## Classical Hartmann test with scanning

Optics Express, Vol. 17, Issue 16, pp. 13959-13973 (2009)

http://dx.doi.org/10.1364/OE.17.013959

Acrobat PDF (3340 KB)

### Abstract

In order to introduce many more evaluation points during the Hartmann test, the scanning of the screen across the pupil is proposed; after each step of the scan a different image of the bright spots is obtained. Basic ideas about how to design radial and square screens for the scanning are presented. Radial screens are scanned by rotation, whereas for square screens a linear inclined scan is enough to introduce many more evaluation points along two independent directions. For square screens it is experimentally shown that the lateral resolution of the test is improved.

© 2009 OSA

## 1. Introduction

2. A. Morales and D. Malacara, “Geometrical parameters in the Hartmann test of aspherical mirrors,” Appl. Opt. **22**(24), 3957–3959 (1983). [CrossRef] [PubMed]

3. V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Point shifting in the optical testing of fast aspheric concave surfaces by a cylindrical screen,” Appl. Opt. **47**(5), 644–651 (2008). [CrossRef] [PubMed]

2. A. Morales and D. Malacara, “Geometrical parameters in the Hartmann test of aspherical mirrors,” Appl. Opt. **22**(24), 3957–3959 (1983). [CrossRef] [PubMed]

## 2. Spiral screen

*n*and the number of holes

*m*along each line (see Fig. 1.a), the full angle that the screen must be rotated for the scan is

*k*, the desired number of new evaluation points (between two originally adjacent holes) to be obtained during the scan, a new set of holes are displaced the distance

### 2.1 Hole size

3. V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Point shifting in the optical testing of fast aspheric concave surfaces by a cylindrical screen,” Appl. Opt. **47**(5), 644–651 (2008). [CrossRef] [PubMed]

*λ*is the wavelength of the light used during the test,

*K*is the conic constant of the mirror under test,

*c*is its curvature at the vertex,

*D*its diameter, and

*L*is the distance from the observation plane to the plane where the reflected rays are focused. For a spherical mirror

*K*= 0, tested under visible light

*λ*= 500 nm,

*r*= 1/

*c*= 600 mm,

*D*= 150mm, the optimum hole size is a function of

*L*; in Fig. 2, a plot of the hole diameter vs.

*L*is shown. It is easy to see that for values of

*L*between 2 and 10 mm, the hole size varies between 10 and 5 mm.

### 2.2 Alternative screen designs

*n*= 4,

*m*= 5, and

*θ*= 90°, is as shown in Fig. 3. It is clear that for spiral screens, the azimuthal distance between adjacent holes depends on the distance to the center of the screen; adding size to the holes, at the center of the screen, they overlap whereas at the rim they are more separated. Then, with this kind of screen the wavefront is not uniformly sampled; scanning the screen increases the number of points but the sampling is still not uniform.

*n*and

*m*, and locating a hole only on these points where the overlap is not present; the other points are not included. For instance, in Fig. 4 a different screen design is obtained for

*n*=

*m*= 20; the small dots are the points obtained with the first proposal above, whereas the circles with the plus signs in the center describe the holes and their centers with a more uniform distribution without overlapping. In Fig. 5, some simulations of scanning the screen in Fig. 4 are shown for the cases with

*t*= 4, 8, 12, 17, and 32; each angular step is given by

*γ*= 2

*π*/ t. Evidently, many new evaluation points are added but at the center and its neighborhood there are large zones without evaluation points.

