Scanning white-light interferometry provides surface profiles by analyzing the interference signal. Studies have been carried out to obtain robust, precise extraction of profiles in accordance with the development of manufacturing processes. In addition to the studies of surface profiling, attempts have been made to obtain an image without any fringe, or to measure additional quantities, such as thickness of thin films and roughness of surface [1-6].

In SWLI, there has been an increasing demand for color area inspections on corroded, annealed, or discolored samples. To perform these various functions, it is necessary to detect the natural colors of the sample. Despite the use of a visible white-light source, most instruments are unable to perform color imaging, because they use a monochrome camera that optimizes the interpretation of interference signals. Several methods have been proposed and utilized to obtain a color image in SWLI. One method is to add a color camera in SWLI and remove color interference fringes; a one-chip camera equipped with a Bayer filter, or a three-chip camera receiving red, green, and blue colors on respective chips, is mainly used. The former has reduced lateral resolution, while the latter is expensive and slow [7]. Another method is to use a switchable RGB light sources instead of a white-light source, or to use side illumination [8, 9]. Using switchable RGB light sources requires an additional measurement sequence of changing the color from the light source to obtain RGB colors. On the other hand, using side illumination requires an extra light source and a color camera. All of these methods require additional hardware for the process.

The purpose of this study was to obtain a color image using the basic hardware configuration of SWLI. An interference signal was decomposed into frequency data via Fourier transform. By analyzing the frequency data, color values of reflected light from a sample can be determined. RGB distance was calculated and peak signal-to-noise ratio (PSNR) performance was carried out, to evaluate and confirm the similarity between the generated image and a color images taken from general microscopy.

2.1. Basic Principle of Interferometry

Figure 1 is a basic schematic of SWLI. The beam splitter separates the beam originating from the light source into two beams, for measurement and reference. The measurement beam is reflected from the sample’s surface, and the reference beam is reflected from the reference mirror. These beams, traveling on different paths, are recombined before reaching the detector. The difference in path between the beams creates interference. At a constant angular wave number k, the intensity of the interference signal g(k,z) is expressed as

Figure 1. Schematic diagram of scanning white-light inter-ferometry.

where h is the height of the surface, z is the scan position, v is the phase change in the reference path, and w is the phase change in the object path. R_mirror and R_sample are the effective reflectance of the reference mirror and sample respectively. This includes reflectance and transmissivity of the beam splitter, which is determined by the type of the interferometry. R_mirror, R_sample, v, and w vary with angular wave number k. For a broadband light source, the total interference signal can be represented as an incoherent superposition of interference signals from a single wave number. This signal, expressed as an integral over all wave numbers, is

where V(k) implies the spectral distribution of the light source, and the spectral responsivity of the detector.

2.2. Frequency-domain Analysis

The Fourier transform decomposes the interference signal into frequency-domain representations [5, 10, 11]. The Fourier transform of Eq. (2) is

By simplifying the expression using the exponential form of the cosine term and the Dirac delta function δ(K), Eq. (3) can be summarized as

where H(K) is the Heaviside step function defined by

The magnitude is related to the reflectance of the sample, while the phase is determined by the height of the sample’s surface and the phase change that occurs in the sample. The spectral reflectance R_sample(K) of a given specimen is obtained from the following relation:

R_ref (K) is the spectral reflectance of the reference sample, the optical constants of which are well known. Generally, bulk materials with specular surfaces are used as reference samples. q_sample(K) and q_ref(K) are the Fourier transforms of interference signals derived from the specimen and the reference samples respectively.

2.3. Generating a True Color Image

Imaging with a color camera in reflected-light microscopy is a common method for true color imaging. In this case the theoretical expression for the camera’s response determined by each RGB color filter E(C) can be described as

where I₀(λ) is the normalized spectral luminous intensity distribution of the input light source, and F_C(λ) is the quantum efficiency of the camera with respect to the color channel C[12]. Since I₀ (λ) and F_C(λ) values are acquired from the product specification, the color response can be mathematically determined by identifying R_sample(λ).

