# Phase Imbalance Optimization in Interference Linear Displacement Sensor with Surface Gratings

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Optical Concept

^{14}= 16,384. However, displacement determination along with interpolation can be accurate if only the original signals are generated without errors. The interpolation error is defined as an error associated with deviations of the working conditions from the quadrature signals and directly depends on the phase imbalance. The interpolation error is cyclic and occurs with each period of the signal, but it does not accumulate or depend on the period value of the diffraction scale. At the last step, before interpolation and determination of displacement, correction of residual ellipticity is carried out, if it is not eliminated in the optical branch, as well as for the correction of random phase deviations in real time. The signal errors depending on the positioning of elements in the optical system in accordance with Abbe errors [24,25] is not the subject of the current research, although it is very important in the practical realization. The Abbe errors need to be aligned in the scheme to avoid interference break in the plane of the photodetectors, where it is necessary to provide an interference strip of infinite wide. Therefore, the durable angle of relative inclination between the scales should not exceed 0.5°. To minimize Abbé errors and the most influential pitching error measurement bench should be provided with high-precision linear guides. In addition, mounting surfaces should be sufficiently machined causing no additional inaccuracies in the system. General recommendations on minimizing the angular errors include moving the heavier load as close as possible to the center of the measurement system.

#### 2.2. Quadrature Signals Generation in the Optical Branch

_{inc}is the incident angle, θ

_{m}is the diffraction angle of the mth order, and λ is the laser wavelength. The ratio of the period of the structure and the wavelength determine the scheme geometry of the circuit.

## 3. Results

_{i}

_{nc}= E

_{0}× exp(I × Φ

_{0}). Then, after the first diffraction three operating waves are formed on the reference grating with amplitudes E

_{11}= η

_{11}× E

_{0}× exp(I × [Φ

_{0}+ Φ

_{11}]), E

_{12}= η

_{12}× E

_{0}·exp(i × [Φ

_{0}+ Φ

_{12}]) and E

_{13}= η

_{13}E

_{0}× exp(I × [Φ

_{0}+ Φ

_{13}]). During the second diffraction on the measurement scale, the dependence on the displacement and the phase shift during diffraction are introduced into the signal; forming four signals with complex amplitudes E

_{21}(x) = η

_{21}× η

_{11}× E

_{0}× exp(i × [Φ

_{0}+ Φ

_{11}+ Φ

_{21}− Ω(x)]), E

_{22}(x) = η

_{22}× η

_{12}× E

_{0}×exp(I × [Φ

_{0}+ Φ

_{12}+ Φ

_{22}+ Ω(x)]), E

_{23}(x) = η

_{23}× η

_{12}× E

_{0}× exp(i × [Φ

_{0}+ Φ

_{12}+ Φ

_{23}− Ω(x)]), E

_{24}(x) = η

_{24}× η

_{13}E

_{0}× exp(I × [Φ

_{0}+ Φ

_{13}+ Φ

_{24}+ Ω(x)]). These signals have become dependent on displacement value. After the third diffraction, each of these waves is subdivided into two, forming four pairs of signals E

_{31}(x) and E

_{34}(x), E

_{32}(x) and E

_{33}(x), E

_{35}(x) and E

_{38}(x), and E

_{36}(x) and E

_{37}(x), which are superimposed due to the interference.

_{PD1}= η

_{32}× η

_{21}× η

_{11}× E

_{0}, B

_{PD1}= η

_{33}× η

_{22}× η

_{12}× E

_{0}, V

_{PD1}= 2 A

_{PD1}× B

_{PD1}/(A

_{PD1}

^{2}+ B

_{PD1}

^{2}) = 2η

_{32}× η

_{21}× η

_{11}× η

_{33}× η

_{22}× η

_{12}/[(η

_{32}η

_{21}η

_{11})

^{2}+ (η

_{33}η

_{22}η

_{12})

^{2}], and Ψ

_{PD1}= Φ

_{32}+ Φ

_{21}+ Φ

_{11}− Φ

_{33}− Φ

_{22}− Φ

_{12}− 2Ω(x).

