## Abstract

The Ronchi test with a LCD amplitude sinusoidal grating is used for testing nominally flat surfaces. We prove that it is possible to measure flat surfaces without using a reference element by modifying the common optical Ronchi set up. The Ronchi rulings are computer generated and displayed on the LCD. By displaying various phase-shifted rulings and capturing the corresponding images, the phase is obtained with the conventional phase-shifting algorithms. Theoretical and experimental results are shown.

©2003 Optical Society of America

## 1. Introduction

Testing the flatness of surfaces generally involves the use of interferometers that have an explicit flat reference surface. For example, a Fizeau interferometer uses a flat at least of the same dimensions than the testing element. With a grazing incidence interferometer the area of the reference element is reduced by a factor that depends of the grazing angle, also it is possible to measure rough surfaces^{1,2,3}, cylindrical lens^{4}, etc. There are also several configurations for making the interference^{5}. The basic principle of this interferometer is that an unpolished surface becomes more reflective when the wavelength of the illuminating beam is increased. This reflection phenomenon is also observed when the surface under test is illuminated with a beam that makes a large angle with the normal to the surface. In this latter case we may speak of an equivalent wavelength. Relative to this equivalent wavelength, the surface appears polished enough to yield interference fringes.

In 2001 we reported a Liquid Crystal Display (LCD) method for testing concave mirrors^{6}, which in 2003 was improved with the introduction of digital sinusoidal gratings instead of binary^{7}. The purpose of this work is to show that with the Ronchi test it is possible the accuracy quantitative evaluation of flat surfaces under a grazing configuration. With this method is not necessary to make a reference beam (neither its optics) or use beam splitting methods to generate the reference. Another advantage is that the phase shifting is done digitally, without moving parts, just by displaying the rulings with different phases on the LCD.

In the following sections we shall describe the Ronchi test under a grazing configuration, we then describe the experimental set-up and also the image processing techniques that we used to implement the digital test. Finally, in the last two sections we present the results and the conclusions respectively.

## 2. Theory of the grazing Ronchi test with an LCD

Figure 1 shows a schematic drawing of the grazing Ronchi set-up. The beam of a frequency stabilized, 4 mW, He-Ne laser is expanded by a weakly divergent lens (not shown) to fill the aperture of a low power microscope objective (MO). The objective focuses the beam at a spatial filter (SF), placed at the focal plane of an achromatic doublet (DB1), with a 62 mm clear diameter. The doublet collimates the beam, which illuminates the surface under test that is placed on top of a metal plate with a 320 mm long by 32.5 mm wide rectangular aperture. The beam that is reflected from this surface is focused at the LCD, placed near the focal plane of a second achromatic doublet (DB2). To image the surface under test through the doublet DB2 a small doublet (DB3) attached to a CCD camera is used. What we needed was to image the entire slot in the metal plate that supports the surface under test within the camera CCD sensor.

The pixelated structure of the LCD replicates the image of the surface under test on the CCD. To eliminate this effect on the Fourier plane of DB2 only the zero order is passed by a mask (*h*).

The basic idea of this method is that the plane wavefront reflected by the surface under test is modified in phase by the surface irregularities. This phase changes modify the shape of the fringes displayed on the LCD giving a deformed fringe pattern called a ronchigram.

Let us denote by *w*(*x*, *y*) the wavefront deformation caused by the surface. The wavefront at the doublet DB2 is stretched in the x-direction by a factor *1*/*cos(θ)*, where θ is the angle of incidence of the object beam on the surface under test respect to its normal. Let *D(x,y)* be the function that accounts for the surface departures from an ideal plane. If the gradient of *D(x,y)* is assumed very small throughout the entire inspection area of the surface under test, we can use the following approximation:

where λ is the wavelength of the light source, and

is the equivalent wavelength of the interferometer. In our case θ=80°, λ=*0.6328 µm* and *λ _{eqν}*=

*3.6441 µm*. It means that the grazing Ronchi test is

*λ*/λ=5.76 times less sensitive than the Ronchi test under normal conditions, i.e., θ=0°.

_{eqν}The wavefront at DB2 is represented by a complex diffracted function *F*
_{0}(*x*
_{0}, *y*
_{0}) which is zero outside the limits imposed by its aperture and

inside, where *x _{0}*=

*x*/

*cos(θ)*and

*y*=

_{0}*y*are the horizontal and vertical axes, respectively, of a right-handed rectangular coordinate system, with its origin on the optical axis.

