## Ronchi test can detect piston by means of the defocusing term

Optics Express, Vol. 12, Issue 16, pp. 3719-3736 (2004)

http://dx.doi.org/10.1364/OPEX.12.003719

Acrobat PDF (1255 KB)

### Abstract

We present simulated results on piston detection applying the classical Ronchi test to a segmented surface. We have found that a piston error in a test segment, induces a change in the transversal aberration, that can be analyzed by mutually comparing the fringes frequency in each segment. We propose that the piston term of the segmented surface can be recovered by geometrically relating the change in transversal aberration with the piston term. To test this, we have simulated some ronchigrams for a known piston error, and we have been able to recover this term for a dynamic range comprised among 57*nm* and 550 *µm*. For piston errors >550*µm* a change in the transversal aberration can be appreciated and measured in the ronchigrams although these large pistons are now classical defocusings. Thus we have demonstrated that the Ronchi test can be an alternative method for the piston detection with a large dynamic range.

© 2004 Optical Society of America

## 1. Introduction

*et al*. [3], uses shearing interferometry, to recover piston errors of the order of a wavelength with random noise.

4. Weiyao Zou, “New phasing algorithm for large segmented telescope mirrors,” Opt. Eng. **41**, 2338–2344,(2002). [CrossRef]

*et al*. [5

5. Achim Shumacher, Nicholas Devaney, and Luzuma Montoya, “Phasing segmented mirrors: a modification of the Keck narrow-band technique and its application to extremely large telescopes,” Appl. Opt. **41**, 1297–1307, (2002). [CrossRef]

*λ*ambiguity in some problematic cases. Díaz-Uribe

*et al*. [6

6. R. Díaz-Uribe and A. Jiménez-Hernádez, “Phased measurement for segmented optics with 1D diffraction patterns,” Opt. Express **12**, 1192–1204, (2004), http:/www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1192 [CrossRef] [PubMed]

*et al*. [8

8. G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the Keck Telescopes: the broadband phasing algorithm,” Appl. Opt. **37**, 140–155, (1998). [CrossRef]

*et al*. [9

9. G. Chanan, C. Ohara, and M. Troy, “Phasing the mirror segments of the Keck Telescopes II: the narrow-band phasing algorithm,” Appl. Opt. **37**, 140–155, (2000). [CrossRef]

*δf*:

*W*(

*x,y*)=

*A*(

*x*

^{2}+

*y*

^{2})

^{2}+

*By*(

*x*

^{2}+

*y*

^{2})+

*C*(

*x*

^{2}+3

*y*

^{2})+

*D*(

*x*

^{2}+

*y*

^{2})+

*δf*; where A, B, C and D are the coefficient for astigmatism, coma, spherical aberration, and defocusing respectively. As the piston term in this polynomial equation is a constant, its first derivative is zero and then the Ronchi test should not detect it. However, the piston term is also contained in the D term, the defocusing (Bai

*et al*. [14

14. J. Bai and Shangyi Cheng Guoguang Yang, “Phase Alignment of segmented mirrors using a digital wavefront interferometer,” Opt. Eng. **36**, 2355–2357 (1997). [CrossRef]

*λ*=632.8 nm.

## 2. Theoretical basis

*R*<0, so

*s*is the object position,

_{o}*s*is the image position and R is the curvature radius measured from the parent surface vertex under test. If we take a reference segment, its image point position is found by setting

_{i}*s*=

_{o}*R*; therefore the position of its image plane is in

*s*=

_{i}*R*. Fig. 1.

*δ*along the optical axis, its position would be at

_{f}*δ*≪

_{f}*R*, that is when the piston error is on the order of a fraction of a wavelength, the term

*s*=

_{i}*R*+

*δ*. This means that the image plane position changes linearly with the piston error whenever

_{f}*δ*≪

_{f}*R*. Now we can estimate values of the defocusing term for which

*δ*can be considered a piston term, see Table 1. For values of piston less than

_{f}*λ*/10 the factor

*µ*m the piston term has a non linear behavior in the image plane.