_{PD2}, I

_{PD3}, and I

_{PD4}arriving to the remaining detectors A

_{PD2}= η

_{31}η

_{21}η

_{11}E

_{0}, B

_{PD2}= η

_{34}η

_{22}η

_{12}E

_{0}, Ψ

_{PD2}= Φ

_{31}+ Φ

_{21}+ Φ

_{11}− Φ

_{34}− Φ

_{22}− Φ

_{12}− 2Ω(x); A

_{PD3}= η

_{35}η

_{23}η

_{12}E

_{0}, B

_{PD3}= η

_{38}η

_{24}η

_{13}E

_{0}, Ψ

_{PD3}= Φ

_{38}+ Φ

_{24}+ Φ

_{13}− Φ

_{35}− Φ

_{23}− Φ

_{12}+ 2Ω(x); A

_{PD4}= η

_{36}η

_{23}η

_{12}E

_{0}, B

_{PD4}= η

_{37}η

_{24}η

_{13}E

_{0}, Ψ

_{PD4}= Φ

_{13}+ Φ

_{24}+ Φ

_{37}− Φ

_{12}− Φ

_{23}− Φ

_{36}+ 2Ω(x). The received four signals I

_{PD1}–I

_{PD4}are shown schematically in Figure 2c. Due to the symmetry condition of a sinusoidal or rectangular profile of the grating, the diffraction efficiencies and optical phase shifts will be equal under symmetric diffraction conditions

_{11}= η

_{13},η

_{22}= η

_{23}, η

_{21}= η

_{24}, η

_{31}= η

_{38}, η

_{32}= η

_{37}, η

_{33}= η

_{36}, η

_{34}= η

_{35};

Φ

_{11}= Φ

_{13}, Φ

_{22}= Φ

_{23}, Φ

_{21}= Φ

_{24}, Φ

_{31}= Φ

_{38}, Φ

_{32}= Φ

_{37}, Φ

_{33}= Φ

_{36}, Φ

_{34}= Φ

_{35}.

_{PD1}and I

_{PD2}to be 90°

_{1-2}= Ψ

_{PD1}− Ψ

_{PD2}= Φ

_{32}+ Φ

_{21}+ Φ

_{11}− Φ

_{33}−Φ

_{22}−Φ

_{12}− 2Ω(x) −

− [Φ

_{31}+ Φ

_{21}+ Φ

_{11}− Φ

_{34}− Φ

_{22}−Φ

_{12}− 2Ω(x)] = Φ

_{32}+ Φ

_{34}− Φ

_{33}− Φ

_{31}= 90°

_{1-4}= Ψ

_{PD1}–Ψ

_{PD4}= Φ

_{11}+ Φ

_{21}+ Φ

_{32}− Φ

_{12}− Φ

_{22}− Φ

_{33}− 2Ω(x) − [Φ

_{13}+ Φ

_{24}+ Φ

_{37}− Φ

_{12}− Φ

_{23}− Φ

_{36}+2Ω(x)] = −4Ω(x). Therefore, Ψ

_{PD4}= Ψ

_{PD1}+ 4Ω(x). ΔΨ

_{2-3}= Ψ

_{PD2}− Ψ

_{PD3}= Φ

_{11}+ Φ

_{21}+ Φ

_{31}− Φ

_{12}− Φ

_{22}− Φ

_{34}− 2Ω(x) − [Φ

_{13}+ Φ

_{24}+ Φ

_{38}− Φ

_{12}− Φ

_{23}− Φ

_{35}+ 2Ω(x)] = −4Ω(x). Therefore, Ψ

_{PD3}= Ψ

_{PD2}+ 4Ω(x). The antiphase signals are determined by various signs in front of the phase term containing the cosine function.

_{cos}∝ I

_{cos}= I

_{PD1}− I

_{PD4}= p + R

_{cos}cos(2Ω(x))

U

_{sin}∝I

_{sin}= I

_{PD2}− I

_{PD3}= q + R

_{sin}sin(2Ω(x) − α)

_{32}η

_{21}η

_{11}E

_{0})

^{2}+ (η

_{33}η

_{22}η

_{12}E

_{0})

^{2}− (η

_{36}η

_{23}η

_{12}E

_{0})

^{2}− (η

_{37}η

_{24}η

_{13}E

_{0})

