## Geometrical optics modeling of the grating-slit test

Optics Express, Vol. 15, Issue 4, pp. 1738-1744 (2007)

http://dx.doi.org/10.1364/OE.15.001738

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### Abstract

A novel optical testing method termed the grating-slit test is discussed. This test uses a grating and a slit, as in the Ronchi test, but the grating-slit test is different in that the grating is used as the incoherent illuminating object instead of the spatial filter. The slit is located at the plane of the image of a sinusoidal intensity grating. An insightful geometrical-optics model for the grating-slit test is presented and the fringe contrast ratio with respect to the slit width and object-grating period is obtained. The concept of spatial bucket integration is used to obtain the fringe contrast ratio.

© 2007 Optical Society of America

## 1. Introduction

1. V. Ronchi, “40 Years of History of Grating Interferometer,” Appl. Opt **3**,437–451 (1964). [CrossRef]

2. T. Yatagai, “Fringe Scanning Ronchi Test for Aspherical Surfaces,” Appl. Opt **23**,3676–3679 (1984). [CrossRef] [PubMed]

4. K. Hibino, D. I. Farrant, B. K. Ward, and B. F. Oreb, “Dynamic range of Ronchi test with a phase-shifted sinusoidal grating,” Appl. Opt **36**,6178–6189 (1997). [CrossRef]

5. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt **13**,2693–2703 (1974). [CrossRef] [PubMed]

6. M. M. Gonzalez and N. A. Ochoa, “The Ronchi test with an LCD grating,” Opt. Commun **191**,203–207 (2001). [CrossRef]

6. M. M. Gonzalez and N. A. Ochoa, “The Ronchi test with an LCD grating,” Opt. Commun **191**,203–207 (2001). [CrossRef]

## 2. Transverse ray theory

*W*at the exit pupil that has a paraxial radius r; the transverse ray aberration at the observation plane is

*TA*

_{Y}

*W*, is usually very small when compared with

*r*and therefore we can neglect

*W*and simplify the equation to,

*TA*

_{Y}is the Y-component of the ray interception position in the observation plane. The transverse ray aberration at the image plane

*TA*

_{Y}is the quantity that is measured in focal plane tests like the Ronchi test.

## 3. Geometrical optics model

*W*can be approximated as independent of the field position

*P*(

*Xa,Ya*) and the resultant geometrical point spread function of the aberrated wavefront is linearly shifted over the small field. This property of linear system invariance over a small field is the basic and important assumption in the grating-slit test theory.

*P*(

*Xa,Ya*) on the sinusoidal grating and assume that the optical magnification ratio of the optics under test is

*m*. The sinusoidal grating is parallel to the Y-direction and it is illuminated uniformly in all angular directions. Light passes through the slit with width d only when the ray interception position FPx at the slit plane satisfies Eq. (3),

*m*is the magnification of the test configuration and

*TAx*is the transverse ray aberration in x direction. With the assumption that the optics under test is a linear invariant system within a small field, the transverse ray aberration

*TAx*is a function independent of the field position (

*x*). By inserting Eq. (2) into Eq. (3) we obtain Eq. (4),

_{A},y_{A}*d*is infinitely small. Therefore, only when the ray interception position in the x-direction is zero, light is set to pass through the slit. In mathematical terms this requires,

*p*, the peak of fringe intensity occurs only when the incoming ray from the object position

*Xa*satisfies Eq. (5),

*N*is an integer number. After combining Eq. (5) and Eq. (6) we obtain a similar result to the obtained for the Ronchi test fringe [8],

*M*is an integer number.

*d*in turn samples and modulates each of the shifted point spread function. Both the point spread function and the slit are spatially coincident. The air slit can be considered to scan linearly the point spread function from different object locations. The lateral position of the spots is modulated into the fringes at the observation plane. The observed fringes of the grating-slit test look just like the observed fringes in the traditional Ronchi test, or so called “Ronchigram”. However, the modulation of the fringes is indeed from the modulation of sinusoidal intensity grating in the object space. This is in contrast to the traditional Ronchi test where the entire Ronchigram is created from a single point source. Although the traditional Ronchi test can be used with an extended light source, each point in the extended source contributes the same whole Ronchigram and is linearly superimposed in intensity with the Ronchigram contributed by other points.

