Three-mirror anastigmat used off-axis in aperture and field,” Proc. SPIE 183, 207–211 (1979).
[Crossref]

Dempewolf, G.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Fan, Z.

Franza, F.

R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

Gu, Z.

Guo, B.

Harnisch, B.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Hawley, S. L.

J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
[Crossref]

Hu, H.

Hubbard, R.

R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006).
[Crossref]

Juranek, H. J.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Kuhn, J. R.

J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
[Crossref]

Kunkel, B.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Litzelmann, A.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Liu, J.

Lu, Z.

Lundgren, M. A.

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
[Crossref]

Noethe, L.

R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

Rakich, A.

Rimmele, T.

R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006).
[Crossref]

Rolland, J. P.

T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
[Crossref] [PubMed]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment of two-mirror astronomical telescopes; the astigmatic component,” Proc. SPIE 7017, 701711 (2008).

Sand, R.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Schillke, F.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Schmid, T.

T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
[Crossref] [PubMed]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment of two-mirror astronomical telescopes; the astigmatic component,” Proc. SPIE 7017, 701711 (2008).

Schmidt, E.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Schweizer, J.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Shack, R. V.

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

Sun, Q.

Thompson, K.

Thompson, K. P.

Upton, R.

R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006).
[Crossref]

Wang, J.

Wang, Y.

Wilson, R. N.

R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

Wolfe, W. L.

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
[Crossref]

Yan, C.

J. Mod. Opt. (1)

R. N. Wilson, F. Franza, and L. Noethe, “Active optics: I. A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34(4), 485–509 (1987).
[Crossref]

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
[Crossref]

Opt. Express (4)

Proc. SPIE (5)

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

L. G. Cook, “Three-mirror anastigmat used off-axis in aperture and field,” Proc. SPIE 183, 207–211 (1979).
[Crossref]

R. Upton, T. Rimmele, and R. Hubbard, “Active optical alignment of the Advanced Technology Solar Telescope,” Proc. SPIE 6271, 62710R (2006).
[Crossref]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment of two-mirror astronomical telescopes; the astigmatic component,” Proc. SPIE 7017, 701711 (2008).

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Publ. Astron. Soc. Pac. (1)

J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
[Crossref]

Other (6)

M. Liang, V. Krabbendam, C. F. Claver, S. Chandrasekharan, and B. Xin, “Active Optics in Large Synoptic Survey Telescope,” Proc. SPIE Astronomical Telescopes + Instrumentation. International Society for Optics and Photonics, 84444Q–84444Q–13 (2012).
[Crossref]

R. Tessieres, “Analysis for alignment of optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).

T. Schmid, “Misalignment Induced Nodal Aberration Fields and Their Use in the Alignment of Astronomical Telescopes,” Ph.D. dissertation (University of Central Florida Orlando, Florida, 2010).

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).

M. L. Lampton, M. J. Sholl, and M. E. Levi, “Off-axis telescopes for dark energy investigations,” Proc. SPIE Astronomical Telescopes + Instrumentation. International Society for Optics and Photonics, 77311G–77311G (2010).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Pupil vector transformation between decentered pupil and parent pupil. The black circle represents the pupil of an on-axis system. The red circle represents the pupil of an off-axis section of the on-axis system.
Fig. 2
Fig. 2 Layout of the main optical system of New Solar Telescope (NST). It can be seen that the offset of aperture stop is 1840mm in the positive direction of Y axis. Then the defined parameter A in Eq. (25) and Eq. (26) can be determined. A = 1840 / 800 .
Fig. 3
Fig. 3 Average RMS WFE before and after alignment for different cases based on NAT method. (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4. Note that the pink spot represents the RMS WFE before alignment. The blue spot represents the RMS WFE after first alignment. The red spot represents the RMS WFE after second alignment. It is the same as the RMA WFE of the nominal design (0.062 waves).
Fig. 4
Fig. 4 Layout of an off-axis Cook-TMA telescope. It can be seen that the offset of aperture stop is −460mm in the negative direction of Y axis. Then the defined parameter A in Eq. (56) and Eq. (57) can be determined. A = 460 / 300 .
Fig. 5
Fig. 5 Average RMS WFE before and after alignment for different cases based on NAT method. (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4. Note that the pink spot represents the RMS WFE before alignment. The blue spot represents the RMS WFE after alignment. The red spot represents the RMS WFE in nominal design (0.072 waves).
Fig. 6
Fig. 6 Average RMS WFE before and after alignment for different cases based on sensitivity table method (SMT). (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4. Note that the pink spot represents the RMS WFE before alignment. The blue spot represents the RMS WFE after alignment. The red spot represents the RMS WFE in nominal design (0.072 waves).

