## Optimal collimation of misaligned optical systems by concentering primary field aberrations |

Optics Express, Vol. 18, Issue 18, pp. 19249-19262 (2010)

http://dx.doi.org/10.1364/OE.18.019249

Acrobat PDF (915 KB)

### Abstract

A collimation method of misaligned optical systems is proposed. The method is based on selectively nullifying main alignment-driven aberration components. This selective compensation is achieved by the optimal adjustment of chosen alignment parameters. It is shown that this optimal adjustment can be obtained by solving a linear matrix equation of the low-order alignment-driven terms of primary field aberrations. A significant result from the adjustment is to place the centers of the primary field aberrations, initially scattered over the field due to misalignment, to a desired common field location. This *aberration concentering* naturally results in recovery of image quality across the field of view. Error analyses and robustness tests show the method’s feasibility in efficient removal of alignment-driven aberrations in the face of measurement and model uncertainties. The extension of the method to the collimation of a misaligned system with higher-order alignment-driven aberrations is also shown.

© 2010 Optical Society of America

## 1. Introduction

*optimal correction*, as it determines the way to compute the corrections as well as the strategy of applying them to the system.

*reverse optimization*is quite often at the core of them [1–4]. The principle is to search for alignment states of individual components, with which the system reproduces the observed wavefront. However, this approach often utilizes non-linear optimization procedures and, as a result, produces a stagnated estimate with significant difference from the true alignment state. One remedy to this is to use wavefronts sampled at multiple fields, but these multi-field samples can be degenerate among themselves and thus the stagnation problem can still persist, especially in multi-element systems [5

5. H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II : Optical system alignment using differential wavefront sampling,” Opt. Express **15**, 15424–15437 (2007). [CrossRef] [PubMed]

5. H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II : Optical system alignment using differential wavefront sampling,” Opt. Express **15**, 15424–15437 (2007). [CrossRef] [PubMed]

*in alignment*, no matter what the actual alignment state is, and this correction can be called

*optimal*. Methods that aim for this definition of optimal correction often use the singular value decomposition of the alignment influence matrix [8

8. H. N. Chapman and D. W. Sweeney, “Rigorous method for compensation selection and alignment of microlitho-graphic optical systems,” Proc. SPIE **3331**, 102–113 (1998). [CrossRef]

9. A. M. Hvisc and J. H. Burge, “Alignment analysis of four-mirror spherical aberration correctors,” Proc. SPIE **7018**, 701819 (2008). [CrossRef]

*aberration concentering*hereafter). This restores the field distribution of aberrations to the nominal and improves the image quality at the same time; (iii) In most of low-order aberration dominant systems, only three alignment-driven terms need to be removed. Thus (maximum) three alignment parameters per axis are required. This can still be true for systems with higher-order alignment-driven aberrations although the aberration concentering may not be achieved. However, adding one more alignment parameter per axis for also removing the fourth term from higher-order aberrations is shown to be effective for further improvement in collimation and aberration concentering quality. In Section 2, details of this approach is described with error analyses. We present the results of case studies and robustness tests in Section 3. The results demonstrate the method’s feasibility in efficient removal of alignment-driven aberrations in the face of measurement and model uncertainties. We finish up this paper with a discussion on how this approach can be useful in collimation of wide-field large aperture multi-surface systems with higher-order field aberrations (Sec. 4).

## 2. Theory

### 2.1. Alignment-driven aberrations

*Coma*,

_{x}*Coma*,

_{y}*Astg*

_{1},

*Astg*

_{2}, and

*Curv*be the coefficients of

*Z*

_{8},

*Z*

_{7},

*Z*

_{6},

*Z*

_{5}, and

*Z*

_{4}, respectively, where

*Z*is the

_{i}*i*-th standard Zernike polynomial [11

11. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. **66**, 207–211 (1975). [CrossRef]

*x, y*) and tilt (

*θ*,

*ϕ*) parameters [12–17]. For a single surface case, these are given as,

*O*

^{(n)}includes terms of order higher than

*n*-1 in field and/or alignment parameters and

*F*

_{0}corresponds to the defocus term at the center of the field. In low-order aberration dominant systems,

*O*

^{(n)}is negligible, but in some wide-field systems these may need to be accounted for as to be discussed later on (Sec. 4).