## 3. Square screen

*c*is given by Eq. (10), and

*b*-1 is the number of columns between P1 and P2. The direction of the scan is defined by the angle

*β*with the

*x*-axis, so it is given by

*s*-1 additional points in each square cell. The total number of points for evaluating the surface is

### 3.1 Separation between points

*j*=

*k*and

*j*=

*k*+ 1, gives the other two distances;

*d*,

_{s}*d*and

_{k}*d*

_{k+1}, are the three smaller distances between the scanned points on the surface. The angle

*α*is given by

_{j}*b*= 3 and

*s*= 4, 5, 6,…, and 15; for each case the number of new points added by the scan is s-1, or 3, 4, 5, and 16, respectively. A hexagon is included to show the nearest points around some particular point; the arrow shows the scan direction. In Table 1, geometrical data for the array of evaluation points are listed. These data is useful to know what are the smallest distances between two points for different number of steps during the scan. Several features of the array of points for different values of s can be drawn from Fig. 8; Table 1, however, quantifies some of them. For instance, Fig. 8.g) seems to be a square array of points, the principal axes of the array being along different directions from the principal axes of the holes on the screen. From Table 1, this assumption can be proved; from the row corresponding to

*s*= 10, it is easy to find that

*ds*=

*dk*, and

*β*+

*α*= 90°, proving that the array is indeed square. Similar data can be computed for other values of b.

_{k}*k*+ 1-th point is not one of the six nearest points; this point must be replaced by the

*k*-1-th point. This is the case when

*k*=

*b*/

*s*, then

*k*is an integer. For this reason, in Table 1, the data for

*d*

_{k-1}and

*α*

_{k-1}are also included.

## 4. Experiment

*F*/4 spherical mirror, with a diameter

*D*= 150 mm, and a radius of curvature

*r*= 600 mm, was conducted. The number of points along a principal diameter of the mirror was chosen as

*m*= 11, according to Eq. (10) the hole separation must be

*c*= 15 mm; to avoid part of the holes at the edge being out of the surface the clear diameter was adjusted to 140 mm, so that

*c*= 14 mm. According to Eq. (11) the total number of points inside the pupil diameter was around 79, however, due to the adjustment of the diameter the real number was increased to 89. In Fig. 9 the screen design is shown. The starting position of the mirror is defined by the circle; as the screen moves to scan the surface, some holes go out of the pupil, but they are substituted by other holes initially outside the pupil, for that reason the screen is larger than a classical Hartmann screen. According to Fig. 2, a convenient diameter for the holes is 5 mm.

*b*= 3 columns, so the inclination angle

*β*was 18.43° and the number of steps of the scan was chosen to be

*s*= 9. The step size was

*d*= 4.92 mm, the full scan distance

_{s}*d*= 44.28 mm.

*s*= 801.

## 5. Evaluation of the normals and integration procedure

*xy*plane and the spot positions on the observing screen are well known, then it is an easy task to find the

*xyz*components of the incident and reflected rays on the mirror for each hole of the screen. In reference to Fig. 13, as the magnitudes of vectors

**V**

_{1},

**V**

_{2}, and

**V**

_{3}are known from the setup (Fig. 10), their directions are easily found. Then the incident ray can be expressed as

**H**and

**S**are, respectively, the vector for a hole on the screen and for its corresponding spot at the observing screen. According to the Reflection Law, the normal to the surface at the point of incidence is

**r**

_{i}and

**r**

_{r}are the unit vectors associated with the incident and reflected rays. Once the normals to the test surface are obtained, the shape of the surface can be found by using the integral

5. R. Díaz-Uribe, “Medium-precision null-screen testing of off-axis parabolic mirrors for segmented primary telescope optics: the large millimeter telescope,” Appl. Opt. **39**(16), 2790–2804 (2000). [CrossRef]

3. V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Point shifting in the optical testing of fast aspheric concave surfaces by a cylindrical screen,” Appl. Opt. **47**(5), 644–651 (2008). [CrossRef] [PubMed]

5. R. Díaz-Uribe, “Medium-precision null-screen testing of off-axis parabolic mirrors for segmented primary telescope optics: the large millimeter telescope,” Appl. Opt. **39**(16), 2790–2804 (2000). [CrossRef]

6. M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen,” Appl. Opt. **48**(5), 1008–1015 (2009). [CrossRef] [PubMed]

*z*. Then, in order to have only one constant, all the integration paths must start at the same point and to reduce the numerical calculation error the paths must be as simple and direct as possible; very involved paths increase the numerical error.

