## Absolute profile measurement of large moderately flat optical surfaces with high dynamic range

Optics Express, Vol. 16, Issue 16, pp. 11975-11986 (2008)

http://dx.doi.org/10.1364/OE.16.011975

Acrobat PDF (627 KB)

### Abstract

We present a novel procedure for absolute, highly-accurate profile measurement with high dynamic range for large, moderately flat optical surfaces. The profile is reconstructed from many sub-profiles measured by a small interferometer which is scanned along the specimen under test. Additional angular and lateral distance measurements are used to account for the tilt of the interferometer and its precise lateral location during the measurements. Accurate positioning of the interferometer is not required. The algorithm proposed for the analysis of the data allows systematic errors of the interferometer and height offsets of the scanning stage to be eliminated and it does not reduce the resolution. By utilizing a realistic simulation scenario we show that accuracies in the nanometer range can be reached.

© 2008 Optical Society of America

## 1. Introduction

2. R. Freimann, B. Dörband, and F. Höller, “Absolute measurement of non-comatic aspheric surface errors,” Opt. Commun. **161**, 106–114 (1999). [CrossRef]

3. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. **45**, 5856–5865 (2006). [CrossRef] [PubMed]

4. M. Beyerlein, N. Lindlein, and J. Schwider, “Dual-wave-front computer-generated holograms for quasi-absolute testing of aspherics,“ Appl. Opt. **41**, 2440–2447 (2002). [CrossRef] [PubMed]

7. F. Simon, G. Khan, K. Mantel, N. Lindlein, and J. Schwider, “Quasi-absolute measurement of aspheres with a combined diffractive optical element as reference,” Appl. Opt. **45**, 8606–8612 (2006). [CrossRef] [PubMed]

11. K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. **74**, 2894–2898 (2003). [CrossRef]

11. K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. **74**, 2894–2898 (2003). [CrossRef]

12. C. Elster, I. Weingärtner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Prec. Eng. **30**, 32–38 (2006). [CrossRef]

10. J. Fleig, P. Dumas, P. E. Murphy, and G.W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces,” Proc. SPIE. **5188**, 296–307 (2003). [CrossRef]

12. C. Elster, I. Weingärtner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Prec. Eng. **30**, 32–38 (2006). [CrossRef]

13. M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. **45**, 060503-1–060503-3 (2006). [CrossRef]

## 2. TMS

*j*(

*j*= 0, …,

*M*−1) of the interferometer is treated as an independent distance sensor suffering from an individual systematic error

*ε*due to an imperfect reference surface of the interferometer. An autocollimator is used for the additional measurement of the tilt angle

_{j}*b*of the scanning stage at each measurement position.

_{i}*j*in the interferometer is denoted by

*s*(

*j*) with

*s*(0) = 0. The lateral coordinate of the first pixel (

*j*= 0) of the interferometer in the measurement position

*i*is named

*p*, the distance between neighboring measurement positions is called

_{i}*d*and the effective pixel distance

_{step}*d*. In each measurement position, the scanning stage introduces a height offset

_{pix}1*a*and a tilt angle

_{i}*b*. For small tilt angles we can use the approximation tan(

_{i}*b*)≈

_{i}*b*. Therefore, in position

_{i}*p*the distance

_{i}*m*between the

_{i,j}*j*-th pixel and the surface can be modeled by

12. C. Elster, I. Weingärtner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Prec. Eng. **30**, 32–38 (2006). [CrossRef]

*d*, the shortest distance

_{step}*d*between neighboring pixels and the distance

_{pix}1*d*between adjacent reconstructed surface points are all the same. As a consequence, all pixels measure the surface

_{s}*f*(

*x*) at the same positions

*x*(

_{k}*k*= 0, ..,

*N*−1) (except for the boundaries), and Eq. (1) reduces to

*x*on (

_{k}*i*,

*j*) has been suppressed. In Eq. (2) the surface enters only at the discrete locations

*x*(

_{k}*k*= 0, ..,

*N*−1), and the reconstruction of the surface heights

*f*(

*x*) from the measurements is a discrete task. Note that Eq. (2) is linear in the unknowns

_{k}*f*(

*x*),

_{k}*ε*,

_{j}*a*and

_{i}*b*provided that the positions

_{i}*s*(

*j*) are known. As shown in [12

**30**, 32–38 (2006). [CrossRef]

*f*(

*x*) can be reconstructed up to an unknown straight line if the tilt angle

_{k}*b*is measured in addition. Note that the tilt angle may be measured with a systematic offset without affecting the reconstruction result. Since no knowledge about the systematic sensor errors

_{i}*ε*and the height offset

_{j}*a*is required for this reconstruction, there is no need to calibrate the systematic errors of the interferometer reference or the scanning stage prior to the measurement.