The LCD-ruling is placed at an axial distance *r* from DB2, almost at its plane of convergence. In this case the complex amplitude distribution at the ruling is given by^{8}

where λ is the illumination wavelength. At the focal plane (*f _{c}*) of the collimating lens, the observation plane, is

where *M*(*x _{r}*,

*y*) is the mathematical representation of the LCD-ruling,

_{r}*x*and

_{r}*y*are the horizontal and vertical axes at the ruling position, respectively. The LCD is considered as a rectangular array of rectangular pixels. The pixels have dimensions and spacing of

_{r}*a*by

_{x}*a*and Δ

_{y}*by Δ*

_{x}*, respectively. With the help of the sampling theorem, the function*

_{y}*M*(

*x*,

_{r}*y*) can be written as

_{r}^{9}

where ** denotes a two-dimensional convolution, *comb* function is an array of delta functions with the same spacing as the pixels and *m _{R}* is the mathematical continuous function representation of the ruling to be displayed on the LCD. For example, an amplitude sinusoidal ruling along the horizontal

*x*-axis is given by

_{r}where 2*p* is the period and *A* the maximum amplitude (in our case *A*=255 grey levels) and *β* is a phase shift parameter determined by the initial position of the grating. The function *M*(*x _{r}*,

*y*), Eq. (6), represents the non-continuous amplitude sinusoidal grating sampled by the pixilated structure of the LCD. The pixelization produces a diffraction pattern in which each diffraction order comprises the laterally sheared images of the pupil, i.e. the ronchigrams.

_{r}Now, the intensity profile of the ronchigram at the detector plane *I*(*x*
_{1}, *y*
_{1})=|*G*(*x*
_{1}, *y*
_{1}) *G**(*x*
_{1}, *y*
_{1})| is calculated from Eqs. (4) and (5) with *M*(*x _{r}*,

*y*) and

_{r}*F*

_{0}(

*x*

_{0},

*y*

_{0}) given by Eqs. (6–7) and Eq. (3) respectively, giving

^{7}

where *C* is a dc term, *V*(*x*
_{1}, *y*
_{1}) is the fringes visibility, Γ(*x*
_{1}, *y*
_{1}) represents undesirable noise term, φ(*x*
_{1}, *y*
_{1}) is the object phase difference to be measured and *γ* is the angle the ruling makes with the *y*
_{1} axis. It is seen in Eq. (8) that shifting the origin of the sinusoidal grating changes the phase of the ronchigrams by the same amount. This shift is done by software, i.e. by changing the origin of the ruling.

As it is seen from Eqs. (7) and (8), displaying sinusoidal gratings with a phase shift *β*, implies a shift of the phase φ by the same amount. Due that the ruling period on the LCD is known, it is known also the discrete shift, i.e. if the period is 24 pixels, shifting the grating by 6 pixels shifts the ronchigrams by π/2 rad without any errors, except the alignments of the pixel during the LCD fabrication.

Thus to recover the phase, and in consequence the wavefront deviations, phase shifting algorithms are a suitable option. If we display four sinusoidal gratings with *β*=*p*/2 and their respective intensities are recorded, the phase is calculated with the Equation

Clearly, the phase obtained is wrapped into the interval [-*π, π*] and an unwrapping process is necessary.

From Eqs. (9) and (10) it is seen that the phase obtained from the Ronchi test give information only of the derivatives of the target function *w*(*x*
_{1}, *y*
_{1}) then an integration procedure is necessary^{6}. To recover unambiguously the deformations is necessary to perform the test twice using rulings with at least two directions, i.e. two values of *γ*. The faster is setting *γ*=0 (vertical ruling) and *γ*=*π*/2 (horizontal ruling). The algorithm we used to perform the integration of the horizontal and vertical unwrapped phases is described in ref [6]. It is based on a fitting procedure where the desired wavefront phase is represented by a 2D polynomial of the kth degree, with k=5. This function is *x* and *y* differentiated to obtain two functions with unknown coefficients. The horizontal and vertical unwrapped phases are used to fit, in the least squares sense, each one of these functions. Once the coefficients of the derivatives are calculated the phase function coefficients are calculated from those.