*linear term*and those in square brackets

*quadratic term*hereafter. The

*linear terms*in

*Coma*are field constant and thus mainly controls the overall magnitude of wavefront error across the system field of view. The

*linear terms*in

*Curv*and

*Astg*effectively determine the overall slope of aberration field and, when exist, produce so-called focus gradient across the image field. The

*quadratic terms*in

*Curv*and

*Astg*are similar to the

*linear terms*in

*Coma*, but usually less significant as

*A*

_{00},

*B*

_{00}, and

*F*

_{00}are much smaller than

*C*

_{0}. These

*linear terms*commonly result in displacement of the centers of the individual aberrations away from the nominal by different amounts. This induces non-intrinsic large asymmetric image quality variations across the field of a system. Thus, the

*linear terms*are those to be removed in the collimation process. It is outside the scope of this paper to give explicit expressions for these coefficients, but one can certainly do this by, for example, a ray-tracing-based numerical sensitivity analysis [16

16. H. Lee, G. B. Dalton, I. A. J. Tosh, and S. Kim, “Computer-guided alignment I : Phase and amplitude modulation of the alignment-influenced wavefront,” Opt. Express **15**, 3127–3139 (2007). [CrossRef] [PubMed]

17. H. Lee, G. Dalton, I. Tosh, and S. Kim, “Computer-guided alignment III: Description of inter-element alignment effect in circular-pupil optical systems,” Opt. Express **16**, 10992–11006 (2008). [CrossRef] [PubMed]

### 2.2. Aberration concentering and optimal collimation by alignment correction

*linear terms*of a misaligned system need be first quantified and this can be done through three steps: (i) measuring wavefront data at a set of field positions, (ii) determining the aberration coefficients by decomposing the wavefront data into aberration functions (such as Zernike polynomials), and (iii) fitting a linear or quadratic function to the distributions of the aberration coefficients across the field.

*H*- and

_{x}*H*-axis) can provide as much information for quantifying the

_{y}*linear terms*as the grid sampling can. This is due to the fact that the

*linear terms*of each aberration only respond to certain parameters associated with one of the two field axes, as easily noticed in Eq. (1), and thus can be split into two groups. For example, all terms of

*Coma*and those in the first line of

_{x}*Astg*

_{1}only respond to

*H*,

_{x}*x*, or

*ϕ*, all of which are associated with the

*H*-axis. Taking this aberration scanning approach, one can obtain a linear curve for

_{x}*Coma*and

*Astg*

_{2}and a quadratic curve for

*Astg*

_{1}and

*Curv*along each field axis after performing step (i) and (ii). By fitting a linear or quadratic function to these scans, as one would do in step (iii), linear fitting coefficients can be obtained. These correspond to the amounts of the

*linear terms*of the aberrations. Note that the aberration scanning naturally requires a fewer wavefront measurements than the grid sampling does and this can simplify and speed up the measurement process.

*X⃗*and

*Y⃗*be vectors containing the measured values of the

*linear terms*of

*Coma*,

*Astg*

_{1},

*Curv*, and

*Astg*

_{2}in the

*H*- and

_{x}*H*-axis, respectively. The field coordinates of the centers of

_{y}*Coma*,

*Astg*

_{1}, and

*Curv*, for example, are given by the following.

*X⃗*and

*Y⃗*is equivalent to

*concentering*the three aberrations at the nominal field center. At the same time, it removes the

*linear terms*from the system aberration field. As a result of these, the aberrations restore their intrinsic distribution patterns across the system field. In order to do this, one needs to apply appropriate alignment corrections (Δ

*x*,Δ

*y*,Δ

*θ*,Δ

*ϕ*) to the alignment parameters (

*x,y,θ,ϕ*) of the system. These corrections can be obtained by solving a set of linear equations given by the measured

*X⃗*and

*Y⃗*and the expressions of the

*linear terms*in Eq. (1). The equations are given in Eq. (3).

