1. Introduction

Light exhibits the phenomena of interference and diffraction [1], with interference being the cornerstone of wave motion theory. Interferometers are powerful tools and widely used in metrology and spectroscopy. Traditional interferometers use plane waves as the incident beam, resulting in straight or circular fringes in the interference pattern. Introducing orbital angular momentum (OAM) beams with helical phase fronts can generate much richer interference patterns such as petal-like patterns, forked and spiral fringes. In recent years, the use of helical phase fronts in OAM interferometers has resulted in some interesting applications. For example, in 2017, Ellie et al. [2] achieved nano-displacement measurement with a naked eye resolution of 44 pm by introducing conjugated twisted beams into a Michelson interferometer (MI). In the same year, Chen et al. [3] achieved nano-displacement and wedge angle measurement by inserting an optical wedge into the interference arm of an OAM interferometer. Pang et al. [4] proposed an OAM interferometer using a Sagnac structure, achieving magnetic field sensing of 3.31%/T in BGO crystal. In 2018, Xia et al. [5] proposed and demonstrated an OAM Mach-Zehnder interferometer (MZI) based temperature sensor, achieving the temperature measurement with a high resolution of 0.005°C. In 2019, Verma et al. [6] theoretically proposed an OAM interferometer and achieved picometer displacement and deformation measurements on solid or liquid surfaces through simulation. In 2022, Li et al. [7] proposed a modified OAM interferometer and achieved nano-displacement measurement, with a measurement range of 50 to 800 nm and a resolution of 50 pm. In the same year, Zhang et al. [8] proposed a computing visual vortex beam interferometer and achieved a displacement measurement with a sensitivity of 0.4 nm, a relative error of 65 ppm, and an uncertainty of 0.18 nm. K. P. Singh et al. [9] proposed a noise self-canceling picoscale twisted interferometer and achieved a sub-100 picometer resolution in measuring a non-repetitive intracavity dynamic event in real-time. In 2023, Gao et al. [10] proposed a modified reversal shearing interferometer using vortex beams for micro-displacement measurement and achieved micro-displacement measurement with a measurement error of less than 0.067 nm. These works are summarized in Fig. 1.

 

Fig. 1. Summary of existing technologies for nano-displacement measurement using OAM beams.

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Currently, all OAM interferometers are based on amplitude-splitting interference (ASI) [11], with physical quantity measurements by calculating the rotation angle of the interferogram caused by the optical path change of the test arm. Therefore, to achieve ultra-high accuracy and sensitive measurement, it is essential to produce a high-robust interference structure with a highly stable and high-quality interferogram. However, most of the OAM interferometers reported to date use MI and MZI structures. The size and complexity of the setup are high which makes the measurement results highly sensitive to external disturbances such as environmental vibrations. The alignment of the optical path also presents a significant challenge. Recently, a common optical path Fizeau interferometer (FI) based on OAM beams (FI-OAM) was reported, which provides a simple and stable interference structure for nano-displacement measurements [12]. In this scheme, the incident $OA{M_{ + l}}$ beam is divided into two beams after passing through the plate BS, the one reflected back is used as the reference beam with the topological charge of $+ \ell $, while the other beam that forward and backward passes through the cylindrical lens and reflected by the reflection mirror, the reflected light with opposite topological charge of $- \ell $, the two conjugated OAM beams interfere on CCD to form petal-like interferograms. Moreover, a phase-demodulation method operated in the domain of the OAM complex spectrum is also proposed, which provides a robust alternative enabling to accurately and quickly extract the phase from the OAM interferogram. However, introducing topological charge transformation through a cylindrical lens in the optical path makes optical alignment difficult, and slight lateral positional deviations can cause the petal-like pattern of conjugated interference to disappear. Therefore, traditional ASI-OAM is challenging to use for high-precision and high-stability measurements in dynamic scenarios. To address this issue, a more robust and compact interference structure such as wavefront-splitting interference (WSI) [13,14] is an alternative choice. WSI is known known for Young’ s experiment [15], and it has also been widely used in optical metrology. Various types of interferometers are applied in different applications. For instance, a Rayleigh interferometer is used to measure two-photon absorption cross-sections in liquid samples rapidly [16]. Fresnel’s biprism is used to measure the spatial intensity and phase of ultrashort laser pulses [17], as well as single-shot ultrafast imaging [18]. Lloyd's prism is used for laser writing to fabricate orthogonal crossed gratings [19,20]. Additionally, WSI is widely used in optical coherence tomography [21,14] since its size is much smaller than MZIs (millimeters), while still maintaining high sensitivity to external parameter changes. It is gradually gaining popularity in practical applications, such as tunable nanofluidic interferometers [22], due to its small size and high sensitivity to external parameter changes. Furthermore, WSI has a special advantage in that it can be made of total reflection optical elements, thus allowing for a wavelength range from far-infrared to soft X-ray.