In SWLI, the spectral reflectance of a sample can be obtained through Fourier magnitude analysis of the interference signal. The color values of the sample can be calculated by substituting Eq. (6) into Eq. (7):

To get the exact reflectance of the sample, q_sample(K) and q_ref (K) must be obtained under the same light intensity, creating an inconvenience in which the reference sample must be measured repeatedly for every specimen. However, supposing that the light intensity only affects the brightness difference among the three elements of color, premeasured reference sample data q_ref (K) can be used. In this case, appropriate brightness can be set in the process of generating a true color image.

3.1. Hardware Configuration

The hardware configuration of the measurement system is described in Fig. 2. Mirau-type interferometry with a white light-emitting diode (LED) and a 10× interference lens was used as the measurement system. A monochrome camera with a CMOS sensor was adopted as the detector. For the color verification process, normal reflected-light microscopy hardware with a 10× objective lens and a color camera with only a Bayer filter added to the monochrome camera was used, for control of the variables. The model of the monochrome camera was the Basler-1300aCA-200um, and the model of the color camera was the Basler-1300aCA-200uc. These cameras have a resolution of 1280 (horizontal) × 1024 (vertical) pixels, with the pixel pitch of 4.8 µm × 4.8 µm. The spectral intensity distribution of the white LED and quantum efficiency of the color camera are shown in Fig. 3.

Figure 2. Hardware configuration for SWLI.

Figure 3. (a) Spectral intensity distribution of the white LED. (b) Quantum efficiency of the color camera.

3.2. Experiment for Generating a True Color Image

An experiment was carried out to verify the validity of the proposed method. The target of the experiment is the four-segmented sample shown in Fig. 4(a). In each region, a thin film of SiO₂ with different thickness is uniformly deposited on the silicon substrate. The difference in reflectance due to the film thickness causes a color difference, which cannot be distinguished from the monochrome interference image. In the measurement, a number of images are taken from various scanning positions, to obtain an interference signal from each pixel. To acquire the full range of interference signals, images were taken with a scanning interval of 72 nm over a scan range of 15 µm. One of the images plus interference signals for P1 to P₄ are shown in Fig. 4.

Figure 4. (a) A monochrome interference image of the four-segmented sample: A thin film of SiO₂ is deposited on the silicon substrate in each region. (b) Interference signals for P₁ to P₄.

Fourier magnitude analysis was performed on the interference data from the measured and reference samples, to calculate the spectral reflectance. A bare silicon wafer was used as the reference sample in the analysis. The reflectance obtained from the Fourier transform was compared to the reference reflectance obtained from measurement by a spectrometer (Ocean Optics Maya2000 Pro). Figure 5 shows that the reflectance curves from the Fourier transform identify with the reference curves. In the red boxed areas, the reflectance from the Fourier transform method shows a large range of error, due to the weak relative intensity of the light source. However, the error is negligible in the color calculation when the reflectance is multiplied by the intensity. To numerically compare the curves, the root-mean-square error (RMSE) and normalized root-mean-square error (NRMSE) values were calculated. The RMSE and NRMSE are defined as

Figure 5. Comparison of the reflectance from the Fourier transform and the spectrometer

and

Table 1 shows the calculated results for the RMSE and NRMSE. Various factors, including sampling error and detection noise, should be considered to reduce errors [13]. The sampling error is caused by the nonlinearity of the scanner, and external vibration. More accurate measurement can be carried out through compensating the scanner’s position using a laser sensor. Detector noise occurs due to not only the noise generated by the sensor itself, but also the inevitable statistical fluctuation of the number of photons converted into photoelectrons. This can be reduced by using a high-performance detector, or averaging multiple images.

TABLE 1.

