For accurate, fast, and nondestructive measurement with nanoscale vertical resolution, interferometric surface profiling is a good technique. It is used for the quality control of a variety of industrial products, including lenses and mirrors. However, conventional interferometric surface profiling methods are highly susceptible to environmental vibration. Moreover, they cannot be used to measure moving objects—plastic film sheets, for example—because they require multiple images with different reference positions.

To address this problem, researchers developed single-shot interferometry, which measures the surface profile of an object from a single image. A typical example is spatial carrier interferometry, in which carrier ‘fringes’ are introduced by tilting the reference mirror. From a single interferogram, the surface profile can be calculated using a variety of fringe analysis methods.^{1–4} However, this technique has the disadvantages of low spatial resolution and a limited measurable slope angle. Consequently, its application has been limited to very smooth surfaces.

Here, we describe a new approach called the global model fitting (GMF) algorithm.^{5} It estimates the surface height at each pixel location from its color information, and, as a result, does not require the introduction of fringes.

Consider the optical configuration for three-wavelength, single-shot interferometry^{6} (see Figure 1), typically used for surface profiling of ink-jet-based color filters for liquid crystal display panels. These optics make it possible to capture an interferometric color image (see Figure 2). This interference color phenomenon is the same as that seen in soap bubbles. The lower graph shows the relationship between the BGR (blue, green, red) signals and the height of the surface (defined as the optical path difference of the two beams).

**Figure 1. **Optics of three-wavelength, single-shot interferometry.

**Figure 2. **Relationship between colors and surface height in three-wavelength interferometry. B: Blue. G: Green. R: Red. d: Optical path difference between the sample and reference light beams.

Figure 2 shows the interferometric intensity that results from the following model: *g*(*i*, *j*)=*a*(*j*)[1+*b*(*j*)cos{Φ(*i*, *j*)}], where *g*(*i*, *j*) is the intensity at point *i*(*i*=1, 2, …, *n*), and wavelength *j*(*j*=B, G, R), *a*(*j*) and *b*(*j*) are the average value and the modulation of the waveform, respectively, and Φ(*i*, *j*) is the phase given by Φ(*i*, *j*)=4π*z*(*i*)/λ_{j}, where *z*(*i*) is the height and λ_{j} is the jth wavelength. The model can be expressed by *g*(*i*, *j*)=*a*(*j*)[1+*b*(*j*) cos {4π*z*(*i*)/λ_{j}}].

We derived this model under the assumption that the waveform parameters a(*j*) and b(*j*) are constant in the field of view and depend only on the wavelength. (This assumption will almost always be valid when the target surface is homogeneous.) Combining the model with the observed intensity data of plural points, we can estimate unknown parameters by the least-squares fitting method. Figure 3 shows the principle of the GMF algorithm. From 3*n* observed data, we estimate (*n*+6) unknown parameters (that is, *n* point heights and six waveform parameters). Since the necessary condition for the solution is that the number of observed data be more than that of the unknown parameters, the number of necessary points is three.

**Figure 3. **Principle of the global model fitting (GMF) algorithm in the case of three wavelengths and *n*points.

The computational cost of the nonlinear least-squares problem is very high. For example, based on our experiment, we estimate the required calculating time for 512×480 pixels to be more than 100s. Consequently, we use the GMF algorithm with a small number of points (e.g., under 100), and then determine the heights of the other points using the following arccosine (ACOS) method: The phases of three wavelengths are calculated by Φ(*i*, *j*)=arccos[{*g*_{ij}&(*j*)−1}/*b*(*j*)*]* from the intensity *g*_{ij} using the waveform parameters obtained by the GMF method. We then obtain the height by *z*(*i*, *j*)=[±Φ(*i*, *j*)/2π+*N*(*i*, *j*)](λ_{j}/2), where *N* is the fringe order (integer). We estimate the fringe order, i.e., phase unwrapping, using the coincidence method^{7} and three phases.

Armed with this new technique, we measured the surface profile of a 50nm step height. The surface was slightly tilted so that the interference image had different colors in the field of view (see Figure 4). From the intensity data of the 20 points shown in Figure 4, we estimated the height of the points using the GMF method. We then calculated the heights for the entire area by ACOS (see Figure 5). The result confirms the prime appeal of the proposed technique: there is no loss in spatial resolution. Another advantage is measurement speed. The total calculation time for 512×480 pixels was about 200ms, including 4ms of GMF fitting using a Windows PC (3.4GHz Core i7 CPU).

**Figure 4. **Captured color image of 50nm step height and 20 sampled data points (shown in white numbers).

**Figure 5. **Estimated heights of the 50nm step in 3D view.

In summary, we have presented a novel algorithm that has several advantages over conventional single-shot interferometry. Its most significant feature is no loss in spatial resolution. That is, the height of each point can be estimated independently without having to know any information about the neighboring points. Other advantages include high-speed measurement, low-cost, simple optics without the requirement for fringe introduction, and no preliminary calibration. We are currently working on extending our technique to develop industrial-type instruments, including a film-thickness measurement system.^{8}

Katsuichi Kitagawa

Toray Engineering Co., Ltd.

Otsu, Japan

Katsuichi Kitagawa received his BE in applied physics (1964) from the University of Tokyo. He worked initially as an instrument engineer with Toray Industries, and in 2000 moved to Toray Engineering Co. He earned a PhD in information science and technology from the University of Tokyo (2011).

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