## 4. Spatial bucket-integration in the geometrical model

*p*is the period of the illuminated sinusoidal grating.

*KdTAx*is the infinitely small solid angle extending from the grating to the observation plane. Since the slit has a finite width

*d*, we should consider the intensity contribution from each point inside the slit. Similar to the concept of temporal bucket integration seen in phase-shifting interferometry, the observed Ronchigram is the integration of the transmitted rays with lateral transverse ray aberration

*TAx*ranging from -

*d*/2 to

*d*/2. We can write the pupil intensity map

*I*(

*Xp , Yp*) as the linearly superimposed intensities from different transverse ray aberration contributions, this is,

*V*is recognized to be,

*mp*must be increased. But to keep a useful fringe contrast ratio, the slit width must be reduced as predicted by Eq. (13). If the slit is not perfectly aligned parallel with the grating, the fringe contrast ratio will be less than its theoretical value. Thus, this property can be used to align the slit with the sinusoidal grating.

## 4. Summary

## References and links

1. | V. Ronchi, “40 Years of History of Grating Interferometer,” Appl. Opt |

2. | T. Yatagai, “Fringe Scanning Ronchi Test for Aspherical Surfaces,” Appl. Opt |

3. | C.-R. Jorge and S. Jose, |

4. | K. Hibino, D. I. Farrant, B. K. Ward, and B. F. Oreb, “Dynamic range of Ronchi test with a phase-shifted sinusoidal grating,” Appl. Opt |

5. | J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt |

6. | M. M. Gonzalez and N. A. Ochoa, “The Ronchi test with an LCD grating,” Opt. Commun |

7. | J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt Acta |

8. | A. Cornejo-Rodriguez, “Ronchi test,” in |

**OCIS Codes**

(080.2720) Geometric optics : Mathematical methods (general)

(220.4840) Optical design and fabrication : Testing

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: November 14, 2006

Revised Manuscript: January 5, 2007

Manuscript Accepted: January 24, 2007

Published: February 19, 2007

**Citation**

Chao-Wen Liang and Jose Sasian, "Geometrical optics modeling of the grating-slit test," Opt. Express **15**, 1738-1744 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1738

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### References

- V. Ronchi, "40 Years of History of Grating Interferometer," Appl. Opt. 3, 437-451 (1964). [CrossRef]
- T. Yatagai, "Fringe Scanning Ronchi Test for Aspherical Surfaces," Appl. Opt. 23, 3676-3679 (1984). [CrossRef] [PubMed]
- C.-R. Jorge and S. Jose, Automatic phase-shifting Ronchi tester with a square Ronchi ruling, O. Wolfgang, and N. Erik, eds. (SPIE, 2004), pp. 199-210.
- K. Hibino, D. I. Farrant, B. K. Ward, and B. F. Oreb, "Dynamic range of Ronchi test with a phase-shifted sinusoidal grating," Appl. Opt. 36, 6178-6189 (1997). [CrossRef]
- J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, "Digital wavefront measuring interferometer for testing optical surfaces and lenses," Appl. Opt. 13, 2693-2703 (1974). [CrossRef] [PubMed]
- M. M. Gonzalez, and N. A. Ochoa, "The Ronchi test with an LCD grating," Opt. Commun. 191, 203-207 (2001). [CrossRef]
- J. L. Rayces, "Exact relation between wave aberration and ray aberration," Opt Acta 11, 85-88 (1964). [CrossRef]
- A. Cornejo-Rodriguez, "Ronchi test," in Optical Shop Testing D. Malacara, ed. (Wiley-Interscience, New York, 1992), p. 321.

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