*δ*≪

_{f}*R*. Then the linearity condition obtained by geometrical optics and the Hopkins theory constitute the theoretical basis that allows considering a small defocusing as a piston term in a segmented surface.

## 3. Relation between the transversal aberration and the piston term

*T*produced by the piston term

*δ*at a segmented surface, shown there for an on-axis source. However the case of off-axis sources is also important. If the source is placed way out off-axis it introduces aberrations that can produce strong effects in the piston measurement. For example, when the off-axis source position in the x direction is >3.0 cm (perpendicular to the fringes), or when the off-axis position in the y direction is >8

_{f}*λ*Fig. 2.

*λ*, while different amounts of sphericity, coma and astigmatism are introduced. Also if air turbulence is present the ronchigrams suffer a degradation effect, Fig. 3. Therefore, for piston detection with the Ronchi test is necessary to take into account these problems.

*W*(

*x,y*) be a mathematical representation of an aberrated wavefront, Malacara [14

14. J. Bai and Shangyi Cheng Guoguang Yang, “Phase Alignment of segmented mirrors using a digital wavefront interferometer,” Opt. Eng. **36**, 2355–2357 (1997). [CrossRef]

*x,y*) are the coordinates on the monolithic surface, and the coefficients

*A, B*and

*C*are the Seidel or third-order aberrations: spherical aberration, coma and astigmatism. The last term, the defocusing

*D*, is the key factor for our analysis, since it is detected by the Ronchi ruling position which is very close to the surface curvature center.

14. J. Bai and Shangyi Cheng Guoguang Yang, “Phase Alignment of segmented mirrors using a digital wavefront interferometer,” Opt. Eng. **36**, 2355–2357 (1997). [CrossRef]

*R*is the curvature radius of the surface under test, and ∇

*W*(

*x,y*) is the wavefront first derivative detected by the Ronchi test. Equation 7 can be expressed in components as,

*T*

_{1}in the reference segment, we have

*F*, and of the position on the mirror and it is reciprocal to the curvature radius.

*D*detected for the ruling depends on the Ronchi ruling position, Δ

*F*, and it is reciprocal to the quadratic curvature radius (O’Neill [17]).

*T*

_{1}, we obtain the transversal aberration for the reference segment as follows,

*T*

_{2}for a displaced segment as function of the piston error,

*δ*, from Fig. 1, where the segment has been displaced along of optical axis by

_{f}*δ*, and its radius of curvature has also been displaced,

_{f}*T*

_{1},

*T*

_{2}, we can obtain the difference in the transversal aberration, Δ

*T*=

*T*

_{2}-

*T*

_{1}detected by the Ronchi test in a segmented surface. After some simplifications we have,

*δ*,

_{f}## 4. Piston term simulation

*z*

_{0}of the order of the wavelength to its sagitta (Fig. 10).

*T*is evaluated in the observation plane in a certain point x on the surface, and is calculated as the difference of the transversal aberrations

*T*

_{1}and

*T*

_{2}, (for the reference and test segment respectively) in the x direction for the central maximums in the Ronchi fringes, Luna E.[18

18. E. Luna, S. Zazueta, and L. Gutiérrez, “An innovative method for the alignment of astronomical telescopes,” PASP , **113**:379–384, (2001). [CrossRef]

*ESC*, since the piston term is obtained in a plane nearby the curvature center of the surface. This plane is obtained for the different convergence image points of each segment. Thus, the width of a Ronchi fringe

*f*must be compared with the transversal aberration of the test segment,

*T*

_{2}then

*f*of a fringe is obtained from the following,

*NLP*is the number of lines per inch of the Ronchi ruling. In this way we evaluate the piston term in a simulated ronchigram from the Ronchi fringes central maximums by means of

*δ*and the simulated introduced piston

_{f}*z*

_{0}, as will be seen next, Table 3.