Tables (15)

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Table 1 Optical Parameters of NST

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Table 2 Wave Aberration Coefficients of SM for NST

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Table 3 Introduced Misalignments of SM for NST

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Table 4 Calculated Misalignments of SM for NST Based on Third-order NAT and Relative Errors between Calculated and Introduced Misalignments

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Table 5 Calculated Misalignments of SM for NST Based on Fifth-order NAT and Relative Errors between Calculated and Introduced Misalignments

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Table 6 Introduced Astigmatic Figure Errors on PM

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Table 7 Calculated Perturbations Based on Fifth-order NAT and Relative Errors between Calculated and Introduced Perturbations

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Table 8 Four Different Cases Considered in Monte-Carlo Simulations

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Table 9 Root Mean Square Deviations (RMSDs) between Introduced and Computed Perturbations

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Table 10 Wave Aberration Coefficients of SM and TM for Three-mirror Parent Telescope and Values of C A & C B & C C

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Table 11 Introduced Misalignments of SM and TM

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Table 12 Calculated Misalignments of SM and TM and Their Relative Errors

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Table 13 Four Different Cases Considered in Monte-Carlo Simulations

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Table 14 RMSDs between Introduced and Calculated Misalignments Based on NAT and STM for Different Cases

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Table 15 Parameter Definitions and Vector Identities

Equations (63)

