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Intensity coupling characteristics of dual-longitudinal mode ring lasers with Ne dual-isotope

Jianning Liu, Lanpeng Guo, Jun Weng, Wenxin Liu, Yuxuan Yang, Kai Zhao, and Yi Zheng

Open Access Open Access

Abstract

The intensity coupling characteristics of Ne dual-isotope inflation and dual-longitudinal-mode operation ring lasers were investigated based on the Lamb theory. Considering the contribution of the Ne isotope system to the polarization of the gain medium and gain saturation effects, the frequency coupling effects were analyzed. Combined with the plasma dispersion function, the optical cavity length is 0.47 m, Ne20: Ne22= 0.53:0.47; the frequency spacing of the adjacent longitudinal mode is 640 MHz, and the intensity tuning curve of the ring laser is simulated. The alterations in the gain self-saturation and mutual saturation coefficients between the four frequencies generated via dual-longitudinal mode splitting are comprehensively discussed. Finally, a detection experiment for the intensity-tuning curve is designed to verify the theoretical analysis.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

A laser gyro is a sensor that utilizes the phase difference between forward and backward waves in a ring laser to measure the angular velocity [1–3]. The ring laser is the core component of a laser gyro [4,5]. A special self-biasing phenomenon [6–8] exists in dual-longitudinal-mode operating laser gyros, which has been proven to be caused by the coupling effect between four frequencies of the same polarization state [9]. Therefore, studying the frequency coupling effect between the dual-longitudinal mode and the four frequencies with the same polarization is essential for stabilizing the self-biasing state. In addition, a mixture of multiple Ne isotopes significantly influences the intensity characteristics of the ring laser, such as the profile of the intensity tuning curve and gain saturation between the longitudinal mode pair. The frequency stability and measurement accuracy of the laser gyro are also affected. To realize stable operation of the self-biasing laser gyro, the intensity coupling characteristics of the dual-longitudinal-mode ring laser were investigated theoretically and experimentally.

This study investigated the intensity coupling characteristics of a dual-longitudinal-mode ring laser based on the Lamb theory, especially considering the saturation effects between four frequencies with the same polarization. In addition, multiple Ne isotope inflations were adopted to avoid overlap of the mirror burning hole and eliminate mode competition [10,11]. Therefore, the contributions of multiple Ne isotopes to the polarization of the gain medium were considered in the derivation. Based on this, the profile of the intensity tuning curve was theoretically calculated. An experimental system for determining the intensity coupling characteristics of a dual-longitudinal-mode ring laser was established. The profile of the intensity-tuning curve detected under the isotope ratio of Ne20: Ne22 was 0.53:0.47, and the inflation pressure was 400 Pa. The experimental results confirmed the theoretical studies.

2. Theory of dual-longitudinal mode ring laser

2.1 Single isotope theory

According to the semiclassical Lamb theory [12,13], Maxwell’s equations are employed to describe the optical frequency electromagnetic field, and quantum theory is employed to describe material particles. The relationships between intensity and phase over time are expressed in the following equations:

$${\dot{E}_n} + \frac{1}{2}({{{{\nu_n}} / {{Q_n}}}} ){E_n} ={-} \frac{1}{2}({{{{\nu_n}} / {{\varepsilon_0}}}} ){\mathop{\rm Im}\nolimits} ({{P_n}} )$$
$${\nu _n} + {\dot{\phi }_n} = {\varOmega _n} - \frac{1}{2}({{{{\nu_n}} / {{\varepsilon_0}}}} ){E_n}^{ - 1}\textrm{Re} ({{P_n}} )$$
where En and Pn denote the intensity amplitude and polarization of the nth frequency split from the dual-longitudinal mode, n = I, II, III, IV; νn, ϕn, and Qn are the nth frequency, phase, and quality factor; Ωn is the resonance frequency of the passive cavity, and ε0 is the permittivity of vacuum.