*X*

_{2},

*Y*

_{2}), (

*X*

_{3},

*Y*

_{3}), and (

*X*

_{4},

*Y*

_{4}) cannot be sensed by on-axis wavefront measurement only. This illustrates the importance of verifying imaging performance across the field.

*linear terms*together, one needs to find a solution to some of the above equations for a given set of correction parameters. For example, in a single surface misalignment case, one can solve the following.

*x*= −

*x*and Δ

*ϕ*= −

*ϕ*remove the

*linear*and

*quadratic terms*altogether.

### 2.3. Description of residual alignment-driven aberrations after alignment correction

*concenter Coma*,

*Astg*

_{1}, and

*Curv*at the nominal center field of the system. Adopting a generic notation of

*x*and

_{i}*y*for the alignment parameters, we can write

_{i}**C**

_{x},

**A**

_{x},

**F**

_{x}are

*N*× 1 vectors of

*H*-axis, respectively, and likewise along

_{x}*H*-axis.

_{y}**M**

_{x}and

**M**

_{y}are 3 ×

*N*matrices.

*x⃗*and

*y⃗*are 2

*N*× 1 vectors. Note that each surface has two alignment parameters per field axis.

**A**

^{T}means the transpose of

**A**.

*linear terms*. However, this may not be sufficient to eliminate the

*quadratic terms*. Let the correction parameters be Δ

*x*and Δ

_{k}*y*with

_{k}*k*= 1,2, 3. As only three parameters are to be adjusted, the influence matrices must be subsets of

**M**

_{x}and

**M**

_{y}. Letting

**m**

_{x}and

**m**

_{y}be the 3 × 3 subset matrices, Δ

*x⃗*and Δ

*y⃗*can be expressed in terms of

*x⃗*and

*y⃗*as,

**m**

^{−1}

_{x}is the inverse of

**m**

_{x}. Although three alignment parameters are sufficient, one may wish to use more alignment parameters in this process for some reason. In that case,

**m**

_{x}and

**m**

_{y}are no longer square matrices and their inverses in Eq. (6) can be replaced by pseudo-inverses via singular-value-decomposition (SVD).

*N*×

*N*unit matrix (

**1**) and a (

*N*− 3) ×

*N*zero matrix (

**0**) as,

*linear terms*vanish, the

*quadratic terms*may still have residuals. The residuals can be expressed in terms of

*x⃗*and

*y⃗*, using Eq. (6) and (7), at the common center (i.e.

*H*=

_{x}*H*= 0) as,

_{y}**F**

_{00},

**A**

_{00}, and

**B**

_{00}are the coefficient matrices of the

*quadratic terms*. Assuming that

*x⃗*and

*y⃗*are random independent variables following Gaussian distributions with zero mean and standard deviations of

*σ*and

_{x}*σ*, the probability distributions of the residual aberrations can be computed. If these distributions happened to be Gaussian, the statistics in Eq. (9) can be used in finding the optimal values of

_{y}*σ*and

_{x}*σ*[19].

_{y}*i*≠ 1,2,3,

*E*[

*A*] is the mean value of

*A*, and

*Var*[

*A*] is the variance of

*A*. The condition of, for example, minimum curvature can be

*Curv*is the allocation to

_{req}*Curv*from the total rms wavefront error budget of a system. Similar conditions can be posed to the other aberrations and one then needs to find the optimum values of

*σ*and

_{x}*σ*that meet these conditions. It should be noted, however, that the distributions are different from one case to another and may significantly deviate from a Gaussian. In such cases, one needs to find the optimal

_{y}*σ*and

_{x}*σ*using more sophisticated optimization procedures.

