Interferometric Surface Profile Measurement Based on Radial Polarization and Wavelength Variation

1. Introduction

In precision manufacturing and engineering applications, surface profile measurement plays a crucial role in quality control and performance evaluation, and is widely used in industrial processes, precision components, and biomedical engineering. Therefore, the development of measurement techniques with both high accuracy and high efficiency is of great importance. Existing surface profilometry methods can be broadly classified into contact and non-contact approaches. Contact probe techniques [1,2] acquire surface topography through mechanical scanning and offer structural simplicity; however, non-contact methods are generally more suitable for delicate or high-precision samples. Optical probe techniques [3,4,5,6] replace mechanical probes with laser scanning and intensity detection, thereby eliminating physical contact. Nevertheless, three-dimensional measurements typically require point-by-point scanning, which may limit measurement speed.

Interferometric techniques have been widely adopted due to their non-contact operation, full-field capability, and high resolution, and have been extensively applied in the measurement of distance, surface roughness, and refractive index [7,8,9,10]. Conventional single-wavelength interferometers are limited in measurement range by the optical wavelength, whereas dual-wavelength techniques [11,12,13] extend the measurable range and enhance phase unwrapping capability using a synthetic wavelength. White-light interferometry [14,15] further enables absolute height determination by exploiting low-coherence characteristics. In addition, fringe projection [16,17,18,19] and Moiré interferometry [20,21,22,23] are commonly employed for three-dimensional surface reconstruction, and can be combined with heterodyne techniques to improve measurement stability [24].

Based on these developments, this study proposes a cylindrical vector beam interferometric profilometry method. A radially polarized cylindrical vector beam is employed as the illumination source, taking advantage of its axisymmetric polarization distribution. The system is implemented using a Twyman–Green interferometer configuration integrated with a dual-wavelength technique to establish a compact and rapid surface profilometry system. Only two interferograms are required to complete phase retrieval and surface reconstruction. Experimental validation using a standard gauge block demonstrates a system resolution of 61.42 μm, with a measurable range extended to the centimeter scale. The proposed method provides non-destructive, full-field, and rapid measurement capabilities with a relatively simple and cost-effective configuration.

2. Principles

Figure 1 schematically depicts the experimental configuration employed in this study. For clarity, the positive z-axis is defined along the direction of light propagation, while the x-axis is oriented perpendicular to the plane of the paper. A linearly polarized laser beam with wavelength λ, whose polarization direction is oriented at 90° with respect to the x-axis, is expanded and collimated by a beam expander (BE). Subsequently, the linearly polarized beam is transformed into a radially polarized symmetric beam using a zero-order vortex half-wave plate (ZR). The corresponding Jones vector can be expressed as

$$\begin{gathered} E_{in}(x,y)=C_R\cdot E_{90°} \\ =\left[\begin{array}{cc}-\sin \theta & \cos \theta \\ \cos \theta & \sin \theta \end{array}\right]\cdot \left[\begin{array}{c}0\\ 1\end{array}\right]e^{i{\omega }_{0}t}=\left[\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right]e^{i{\omega }_{0}t}, \end{gathered}$$

(1)

where ω₀ denotes the angular frequency of the light source, and θ represents the azimuthal angle at a given position in the beam cross section. The radially polarized beam is then divided into transmitted and reflected components by a beam splitter (BS). The transmitted beam is normally incident on the test specimen and is reflected back along its original optical path. The associated Jones vector is given by

$$\begin{gathered} E_t(x,y)=B{S}_r\cdot S\cdot B{S}_t\cdot E_{in} \\ ={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\cdot \left[\begin{array}{cc}-e^{i\phi (x,y)}& 0\\ 0& e^{i\phi (x,y)}\end{array}\right]\cdot {\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\cdot \left[\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right]e^{i{\omega }_{0}t} \\ ={\displaystyle \frac{e^{i\phi (x,y)}}{2}}\left[\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right]e^{i{\omega }_{0}t}, \end{gathered}$$

(2)

where φ(x, y) represents the phase difference distribution introduced by the specimen. Meanwhile, the reflected beam is directed toward a mirror (M) and retraces its original path after reflection. Its Jones vector can be written as

$$\begin{gathered} E_r(x,y)=B{S}_t\cdot M\cdot B{S}_r\cdot E_{in} \\ ={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\cdot \left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\cdot {\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\cdot \left[\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right]e^{i{\omega }_{0}t} \\ ={\displaystyle \frac{1}{2}}\left[\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right]e^{i{\omega }_{0}t}. \end{gathered}$$

(3)

The two beams are recombined at the BS and subsequently detected by a CMOS camera (C). The resulting Jones vector at the CMOS plane is therefore expressed as

$$\begin{gathered} E(x,y)=(E_r(x,y)+E_t(x,y)) \\ ={\displaystyle \frac{1+e^{i\varphi (x,y)}}{2}}\left[\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right]e^{i{\omega }_{0}t}. \end{gathered}$$

(4)

Accordingly, the intensity recorded by the CMOS camera can be derived as

$$I(x,y)={\left|E(x,y)\right|}^2={\displaystyle \frac{1+\cos \varphi (x,y)}{2}}.$$

(5)

From Eq. (5), the phase difference distribution φ(x, y) can be retrieved as

$$\phi (x,y)={\cos }^{-1}(2I(x,y)-1).$$

(6)

