Digital speckle pattern interferometry applied to a surface roughness study

Juan A. Pom arico Roberto D. Torraba Centro de Investigaciones Opticas (ClOp) C. C. 124 1900 La Plata, Buenos Aires, Argentina and Universidad Nacional de La Plata Facultad de Ingeniería La Plata, Buenos Aires, Argentina Abstract. Surface roughness determination is of great interest for many applications. Several methods can be found in the literature, but most of them rely on indirect evaluation of the information, such a s photographic techniques. W e propose a method to measure surface roughness that takes advantage of digital speckle pattern interferometry for obtaining the data, and of digital image processing for evaluating it. After defining the problem, a theoretical description is presented, and finally it is com pared with experimental results, showing good agreement.


pattern interferometry (DSPI), normally used in various me trological applications.7 9
In DSPI a fringe pattern is obtained using video processing by storing an initial reference image of the surface in a frame grabber and then subtracting it from the one being acquired, displaying the modulus of the result on a TV monitor.Fringes, similar to those obtained in holographic interferometry, then appear when a phase change is introduced between the ref erence image and the subsequent ones.As noted by Wykes,10 the rms difference of the speckle intensities from different wavelengths is directly related to the degree of speckle correlation.
In this work we apply the DSPI technique on a single wavelength basis, combined with the ASC method, to show that the visibility of the correlation fringes in this case also depends on the roughness of the object surface.A theoretical approach to the proposed technique is also given.

Theoretical D escription
Let us suppose the situation sketched in Fig. 1.A plane wave £ illuminates a rough surface S at an angle of incidence 0 with respect to the normal to the surface.In our case, the surface is considered to be formed from a

set of n identical scattering elements of random height following a normal distribution of standard deviation a and zero mean. A CCD camera images the surface on the detector array. A separate reference beam is added to the first one via the beamsplitter BS. Since the DSPI detects a quantity proportional to |<7, 72)2)I/2|, we examine how this quantity varies with a tilt of the beam illuminating the surface.
To do that we assume to be the field amplitude detected by the CCD array when the surface is illuminated at an angle 0. In this equation the first term denotes the speckle pattern amplitude field, where A 0 is the light amplitude scattered by one element, Z, is the height of the j t h scattering element, and 4>0 allows for the constant phase of the incident light.In the second term, A R is the amplitude of the reference beam and its phase.In the following, all calculations are assumed to be made at the same point (x,y).
Similarly, if the illumination direction is changed by an amount A0, the new field amplitude can be described by (2) Equations ( 1) and (2) can be rewritten as the following more compact expressions: (3) ( 4) where e 1* is the common phase term expi&ircjio/X], A R is the amplitude of the reference beam, and a and y are abbre viations for (2Tr/X)(l+cos0) and (2tt/X)[1+cos(0 +A0)], respectively.
To obtain the third term in Eq. ( 5) we need now to calculate the mean value of the product 7, l2: where £ a 7 (2TT/\)[cos0 cos(0 + A0)] «(2'ir/X.)sin0 A0.From Eqs. ( 11) and ( 16) the following expression for ((7, 72)2) can be obtained: The number of scatterers producing the speckle field, n, is very large, and thus n -1 «n.In addition, we suppose that nA\=A2 represents the total light intensity emanating from the surface and AR can be chosen to be equal to As.With these assumptions, the final expression for ((7, 72)2) results: (18) We are primarily concerned with the determination of the local visibility VL and its variation with respect to A0.By using the substitution ((7, 72)2) (72), the local visibility is defined as In practice, it is equivalent to measure the global visibility through the fringe pattern that arises because of the tilt A0 of the illuminating beam, defined by the equation where 7max and 7m in are the maximum and minimum intensity values present in the fringe pattern, respectively.
For the case of A0 O, we have ((7, 7 2)2)'/2 0 (no fringes are present), and Eq. ( 19) becomes undefined.The corresponding visibility can be taken as one, as for all other cases, having A0^O, lower values are obtained.
Figure 2 shows the variation of the local visibility defined in Eq. ( 19) versus the rotation A0 for ct 6.3 |xm.