^{2}→0; R

_{cos}= 2E

_{0}

^{2}(η

_{32}η

_{21}η

_{11}η

_{33}η

_{22}η

_{12}+ η

_{36}η

_{23}η

_{12}η

_{37}η

_{24}η

_{13}) → 4η

_{32}η

_{21}η

_{11}η

_{33}η

_{22}η

_{12}E

_{0}

^{2}; q = (η

_{31}η

_{21}η

_{11}E

_{0})

^{2}+ (η

_{34}η

_{22}η

_{12}E

_{0})

^{2}− (η

_{35}η

_{23}η

_{12}E

_{0})

^{2}− (η

_{38}η

_{24}η

_{13})

^{2}→0; R

_{sin}= 2E

_{0}

^{2}(η

_{31}η

_{21}η

_{11}η

_{34}η

_{22}η

_{12}+ η

_{35}η

_{23}η

_{12}η

_{38}η

_{24}η

_{13}) → 4 η

_{31}η

_{21}η

_{11}η

_{34}η

_{22}η

_{12}E

_{0}

^{2}; and α = |Φ

_{32}+ Φ

_{34}− Φ

_{33}− Φ

_{31}− 90°|→0°.

_{cos}and R

_{sin}are the amplitudes of the signals, which relation defines the needed amplification in the channels; p and q are the offset levels of the sine and cosine channels, respectively; and α is the quadrature error or phase imbalance. When the condition in Equation (5) is fulfilled, the signal amplitude doubles, and the offset level is zeroed (p →0, q →0). The advantage of this procedure, in addition to zeroing and equalizing the basic signal levels, is that it also consists in an increase in the signal-to-noise ratio. The period of the signals decreases by half relative to the period of the measurement scale; p and q are different if the diffraction efficiencies are different for different diffraction conditions. α expresses the residual ellipticity (imperfect quadrature alignment) that is zeroed out in the practical implementation of the condition in Equation (5)—successful adjustment of the phase imbalance due to proper manufacturing of the static reference scale. In electronic processing, we no longer work with the optical signals I

_{cos}and I

_{sin}, but with the proportional voltage signals U

_{cos}and U

_{sin}, which are schematically shown in Figure 2d.

#### 3.1. Elliptical Correction Phase Analysis

_{cos}

^{perf}= R cos(Ψ), U

_{sin}

^{perf}= R sin(Ψ), where = 2Ω(x)

**=**4π x/d, R = R

_{cos}= G R

_{sin}is the radius of the Lissajous figure in the form of a circle, and G = R

_{cos}/R

_{sin}is the—the amplification coefficient. Measurements of the instantaneous coordinate values based on instantaneous signals U

_{cos}

^{perf}и U

_{cos}

^{perf}are possible by the interpolation between the calculated points of zero intersections x in the Lissajous figure. These intersections correspond to a quarter of the period of the quadrature signal. Fractional distance values in phase representation Ψ are obtained from the arctangent function of the ratio of signal amplitudes. The main difficulty in digital adjustment of phase imbalance is to determine the coefficients r, p, q, and α from the experimental signal (Equation (6)), assuming that these are the only significant demodulation errors. To provide it, a sufficiently large distance x is covered, accumulating experimental data to determine these four components. This operation is carried out by selecting the least square method.

_{cos}, U

_{sin}) is not located on a circle, but on a distorted ellipse, which leads to significant errors in determining the fractional part Ψ of a quarter of the period. The circle (U

_{cos}

^{perf})

^{2}+ (U

_{cos}

^{perf})

^{2}= R

^{2}turns into an ellipse (U

_{cos})

^{2}+ (U

_{sin})

^{2}= R

^{2}. The trajectory of the end of the vector (U

_{cos}, U

_{sin}) representing a distorted ellipse can be described according to

_{cos}and U

_{sin}(Figure 2d) can be corrected and reduced to the form U’

_{cos}and U’

_{sin}(Figure 2e), respectively

_{32}η

_{33}/η

_{31}η

_{34}→1; α = |Φ

_{32}+ Φ

_{34}− Φ

_{33}− Φ

_{31}− 90°|→0°.

_{32}η

_{33}/η

_{31}η

_{34}→1; α = |Φ

_{32}+ Φ

_{34}− Φ

_{33}− Φ

_{31}− 90°|→ 0°

_{31}, η

_{32}, η

_{33}, η

_{34}, Φ

_{31}, Φ

_{32}, Φ

_{33}, and Φ

_{34}were carried out using MATLAB environment (ETMC Exponenta Ltd., Moscow, Russian Federation). To simulate the diffraction in sinusoidal grating, the RCWA method with the built-in method of curvilinear coordinates was used [30,31].

