High-performance arduino-based interferometric quadrature phase-shift detection system with 1 nm resolution

Optical metrology is widely used in all branches of science in order to extract information by measuring small displacements. Additionally, it plays a vital role in today’s manufacturing industry, both for precise positioning and for quality control.¹ The 1887 introduced Michelson interferometer² has proven to be a particularly popular interferometer setup within optical metrology that is still widely used today. The working principle relies in essence on detecting the phase of an optical wave, inherited in an interference pattern. This can not only provide information about the object of interest regarding displacements or external parameters, e.g., pressure³ or temperature,⁴ but can also be used for medical imaging^5,6 or surface inspection.⁷ The accuracy of the interferometer is mainly determined by the precision with which the phase of the interference pattern can be measured. In the case of fast processes, the sampling frequency at which the interference pattern is measured can be another limiting factor. Today, highly accurate interferometers with resolutions of 10 pm have been realized using common HeNe lasers,⁸ and advances have recently even led to the detection of gravitational waves.⁹ However, those high-tech interferometers require a massive time investment and usually come at a high monetary price.

Here, we present a detection system for interferometry based on an affordable microcontroller and two photodiodes. The detection system can be used for multiple interferometric applications. Its capabilities are demonstrated by monitoring the trajectory of a high-precision translation stage.

The precision of the movement of the translation stage was investigated using a Michelson interferometer as illustrated in Fig. 1. The polarized beam of a HeNe laser (Thorlabs HNL050LB) operating at a wavelength of λ = 632.8 nm was split equally by a beam splitter (BS) into the reference arm and the probe arm. The reference beam was reflected by a fixed retroreflector (RR). The probe beam was reflected by a second RR that was mounted on top of a translation stage (Aerotech ALS130H-50) with a resolution of 0.5 nm and an accuracy of ±1 µm. This translation stage was programmed to perform either sinusoidal or triangular movements using the Aerotech Ensemble Software. The two reflected beams overlapped, and the resulting interference pattern was incident on two detectors S₁ and S₂. A sketch of the interference pattern with the sensor size and position is shown as the inset in Fig. 1. The laser beam was slightly focused by a spherical lens (f = 150 mm) to add some curvature to the wave front, so the phase modulation of the interference pattern could be resolved spatially. The lens was positioned such that the beam passed its focal point before it was split into reference and probe beam at BS₁. The starting point of the translation stage was chosen such that the reference arm and probe arm had comparable optical path lengths (OPLs).

The essence of the measurement is the determination of the relative phase of the interference pattern. The relative phase is directly related to the different optical path lengths (OPLs) of the beam in the probe arm and the beam in the reference arm and is given by

where λ is the wavelength of the probing laser beam and ΔOPL is the difference between the OPL of the probe arm and that of the reference arm. Since the reference arm is fixed, it follows that ΔOPL is fully determined by the change of the OPL of the probe arm. The latter is the product of the refractive index in air n ≈ 1 and the physical length of the path in the probe arm x(t). Using paraxial approximation, the change of the relative phase is directly proportional to the change of the length of the probe arm and thereby the position of the translation stage as follows:

Here, a factor 2 is added to take into account that the light in the probe arm is traveling twice its length. For the initial position x₀ = 0, the initial relative phase ϕ₀ can be set to zero. Hence, the phase change Δϕ is identical to the relative phase ϕ. Moving the translation stag away from the BS, the fringes of the interference pattern move outward and the phase difference increases, while approaching the BS, the fringes of the interference pattern move inward and the phase difference decreases.

The phase-dependent intensity profile of the interference pattern can be derived assuming two electromagnetic waves with an equal amplitude but a different phase. This leads to the following equation:

where I₀ is the intensity of the initial laser beam. The information about the position of the translation stage is carried in the relative phase of the interference response function.