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W = j p n m ( W k l m ) j ( H H ) p ( ρ ρ ) n ( H ρ ) m , k = 2 p + m , l = 2 n + m ,
ρ = B ρ + h ,
h = h x i + h y j , B = r o r O , h x = o x r O , h y = o y r O ,
W = j p n m ( W k l m ) j ( H H ) p [ ( B ρ + h ) ( B ρ + h ) ] n [ H ( B ρ + h ) ] m .
H A j = H σ j ,
W = j p n m ( W k l m ) j ( H A j H A j ) p [ ( B ρ + h ) ( B ρ + h ) ] n [ H A j ( B ρ + h ) ] m .
W = 1 2 j W 222 j [ H A j 2 ( B ρ + h ) 2 ] + j W 131 j [ H A j ( B ρ + h ) ] [ ( B ρ + h ) ( B ρ + h ) ] + j W 040 j [ ( B ρ + h ) ( B ρ + h ) ] 2 + j W 240 M j ( H A j H A j ) [ ( B ρ + h ) ( B ρ + h ) ] 2 + j W 331 M j ( H A j H A j ) [ H A j ( B ρ + h ) ] [ ( B ρ + h ) ( B ρ + h ) ] + 1 2 j W 422 j ( H A j H A j ) [ H A j 2 ( B ρ + h ) 2 ] + 1 4 j W 333 j [ H A j 3 ( B ρ + h ) 3 ] + 1 2 j W 242 j [ H A j 2 ( B ρ + h ) 2 ] [ ( B ρ + h ) ( B ρ + h ) ] + j W 151 j [ H A j ( B ρ + h ) ] [ ( B ρ + h ) ( B ρ + h ) ] 2 + j W 060 j [ ( B ρ + h ) ( B ρ + h ) ] 3 .
W = i C i ( H x , H y ) Z i ( ρ , φ ) .
W = 1 2 j W 222 j H A j 2 ( B ρ + h ) 2 ,
W = 1 2 j W 222 j [ B 2 ( H A j 2 ρ 2 ) + Δ ] ,
B 2 [ H x 2 H y 2 H x H y 1 0 2 H x H y H y H x 0 1 ] [ W 222 2 A 222 , x 2 A 222 , y B 222 , x 2 B 222 , y 2 ] = 2 [ C 5 w 222 C 6 w 222 ] .
W = j W 131 j [ H A j ( B ρ + h ) ] [ ( B ρ + h ) ( B ρ + h ) ] ,
W = j W 131 j [ B 3 ( H A j ρ ) ( ρ ρ ) + B 2 ( H A j h ρ 2 ) + Δ ] ,
B 3 [ H x 1 0 H y 0 1 ] [ W 131 A 131 , x A 131 , y ] = 3 [ C 7 w 131 C 8 w 131 ] ,
B 2 [ h y 0 0 h y ] [ H y 0 1 H x 1 0 ] [ W 131 A 131 , x A 131 , y ] = [ C 5 w 131 C 6 w 131 ] .
C 5 w 131 = 3 h y B C 8 w 131 C 6 w 131 = 3 h y B C 7 w 131 .
W = j W 040 j [ ( B ρ + h ) ( B ρ + h ) ] 2 ,
W = j W 040 j [ B 4 ( ρ ρ ) 2 + 4 B 3 ( h ρ ) ( ρ ρ ) + 2 B 2 ( h 2 ρ 2 ) + Δ ] ,
B 4 W 040 = 6 C 9 w 040 ,
4 B 3 h y W 040 = 3 C 8 040 ,
2 B 2 h y 2 W 040 = C 5 w 040 .
C 8 w 040 = 8 h y B C 9 w 040 C 5 w 040 = 12 h y B C 9 w 040 .
{ C 5 w 222 = C 5 C 5 w 131 C 5 w 040 C 6 w 222 = C 6 C 6 w 131 .
{ C 7 w 131 = C 7 C 8 w 131 = C 8 C 8 w 040 .
{ C 5 w 222 = C 5 + 3 A C 8 + 12 A C 9 24 A 2 C 9 C 6 w 222 = C 6 3 A C 7 ,
{ C 7 w 131 = C 7 C 8 w 131 = C 8 8 A C 9 .
B 2 [ H x 2 H y 2 H x H y 1 0 2 H x H y H y H x 0 1 ] [ W 222 2 A 222 , x 2 A 222 , y B 222 , x 2 B 222 , y 2 ] = 2 [ C 5 + 3 A C 8 + 12 A C 9 24 A 2 C 9 C 6 3 A C 7 ] ,
B 3 [ H x 1 0 H y 0 1 ] [ W 131 A 131 , x A 131 , y ] = 3 [ C 7 C 8 8 A C 9 ] .
{ A 222 x = W 222 , S M s p h σ S M , x s p h + W 222 , S M a s p h σ S M , x a s p h A 222 y = W 222 , S M s p h σ S M , y s p h + W 222 , S M a s p h σ S M , y a s p h A 131 x = W 131 , S M s p h σ S M , x s p h + W 131 , S M a s p h σ S M , x a s p h A 131 y = W 131 , S M s p h σ S M , y s p h + W 131 , S M a s p h σ S M , y a s p h ,
{ X D E S M = u ¯ P M d 1 σ S M , x a s p h Y D E S M = u ¯ P M d 1 σ S M , y a s p h A D E S M = u ¯ P M ( 1 + c S M d 1 ) σ S M , y s p h c S M Y D E S M B D E S M = u ¯ P M ( 1 + c S M d 1 ) σ S M , x s p h + c S M Y D E S M ,
