## Abstract

The feasibilities for optical correlation diagnostics of rough surfaces with large surface inhomogeneities by determining the transformations of the longitudinal coherence function of the scattered field are substantiated and implemented.

© 2006 Optical Society of America

## 1. Introduction

The optical techniques are non-contact, non-destructive and highly efficient for diagnostics of rough surfaces. The optical techniques may be divided into profile interference and heterodyning techniques [1, 2], techniques based on measuring of the angular distribution of scattered radiation [1, 3], and optical correlation techniques [4, 5]. High operation rates, data processing in the optical channel, high accuracy of measurements, feasibilities for real-time data processing, and small sizes of the measuring devices are the main advantages provided by optical correlation diagnostic methodology [6]. However, when the height of surface inhomogeneities exceeds the wavelength of the probing beam, and the specular component of the reflected radiation is absent, the unambiguous connection between the statistical parameters of the roughness and of the scattered field is lost. The new approaches of fractal and singular optics are consequently needed for diagnostics of such surfaces [7, 8]. At the same time, these approaches so far provided only classification of rough surfaces into random and fractal ones.

In this paper, we promote the optical correlation technique for determining the height distribution function for surfaces with large surface inhomogeneities (roughness) and for the reconstruction of the relief structure of such a surface based on the results of measuring and processing of the longitudinal coherence function for the scattered field. For diagnostics of surfaces with large inhomogeneities one often uses the techniques of speckle optics [4, 9, 10]. In these cases, however, the statistics of the height distribution of the inhomogeneities influences the results of measurements, and the depth of roughness must not exceed 3 microns.

The tilt fringe technique and the phase-shifting interferometry applied to optical surfaces [11, 12] are the closest techniques to the one proposed in this paper. Similarly to the proposed technique, measuring of the relative height variations of a surface is provided by the mentioned methods. The mentioned techniques are based on interfering the phase-modulated, backscattered object beam with a coherent reference beam having a simple wave front. The interferogram obtained using completely coherent radiation contains data on the structure of rough surfaces with large inhomogeneities. Such data are here extracted from the interference fringes obtained in temporally incoherent light and their associated visibility.

The advantages of the proposed technique will be considered below.

## 2. Theory

Illumination of a rough surface with the relief structure *h*(*x*,*y*) by a normally incident monochromatic plane wave results in a spatial phase modulation of the reflected wave at the boundary field, 4*πh*(*x*,*y*)/ *λ*. The longitudinal coherence function of the scattered field is not assumed to be spatially restricted. If the same surface is illuminated by a polychromatic beam with a finite longitudinal coherence length, reflection from a rough surface is accompanied by spreading of the longitudinal coherence function of the resulting field [5, 6] due to various delays of the partial signals, which are determined by the roughness relief. As a result, the modulus of the complex degree of coherence of the field decreases. The longitudinal coherence function of the field reflected from a rough surface is determined from the Fredholm’s integral equation of the 1^{st} kind:

where Γ_{0}(∆*z*) is the longitudinal coherence function of the probing beam, and *f*(*z*) ≡ *f*(*h*)
is the function of the partial signal delays determined by the height distribution function of the surface inhomogeneities. It has earlier been shown [13] that one can determine the distribution of the partial signal delays, *f*(*z*), based on the experimentally obtained coherence function of
the scattered field and the known coherence function of the probing beam. Here, we use the techniques given in [14, 15] for solving the Fredholm’s equation of the 1^{st} kind for determining the height distribution for the surface inhomogeneities, *f*(*h*).

Experimental determination of the longitudinal coherence function of a polychromatic light field scattered by a rough surface is hampered by the non-uniformity of the visibility of the resulting interference pattern arising when the reference beam is superimposed on the test beam. This difficulty follows from the fact that the phase modulation depth caused by the relief structure can exceed the coherence length of the probing polychromatic radiation. We propose the following procedure for determining Γ(*z*). In the Michelson interferometer, see Fig. 1, a monochromatic or a polychromatic field formed by a micro-objective at the plane of the image of a rough surface is mixed with a monochromatic or a polychromatic reference field, respectively.

Subsequently, one determines the correlation between the monochromatic interference image of a rough surface with high visibility with the set of polychromatic interference images with non-uniform visibility obtained with varying path differences in the interferometer. This algorithm provides determination of the longitudinal covariance function of the images and thus deriving the longitudinal coherence function of a polychromatic field scattered from the object, Γ(*z*).