In this manuscript, we proposed a nano-displacement measurement system based on WSI-OAM, it uses only two pieces of mutually glued right-angle prisms to achieve two half-conjugated OAM beams interference. Figure 1 is a summary of existing technologies for nano-displacement measurement using OAM beams. From this summary, we can see that WSI-OAM is a viable method in terms of setup and elements number needed. Experiments proved that this WSI-OAM is more compact and stable than OAM interferometers with other structures (MI, MZI, and FI). The resolution of nano-displacement is about 0.1 nm with an uncertainty of 0.013 nm. Therefore, this method can be used for nano-displacement measurement with ultra-high precision and sensitivity.

2. Principle

Figure 2 shows the principle of wavefront-splitting conjugated interference of the OAM beam. The incident light is an OAM beam with a phase front of $\phi = \ell \theta ({\theta \in ({0,{\; }2\pi } ]} )$, whose phase front is split at the interface between beam splitter (BS) and penta prism (PP) into ${\phi _a}$ and ${\phi _b}$. The phase front ${\phi _a}$ changes to $\phi _a^\mathrm{\ast } = {({ - 1} )^1}\ell \vartheta ={-} \ell \vartheta $ through a single reflection inside the BS and the phase front ${\phi _b}$ changes to ${[{\phi_b^\mathrm{\ast }} ]^\mathrm{\ast }} = {({ - 1} )^2}\ell \vartheta = \ell \vartheta $ through twice reflections inside the PP, where $\vartheta \in ({0,\pi } ]$. The electric field of the incident OAM beam is ${E_{in}} = A\textrm{exp}({i\ell \theta } )$, where A is the amplitude. The electric fields of the double-output light beams after wavefront splitting are ${E_a} = A/2\textrm{exp}[{i({ - \ell \vartheta + {\varphi_a}} )} ]$ and ${E_b} = A/2\textrm{exp}[{i({\ell \vartheta + {\varphi_b}} )} ]$, respectively. Therefore, the interference light intensity $I = {A^2}/2 + {A^2}/2\textrm{cos}({ - 2\ell \vartheta + \Delta \varphi } )$, and $\Delta \varphi = {\varphi _a} - {\varphi _b}$. Therefore, the interference light intensity is a half-petal-like pattern, satisfies the cosine function in the angular direction, and has $\ell $ maximum values. The half-petal-like pattern rotates with the change of optical path difference between the two optical paths. The principle for nano-displacement measurement based on WSI-OAM is derived as follows:

 

Fig. 2. Wavefront-splitting conjugated interference of the OAM beam. BS: beam splitter; PP: penta prism.

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The incident plane wave passes through a spiral phase plate (SPP) and is converted into an OAM beam with the topological charge $\ell $ is

The OAM beam splits into two beams within the prism, the test beam is expressed as

After subtracting the background light intensity term $A_r^2$ and $A_t^2$ from Eq. (4), we get

Mark $\phi = k({z^{\prime} - z} )+ 2({\ell \vartheta + k\Delta z} )$, it represents half-helical phase front and is given by

Therefore, the displacement $\Delta z$ is

Equation (8) shows that the displacement $\Delta z$ is proportional to the rotation angle of the half-helical phase front $\Delta \vartheta $, and displacement $\Delta z$ can be measured by identifying the angles of the dislocation line of the half-helical phase front.

3. Simulation and experiment

Figure 3 shows the simulation of WSI-OAM in OpticStudio. The incident light is a plane wave with a wavelength of 633 nm, and the SPP with a topological charge of 6. After passing through the SPP, the light carries a helical phase front, see Fig. 3(a)(b). Figure 3(c) shows the phase of SPP. The Coherent Irradiance on the detector is a half-petal-like pattern, as shown in Fig. 3(d), and the number of petals is equal to the topological charge number $\ell $. Figure 4 shows the half-petal-like interferograms in simulation and experiment. The differences between the experimental interferograms and the simulated ones are due to the inevitable light diffraction at the edge of the prism in the wavefront-splitting process.

 

Fig. 3. Simulation of WSI-OAM in OpticStudio. (a) NSC 3D Layout; (b)NSC Shade Model; (c)Phase of SPP; (d) Detector Image: Coherent Irradiance. SPP: spiral phase plate.

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Fig. 4. Half-petal-like interferograms in simulation and experiment for $\ell $ = 2, 4 and 6.

