# Sagnac Effect Compensations and Locked States in a Ring Laser Gyroscope

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## Abstract

**:**

## 1. Introduction

## 2. Sagnac Effect for the RLG

**c**is the speed of light. If the Sagnac effect is generalized to a structure in the form of an arbitrary closed optical path rather than an ideal circular optical path, it can be verified that the round-trip optical path difference owing to the rotational physical quantity increases in proportion to the area enclosed by the closed path, as expressed in Equation (2).

**L**is the perimeter of the light path and

**λ**is the wavelength of the laser light.

## 3. Frequency Lock-In Dynamics in the RLG

**ν**is the optical frequency that is obtained with the dispersive effects of the active medium; ${\mathit{\chi}}_{0}$ and ${\mathit{\epsilon}}_{0}$ are the dielectric susceptibility and vacuum permittivity, respectively; and ${\mathit{r}}_{\mathit{s}}$ is the amplitude of the coupling factor owing to backscattering. Equation (4) can be analytically divided into two parts and examined according to the relationship between $\mathit{a}$ and $\mathit{b}$ when determining the value of $\mathit{\psi}$. That is, when $\mathit{a}$ is smaller than

**b**, the right side of Equation (4) becomes zero, regardless of the value of $\mathit{a}$, even though $\mathit{a}$ is not zero. In this case, the frequencies between the two beams traveling in opposite directions inside the ring laser resonator are locked to the same value. By contrast, when $\mathit{a}$ is larger than $\mathit{b}$, the right side of Equation (4) is outside the frequency-lock state and has a nonzero value corresponding to $\mathit{a}$. If the average period

**T**for $\dot{\mathit{\psi}}$ is defined, it can be expressed as Equation (6).

**a**shows a nonlinear relationship characteristic owing to the square root function relationship. However, when the difference between the $\mathit{a}$ and $\mathit{b}$ values is sufficiently large, the average angular frequency difference converges to $\mathit{a}$, showing a linear relationship.

**,**according to the possible practical situations. The solution of Equation (8), $\mathit{\psi}$

**,**can be expressed in the form of Equation (9).

**J**is the

_{m}**m**th Bessel function of the first kind. The sum of Equation (10) has a special meaning when $\mathit{a}$ is near an integer multiple of ${\mathit{\omega}}_{\mathit{D}}$, as expressed in Equation (11).

**,**according to the change in $\mathit{a}$ value, can be calculated using Equation (13) [16], and the RLG output $\langle {\mathit{f}}_{\mathit{o}\mathit{u}\mathit{t}}\rangle {}_{\Delta \mathit{t}}$in Figure 1 corresponds to the calculated result for $\Delta \mathit{t}$ = 10 s,

## 4. Interpretation of Lock-In Mechanism with Sagnac Effect Compensation

**,**where ${\mathit{F}}^{*}$ is the complex conjugate of $\mathit{F}$. The frequency difference $\dot{\mathit{\psi}}/2\mathit{\pi}$ was extracted from the measured interference signal. A complex error function $\mathit{\u03f5}$ is defined as

## 5. Analysis of Sagnac Effect Compensations with the Error Function

**ϵ**defined in Section 4.

**,**for $\mathit{\psi}$- and ${\mathit{\psi}}_{\mathit{i}}$- related data, the same conditions as in Figure 1 were used. The frequency component that dominates ${\mathit{R}}_{\mathit{a}}$ increased in proportion to the increases in $\mathit{a}$ in the frequency lock-in region, whereas it was generally lowered in inverse proportion to the increase in $\mathit{a}$ when it was outside the frequency lock-in region. When recalling the typical frequency lock-in characteristic that the RLG has a zero-output value when $\mathit{a}$ is within the frequency lock-in region, whereas the outputs of the RLG approach $\mathit{a}$ as a result of increases in $\mathit{a}$ when it is outside the frequency lock region, it was inferred that the frequency component that dominates ${\mathit{R}}_{\mathit{a}}$ acted in a form that compensates for the given Sagnac effect. It is particularly noteworthy from Figure 2 and Figure 3 that, outside the frequency lock-in region, different frequency components are mixed, unlike inside the frequency lock-in region. It was observed that phase transitions occurred based on the frequency lock-in threshold, and the mechanism for compensating the Sagnac effect outside the frequency lock-in region was not simple, unlike inside the frequency lock-in region where the Sagnac effect was completely compensated by a single frequency component.

**a**is outside the frequency lock-in region, so that ${\mathit{f}}_{\mathit{m}}$ is equivalent to ${\mathit{f}}_{\mathit{a}}$ and ${\mathit{f}}_{\mathit{e}}$. However, at ${\mathit{f}}_{\mathit{a}}$ = 4.14 Hz, which is near the frequency lock-in threshold, ${\mathit{f}}_{\mathit{m}}$ still shifts linearly with the change in $\mathit{a}$ and coincides with ${\mathit{f}}_{\mathit{a}}$, while ${\mathit{f}}_{\mathit{e}}$ in Figure 4 has a lower value of ~2.69 Hz. This difference is due to the combined action of the following two characteristics.

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Numerically calculated RLG outputs $\langle {\mathit{f}}_{\mathit{o}\mathit{u}\mathit{t}}\rangle {}_{\Delta \mathit{t}}$ with sinusoidal dithering as a function of ${\mathit{f}}_{\mathit{a}}=\mathit{a}/2\mathit{\pi}$.

**Figure 4.**Sagnac effect compensations with ${\mathit{f}}_{\mathit{m}}$ and ${\mathit{f}}_{\mathit{e}}$ as a function of ${\mathit{f}}_{\mathit{a}}=\mathit{a}/2\mathit{\pi}$.

**Figure 5.**Analyzed frequency components of ${\mathit{R}}_{\mathit{a}}$ with changes in ${\mathit{f}}_{\mathit{a}}=\mathit{a}/2\mathit{\pi}$.

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Choi, W.-S.; Shim, K.-M.; Chong, K.-H.; An, J.-E.; Kim, C.-J.; Park, B.-Y.
Sagnac Effect Compensations and Locked States in a Ring Laser Gyroscope. *Sensors* **2023**, *23*, 1718.
https://doi.org/10.3390/s23031718

**AMA Style**

Choi W-S, Shim K-M, Chong K-H, An J-E, Kim C-J, Park B-Y.
Sagnac Effect Compensations and Locked States in a Ring Laser Gyroscope. *Sensors*. 2023; 23(3):1718.
https://doi.org/10.3390/s23031718

**Chicago/Turabian Style**

Choi, Woo-Seok, Kyu-Min Shim, Kyung-Ho Chong, Jun-Eon An, Cheon-Joong Kim, and Byung-Yoon Park.
2023. "Sagnac Effect Compensations and Locked States in a Ring Laser Gyroscope" *Sensors* 23, no. 3: 1718.
https://doi.org/10.3390/s23031718