Since the phase difference induced by the specimen is related to its surface height distribution by φ(x, y) = 4πD(x, y)/λ, Eq. (6) can be further rewritten as

$$D(x,y)={\displaystyle \frac{\lambda }{4\pi }}{\cos }^{-1}(2I(x,y)-1).$$

(7)

where D(x, y) denotes the surface height distribution of the specimen. To extend the unambiguous measurement range, the illumination wavelength is changed to λ′, leading to a modified phase difference distribution

$${\phi }^{\prime }(x,y)={\cos }^{-1}(2I^{\prime }(x,y)-1).$$

(8)

The resulting phase difference variation can therefore be expressed as

$$\begin{gathered} \Delta \phi (x,y)=\phi (x,y)-{\phi }^{\prime }(x,y) \\ ={\cos }^{-1}\left[2I(x,y)-1\right]-{\cos }^{-1}\left[2I^{\prime }(x,y)-1\right]. \end{gathered}$$

(9)

This phase variation can also be represented in terms of an equivalent wavelength Λ as

$$\Delta \phi (x,y)=4\pi D(x,y){\displaystyle \frac{1}{\mathsf{\Lambda }}},$$

(10)

where $\mathsf{\Lambda }={\displaystyle \frac{\lambda {\lambda }^{\prime }}{\left|\Delta \lambda \right|}}$. Consequently, the surface height distribution D(x, y) of the specimen can be determined as

$$D(x,y)={\displaystyle \frac{\mathsf{\Lambda }}{4\pi }}\left\{{\cos }^{-1}\left[2I(x,y)-1\right]-{\cos }^{-1}\left[2I^{\prime }(x,y)-1\right]\right\}.$$

(11)

As indicated by Eq. (11), accurate measurements of the intensities I and I′ enable quantitative retrieval of the specimen surface height. By applying this procedure to each pixel of the CMOS image, the full surface profile of the tested object can be reconstructed.

3. Experimental Results and Discussions

To assess the feasibility of the proposed method, the first to fifth steps of a standard step gauge were measured, and the specifications are shown in Figure 2. The experimental system consisted of a tunable diode laser (Model 6304, New Focus) operating at wavelengths of λ = 632.8 nm and λ′ = 632.82 nm, corresponding to a wavelength difference of Δλ = 0.02 nm. The optical configuration further included a beam expander equipped with a 40× objective, a 5 μm pinhole, an achromatic lens with a focal length of 70 mm, and a zero-order vortex half-wave plate (ZR, WPV10L-633, Thorlabs). The interference patterns were recorded using a CMOS camera (Basler A504k, Basler AG) with an 8-bit gray level and a spatial resolution of 1280 × 1024 pixels. All captured images were processed and analyzed using MATLAB on a personal computer.

The interference patterns captured at wavelengths of 632.8 nm and 632.82 nm are shown in Figure 3. For computational convenience, the recorded interferograms were normalized in intensity using a MATLAB program, as illustrated in Figure 4. Subsequently, Eq. (9) was employed to convert the normalized intensity distribution into a three-dimensional phase distribution, as shown in Figure 5, where the phase values range from 15° to 45°. Finally, the surface profile of the standard step gauge was reconstructed using Eqs. (10) and (11), and the resulting contour map is presented in Figure 6.

The reconstructed data of each step were averaged to obtain the recovered step heights of layers i–v, which were 6.96 mm, 3.95 mm, 3.07 mm, 2.80 mm, and 2.71 mm, respectively. The corresponding step height differences were 3.01 mm, 0.88 mm, 0.27 mm, and 0.09 mm. These results are summarized in Table 1. As shown in Table 1, the reconstructed values exhibit good agreement with the standard values, thereby demonstrating the feasibility of the proposed method.

According to Eq. (11), the height error ∆D_err associated with the proposed method can be expressed as

$$\Delta D_{err}=\left|{\displaystyle \frac{\partial D}{\partial I}}\cdot \Delta I_{err}\right|+\left|{\displaystyle \frac{\partial D}{\partial I^{\prime }}}\cdot \Delta {I^{\prime }}_{err}\right|+\left|{\displaystyle \frac{\partial D}{\partial \lambda }}\cdot \Delta {\lambda }_{err}\right|+\left|{\displaystyle \frac{\partial D}{\partial {\lambda }^{\prime }}}\cdot \Delta {{\lambda }^{\prime }}_{err}\right|,$$

(12)

where ΔI_err and ΔI′_err denote the intensity errors of the CMOS camera, and ∆λ_err and ∆λ′_err represent the wavelength uncertainties of the tunable laser source. Considering the intensity resolution of the CMOS camera, the values of ΔI_err and ΔI′_err were estimated to be 0.0039. In addition, based on the wavelength resolution of the tunable diode laser (Model 6304, New Focus), both ∆λ_err and ∆λ′_err were determined to be 0.02 nm. Substituting these parameters into Eq. (12) yields a height error ΔD_err of approximately 61.42 μm. To further reduce the measurement uncertainty, a CMOS camera with a 16-bit gray level was employed. Under the same experimental conditions, the corresponding height error was reduced to 36.62 μm. This error analysis confirms that the proposed method provides high measurement accuracy and enhanced resolution.

4. Conclusions

A radial-polarization-based interferometric system for surface profile measurement is proposed, which integrates the properties of radially polarized beams, a modified Twyman–Green interferometer, and dual-wavelength measurement technology. The feasibility of the proposed approach was experimentally verified by measuring a convex mirror, resulting in a height error of approximately 61.42 μm. The experimental results demonstrate that the proposed method effectively combines the advantages of these techniques, including a simple optical configuration, rapid measurement capability, high measurement accuracy, and enhanced spatial resolution.