#### 3.2. Phase Imbalance and Amplification in Quadrature Signals

_{1}from 25° to 45° (Figure 2), which determines the periods for a laser wavelength using Equation (1). Different polarization conditions are considered; TE polarization is recommended. The groove depth in the analysis is limited by the maximum value of the period.

_{B}= 0.751 μm, h

_{B}= 0.67 μm for λ

_{B}= 450 nm; d

_{G}= 0.875 μm, h

_{G}= 0.785 μm for λ

_{G}= 0.52 μm; and d

_{R}= 1.123 μm, h

_{R}= 1.125 μm for λ

_{R}= 0.66 μm. Furthermore, to analyze the contribution of the instrumental uncertainty in linear displacement encoder under study, the influence of manufacturing errors of reference scale and the stability of operating conditions are considered. It should be noted that, if these parameters do not match the calculated ones, it will be necessary to use an ellipticity correction in the interface electronics.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The scheme of the linear displacement sensor in conjunction with a moving stage (

**a**); signal’s view after the elliptical correction (

**b**); and optical system of double grating interferometer (

**c**).

**Figure 2.**Signal transformation in the optical system: (

**a**) the formation of signals due to ray path; (

**b**) alignment for interference band of infinite width; (

**c**) uncompensated signals; (

**d**) signals after antiphase subtraction; and (

**e**) signals after ellipticity correction.

**Figure 3.**Phase imbalance α (

**a**) and the amplification G between the channels (

**b**) via period and groove depth of the reference grating at λ

_{B}= 450 nm.

**Figure 4.**Phase imbalance α and amplification G of quadrature signals via period of: reference grating (

**a**); groove depth of reference grating (

**b**); laser wavelength (

**c**); and refractive index (

**d**). Dotted lines indicate optimal conditions.

1st diffraction on reference scale | ||||||||
---|---|---|---|---|---|---|---|---|

Order | −1 | 0 | +1 | |||||

Diff. eff. | η_{11} | η_{12} | η_{13} | |||||

Phase shift | Φ_{11} | Φ_{12} | Φ_{13} | |||||

2nd diffraction on measurement scale | ||||||||

Order | −1 | +1 | −1 | +1 | ||||

Diff. eff. | η_{21} | η_{22} | η_{23} | η_{24} | ||||

Phase shift | Φ_{21} − Ω(x) | Φ_{22} + Ω(x) | Φ_{23} − Ω(x) | Φ_{24} + Ω(x) | ||||

3rd diffraction on reference scale | ||||||||

Order | 0 | +1 | 0 | −1 | +1 | 0 | −1 | 0 |

Diff. eff. | η_{31} | η_{32} | η_{33} | η_{34} | η_{35} | η_{36} | η_{37} | η_{38} |

Phase shift | Φ_{31} | Φ_{32} | Φ_{33} | Φ_{34} | Φ_{35} | Φ_{36} | Φ_{37} | Φ_{38} |

Detector | PD2 | PD1 | PD1 | PD2 | PD3 | PD4 | PD4 | PD3 |

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**MDPI and ACS Style**

Odinokov, S.; Shishova, M.; Kovalev, M.; Zherdev, A.; Lushnikov, D. Phase Imbalance Optimization in Interference Linear Displacement Sensor with Surface Gratings. *Sensors* **2020**, *20*, 1453.
https://doi.org/10.3390/s20051453

**AMA Style**

Odinokov S, Shishova M, Kovalev M, Zherdev A, Lushnikov D. Phase Imbalance Optimization in Interference Linear Displacement Sensor with Surface Gratings. *Sensors*. 2020; 20(5):1453.
https://doi.org/10.3390/s20051453

**Chicago/Turabian Style**

Odinokov, Sergey, Maria Shishova, Michael Kovalev, Alexander Zherdev, and Dmitrii Lushnikov. 2020. "Phase Imbalance Optimization in Interference Linear Displacement Sensor with Surface Gratings" *Sensors* 20, no. 5: 1453.
https://doi.org/10.3390/s20051453