_{y}### 2.4. Error analysis

*w*and

_{i}*δw*be the average and error of a particular aberration coefficient, inferred from

_{i}*M*measurements at the

*i*-th field locations

*H*. The curve fit coefficients

_{i}*p⃗*of the aberration scans can be computed by a least-square analysis [18] as,

**R**

^{T}

**R**]

^{−1}=

**C**is the covariance matrix of the fit coefficients so that the variance of

*p*equals to the (

_{i}*i,i*) element of

**C**(

*p*is the

_{j}*i*-th element of

*X⃗*, then

*x⃗*, if the true influence matrix of the system differs from what we think it is (

**M**

_{x}) by

*δ*

**M**

_{x}, the corrections are expressed as,

*δM*and

_{ij,x}*x*, respectively, the expected variance of the alignment corrections can be approximated by the following.

_{i}*n*is the (

_{ij,x}*i,j*) element of

**n**

_{x}. This outcome obviously depends on

*unknown*true alignment state. However, this can be substituted for its expected variance to set the expected

*upper*limit on the alignment correction uncertainty.

## 3. Case study

### 3.1. Two-mirror finite conjugate system

*x*

_{2}=+0.125mm,

*y*

_{2}=−0.515mm and in tilt by

*ϕ*

_{2}=−0.045deg,

*θ*

_{2}=0.185deg. The decenter and tilt of M2 are used as the correction terms for concentering coma and astigmatism at the nominal field center. The curve fit to the initial scans shown in Fig. 1 locates the centers of the three aberration fields at (+0.092,+0.698) for

*Coma*, (−0.826,+3.390) for

*Astg*

_{1}, and (−0.239,+0.980) for

*Curv*in normalized field coordinates.

*x*

_{2}=−0.1248mm, Δ

*y*

_{2}=0.5148, Δ

*ϕ*

_{2}=0.04493deg, and Δ

*θ*

_{2}=+0.18493deg. These corrections are close to the actual misalignment as expected. As a result of these corrections, the field aberrations show symmetric distributions around the nominal field center and the system performance is fully restored (Fig. 2).

### 3.2. Three-mirror camera and robustness test

*ϕ*=−0.017deg,

*θ*=0.035deg for M2 and x=0.095mm, y=−0.100mm,

*ϕ*=−0.029deg,

*θ*=0.031deg for M3. The initial field scans of the aberration is shown in Fig. 4. Notice the weak sign of higher-order field aberrations in the coma field scans [Fig. 4(A)].

*linear terms*in

*Coma*,

*Astg*

_{1}, and

*Curv*. Three parameters per axis are sufficient for the correction. The required corrections are Δx

_{2}=0.091mm, Δy

_{2}=-0.117mm, Δ

*θ*

_{2}=0.036deg, Δ

*ϕ*

_{2}=−0.016deg, Δ

*θ*

_{3}=0.029deg, Δ

*ϕ*

_{3}=0.031deg. The field scans after the correction are shown in Fig. 5. The correction indeed concentered the field aberrations at the desired field location and restored the system performance close to nominal.

## 4. Discussion: Wide-field large aperture multi-surface systems with higher-order alignment-driven aberrations

*Coma*shows field quadratic terms and, when scanned along

_{x}*H*axis, can be approximated by the following functions.

_{x}*dh*and

_{x}*dh*are linear functions of (

_{y}*x, ϕ*) and (

*y,θ*), respectively. Here, the field constant term is still substantial and the linear coefficients in this term are much larger than the cubic ones. Therefore, it should be possible to reduce the field constant term in the same way as used in the previous cases. However, in the presence of higher-order aberrations, the new field quadratic term (coupled with alignment parameters) can substantially contributes to

*Coma*. The major influence of this term is to deform

*Coma*field scans into quadratic shape, effectively breaking the oddness of the original functional form of

*Coma*. Therefore, in this case, the reduction of the field constant term of

*Coma*does not necessarily place its center at a desired field location. A similar effect also occurs in

*Astg*and

*Curv*, where substantial amounts of field cubic terms, coupled with alignment parameters, can appear.