W = 1 2 j W 422 j ( H A j H A j ) [ H A j 2 ( B ρ + h ) 2 ] ,
W = 1 2 j W 422 j [ B 2 ( H A j H A j ) ( H A j 2 ρ 2 ) + Δ ] ,
B 2 [ H x 4 H y 4 4 H x 3 4 H y 3 2 H x H y ( H x 2 + H y 2 ) 6 H x 2 H y 2 H y 3 2 H x 3 6 H x H y 2 ] [ W 422 A 422 , x A 422 , y ] = 2 [ C 5 w 422 C 6 w 422 ] ,
W = j W 331 M j ( H A j H A j ) [ H A j ( B ρ + h ) ] [ ( B ρ + h ) ( B ρ + h ) ] ,
W = j W 331 M j [ ( H A j H A j ) ( H A j ρ ) ( ρ ρ ) + ( H A j H A j ) ( H A j h ρ 2 ) + Δ ] ,
B 3 [ H x 3 + H x H y 2 3 H x 2 H y 2 2 H x H y H x 2 H y + H y 3 2 H x H y H x 2 3 H y 2 ] [ W 331 M A 331 M , x A 331 M , y ] = 3 [ C 7 w 331 M C 8 w 331 M ] ,
B 2 [ h y 0 0 h y ] [ H x 2 H y + H y 3 2 H x H y H x 2 3 H y 2 H x 3 + H x H y 2 3 H x 2 H y 2 2 H x H y ] [ W 331 M A 331 M , x A 331 M , y ] = [ C 5 w 331 M C 6 w 331 M ] ,
C 5 w 331 M = 3 h y B C 8 w 331 M C 6 w 331 M = 3 h y B C 7 w 331 M .
W = 1 4 j W 333 j H A j 3 ( B ρ + h ) 3 ,
W = 1 4 j W 333 j [ B 3 ( H A j 3 ρ 3 ) + 3 B 2 ( H A j 3 h ρ 2 ) + Δ ] ,
C 5 w 333 = 3 h y B C 11 w 333 C 6 w 333 = 3 h y B C 10 w 333 .
W = j W 240 M j ( H A j H A j ) [ ( B ρ + h ) ( B ρ + h ) ] 2 ,
W = j W 240 M j [ B 4 ( H A j H A j ) ( ρ ρ ) 2 + 4 B 3 ( H A j H A j ) ( h ρ ) ( ρ ρ ) + 2 B 2 ( H A j H A j ) ( h 2 ρ 2 ) + Δ ] ,
C 8 w 240 M = 8 h y B C 9 w 240 M C 5 w 240 M = 12 h y B C 9 w 240 M .
W = 1 2 j W 242 j [ H A j 2 ( B ρ + h ) 2 ] [ ( B ρ + h ) ( B ρ + h ) ] ,
W = 1 2 j W 242 j [ B 4 ( H A j 2 ρ 2 ) ( ρ ρ ) + B 3 ( h H A j 2 ρ 3 ) + 3 B 3 ( H A j 2 h ρ ) ( ρ ρ ) + 3 B 2 ( h h ) ( H A j 2 ρ 2 ) + Δ ] ,
{ C 10 w 242 = 4 h y B C 13 w 242 C 11 w 242 = 4 h y B C 12 w 242 C 7 w 242 = 4 h y B C 13 w 242 C 8 w 242 = 4 h y B C 12 w 242 C 5 w 242 = 12 h y 2 B 2 C 12 w 242 C 6 w 242 = 4 h y 2 B 2 C 13 w 242 .
W = j W 151 j [ H A j ( B ρ + h ) ] [ ( B ρ + h ) ( B ρ + h ) ] 2 ,
W = j W 151 j [ B 5 ( H A j ρ ) ( ρ ρ ) 2 + 2 B 4 ( H A j h ρ 2 ) ( ρ ρ ) + B 3 ( H A j h 2 ρ 3 ) + 3 B 4 ( H A j h ) ( ρ ρ ) ( ρ ρ ) +6 B 3 ( h h ) ( H A j ρ ) ( ρ ρ ) + 3 B 3 ( h 2 H A j ρ ) ( ρ ρ ) + 2 B 2 ( h h ) ( H A j h ρ 2 ) + 2 B 2 ( H A j h ) ( h 2 ρ 2 ) + Δ ] ,
{ C 12 w 151 = 5 h y B C 15 w 151 C 13 w 151 = 5 h y B C 14 w 151 C 10 w 151 = 10 h y 2 B 2 C 14 w 151 C 11 w 151 = 10 h y 2 B 2 C 15 w 151 C 9 w 151 = 5 h y B C 15 w 151 C 7 w 151 = 10 h y 2 B 2 C 14 w 151 C 8 w 151 = 30 h y 2 B 2 C 15 w 151 C 5 w 151 = 40 h y 2 B 2 C 15 w 151 C 6 w 151 = 20 h y 2 B 2 C 14 w 151 .
W = j W 060 j [ ( B ρ + h ) ( B ρ + h ) ] 3 ,
W = j W 060 j [ B 6 ( ρ ρ ) 3 + 6 B 5 ( ρ ρ ) 2 ( ρ h ) + 6 B 4 ( ρ ρ ) ( ρ 2 h 2 ) + 2 B 3 ( ρ 3 h 3 ) + 9 B 4 ( ρ ρ ) 2 ( h h ) + 18 B 3 ( ρ ρ ) ( ρ h ) ( h h ) + 6 B 2 ( ρ 2 h 2 ) ( h h ) + Δ ] ,
{ C 5 w 060 = 20 h y 4 B 4 C 16 w 060 C 8 w 060 = 120 h y 3 B 3 C 16 w 060 C 9 w 060 = 30 h y 2 B 2 C 16 w 060 C 11 w 060 = 40 h y 3 B 3 C 16 w 060 C 12 w 060 = 30 h y 2 B 2 C 16 w 060 C 15 w 060 = 12 h y B C 16 w 060 .