_{i}## 3. Extended TMS

*x*. However, positioning errors or distortion of the interferometer lead to lateral measurement locations which differ from the

_{k}*x*which then results in reconstruction errors for TMS [14

_{k}14. A. Wiegmann, C. Elster, R.D. Geckeler, and M. Schulz, “Stability analysis for the TMS method: Influence of high spatial frequencies,” Proc. SPIE. **6616**, 661618 (2007). [CrossRef]

*d*

_{pix1},

*d*and

_{s}*d*are no longer identical. Exact positioning of the interferometer is more difficult to realize than the determination of its actual position. For this, we propose to extend the TMS measurement set-up by an additional measurement of the lateral position of the interferometer, which can be done with the help of a displacement interferometer, cf. Fig. 1 for a schematic and Fig. 2 for a prototype set-up. The proposed set-up now allows for more flexibility in the scanning steps with the actual positions of the scanning stage being accounted for. The reconstruction algorithm underlying TMS is based on the fact that the surface is measured at the discrete lateral positions

_{step}*x*repeatedly by many different interferometer pixels. This is utilized to account for the additional unknowns in Eq. (2), i.e. offsets

_{k}*a*and systematic errors

_{i}*ε*. Since for the extended TMS the surface may be measured at many more positions, several measurements by different interferometer pixels at the same surface position may no longer be available. As a consequence, immediate application of the algorithm underlying TMS is not possible. In contrast to common stitching techniques, interpolation of the measured values in every sub-profile at the grid points

_{j}*x*as a preprocessing to the TMS algorithm is not useful because both, topography heights and interferometer errors would be affected by such an interpolation. Now, the idea is to still model the surface at a set of positions

_{k}*x*, but to use an interpolation scheme to account for the measurements made at locations different from the

_{k}*x*. In this way, the TMS data analysis scheme can still be applied and the surface be reconstructed at the positions

_{k}*x*. Effectively, a continuous reconstruction of the surface is being carried out in this way, and the three values

_{k}*d*

_{pix1},

*d*and

_{step}*d*need no longer be identical.

_{s}*f*(

*x*) by applying Lagrange interpolation [15]. The surface height

_{k}*f*(

*x*̃) at position

*x*̃(

*x*̃ ≠

*x*;

_{k}*k*= 0, ..,

*N*−1) is given as a linear combination of the surface values

*f*(

*x*) in the neighborhood of

_{k}*x*̃, and Eq. 2 now reads:

*o*is the chosen degree of the interpolation polynomial,

**c**

*the vector with all interpolation coefficients and*

_{N}*f*

*the vector with the desired surface values. The coefficients*

_{N}*c*(

_{k}*x*̃) can be calculated independent of the

*f*. When a measurement position matches one of the

_{N}*x*’s, all the coefficients

_{k}*c*(

_{k}*x*̃) would be zero apart from one and Eq. (3) reduces to Eq. (2).

*f*(

*x*),

_{k}*ε*,

_{j}*a*and

_{i}*b*. Therefore, the least-squares estimation procedure described in [12

_{i}**30**, 32–38 (2006). [CrossRef]

*x*̃ can be calculated from the measured positions

*p*and the known pixel positions

_{i}*s*(

*j*).

*x*are chosen only for the purpose of analysis (and not for the design of the experiment), we have the freedom to determine them in some optimal manner. To this end (and prior to the analysis), we choose the

_{k}*x*equidistantly and place this grid in an optimal way: we shift the whole set of

_{k}*x*such that the accumulated squared distances between every actual measurement position

_{k}*x*̃ and its particular nearest neighbor from the set of

*x*’s is minimum.

_{k}*o*+1 points used for the interpolation are chosen in such a manner that

*x*̃, and hence

*o*has to be an odd number. A transfer function can be associated with the interpolation scheme [16] and this transfer function depends on the chosen degree of the interpolation polynomial. Figure 3 shows that the higher the degree of the interpolation polynomial, the more rectangular the shape of the transfer function. Ideally, a rectangular transfer function is sought which correctly interpolates all continuous functions containing only spatial frequencies below the Nyquist frequency. The drawback of a high interpolation degree is increasing computing time and memory requirements due to a decrease in the sparsity of the design matrix implied by Eq. (3). In any case, a too high degree is not required since other error influences such as errors of the lateral position measurements then become the dominant error source, cf. section 4.

