## Abstract

We observe a non-complementary dark-space produced when two Ronchi-grams, at zero-phase and π -phase, are overlapped and use these dark spaces to quantify Ronchi-grams. Diffraction and multiple beam interference effects narrow the Ronchi fringes created with a coherent point source illumination and prevent accurate determination of the geometrical fringe edges. The dark spaces created when the intensity of two Ronchi grams is added allow assessing the geometrical edge at the dark space middle providing a way to reduce measurement errors. We re-construct the wavefront deformation in a test beam with a 35-term Zernike polynomial. Experimental results are presented.

©2009 Optical Society of America

## 1. Introduction

The Ronchi test [1] is one of the simplest and convenient methods used for evaluating aberrations qualitatively. Several problems in quantifying the Ronchi test have been discussed in the literature and they pertain to coherence, fringe visibility, diffraction, and multiple beam interference effects. Although, some Ronchi tests that include phase shifting provide quantitative results [2, 3], it is still desirable to count on an inexpensive and simple method for quantifying Ronchi-grams. To date there is still not a quantitative Ronchi test methodology that has been broadly accepted or implemented commercially.

To reduce interference and diffraction effects and increase illumination, the use of quasi-monochromatic, extended, and incoherent light sources has become the choice for Ronchi testing. One method for quantifying the Ronchi test is to determine the fringe edge position and relate it to the Transverse Ray Aberrations (TRA). Accurate Ronchi fringe edge determination poses several problems. The first one is assessing where is the actual geometrical fringe edge in relation to the location of the measured fringe intensity distribution. The geometrical fringe edge and the actual intensity edge may not coincide due to diffraction and interference effects. The error can be a significant fraction of a fringe width and amounting to tens of pixels in a CCD image when determining fringe edge position.

In performing Ronchi testing with a coherent point light source we noticed the substantial narrowing of the bright Ronchi fringes that takes place. This effect is illustrated in Fig 1(a), called as 0-phase, and shows that the actual illumination fringe edge is not at the expected geometrical location. Figure 1(b) shows a second Ronchigram taken with the grating moved half a period, called as π-phase. The addition of the intensities in these two Ronchi-grams is shown in Fig. 1(c) which clearly shows that both Ronchi-grams are non-complementary and leave dark spaces. This effect is not due to the duty cycle of the binary Ronchi used which was measured to be 50%. The dark spaces provide a better indication of where the Ronchi fringe geometrical edge is located. The center of the dark spaces is believed to locate the very fringe edge that we want to extract. We used these dark spaces for Ronchigram quantification as discussed below.

## 2. Testing setup

The Ronchi test setup we used is shown in Fig. 2. We used a He-Ne laser as a light source (λ= 632.8 nm) whose spatial frequencies are filtered with a pinhole of 5 μm in diameter. We used a thick positive lens as a test optics (Edmund optics DCX 50×100) and a Ronchi grating (50% duty cycle) with a period of 200 μm placed near the Gaussian image point produced by the lens. The light transmitted by the grating was collected by a CMOS camera (Mightex Systems). The collected image is 640 by 512 pixels and each pixel is 10.4 μm by 10.4 μm. Images were stored in a bitmap format with a 255 gray scale. Black color has a value of zero and white a value of 255. After some preliminary study, the camera with an imaging lens is placed at a fixed location for which the image becomes a circle with radius of 220 pixels. The grating was mounted on a 3-axes translation stage and each direction could be controlled with a resolution of 2 μm. We also used a rotation stage to rotate the ruling which could be set in vertical and horizontal directions within a few tenths of degrees.

In order to test the procedure to determine the wavefront aberration of the test lens, the grating was placed at various axial locations and the distance between each location was fixed at 0.5 mm. At each location of the grating two images with two different phases were collected with the ruling in the vertical and horizontal directions, respectively. The images were processed with the steps, described in the following section, to obtain the 35 Zernike polynomial coefficients.

## 3. Test procedure to analyze images

Because of the coherent point light source, diffraction, multiple reflections from various surfaces and speckle effects are substantial in the Ronchigrams. We followed the next steps to acquire the desired geometrical fringe edge information and then a Zernike representation of the wavefront deformation in the test beam. 1) A median filtering process is used to smooth the fast oscillating intensity distribution of the combined Ronchigram, 2) The Ronchigram fringe edges are found at the minimums of the combined Ronchigram dark spaces by using least square fitting method with a quadratic equation, 3) Transverse ray aberrations are then assigned to the fringe edges, 4) A linear fitting method is used to determine 35 Zernike terms from two orthogonal sets of Ronchi-grams.