The intensity I(ϕ) at fixed positions in the interference pattern oscillates when the translation stage is moving. By placing two detectors, S₁ and S₂, with a 90° phase difference in the interference pattern, the quadrature phase-shift can be measured. To illustrate the technique, typically measured data for a sinusoidal movement of the translation stage are shown in Fig. 2. The two raw signals from detectors S₁ and S₂ are shown in Fig. 2(a). The measured signals oscillate around their respective centers, S₁ ≈ 2.2 V and S₂ ≈ 1.7 V. For clarity reasons, only a small section of the recorded data is shown. The different center and peak-to-peak voltages are due to differing intensity responses of the two sensors as well as variation of light intensities reflected from the retroreflectors. Figure 2(b) shows the identical data but represented in a Lissajous plot. This plot can be used during alignment to position the detectors in the interference pattern ensuring the 90° phase difference between them. For 0° phase difference, the Lissajous plot would show a diagonal line. Slowly moving one detector perpendicular to the laser beam to alter the phase difference will transform this diagonal line first to an ellipse and finally into a circle once the 90° phase difference has been reached. To position the sensors exactly at a 90° phase difference can be challenging, but it has been shown that phase shift errors as high as 10° are not significant to the measurement error.⁷ The ellipticity of the raw data shown in Fig. 2(b), caused by the different intensity responses of the sensors and alignment of the sensors, is compensated for in a normalization step after the raw data are offset around zero. This is done during post-processing or in situ by finding suitable offset parameters a priori. It is important to note that normalizing the data does not affect the measurement, which relies purely on the phase relation between the two sensors. The normalized and offset data are shown as a Lissajous plot in Fig. 2(c). Figure 2(d) shows the identical data but with the added time axis. Next, by applying a simple arctangent function to the normalized data points, the wrapped phase change shown in Fig. 2(e) is obtained. Finally, the unwrapped phase change is calculated by compensating for the phase jumps in the arctangent function. The measured unwrapped phase change Δϕ is plotted in Fig. 2(f).

The quadrature phase-shift detection and the data processing of the raw data in order to obtain the unwrapped phase change are explained in more detail in Ref. 7.

The quadrature phase-shift detection is executed by the developed detection unit shown in Fig. 3. Its main components are two light-to-voltage converters (TAOS TSL257R) and a microcontroller (PJRC Teensy 4.0). The two light-to-voltage converters S₁ and S₂, consisting of a photodiode and an operational amplifier, act as the detectors. The ends of two 1 mm diameter flexible waveguides [indicated by the arrows in Fig. 3(a)] are positioned with a 90° phase difference in the interference pattern and guide light to the photodiodes. At the position of the waveguides, the fringe size of the interference pattern is bigger than the waveguide diameter. All electrical components are soldered to a custom printed circuit board (PCB) mounted inside an opaque 3D-printed housing [see Fig. 3(b)]. The exact layout of the PCB, a list of all used components, and the .stl-files for the 3D-print can be found in the GitHub repository “QPSdetectionsystem.”¹⁰ The advantage of guiding the light to the detectors is an easy access to the interference pattern, while the detection unit box can be placed at a convenient place outside the actual measurement setup. The output voltage of the detectors is directly proportional to the light incident on the photodiode. The voltage is applied to the analog inputs of the microcontroller [see Fig. 3(c)], where it is sampled and transmitted to a computer via a serial connection. On the computer, a python script is post-processing the data as described in Sec. II C. A working example of the arduino code running on the microcontroller, executing the data acquisition, as well as the python script, executing the data processing, is provided in Ref. 10. In an additional step, the position of the translation stage is calculated from the measured phase change using Eq. (2).

The detection unit was characterized by monitoring the movement of the translation stage. The travel range was limited to ±1 mm to minimize systematic errors due to slight misalignment. Sinusoidal as well as triangular translation stage trajectories were investigated for different frequencies. The detection unit was set to sample at a frequency of f_s = 5 kHz. A typical result of the measurement for both the sinusoidal as well as the triangular case is shown in Fig. 4.

In the presented sinusoidal example, the translation stage is set to perform sinusoidal oscillations with an amplitude of 1 mm at a set frequency of f_set = 75 mHz, corresponding to a maximum velocity of 471 μm s⁻¹. This means that sampling is done at least every 100 nm of displacement. The data acquisition, the transmission to the computer, and the movement of the translation stage are initiated by an external trigger signal. The measured time-dependent position of the translation stage is plotted in Fig. 4(a) together with a sinusoidal fit. The data are fitted by the following function:

where x_fit(t) is the time-dependent position of the stage, A is the amplitude of the oscillation, and f is the frequency of the sinusoidal movement. The fit yields an amplitude of A = 1.001 mm and a frequency of f = 74.98 mHz. The root mean square error (RMSE) is 143 nm. Figure 4(c) shows the deviation of the measured position from the ideal sinusoidal fit, which is below 0.6 µm.