_{o}**47**(5), 644–651 (2008). [CrossRef] [PubMed]

*M*= [

*d*

^{2}

*f*(

*x*)/

*dx*

^{2}]

_{max}, is the maximum value of the second derivative of the integrating function

*f*(

*x*). Then, as a sphere is a second degree surface, it can be expressed in the

*zx*–plane as

*A*,

*B*and

*C*are arbitrary constants (

*A*≠ 0). The normal to this curve is obtained through the derivative of Eq. (22),

*f*(

*x*) in Eq. (24) is null and so is

*M*in Eq. (21); in consequence, the truncation error in Eq. (21) is negligible not only for spheres, but also for all the conic surfaces, when the trapezoid rule is used. The error is not zero because, this is true only for ideal conics. Important truncation errors are introduced for very fast surfaces where the fabrication errors make the surface depart appreciably from the ideal surface.

## 6. Conclusions

## Acknowledgments

## References and links

1. | D. Malacara-Doblado and E. Ghozeil, “Hartmann, Hartmann-Shack and other screen tests,” in Optical Shop Testing, Third Edition, Edited by Daniel Malacara (John Wiley and Sons, 2007), Chapter10. |

2. | A. Morales and D. Malacara, “Geometrical parameters in the Hartmann test of aspherical mirrors,” Appl. Opt. |

3. | V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Point shifting in the optical testing of fast aspheric concave surfaces by a cylindrical screen,” Appl. Opt. |

4. | D. Liu, H. Huang, B. Ren, A. Zeng, Y. Yan, and X. Wang, “Scanning Hartmann test method and its application to lens aberration measurement,” Chin. Opt. Lett. |

5. | R. Díaz-Uribe, “Medium-precision null-screen testing of off-axis parabolic mirrors for segmented primary telescope optics: the large millimeter telescope,” Appl. Opt. |

6. | M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen,” Appl. Opt. |

**OCIS Codes**

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.4800) Instrumentation, measurement, and metrology : Optical standards and testing

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

(220.4840) Optical design and fabrication : Testing

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 26, 2009

Revised Manuscript: June 27, 2009

Manuscript Accepted: July 3, 2009

Published: August 3, 2009

**Citation**

Rufino Díaz-Uribe, Fermín Granados-Agustín, and Alejandro Cornejo-Rodríguez, "Classical Hartmann test with scanning," Opt. Express **17**, 13959-13973 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13959

Sort: Year | Journal | Reset

### References

- D. Malacara-Doblado and E. Ghozeil, “Hartmann, Hartmann-Shack and other screen tests,” in Optical Shop Testing, Third Edition, Daniel Malacara, ed., (John Wiley and Sons, 2007), Chapter10.
- A. Morales and D. Malacara, “Geometrical parameters in the Hartmann test of aspherical mirrors,” Appl. Opt. 22(24), 3957–3959 (1983). [CrossRef] [PubMed]
- V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Point shifting in the optical testing of fast aspheric concave surfaces by a cylindrical screen,” Appl. Opt. 47(5), 644–651 (2008). [CrossRef] [PubMed]
- D. Liu, H. Huang, B. Ren, A. Zeng, Y. Yan, and X. Wang, “Scanning Hartmann test method and its application to lens aberration measurement,” Chin. Opt. Lett. 4, 725–728 (2006).
- R. Díaz-Uribe, “Medium-precision null-screen testing of off-axis parabolic mirrors for segmented primary telescope optics: the large millimeter telescope,” Appl. Opt. 39(16), 2790–2804 (2000). [CrossRef]
- M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen,” Appl. Opt. 48(5), 1008–1015 (2009). [CrossRef] [PubMed]

## 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.

### Figures

Fig. 1. |
Fig. 2. |
Fig. 3. |

Fig. 4. |
Fig. 5. |
Fig. 6. |

Fig. 7. |
Fig. 8. |
Fig. 9. |

Fig. 10. |
Fig. 11. |
Fig. 12. |

Fig. 13. |
Fig. 14. |
Fig. 15. |

« Previous Article | Next Article »

OSA is a member of CrossRef.