In order to obtain the edge information we scanned every row of the processed image horizontally for the vertical fringe Ronchigram and every column vertically for the horizontal fringe Ronchigram. For every pixel in each row or in each column, we determined local minimum location, not maximum, by fitting a quadratic equation to a determined number of pixels in the image of the dark space. Once we obtained edge data we assigned transverse ray aberration to the edges with a given value of grating period. To do this, we first assigned edge order numbers. The fringe edges were assigned a positive integer to the right, and a negative integer to the left of the center of the image for the image taken with the vertically oriented Ronchi grating. Positive integers were assigned to the bottom, and negative integers to the top from the center of the image taken with the horizontally oriented Ronchi grating. For pixels with right edge order numbers we used the following equation to assign the value of TRA,

where *p* is the period of the grating and *m* is the edge order number. For positive integer *m*, the negative sign is used, and the positive sign is used for negative integer.

## 4. Reconstruction wavefront aberration with transverse ray aberrations

The transverse ray aberrations are described as a sum of Zernike polynomials [4],

$$\epsilon {\left(i\right)}_{y}=-2F/\#\sum _{k=1}^{35}{a}_{k}\frac{\partial Z{\left(i\right)}_{k}}{\partial y}$$

where F/# is the image-side paraxial F-number of the system. The best fit of the polynomials coefficients can be determined by defining the difference between experimental values and the corresponding theoretical values as

By setting $\frac{\partial \chi}{\partial {a}_{j}}=0$ = 0 for *j*=1…35 provides a best fit of the coefficients and 35 linear equations can be written as a simple matrix equation,

where *A* is a row vector comprising the 35 coefficients,

The entity *E* is a row vector comprising the *N*
_{x}+*N*
_{y} experimental data collected from edge information for each pair of Ronchi-grams for vertical and horizontal grating positions,

and *Z* is the matrix, of 35 rows and *N*
_{x}+*N*
_{y} columns,

The solution, a set of Zernike polynomial coefficients, is then determined by,

## 5. Test results

For each location of the ruling along the z-axis four Ronchi-grams were processed as described above and a set of 35 Zernike polynomial coefficients were determined and the best fit values are listed in Table 1 in unit of wavelengths. The values of Xo, where Xo is the horizontal center pixel of the circular image, were determined to make the coefficients of Z_{1} zero. Similarly, the values of Yo, where Yo is the vertical center pixel of the circular image, were determined to make the coefficients of Z_{2} zero. Those not listed in the table are equal to or less than 0.2 in their magnitudes. The test wavefront error consists of not only primary spherical aberration but also higher-orders of spherical aberration, as expected. The values of Z represent relative grating positions in mm and the location at Z=0 is expected to be the paraxial image plane. The constant grating displacement is revealed as a constant variation of 2.5 waves in the coefficient *a*
_{3} for defocus term at F/#=4. This variation is consistent even though there are many image artifacts involved during the geometrical fringe edge acquisition process.

The primary spherical aberration is determined as 5.27±0.20 waves and it agrees so well with the value, 5.26 determined by a lens design software, ZEMAX [5]. A slight disturbance of the values of *a*
_{2} is due to many factors of experimental procedure, the quality and structural parameters of the imaging lens and the location of the detector, etc. However, the overall constant value of *a*
_{8} is clearly strong indicative for the Ronchi test as a quantifying wavefront aberration for any optical system.

6. Summary We show a technique to acquire data for Ronchi test analysis where diffraction, interference, and other image artifacts are suppressed. The addition of two Ronchi-grams, at zero-phase and p-phase, produces non-complementary dark-spaces and we used these dark spaces to determine Ronchi-gram geometrical fringe edges and thus re-construct a wavefront deformation with a 35-term Zernike polynomial fit. We illustrated our technique by determining the spherical aberration of a test lens to within 0.2 waves.

## Acknowledgments

S. Lee thanks to the College of Optical Sciences, University of Arizona, Tucson, AZ, for the warm hospitality during his sabbatical leave. This work was supported by Inha University Research Grant.

## References and links

**1. **V. Ronchi, “Forty Years of History of a 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]

**3. **J. Castro-Ramos and J. Sasian, “Automatic phase shifting Ronchi tester with a square Ronchi ruling,” SPIE **5532**, 199–210 (2004). [CrossRef]

**4. **
There are many different versions, but we adopted one. J. C. Wyant and K. Creath, “Basic Wavefront Aberration Theory for Optical Metrology,” in *Applied Optics and Optical Engineering*, XI, 1992, Academic Press, Inc.

**5. **
ZEMAX Optical Design Program, ZEMAX Development Corporation, www.zemax.com.