## 3. Experiments

In our experiments, we use a solid-state laser with a wavelength λ = 532 nm and a coherence length of 4 cm (considerably exceeding the maximal span of heights of the surface inhomogeneities) as the monochromatic source. In such a manner, we provide high visibility of an interference pattern over the observed area of a rough surface, see Fig. 2. The intensity of the resulting field at the interferometer output becomes

where ${I}_{0}^{m}$ is the reference wave intensity, and ${I}_{s}^{m}$
(*x*,*y*) is the intensity distribution at the surface image in monochromatic light.

The use of a temporal source with a finite coherence length in the Michelson interferometer enables a longitudinal scanning of the rough surface relief, e.g. the step-by-step interference selection of cross-sections of the relief, which are parallel to a mean surface line. Color intensity distributions of the resulting interference fields are shown in Fig. 3 for various optical path differences in the interferometer legs.

The experimentally obtained resulting white-light interference pattern for the optical path difference *z* is described by the following relation:

where *I*
_{0} is the reference wave intensity, *I*_{s}
(*x*,*y*) is the intensity distribution at the surface image in polychromatic light, and *z*
_{0} is an arbitrary starting position.

The mutual longitudinal intensity correlation function (MLICF) of *I*^{m}
(*x*,*y*,*z*) and *I*(*x*,*y*,*z*) is determined by the following relation:

Averaging is an ensemble average carried out over the entire observed area of the rough surface.

A typical form of the MLICF is shown in Fig. 4. The distance between the maxima is *λ*/2 (266 nm, in our case), which is convenient for the use as marking for the coherence function with respect to the roughness height parameter. The degree of coherence of the field for the relative longitudinal shift *z* of the corresponding distributions *I*^{m}
(*x*,*y*,*z*) and *I*(*x*,*y*,*z*) is determined by the magnitudes of the extrema for the MLICF, namely:

From these data, one can reconstruct the longitudinal coherence function of the object field, see Fig. 5.

Experimental studies have been carried out using the arrangement shown in Fig. 1. We used a Nd:YAG laser, L, operating at a wavelength of 532 nm and a temporal source, S, as the sources of probing radiation. Radiation of the source S is projected onto the 20 *μm* - diameter diaphragm, D, positioned at the focus of the objective O3 (focal length 200 mm) by the objectives O1 and O2. In this way, we form a white-light beam with a degree of spatial (transverse) coherence of 95%. If the mirror M is used, the laser beam is spread using the telescope T and focused on the same diaphragm D. The beamsplitter BS divides the incident beam into the reference and the object beam. The images of the surface of interest, O, and the reference mirror, M2, are projected onto the CCD-camera by the micro-objectives MO1 and MO2, and the objective O4. The mirror M2 with the micro-objective MO2 is shifted along the direction of propagation of the beam, facilitating a change in the optical path delay in the interferometer. Optical phase difference (within a wavelength) is provided by a piezo-ceramic translator, PC, mounted with a mirror M2. By moving the surface of interest along the direction of propagation of the beam, one can obtain an image of the surface using a monochrome CCD-camera.

We used a CCD-camera with a maximum spectral sensitivity at 532 nm, and studied the roughness of grinded plates of mono-crystalline silicon. Initially, we record two video sequences. The fist of them reflects the evolution of the monochromatic interference field while changing the optical path difference between the object and the reference beam within one wavelength, cf. Fig. 6, avi_1. The second sequence reflects the evolution of the white-light interference field for changed optical path delays within the maximal depth of the rough surface relief, cf. Fig. 7, avi_2. Then, one frame of the monochromatic interference field is correlated with each frame of the polychromatic interference field.

The resulting mutual intensity covariance function is determined from the relation:

where *n* is the frame number, *N* and *M* are the number of pixels in the two directions, ${I}_{\mathit{\text{ij}}}^{m}$
and *I*_{ij}
(*n*) are the intensities of the pixels with the coordinates *i*,*j* of the corresponding frames of the specified frame of the monochromatic sequence and the *n*-th frame of the polychromatic frame, respectively.

So, the use of the reference frame of the monochromatic interference distribution enables one, on the one hand, to connect the coordinate of a polychromatic interference distribution to the corresponding surface relief height, and, on the other hand, to detect the transversal distribution of the intensity extrema for the polychromatic interference field as a function of the path difference between the reference and the object polychromatic beams ranging within the maximal span of heights of the surface inhomogeneities. The frame with a monochromatic interference pattern here serves as the matched spatial amplitude filter through which the polychromatic interference pattern is observed and analyzed for various optical path differences. We notice that such a procedure has recently been introduced for diagnostics of phase singularities in quasi-monochromatic vortex beams [16]. The experimentally obtained normalized mutual intensity correlation function of monochromatic and polychromatic fields reflected from a rough surface of a sample of germanium is shown in Fig. 4. The longitudinal coherence function of a polychromatic object field can be determined from the magnitudes of the extrema of this function using Eq. (6). The spacing between an adjacent maximum and minimum of the mutual intensity coherence function is *λ*/4 , giving a spacing of 133 nm in Fig. 4.