*H*scans, in Table 1, clearly show the existence of significant quadratic term for

_{x}*Coma*and cubic terms for

*Astg*and

*Curv*, whereas the terms intrinsic to each aberration (e.g. odd functions of

*H*for Coma) have been changed by only small amounts, meaning relatively weak alignment-influence in these terms. Note, however, that the cubic terms in

_{x}*Astg*and

*Curv*are still less significant than the linear terms by many factors. A clear indication from this is that selective reduction of the

*H*

^{0}and

*H*

^{2}terms of

*Coma*and the

*H*

^{1}terms of

*Astg*and

*Curv*should restore the distributions of the aberrations to their nominal over the field of view.

*H*

^{2}term in

*Coma*(Method II). In Method II, total four alignment parameters per axis are required to completely correct the four alignment-driven terms.We have chosen M4 decenter/tilt, M5 decenter, and PFC tilt, and these are also used in Method I. We use the initial scan data in Fig. 7.

*Astg*and

*Curv*in particular (Fig. 8). At the edge of the field, the RMS wavefront error is reduced from 11 wv to 2.5 wv. However, as discussed, the distribution of

*Coma*is still off-set from the desired nominal center field, showing large asymmetry over the field. The distributions of

*Astg*and

*Curv*are also off-centered, but by smaller amounts than

*Coma*. Though the individual aberrations are not quite centered at the common field, the overall wavefront error, across the field, becomes close to the nominal. Note that almost identical result is obtained when only M4 decenter/tilt and PFC tilt are used.

*H*

^{2}term is also corrected, Method II produced a set of alignment corrections that exactly concentered the field distributions of

*Coma*,

*Astg*, and

*Curv*at the nominal center field (Fig. 9).

*Coma*scans follow the original odd function in

*H*. The amount of asymmetry in all aberrations is negligible and the overall wavefront error is nearly identical to the nominal. In comparison to Method I’s results, the RMS wavefront error at the edge of the field was reduced from 2.5wv to 2.2wv (roughly 10% improvement) by Method II. Although Method I can be effective in removing the alignment-driven aberrations, the full field ray-spot distribution, shown in Fig. 10, clearly demonstrates that Method II can further improve the quality of aberration concentering and collimation in the presence of higher-order aberrations, towards the edge of the field in particular.

*Astg*and

*Curv*, two more alignment parameters per field axis are likely to be necessary for concentering the three aberrations. Even so, however, not all required alignment parameters may be necessary depending on the amount of alignment-driven optical performance degradation. If a system is in a late stage of its commissioning and the performance is not far a way from the nominal, one may use some of the alignment parameters to efficiently reduce alignment-driven aberrations. If the degradation is still large, it would be necessary to use all of the required alignment parameters. In any case, the proposed method can be a useful way to test the alignment state of a system and to determine the optimal next adjustment.

## 5. Conclusion

## Appendix: The design prescription of the three-mirror system used in Section 3.2

## Acknowledgements

## References and links

1. | H. J. Jeong, G. N. Lawrence, and K. B. Nahm, “Auto-alignment of a three mirror off-axis telescope by reverse optimization and end-to-end aberration measurements,” Proc. SPIE |

2. | M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. |

3. | W. Sutherland, |

4. | S. Kim, H.-S. Yang, Y.-W. Lee, and S.-W. Kim, “Merit function regression method for efficient alignment control of two-mirror optical systems,” Opt. Express |

5. | H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II : Optical system alignment using differential wavefront sampling,” Opt. Express |

6. | H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Practical implementation of the complex wavefront modulation model for optical alignment,” Proc. SPIE |

7. | H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Implementation of differential wavefront sampling in optical alignment of pupil-segmented telescope systems,” Proc. SPIE |

8. | H. N. Chapman and D. W. Sweeney, “Rigorous method for compensation selection and alignment of microlitho-graphic optical systems,” Proc. SPIE |

9. | A. M. Hvisc and J. H. Burge, “Alignment analysis of four-mirror spherical aberration correctors,” Proc. SPIE |

10. | D. O’Donoghue, South African Large Telscope, Observatory, 7935, South Africa (Personal communication, 2009). |

11. | R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. |

12. | R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” in |

13. | B. McLeod, “Collimation of Fast Wide-Field Telescopes,” Publ. Astron. Soc. Pac. |