In the presented triangular example, the translation stage is set to perform a triangular movement with an amplitude of 1 mm at a set frequency of f_set = 75 mHz, corresponding to a maximum velocity of 300 μm s⁻¹. Therefore, sampling is done at least every 60 nm of displacement. The data acquisition is done analogously to the sinusoidal case. The measured position of the translation stage is plotted in Fig. 4(b) together with a triangular fit. The data are fitted by the following function:

where x_fit(t) is the time-dependent position of the stage, A is the amplitude of the oscillation, and f is the frequency of the triangular movement. The fit yields an amplitude of A = 1.002 mm and a frequency of f = 74.91 mHz. The RMSE is 368 nm. Figure 4(c) shows the deviation of the measured position from the fitted function and is below 3 µm.

The beam of the HeNe laser was focused by a spherical lens. This allowed the interference pattern to be resolved spatially. It is important to make sure that the focal length is sufficiently long so that the paraxial approximation is valid. However, it needs to be short enough to apply sufficient wave front curvature so the fringe size of the interference pattern at the position of the detectors is large, i.e., a period of the interference pattern is significantly larger than the detector size as illustrated in the inset of Fig. 1. Generally, there is flexibility in the exact choice of the focal length of this lens. Here, the focal length f = 150 mm was empirically determined to be ideal for the specific setup and detector dimensions.

The noise of the detection unit was determined for a fixed stage and is ϕ = ±0.006π, corresponding to a spatial resolution of ±1 nm, more than two orders of magnitude lower than the measured RMSE. One factor contributing to the observed noise is the detector size. Generally, larger detectors average the detected signal. This causes a higher offset of the signal and thus a lower signal-to-noise ratio. Smaller detectors average the detected signal less. However, in this case, it is important to match the laser power to the dynamic range of the detector to optimize the signal-to-noise ratio. In practice, a balance has to be found for the detector size to have a good spatial resolution while still being able to detect light. Another approach to reduce noise would be to monitor the entire interference pattern with a camera. While this would lead to a higher spatial resolution of the phase change, this approach would be limited in time compared to the quadrature phase-shift detection due to a lower sampling frequency. According to the Nyquist–Shannon sampling theorem, the sampled signal is fully determined when the sampling frequency f_s is faster than or at least equal to twice the frequency f_i at which the phase ϕ of the interference pattern oscillates. The time derivative of the phase change can be approximated by

From Eqs. (2) and (6), it follows that

where λ is the wavelength of the probing laser beam and v = Δx/Δt is the speed at which the translation stages moves. The fastest speed of the translation stage was set to v = 471 μm s⁻¹ corresponding to a maximum frequency of the oscillating interference pattern of f_i ≤ 1.5 kHz well below the sampling frequency f_s = 5 kHz.

The actual measured movement of the translation stage is in good agreement with the set trajectory and could be determined with sub-micrometer resolution. The measured frequency is within a relative error of 0.12% of the set frequency of f_set= 75 mHz. The RMSE (≤368 nm) is far below the specified accuracy of the translation stage of ±1 µm. Peaks in deviation are visible in Figs. 4(a) and 4(b) that are substantially larger than the average deviation. Those peaks in deviation are related to a change in the direction of the translation stage. Especially in the case of the triangular movement, this leads to distinct momentarily deviations. This can be an explanation for the relatively large RMSE compared to the noise level.

To conclude, we have demonstrated an affordable system that is able to detect rapid phase changes accurately for applications relying on fast in situ monitoring. While the system cannot compete with the specifications of other state of the art systems, its beauty lies in simplicity, availability, and price. Furthermore, the application of the detection system is not limited to displacement measurements, but can be used in any interferometer to detect phase changes and thus serve, for example, as an interferometric thermometer.⁴ 

This work was funded by the Swedish Foundation for Strategic Research (Grant No. RMA15-0135).

The data that support the findings of this study are openly available in the GitHub repository at https://github.com/mukogit/QPSdetectionsystem.¹⁰ In the same repository, all additional information is available, which is necessary to rebuild the detection system. This includes: arduino code for the microcontroller, python code for post-processing, .stl-files for the 3D-prints, and the layout of the PCB.