The longitudinal coherence function of the probing polychromatic beam, Γ_{0} (∆*z*), is determined using the same arrangement, cf Fig. 1, by measuring the visibility of the resulting interference field at the infinitely extended interference fringe for varying optical path difference in the interferometer. In this case, the tested rough surface is replaced with a high-quality mirror. The experimentally found longitudinal coherence function of the probing beam fitted with a Gaussian function is shown in Fig. 8 and is adequately represented by

where **α** = 0.83*ρ*/ *π*, *ρ* being the coherence length of the probing beam, which in our case was equal to 0.8 *μm* . Thus, Eq. (1) can be written in the form:

The solution of this equation is found using the regularization technique with functional minimization [10]. The distribution function for the delays of partial signals derived from the measured data is shown in Fig. 9. The reconstruction accuracy of this function is determined by the accuracy of interference measurements, in our case not being worse than 5%.

When the height of surface inhomogeneities do not considerably exceed *λ*, the number of points determining the coherence function is small. To increase the accuracy in determination of *f*(*z*) in this case, one can specify the points of the longitudinal coherence function for smaller separation of the frames, or even for subsequent ones. Therefore, one detects the frames recorded in white light and finds the covariance with the flow recorded in a monochromatic field. While the monochromatic flow is recorded for scanning of the reference wave within a wavelength, the mutual covariance function obviously has maxima and minima, based on which one determines the degree of coherence of the object field. The dependence of the degree of coherence on the frame number is scaled on the mutual intensity correlation function of the field, cf. Fig. 4, where the position of the extremum is determined by the frame number, and the spacing between the adjacent maximum and minimum equals 133 nm.

Two video flows, being recorded using a CCD-camera and reflecting a coherent monochromatic interference field for scanning the reference beam within a wavelength (avi_1) and the interference field in a white light for scanning by the reference beam within the maximal span of the heights of surface inhomogeneities (avi_2), cf. Fig. 6, thus facilitates the reconstruction of the relief structure of the rough surface.

Firstly, one uses the interference field obtained using scanning with white light (Fig. 7, avi_2). For each frame of the video flow, one determines the coordinates of the interference fringes with the lowest intensity and prescribes the height magnitude corresponding to the frame number. Notice that this magnitude can be associated with each individual pixel. The coordinates and the number magnitudes are stored in a separate file, where the data of the surface relief are formed. Subsequently, the obtained data are scaled on the MLICF, cf. Fig. 4, and the position of the extremum is specified by the frame number, while the spacing between the adjacent maxima and minima is *λ*/4 (or 133 nm). The surface relief structure reconstructed by processing of the videos avi_1 and avi_2 is shown in Fig. 10.

## 4. Conclusions

Thus, we have showed the feasibility for diagnostics of rough surfaces with large surface inhomogeneities exceeding the wavelength of the probing radiation by transformation of the longitudinal coherence function for a polychromatic field. By solving the Fredholm’s integral equation of the first kind, one can determine the height distribution function of the surface inhomogeneities and reconstruct the relief structure of the surface of interest. Implementation of the proposed approach is based on correlation comparison of interference distributions for monochromatic and polychromatic radiation, with imposing the reference wave on the image of a rough surface. The accuracy of the correlation measurements is determined by the accuracy provided by the Michelson’s interferometer. The proposed diagnostic methodology facilitates a non-contact fast-operating measuring device for control of rough surfaces with large surface inhomogeneities.

Let us emphasize the advantages of the proposed technique. First of all, data processing takes place in real-time in the optical channel including spatial averaging, and thus provides high reproducibility of the measurement results. Secondly, the height distribution function of the surface inhomogeneities is obtained directly, without intermediate stages of data processing, i.e. obtaining the relief of a surface or the map of height. Further, usage of low-coherent (white-light) radiation enables to scan the relief of a surface with a single crossing plane rather than with a set of such planes multiple to 2π, as is the case of using completely coherent radiation. This simplifies the algorithm of data processing. Finally, the upper limit of applicability of this technique is determined by the depth of focus of the used projecting objective. Accuracy and sensitivity of the proposed technique and conventional interference techniques are identical, only depending on the provided accuracy of phase shifts under scanning.

## Acknowledgments

The authors are in debt to Dr A. Shumelyuk for technical assistance of this study.

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