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To verify the measurement accuracy of the proposed method, we use OAM beams with topological charges of 2, 4 and 6 for nano-displacement measurement. The measurement device is shown in Fig. 5. The laser wavelength is 633 nm, and the DS (P-733.3DD-Linearity, 0.10 nm resolution, PI) drives the mirror to move 105.50 nm along the optical axis. Figure 6 shows half-petal-like interferograms and half-helical phase fronts and dislocation lines before and after displacement. We use the robust m-regression fitting method [24–26] to fit the dislocation lines of half-helical phase front before and after displacement, and calculate the rotation angles of the half-helical phase front as $\Delta {\vartheta _{\ell = 2}} = 30.07^\circ $, $\Delta {\vartheta _{\ell = 4}} = 15.02^\circ $, and $\Delta {\vartheta _{\ell = 6}} = 7.51^\circ $, respectively. Thereby, we obtain the nano-displacement values of $\Delta {z_{\ell = 2}} = 105.73nm$, $\Delta {z_{\ell = 4}} = 105.66nm$, and $\Delta {z_{\ell = 6}} = 105.64nm$, respectively. Compared with the actual displacement of 105.50 nm, it is found that the absolute error is 0.23 nm, 0.16 nm and 0.14 nm, respectively, and the measurement accuracy is 99.78%, 99.85% and 99.87%. The main reason for the large displacement measurement error when using an OAM beam with a topological charge of 2 is that the dislocation line shown in Fig. 6 is curved, which will lead to the angle error of the dislocation line. Therefore, it is necessary to align the mirror to be strictly perpendicular to the optical axis before the experiment, and using OAM beams with large topological charges is more conducive to improving measurement accuracy.

 

Fig. 5. Schematic diagram of the optical path for nano-displacement measurement based on WSI-OAM. BE, beam expander; SPP, spiral phase plate; DS, displacement stage.

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Fig. 6. Experimental results of a single nano-displacement measurement for $\ell $=2, 4 and 6.

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To demonstrate the vibration resistance of the proposed method, small-range experiments were done in this study. We use an OAM beam with a topological charge of 6 and a DS to drive the mirror with a step-wise of 0.1 nm and delayed 0.5 s after a single step. While the control group uses a FI-OAM, as shown in Ref. [12]. The vibration source is a mobile phone in the ringing vibration mode. We use a camera (BFS-U3-28S5M-C Blackfly S, 130FPS, FLIR) to collect interferograms, and the measurement results of nano-displacement and measurement error curves are shown in Fig. 7(a) (b). The measurement results obtained with our method coincided well with the actual displacement of DS. In Fig. 7(b), the standard deviations (STD) of the FI-OAM method and WSI-OAM method are 0.028 nm and 0.013 nm, respectively, and the peak-valley value (PV) is 0.068 nm and 0.031 nm, respectively. Therefore, using WSI-OAM, the measurement error of nano-displacement is smaller in the presence of vibration. Compared with the control group, the proposed method has less fluctuation in displacement measurement error, and due to its ultra-simple structure, it has vibration resistance. Next, we use OAM beams with topological charges of 2, 4, and 6 for the reliability experiment. The scanning distance is 633 nm and the scanning step is 1 nm, and the scanning speed is 10 nm/s. The experimental device is installed on a vibration isolation platform. Figure 7(c) shows the curve between the rotation angle of the dislocation line and the displacement. It can be seen from the figure that there is an almost ideal linear relationship between the nano-displacement and the rotation angle of the dislocation line. The R² (fitting degree) of the three fitting lines is 99.06%, 98.99%, and 98.96%, respectively, all close to 99%, indicating that the proposed WSI-OAM is highly stable and reliable. Figure 7(d) shows that the measurement accuracy dropped to 90% as the displacement decreased to a few nanometers, and the experiment results show that the measurement accuracy decreases with the decrease of nano-displacement.

 

Fig. 7. (a) the measurement results for the step-wise motion with a step of 0.1 nm. (b) measurement error of WSI-OAM method compared with FI-OAM method. (c) relationship between the rotation angle of the dislocation line and displacement. (d) relationship between the measurement accuracy and displacement.

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4. Conclusions

In conclusion, we propose and demonstrate a robust WSI-OAM interference measurement method. It uses two conjugated half-OAM beams for interference. The control experiment shows this scheme can ensure measurement accuracy and is vibration insensitive compared with ASI-OAM. The resolution of nano-displacement is about 0.1 nm with an uncertainty of 0.013 nm and the measurement accuracy higher than 99.90%. WSI-OAM is composed of only two right-angle prisms, which makes it simpler, more compact, and easier to align. The proposed phase demodulation method provides a robust and efficient way to extract rotation angles from petal-like interferograms. WSI-OAM can not only be used for nano-displacement measurement, but also has potential application value in optical surface shape measurement. It provides another way to interpret interference phenomenon using OAM beam.