14. | R. N. Wilson and B. Delabre, “Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT,” Publ. Astron. Soc. Pac. |

15. | L. Noethe and S. Guisard, “Analytic expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” Acta Anat. Suppl. |

16. | H. Lee, G. B. Dalton, I. A. J. Tosh, and S. Kim, “Computer-guided alignment I : Phase and amplitude modulation of the alignment-influenced wavefront,” Opt. Express |

17. | H. Lee, G. Dalton, I. Tosh, and S. Kim, “Computer-guided alignment III: Description of inter-element alignment effect in circular-pupil optical systems,” Opt. Express |

18. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

19. | G. D’Agostini, |

**OCIS Codes**

(220.1000) Optical design and fabrication : Aberration compensation

(220.1140) Optical design and fabrication : Alignment

(220.4840) Optical design and fabrication : Testing

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: April 1, 2010

Revised Manuscript: May 23, 2010

Manuscript Accepted: June 28, 2010

Published: August 26, 2010

**Citation**

Hanshin Lee, "Optimal collimation of misaligned optical systems by concentering primary field aberrations," Opt. Express **18**, 19249-19262 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-19249

Sort: Year | Journal | Reset

### References

- H. J. Jeong, G. N. Lawrence, and K. B. Nahm, “Auto-alignment of a three mirror off-axis telescope by reverse optimization and end-to-end aberration measurements,” Proc. SPIE 818, 419–430 (1987).
- M. A. Lundgren, and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30, 307–311 (1991). [CrossRef]
- W. Sutherland, Alignment and Number of Wavefront Sensors for VISTA, VIS-TRE-ATC-00112–0012 (Technical report, Astronomy Technology Center, UK, 2001).
- S. Kim, H.-S. Yang, Y.-W. Lee, and S.-W. Kim, “Merit function regression method for efficient alignment control of two-mirror optical systems,” Opt. Express 15, 5059–5068 (2007). [CrossRef] [PubMed]
- H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II: Optical system alignment using differential wavefront sampling,” Opt. Express 15, 15424–15437 (2007). [CrossRef] [PubMed]
- H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Practical implementation of the complex wavefront modulation model for optical alignment,” Proc. SPIE 6617, 66170N (2007). [CrossRef]
- H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Implementation of differential wavefront sampling in optical alignment of pupil-segmented telescope systems,” Proc. SPIE 7017, 70171T (2008). [CrossRef]
- H. N. Chapman, and D. W. Sweeney, “Rigorous method for compensation selection and alignment of microlithographic optical systems,” Proc. SPIE 3331, 102–113 (1998). [CrossRef]
- A. M. Hvisc, and J. H. Burge, “Alignment analysis of four-mirror spherical aberration correctors,” Proc. SPIE 7018, 701819 (2008). [CrossRef]
- D. O’Donoghue, South African Large Telescope, Observatory, 7935, South Africa (Personal communication, 2009).
- R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1975). [CrossRef]
- R. V. Shack, and K. 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).
- B. McLeod, “Collimation of Fast Wide-Field Telescopes,” Publ. Astron. Soc. Pac. 108, 217–219 (1996). [CrossRef]
- R. N. Wilson, and B. Delabre, “Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT,” Publ. Astron. Soc. Pac. 109, 53–60 (1997). [CrossRef]
- L. Noethe, and S. Guisard, “Analytic expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” Acta Anat. Suppl. 144, 157–167 (2000).
- H. Lee, G. B. Dalton, I. A. J. Tosh, and S. Kim, “Computer-guided alignment I: Phase and amplitude modulation of the alignment-influenced wavefront,” Opt. Express 15, 3127–3139 (2007). [CrossRef] [PubMed]
- H. Lee, G. Dalton, I. Tosh, and S. Kim, “Computer-guided alignment III: Description of inter-element alignment effect in circular-pupil optical systems,” Opt. Express 16, 10992–11006 (2008). [CrossRef] [PubMed]
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C 2nd ed. (Cambridge, 2002).
- G. D’Agostini, Bayesian Reasoning in Data Analysis (World Scientific, 2005).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.