## 5. Dynamic range of the test

*λ*(≈57nm) results in seven fringes. Random noise of 10% has been added to the simulation to make it more real. Here, we just barely miss to observe a discontinuity in the ronchigrams and their fringes are aligned to each other perfectly. Then the minimum limit of piston detection is obtained by the resolving power of each Ronchi ruling. In Fig. 6(b) we show a large change in the fringes frequency as a result of a large piston term (note that according to Eq. (3) this is still a piston term). In this case (for a surface with R=1200 mm, and D=200 mm) the upper limit of the detection range is 550

*µ*m. The upper limit changes in accordance with the radius of curvature of the surface. In Fig. 6(c), we show the effect of piston values larger than 550

*µ*m in the surface plane that can still be seen with the Ronchi test, although this case rapidly becomes non linear as can be seen from Fig. 1 and Eq. (3). That is, the Ronchi test can continue to be used to detect lack of co-phasing, although it may be that the actual amount cannot be correctly evaluated for large piston errors.

*µ*m to 57 nm for seven and three fringes in each segment. We have used a reference wavelength of 632.8 nm.

*δ*agrees with the introduced piston

_{f}*z*

_{0}to better than 1/1000. The knowing of

*δ*will allow the active correction of the piston term by bringing equal the two transversal aberrations

_{f}*T*

_{1}and

*T*

_{2}.

## 6. Conclusions

*λ*with a same Ronchi ruling and with the same measurement algorithm, Salinas [19].

## Acknowledgments

## References and links

1. | D.J. Shroeder, |

2. | V. Orlov, “Co-phasing of segmented Mirror telescopes,” (Large Ground based telescopes projects and instrumentation, Workshop, Leiden 2000), pp. 391–396. |

3. | V. Voitsekhovich, S. Bara, and V.G. Orlov, “Co-phasing of segmented telescopes: A new approach to piston measurements,” A & A, |

4. | Weiyao Zou, “New phasing algorithm for large segmented telescope mirrors,” Opt. Eng. |

5. | Achim Shumacher, Nicholas Devaney, and Luzuma Montoya, “Phasing segmented mirrors: a modification of the Keck narrow-band technique and its application to extremely large telescopes,” Appl. Opt. |

6. | R. Díaz-Uribe and A. Jiménez-Hernádez, “Phased measurement for segmented optics with 1D diffraction patterns,” Opt. Express |

7. | G. Chanan, M. Troy, and E. Sirko, “Phase discontinuity sensing: a method for phasing segmented mirrors in the infrared,” Appl. Opt. |

8. | G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the Keck Telescopes: the broadband phasing algorithm,” Appl. Opt. |

9. | G. Chanan, C. Ohara, and M. Troy, “Phasing the mirror segments of the Keck Telescopes II: the narrow-band phasing algorithm,” Appl. Opt. |

10. | J. Salinas, E. Luna, L. Salas, A. Cornejo, I. Cruz, and V. Garcia, “The Classical Ronchi test for piston detection,” in Large Ground, Based Telescopes, Jacobus M. Oschmann and Larry M. Stepp, eds., Proc. SPIE4837, 758–763 (2003). |

11. | J. Salinas-Luna, “Cofaseo de una superficie segmentada,” PHD thesis, INAOE, Puebla, México, (2002). |

12. | Javier Salinas-Luna, Esteban Luna-Aguilar, and Alejandro Cornejo-Rodríguez, “Detección de pistón por polarimetría,” Rev. Mex. de Fís. (to be published). |

13. | D. Malacara, |

14. | J. Bai and Shangyi Cheng Guoguang Yang, “Phase Alignment of segmented mirrors using a digital wavefront interferometer,” Opt. Eng. |

15. | Hecht-Zajac, |

16. | H.H. Hopkins, |

17. | Edward L. O’ Neill, |

18. | E. Luna, S. Zazueta, and L. Gutiérrez, “An innovative method for the alignment of astronomical telescopes,” PASP , |

19. | IAUNAM, OAN, Apdo. Postal 877, Ensenada B. C. México, c.p. 22830 and Javier Salinas-Luna et al. are preparing a manuscript to be called “The classical Ronchi test for piston detection:experimental part.” |

**OCIS Codes**

(110.6770) Imaging systems : Telescopes

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(220.1140) Optical design and fabrication : Alignment