{ C 5 w 222 + C 5 w 422 = C 5 s u m C 5 w 060 C 5 w 151 C 5 w 242 C 5 w 040 + w 240 M C 5 w 131 + w 331 M C 5 w 333 C 6 w 222 + C 6 w 422 = C 6 s u m C 6 w 151 C 6 w 242 C 6 w 131 + w 331 M C 6 w 333 .
{ C 7 w 131 + C 7 w 331 M = C 7 s u m C 7 w 151 C 7 w 242   C 8 w 131 + C 8 w 331 M = C 8 s u m C 8 w 060 C 8 w 151 C 8 w 242 C 8 w 040 + w 240 M   .
{ C 5 w 222 + C 5 w 422 = C 5 + 40 A 2 C 15 + 3 A C 8 12 A 2 C 9 3 A C 11 + 12 A 2 C 12 + 360 A 4 C 16 480 A 3 C 16 C 6 w 222 + C 6 w 422 = C 6 3 A C 7 20 A 2 C 14 + 3 A C 10 + 12 A 2 C 13
{ C 7 w 131 + C 7 w 331 M = C 7 4 A C 13 + 10 A 2 C 14 C 8 w 131 + C 8 w 331 M = C 8 8 A C 9 + 4 A C 12 + 30 A 2 C 15 + 120 A 2 C 16 240 A 3 C 16
B 2 [ H x 2 H y 2 2 H x H y H x H y H y H x 1 0 0 1 H x 4 H y 4 2 H x H y ( H x 2 + H y 2 ) 4 H x 3 6 H x 2 H y 2 H y 3 4 H y 3 2 H x 3 6 H x H y 2 ] T [ W 222 2 A 222 , x 2 A 222 , y B 222 , x 2 B 222 , y 2 W 422 A 422 , x A 422 , y ] = 2 [ C 5 w 222 + C 5 w 422 C 6 w 222 + C 6 w 422 ] ,
B 3 [ H x H y 1 0 0 1 H x 3 + H x H y 2 H x 2 H y + H y 3 3 H x 2 H y 2 2 H x H y 2 H x H y H x 2 3 H y 2 ] T [ W 131 A 131 , x A 131 , y W 331 M A 331 M , x A 331 M , y ] = 3 [ C 7 w 131 + C 7 w 331 M C 8 w 131 + C 8 w 331 M ] ,
B 222 , F i g 2 = B 2 ( B 222 2 B 222 , M i s 2 )
R M S D i = 1 150 n = 1 150 [ X i ( n ) x i ( n ) ] 2
{ H T M , x = C A σ S M , x s p h H T M , y = C A σ S M , y s p h H I M A G E , x = C B σ S M , x s p h + C C σ T M , x s p h H I M A G E , y = C B σ S M , y s p h + C C σ T M , y s p h A 222 x = W 222 , S M s p h σ S M , x s p h + W 222 , S M a s p h σ S M , x a s p h + W 222 , T M s p h σ T M , x s p h + W 222 , T M a s p h σ T M , x a s p h A 222 y = W 222 , S M s p h σ S M , y s p h + W 222 , S M a s p h σ S M , y a s p h + W 222 , T M s p h σ T M , y s p h + W 222 , T M a s p h σ T M , y a s p h A 131 x = W 131 , S M s p h σ S M , x s p h + W 131 , S M a s p h σ S M , x a s p h + W 131 , T M s p h σ T M , x s p h + W 131 , T M a s p h σ T M , x a s p h A 131 y = W 131 , S M s p h σ S M , y s p h + W 131 , S M a s p h σ S M , y a s p h + W 131 , T M s p h σ T M , y s p h + W 131 , T M a s p h σ T M , y a s p h
{ X D E S M = u ¯ P M d 1 σ S M , x a s p h Y D E S M = u ¯ P M d 1 σ S M , y a s p h A D E S M = u ¯ P M ( 1 + c S M d 1 ) σ S M , y s p h c S M Y D E S M B D E S M = u ¯ P M ( 1 + c S M d 1 ) σ S M , x s p h + c S M Y D E S M X D E T M = [ d 2 + d 1 ( 2 c S M d 2 1 ) ] u ¯ P M σ T M , x a s p h + 2 d 2 ( B D E S M + c S M X D E S M ) Y D E T M = [ d 2 + d 1 ( 2 c S M d 2 1 ) ] u ¯ P M σ T M , y a s p h + 2 d 2 ( A D E S M + c S M X D E S M ) A D E T M = [ c T M ( d 2 d 1 ) + 2 c S M ( c T M d 1 d 2 + d 1 ) + 1 ] u ¯ P M σ T M , y s p h + 2 ( 1 + c T M d 2 ) ( c S M Y D E S M + A D E S M ) c T M Y D E T M B D E T M = [ c T M ( d 2 d 1 ) + 2 c S M ( c T M d 1 d 2 + d 1 ) + 1 ] u ¯ P M σ T M , x s p h 2 ( 1 + c T M d 2 ) ( c S M X D E S M B D E S M ) + c T M X D E T M

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