*f*(

*x*) =

*f*(

*x*+

*Nd*)), axial symmetric (

_{s}*f*(

*x*

_{0}−Δ

*x*) =

*f*(

*x*

_{0}+Δ

*x*),

*f*(

*x*

_{N−1}−Δ

*x*) =

*f*(

*x*

_{N−1}−Δ

*x*)), point symmetric (

*f*(

*x*

_{0}−Δ

*x*) = 2

*f*(

*x*

_{0}) −

*f*(

*x*

_{0}+Δ

*x*),

*f*(

*x*

_{N−1}+Δ

*x*) = 2

*f*(

*x*

_{N−1})−

*f*(

*x*

_{N−1}−Δ

*x*)) or surrounded by zeros. To avoid such assumptions, it is also possible to use an asymmetric interpolation scheme or to reduce the degree of interpolation near the margins down to a nearest neighbor interpolation. From our experience we found that the assumption of point symmetry appears to perform best and subsequent results are based on this assumption. To account for the boundary conditions, Eq. (3) has to be modified according to

**A**in Eq. (5) is a

*N*×(

*N*+

*o*) matrix with a

*N*×

*N*identity matrix inside. Vector

**c**

_{N+o}represents now also the interpolation coefficients for the external surface points. The matrices

**L**and

**R**are designed in such a way, that the desired boundary conditions are fulfilled.

**30**, 32–38 (2006). [CrossRef]

*f*

_{0}=

*f*

_{N−1}= 0. Note finally, that one of the systematic sensor errors still has to be set to zero, e.g.

*ε*

_{0}= 0 as explained in [12

**30**, 32–38 (2006). [CrossRef]

## 4. Results

*M*= 100,

*d*

_{pix1}= 18.9μm) by successively setting the value of each error parameter to zero. As a result, the lateral positioning error characterized by

*σ*

_{pos1}(for TMS), the lateral position measurement error

*σ*

_{pos2}(for extended TMS) and the assumed pixel distance

*d*

_{pix2}(cf. section 4.3) for the reconstruction process turned out to be the most important error sources.

### 4.1. Simulation scenario and quality assessment

*b*and offset

_{i}*a*of the interferometer lead to lateral displacements in real measurements and this has been considered in the simulation. The interferometer was assumed to be approximately 5mm above the surface. The point of rotation was set to the first interferometer pixel. The measurement directions of the interferometer pixels are assumed to be parallel (according to a planar wavefront). These measurement directions change with a tilt of the interferometer and lead to lateral displacements on the surface. These displacements were considered when simulating the measurements by calculating the geometric distances between every pixel and the surface under test in the direction of the tilted interferometer. Note that for the analysis these lateral displacements were not accounted for, cf. Eqs. (1)-(3). The interferometer is not capable to measure the complete geometric distance, and hence from every sub-profile the mean value was subtracted. Finally, the systematic interferometer errors were added to these sub-profiles. Autocollimator and displacement measurements suffer from adjustment errors. To account for this, a constant offset angle bsys was added to the autocollimator measurements

_{i}*b*̃

*and a systematic adjustment error*

_{i}*p*was considered for the measurements of the displacement interferometer

_{sys}*p*̃

*. Table 1 gives a summary of all parameters used for the simulations. Due to computing time, for some simulations the number of pixels was reduced to 10 and the pixel distance set to 189*

_{i}*μ*m instead of 18.9

*μ*m. Furthermore, the maximum lateral dimension of the measured profile was limited to 80mm. Since TMS does not suffer from accumulation of errors, the maximum length of the profile is not limited by the algorithm.

### 4.2. Accuracy and resolution

*M*= 10,

*d*

_{pix1}=

*d*

_{pix2}= 189

*μ*m,

*σ*

_{pos2}= 250nm. The size

*σ*

_{pos1}of the positioning error is given in units of the reconstruction distance

*d*which eases comparison with other set-ups. For low positioning errors the normal and extended TMS algorithm lead to nearly identical results. When

_{s}*σ*

_{pos1}is larger than

*μ*m here), the results of TMS become worse, while the extended TMS is not affected at all. Hence, Fig. 6 demonstrates that the extended TMS algorithm does not suffer from positioning errors of the scanning stage (provided that the actual measurement locations have been measured sufficiently accurate).