The spectral intensity distribution of the light reflected from the sample was obtained through multiplying by the spectral distribution of the input light source. The intensity distribution of the reflected light at each point is shown in Fig. 6(a). The shapes of the intensity distributions for the four parts of the sample are different. The RGB values are determined by summation over the entire wavelength range for each color channel, as proposed in Eq. (8). Figure 6(b) shows an image generated with the RGB values obtained from every pixel. The intensity scale of the image is set equal to the average intensity of the interference signal. Since all components of the beam path have been considered, the values have the same RGB ratios as those of the output values from the sensor of the color camera.

Figure 6. (a) Calculated distributions of the reflected light from the interference signal, for P₁ through P₄. (b) Generated color image of the 4-segmented sample.

The values acquired from the sensor of the color camera are different from the RGB values observed by the human eye. To render the acquired values closer to those of human observation, a color-correction process [14] is generally applied to a color camera. The image generated by the proposed method should also be corrected by applying white balance and color transformation, just as in the post-processing of color cameras. Figure 7 shows the comparison of post-processed images generated from this method and the images from a color camera, for various samples. Image brightness was adjusted for visibility.

Figure 7. Images from a color camera, and images generated by the proposed method with a color-correction process.

Table 2 shows the results for PSNR and average RGB distance, which evaluate the quality of the reconstruction from image generation. The RGB distance and PSNR are defined as

TABLE 2.

and

The RGB distance is the Euclidean distance in RGB color space, showing the color difference. To confirm the color difference of the entire image, the average distance was calculated for all of the pixels of the image. However, for both RGB distance and PSNR analysis, pixels with weak interference signals were excluded from the calculation. The values of RGB distance show that the color difference between a generated image and the color image is within 7 percent. The PSNR results for images were around 30 dB. Generally for an image of 8 bits in depth, a PSNR value between 30 and 50 dB is acceptable for video compression and a lossy image [15, 16]. The results of PSNR analysis are nearly in agreement with the acceptable range.

A three-dimensional (3D) surface with color information can be obtained from monochrome interference images through integrating the generated true color image with 3D surface-profile data. The integrated surface-profile images are shown in Fig. 8.

Figure 8. Reconstructed 3D color images of four samples.

This paper suggests a frequency analysis-based methodology to acquire a true color image in SWLI using a monochrome camera. This study is summarized as follows:

(1) In the SWLI system, which consists of a monochrome CMOS camera, Mirau-type objective, and white LED, the interference signal was transformed into spectral data through frequency-domain analysis.

(2) The spectral reflectance of the sample was derived from the spectral data. By considering the light-source distribution and spectral reflectance, the spectral intensity of the light reflected from the sample was obtained.

(3) The RGB ratio was determined by calculating the spectral intensity of the sample and the quantum efficiency of the color camera.

(4) After generating the true color image through color correction, the color images from the proposed method and those from reflected-light microscopy were compared, based on PSNR analysis and RGB distance.

This study suggests that the frequency-domain analysis of an interference signal enables us to generate a true color image. Visually, we confirmed that the colors of the generated images are compatible with those from a color camera. Numerically, we have assessed that the results for the RGB distance lie within 7 percent of the reference. The values from PSNR analysis were about 30 dB, which is acceptable for video compression and a lossy image. The results of these evaluations demonstrate that this method can supplant existing methods for color-image acquisition in SWLI.

The point of this study is that a better intuitive understanding of a sample can be achieved by adopting the proposed methodology within the existing SWLI, without any additional hardware. Here the generated images were restored based on the intensity distribution of the white LED, to compare them to color images from reflected-light microscopy, but a color image can be generated by applying other light sources or color-space conversions to facilitate observation, since the true color image is generated by spectral reflectance.

Furthermore, to the best of our knowledge, obtaining a color image through additional hardware results in the loss of lateral resolution, or expansion of data. At the same time, adopting an additional camera could lead to misalignment between the color image and surface profile. However, this study uses pixel data from the original image, from the measurement. Therefore, definite coincidence of the generated color image and surface profile can be achieved.