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 15, 2004

Revised Manuscript: July 12, 2004

Published: August 9, 2004

**Citation**

J. Salinas-Luna, E. Luna, L. Salas, I. Cruz-González, and A. Cornejo-Rodríguez, "Ronchi test can detect piston by means of the defocusing term," Opt. Express **12**, 3719-3736 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-16-3719

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### References

- D.J. Shroeder, Astronomical Optics, Multiple-Aperture telescopes (Academic Press, 1987).
- V. Orlov, ???Co-phasing of segmented Mirror telescopes,??? (Large Ground based telescopes projects and instrumentation, Workshop, Leiden 2000), pp. 391-396.
- V. Voitsekhovich S. Bara and V.G. Orlov, ???Co-phasing of segmented telescopes: A new approach to piston measurements,??? A & A, 382, 746-751, (2002).
- Weiyao Zou, ???New phasing algorithm for large segmented telescope mirrors,??? Opt. Eng. 41, 2338-2344,(2002). [CrossRef]
- Achim Shumacher, Nicholas Devaney and Luzuma Montoya, ???Phasing segmented mirrors: a modification of the Keck narrow-band technique and its application to extremely large telescopes,??? Appl. Opt. 41, 1297-1307, (2002). [CrossRef]
- R. Díaz-Uribe, A. Jiménez-Hernádez, ???Phased measurement for segmented optics with 1D diffraction patterns,??? Opt. Express 12, 1192-1204, (2004), <a href="http:/www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1192">http:/www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1192</a> [CrossRef] [PubMed]
- G. Chanan M. Troy and E. Sirko, ???Phase discontinuity sensing: a method for phasing segmented mirrors in the infrared,??? Appl. Opt. 38, 704-713, (1999). [CrossRef]
- G. Chanan M. Troy F. Dekens S. Michaels J. Nelson T. Mast D. Kirkman, ???Phasing the mirror segments of the Keck Telescopes: the broadband phasing algorithm,??? Appl. Opt. 37, 140-155, (1998). [CrossRef]
- G. Chanan C. Ohara M. Troy, ???Phasing the mirror segments of the Keck Telescopes II: the narrow-band phasing algorithm,??? Appl. Opt. 37, 140-155, (2000). [CrossRef]
- J. Salinas, E. Luna, L. Salas, A. Cornejo, I. Cruz and V. Garcia, ???The Classical Ronchi test for piston detection,??? in Large Ground, Based Telescopes, Jacobus M. Oschmann, Larry M. Stepp, eds., Proc. SPIE 4837, 758-763 (2003).
- J. Salinas-Luna,???Cofaseo de una superficie segmentada,??? PHD thesis, INAOE, Puebla, México, (2002).
- Javier Salinas-Luna, Esteban Luna-Aguilar & Alejandro Cornejo-Rodríguez, ???Detección de pistón por polarimetría,??? Rev. Mex. de Fís. (to be published).
- D. Malacara, Optical Shop testing, ???Ronchi test,??? (Academic Press 1992), Chap 9.
- J.Bai, Shangyi Cheng Guoguang Yang , ???Phase Alignment of segmented mirrors using a digital wavefront interferometer,??? Opt. Eng. 36, 2355-2357 (1997). [CrossRef]
- Hecht-Zajac, OPTICA, ??ptica geométrica, teoría paraxial, (Addison Wensley Longman, 1998).
- H.H. Hopkins, Wave Theory of Aberrations, Wave and Ray Aberrations, (the Clarendon Press, 1950).
- Edward L. O??? Neill, Introduction to Statistical Optics, The Geometrical theory of aberrations, (Dover Publications, 1992).
- E. Luna, S. Zazueta and L. Gutiérrez, ???An innovative method for the alignment of astronomical telescopes,??? PASP, 113:379-384, (2001). [CrossRef]
- IAUNAM, OAN , Apdo. Postal 877, Ensenada B. C. México, c.p. 22830 and Javier Salinas-Luna et al. are preparing a manuscript to be called ???The classical Ronchi test for piston detection:experimental part.???

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