*M*= 10,

*d*

_{pix1}=

*d*

_{pix2}= 189

*μ*m,

*σ*

_{pos2}= 250nm). The number of realizations was chosen that the standard deviation for every parameter set is about a tenth of the mean value. While TMS suffers from positioning errors for high spatial frequencies, the extended TMS procedure is almost independent of this error source. Hence, the extended TMS method has an improved resolution as compared to TMS when positioning errors emerge, provided that their position is determined accurately. Without any measurement errors the transfer function of the entire algorithm would be identical to the transfer function of the applied interpolation (cf. Fig. 3). Realistic simulation scenarios (

*M*= 100,

*d*

_{pix1}=

*d*

_{pix2}= 18.9

*μ*m,

*σ*

_{pos1}= 5μm) have shown, that the lateral positioning measurement error is the most important error source regarding lateral resolution (Fig. 8). For high spatial frequencies and large lateral position measurement errors the reconstructed surface is nearly averaged to zero and the rms error converges to

*σ*

_{pos2}of the lateral position measurement error exceeds a tenth of the distance ds between neighboring reconstruction points, the surface reconstruction for short wavelength is limited by this error rather than by the degree of interpolation. When

*σ*

_{pos1}is larger than half of the reconstruction distance, several peaks occur for the reconstruction error. These peaks correspond to peaks of the condition number of the design matrix of the system of linear equations. Avoiding this instability demands a lateral position measurement error smaller than half of the reconstruction distance.

### 4.3. Optimization and limitations

*x*were shifted by Δ

_{k}*relative to their optimal position as displayed in Fig. 9. As expected, the rms error increases with increasing Δ*

_{pos}_{pos}, cf. Fig. 10; the minimal rms error is about 2.9nm (simulation parameters:

*M*= 10,

*d*

_{pix1}=

*d*

_{pix2}= 189μm,

*σ*

_{pos1}= 5μm,

*σ*

_{pos2}= 250nm). When the grid is shifted only by ±0.5ds, the rms error increases by 1nm. For higher shifts the increase of the rms error is caused by effects at the surface margins.

*d*

_{pix1}, which is the distance of CCD pixels traced through the optical system of the interferometer, has to be known. Errors of this parameter lead to a scaling of the reconstructed surface perpendicular to the scanning direction; Fig. 11 shows results of extended TMS in dependence on the relative error of the pixel distance

*d*

_{pix1}as introduced in table 1. In contrast to Fig. 10, Fig. 11 shows not the rms errors, but the reconstruction error at each position of the reconstructed surface for one typical simulation. When the assumed pixel distance is too large, the reconstructed surface is scaled down in

*y*-direction. If an error below 10nm is requested for the topography displayed on the left hand side of Fig. 5, a relative error for the effective pixel distance below 10–3 is required.

*μ*m range and not displayed here. The remaining peaks of the reconstruction error at the positions 10mm, 20mm and 70mm in Fig. 11 are caused by the linear approximation in Eq. 1. The tilt of the sensor leads to a displacement in the lateral position which is not considered in Eq. 1. Especially at positions with large surface slope this modeling error leads to deviations for the reconstructed surface.

17. B. Doerband and J. Hetzler , “Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF),” Proc. SPIE. **5878**, 587806 (2005). [CrossRef]

*d*

_{pix1}(pixel distance),

*d*

_{step}(scanning step) and

*d*(reconstruction distance) need not be identical. However, there may still exist some constraints. We found that the reconstruction distance should not be smaller than half of the distance between two neighboring pixels. Furthermore, the scanning step has to be smaller than the reconstruction distance. In case of an equidistant sensor array this leads to the conditions

_{s}## 5. Conclusion

## 6. Acknowledgments

## Footnotes

All errors except for the piston errors were simulated by drawing zero mean Gaussian random numbers with standard deviations given by the values of the parameters. The piston errors were simulated as 5 mm to which a zero-mean Gaussian random number was added with standard deviation 5 μm. |

## References and links

1. | D. Malacara, Optical Shop Testing, (John Wiley & Sons Inc, 1992). |

2. | R. Freimann, B. Dörband, and F. Höller, “Absolute measurement of non-comatic aspheric surface errors,” Opt. Commun. |

3. | U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. |

4. | M. Beyerlein, N. Lindlein, and J. Schwider, “Dual-wave-front computer-generated holograms for quasi-absolute testing of aspherics,“ Appl. Opt. |

5. | S. Reichelt and H. J. Tiziani, “Twin-CGHs for absolute calibration in wavefront testing interferometry,” Opt. Commun. |

6. | S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute testing of aspheric surfaces,” Optical Fabrication, Testing, and Metrology |