## 3. Experimental results

The optical set-up that we used for carrying out our experiments is depicted in Fig. 1. We place a piece of common glass (window glass) on the top plate and by means of adjustment screws the image is centred on the CCD (the area under test was about 32mm×130mm). The collimated light picked up by the CCD was transferred to a frame grabber installed in a PC and displayed on its monitor. Now, the sinusoidal grating, *m _{R}* was computer generated and displayed on the LCD: the deformed grating image (ronchigram) was observed on the computer screen. In general the frequencies of the rulings displayed by the PC monitor did not match with those displayed by the LCD, so a calibration procedure is necessary. We did it with the help of a 100× microscope (VZM 1000).

The LCD screen used was a pocket KOPIN CiberDisplay™320, 0.24″ diagonal size, and 320×240 square pixels of 15µm×15 µm.

The frame grabber used to capture the ronchigrams was the VFG-100 from Imaging Tech. and our images had 256×240 pixels and 256 grey levels.

A vertical digital sinusoidal ruling (*2p*=*0.36 mm*) was displayed on the LCD. An initial ronchigram (*I*
_{1}) was grabbed and stored in the computer [Fig. 2(a)]. The digital ruling was shifted causing the ronchigram to be shifted by π/2 rad in phase (*I _{2}*). The same process followed for the other two images. Once the four ronchigrams were captured, they were smoothed by a convolution operation with a 5×5 matrix. Then Eq. (10) was applied, resulting in the wrapped phase shown in Fig. 2b, which was later unwrapped

^{11}.

By displaying horizontal sinusoidal rulings and following the same capturing and data processing followed for vertical rulings, we obtain the horizontal wrapped phase (not shown). The deformations *D(x,y)* [Eq. 1] of the glass with respect to an ideal plane were calculated by integrating the corresponding horizontal and vertical unwrapped phases^{6}, having a Peak to Valley and RMS deviations of 2.03λ and 0.29λ respectively [Fig. 3]. In order to assess the proposed method, the same optical element was tested with a commercial Fizeau interferometer [Fig. 4]. We found a Peak to Valley and RMS deviations of 1.98λ and 0.34λ respectively.

It is appropriate to mention that although numerically agreement was good, the integration procedure we have used smoothed the wavefront shape. This smoothing process is better seen in Figs. 5(a) and 5(b), where Figs. 3 and 4 were converted into fringes digitally.

## 5. Conclusions

A method to evaluate flat surfaces by means of a grazing configuration of an LCD Ronchi test has been described. The liquid crystal display was used as a grating and a phase shifting device. Even that under a grazing configuration the sensitivity of the test is reduced by a factor of 6 or more we have obtained experimental results similar to those obtained with a commercial Fizeau interferometer under normal configuration. We believe that the main differences rest on the least squared integration method that we used which smoothed the calculated surface.

## Acknowledgements

The authors would like to thank Mr. José de la Luz Hurtado for their contributions to the developments of this work. Mr. Mora would like also to acknowledge the financial support from Centro de Investigaciones en Óptica, México.

## References and links

**1. **P. Hariharan, “Improved Oblique-Incidence Interferometer,” Opt. Eng. **14**, 257–258 (1974).

**2. **M.V.R.K. Murty and R.P. Shukla, “An Oblique Incidence Interferometer,” Opt. Eng. **15**, 461–463 (1976).

**3. **D. Boebel, B. Packroβ, and H.J. Tiziani, “Phase shifting in an oblique incidence interferometer,” Opt. Eng. **30**, 1910–1914 (1991). [CrossRef]

**4. **H. Nürge and J. Schwider, “Testing of cylindrical lenses by grazing incidence interferometry,” Optik **111**, 545–555 (2000).

**5. **Peter de Groot, “Diffractive grazing-incidence interferometer,” Appl. Opt. **39**, 1527–1530 (2000). [CrossRef]

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

**7. **M. Mora-González and N. Alcalá Ochoa, “Sinusoidal liquid crystal display grating in the Ronchi test,” Opt. Eng. **42**, 1725–1729 (2003). [CrossRef]

**8. **R. Barakat, “General Diffraction Theory of Optical Aberration Tests, from the Point of View of Spatial Filtering,” J. Opt. Soc. Am. **59**, 1432–1439 (1969). [CrossRef]

**9. **J. E. Greivenkamp and J. H. Bruning, “Phase Shifting Interferometry,” Chapter 14 in *Optical Shop Testing*, D. Malacara, Ed., pp. 548–551, Wiley, New York, (1992).

**10. **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]

**11. **R.M. Goldstein, H.A. Zebker, and C.L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Science , **23**, 713–720 (1988). [CrossRef]