Exp erim en tal R e s u lt s
The experimental arrangement of Fig. 3 was used to repro duce the conditions used in the theoretical approach.A 10 mW HeNe laser beam was divided by the beamsplitter BS1.One of the beams obtained in this way was expanded, collimated, and used to illuminate the rough surface under study at an angle 0 with its normal.The second beam was overlapped with the image of the surface, which is a speckle pattern, at the detector array of the CCD camera by using the beamsplitter BS2.It is clear that the optical system does not resolve the individual scatterers.The video signal from the camera is sent to the frame grabber for further processing.
At the frame grabber the following operation, which can be accomplished at near-real-time rates, is performed: a first image of the surface is acquired and stored as a reference, the subsequent images coming from the camera are subtracted from the first one, and the result is then displayed, in modulus, on a TV monitor.If an image is obtained under the same illumination conditions as the reference, then no differences are present between them, and the TV monitor appears dark.On the other hand, when the beam illuminating the surface is rotated by an angle A0, a linear phase change occurs, altering the speckle pattern generated by the surface.At points where this phase change equals Inru, with m an integer, the speckle pattern reproduces itself and the TV monitor shows a dark spot.On the contrary, for phase changes other than 2m TT, the subtraction gives a nonzero value and the monitor shows a spot whose gray level varies accordingly to the phase change.Thus the monitor shows a set of fringes immersed in speckle noise and following the zones of constant phase change.Both the spatial frequency and the visibility of these fringes depends on the rotation A0 of the beam illuminating the surface, as demonstrated above.
In our experiment, the rotation A0 was obtained by a lateral shift Ax of the collimating lens LI, which was mounted on a translator driven by a stepper motor capable of a minimum displacement of 2 p.m. Lens LI was chosen to have a focal length of 22 cm.Thus, the rotation A0 can be obtained from the simple equation with Ax varying from several microns to a few millimeters.
Although this produces not only a rotation but also a dis placement of the object beam, using plane waves ensures that all phase changes are only a consequence of the rotation, provided that the whole field of view of the camera (which can always be chosen smaller than the illuminated area) re mains unchanged.
The curves plotted in Fig. 4 (fringe visibility as a function of A0) were obtained for three different values of the surface roughness, a 6.

C o n c lu s io n s
A method for surface roughness determination using digital speckle pattern interferometry has been described theoreti cally and tested experimentally, showing good agreement.The experimental setup requires an image processor but not sophisticated optical components, making the method suit able, for example, for industrial applications.In addition, the whole process can be accomplished digitally, thus reducing the time needed for a measurement.The range of operation can be controlled by adjusting the parameters.

1 Introduction
Since the fundamental work of Beckmann et al.1 dealing with the scattering of electromagnetic waves from rough surfaces, their results have been extensively used to obtain information about surface roughness from the scattered light in the far field region.The correlation fringes of two speckle patterns, obtained from a rough surface under different angles of illumination, bear information about the roughness.Several optical roughness measurements techniques involve the use of a doubleexposure specklegram method.The basic approach is to pro duce a shift in the incidence angle of the illumination beam between exposures, in order to produce, after a Fourier trans formation, a set of interference fringes whose visibility depends on the rms roughness of the scattering surface.23This technique is known as angular speckle correlation (ASC).Tribillon4 showed that the degree of correlation between speckle patterns generated by two different wavelengths depends strongly on the surface roughness.The method based on this effect is known as spectral speckle correlation (SSC).5All of the above-mentioned techniques suffer from the disadvantages of a photographic process, besides introducing a second step in the use of an optical Fourier transform.There also exist statistical methods that derive roughness infor mation from the degree of decorrelation of imaged speckle patterns as a function of the wavelength,6 where electronic detection and processing replaces the conventional photographic record.Electronic correlation of speckle patterns is currently accomplished by a technique called digital speckle Current affiliation: Universidad de Bahia Blanca, Buenos Aires, Argentina.Paper 09074 received July 12, 1994; revised manuscript received Nov. 1, 1994; accepted for publication Nov. 1, 1994.© 1995 Society of Photo-Optical Instrumentation Engineers.0091-3286/95/56.00.

F ig - 1
Schematic representation of two plane waves, X and X , il luminating a rough surface at two different angles, 6 and 6 + AO.

( 10 )
OPTICAL EN G IN EERING /April 1995/Vol.34 No. 4/1149 we obtain, taking into account11 that the mean value of the cosine function evaluates to zero when the phase is uniformly distributed in the interval (ID and ((Zj-Zk)2) = 2a2(l -rJk), where rjk is the normalized surface correlation.Since from Eq. (12) we have j¥=k, and if the individual steps Zy are uncorrelated with each other, then Eq.(13)reduces to(15)

4 439Fig. 2
Fig. 2 Theoretical variation of the visibility a s a function of the change in the illumination direction, AO.

Fig. 3
Fig. 3 Experimental D S P I setup for obtaining speckle correlation fringes and measuring the angular variation AO.
Fig. 4 Visibility of the fringe pattern a s a function of A0 for three different values of the surface roughness.Solid curves represent the best fit to the experimental data.

Fig. 5
Fig. 5 Best-fit curves for the visibility of the fringe pattern a s a func tion of A0 for three different values of the initial illumination angle 0 and for a fixed value of the surface roughness, a 6.3 p.m.