7. | F. Simon, G. Khan, K. Mantel, N. Lindlein, and J. Schwider, “Quasi-absolute measurement of aspheres with a combined diffractive optical element as reference,” Appl. Opt. |

8. | T.W. Stuhlinger “Subaperture optical testing - Experimental verification,” Proc. SPIE. |

9. | M. Sjoedahl and B. F. Oreb, “Stitching interferometric measurement data for inspection of large optical components,” Opt. Eng. |

10. | J. Fleig, P. Dumas, P. E. Murphy, and G.W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces,” Proc. SPIE. |

11. | K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,“ Rev. Sci. Instrum. |

12. | C. Elster, I. Weingärtner, and M. Schulz, “Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors,” Prec. Eng. |

13. | M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. |

14. | A. Wiegmann, C. Elster, R.D. Geckeler, and M. Schulz, “Stability analysis for the TMS method: Influence of high spatial frequencies,” Proc. SPIE. |

15. | W. H. Press, P. B. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C : The Art of Scientific Computing, (Cambridge University Press, 1992). |

16. | B. Jähne Digitale Bildverarbeitung, (Springer, 2005). |

17. | B. Doerband and J. Hetzler , “Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF),” Proc. SPIE. |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

(100.3008) Image processing : Image recognition, algorithms and filters

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: April 14, 2008

Revised Manuscript: June 4, 2008

Manuscript Accepted: June 4, 2008

Published: July 25, 2008

**Citation**

A. Wiegmann, M. Schulz, and C. Elster, "Absolute profile measurement of large moderately flat optical surfaces with
high dynamic range," Opt. Express **16**, 11975-11986 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-11975

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### References

- D. Malacara, Optical Shop Testing, (John Wiley & Sons Inc, 1992).
- R. Freimann, B. Dorband, and F. Holler, "Absolute measurement of non-comatic aspheric surface errors," Opt. Commun. 161, 106-114 (1999). [CrossRef]
- U. Griesmann, "Three-flat test solutions based on simple mirror symmetry," Appl. Opt. 45, 5856-5865 (2006). [CrossRef] [PubMed]
- M. Beyerlein, N. Lindlein and J. Schwider, "Dual-wave-front computer-generated holograms for quasi-absolute testing of aspherics," Appl. Opt. 41, 2440-2447 (2002). [CrossRef] [PubMed]
- S. Reichelt and H. J. Tiziani, "Twin-CGHs for absolute calibration in wavefront testing interferometry," Opt. Commun. 220, 23-32 (2003). [CrossRef]
- S. Reichelt, C. Pruss and H. J. Tiziani, "Absolute testing of aspheric surfaces," Optical Fabrication, Testing, and Metrology 45, 252-263 (2004).
- F. Simon, G. Khan, K. Mantel, N. Lindlein and J. Schwider, "Quasi-absolute measurement of aspheres with a combined diffractive optical element as reference," Appl. Opt. 45, 8606-8612 (2006). [CrossRef] [PubMed]
- T. W. Stuhlinger, "Subaperture optical testing - Experimental verification," Proc. SPIE. 656, 118-127 (1986).
- M. Sjoedahl and B. F. Oreb, "Stitching interferometric measurement data for inspection of large optical components," Opt. Eng. 41, 403-408 (2002). [CrossRef]
- J. Fleig, P. Dumas, P. E. Murphy and G. W. Forbes, "An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces, "Proc. SPIE. 5188, 296-307 (2003). [CrossRef]
- K. Yamauchi,K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa and Y. Mori, "Microstitching interferometry for x-ray reflective optics," Rev. Sci. Instrum. 74,2894-2898 (2003). [CrossRef]
- C. Elster, I. Weingrtner and M. Schulz, "Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors," Prec. Eng. 30,3 2-38 (2006). [CrossRef]
- M. Schulz and C. Elster, "Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution," Opt. Eng. 45, 060503-1-060503-3 (2006). [CrossRef]
- A. Wiegmann, C. Elster, and R. D. Geckeler and M. Schulz, "Stability analysis for the TMS method: Influence of high spatial frequencies," Proc. SPIE. 6616, 661618 (2007). [CrossRef]
- W. H. Press, P. B. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C : The Art of Scientific Computing, (Cambridge University Press, 1992).
- B. Jahne, Digitale Bildverarbeitung, (Springer, 2005).
- B. Doerband and J. Hetzler, "Characterizing lateral resolution of interferometers: the Height Transfer Function (HTF)," Proc. SPIE. 5878, 587806 (2005). [CrossRef]

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