Based on density matrix theory [14], the first-order polarization is given by

$${P_n}^{(1 )}(t )={-} \wp \textrm{ }\bar{N}{({\hbar Ku} )^{ - 1}}{E_n}(t )Z[{\gamma + i({\omega - {\nu_n}} )} ]$$
where ℘ is the matrix element of the electric-dipole moment; $\bar{N}$ is the average inverted population; $\hbar$ is reduced Planck constant; Ku is the half-width of Doppler broadening; γ is the decay coefficient of the off-diagonal elements; ω is the resonance frequency of Ne20 atoms.

In addition, the plasma dispersion function is expressed as [15]

$$Z(\zeta )= iK{\mathrm{\pi }^{{{ - 1} / 2}}}\int_{ - \infty }^\infty {\exp [{ - {{({{V / u}} )}^2}} ]} {({\zeta + ikV} )^{ - 1}}dV$$
$$\zeta = \gamma + i({\omega - {\nu_n}} )$$
where K = 2π/λ is the wave number, V is the average velocity of the atoms, and u is the most probable speed of thermal motion of atoms.

According to third-order perturbation theory, the third-order polarization intensity Pn(3) is

$${P_n}^{(3 )} = 2({{{{\varepsilon_0}} / {{\nu_n}}}} )\sum {\sum {\sum {{E_\mu }{E_\rho }{E_\sigma }} } } {\theta _{n\mu \rho \sigma }}{e^{i{\psi _{n\mu \rho \sigma }}}}$$
where subscript n is the frequency number, n = I, II, III, IV; subscripts μ, ρ, σ are substitutes for n generated from the three iterations process of solving the polarization intensity. ${\theta _{n\mu \rho \sigma }}$ is the saturation coefficient; the detailed values and determining methods are given in the appendix; ${\psi _{n\mu \rho \sigma }} = ({{\nu_n} - {\nu_\mu } + {\nu_\rho } - {\nu_\sigma }} )t + {\phi _n} - {\phi _\mu } + {\phi _\rho } - {\phi _\sigma }$ is the relative phase. Because ${\psi _{n\mu \rho \sigma }}$ changes more rapidly than En, only the terms ${\psi _{n\mu \rho \sigma }}\textrm{ = }0$ are retained. Considering that the laser operates in a steady state, the following two groups of mode indices satisfy the above requirements [16].
$$\begin{aligned} &(a)\quad n = \mu ,\quad \rho = \sigma = m,\\ &(b)\quad n = \sigma ,\quad \mu = \rho = m. \end{aligned}$$
where n and m are the ordinal numbers of the longitudinal modes.

The total polarization, ${P_n}\textrm{ = }{P_n}^{(\textrm{1} )}\textrm{ + }{P_n}^{(3 )}$, and Eqs. (3) and (7) were substituted into Eq. (1), and ${E_n}{({{\wp / \hbar }} )^2}{({{\gamma_a}{\gamma_b}} )^{ - 1}}$ was multiplied by the equation to obtain the motion equations of intensity.

$${\dot{I}_n} = 2{I_n}\left( {{\alpha_n} - \sum\limits_m {{\theta_{nm}}{I_m}} } \right)$$
$${I_n} = \frac{1}{2}{({{{\wp {E_n}} / \hbar }} )^2}{({{\gamma_a}{\gamma_b}} )^{ - 1}}$$
where αn is the net gain coefficient of the nth mode; θnm is the mutual saturation coefficient. when m = n, θnn is the self-saturation coefficient, and to distinguish from the mutual saturation coefficient, it is denoted as βn in the following. Where ${\gamma _a}$ and ${\gamma _b}$ is the decay constants of the atomic energy levels. The net gain coefficient is expressed as follows:
$${\alpha _n} = {Z_i}[{\gamma + i({\omega - {\nu_n}} )} ]{\nu _n} {\Re} {[{2Q{Z_i}(\gamma )} ]^{ - 1}} - {\omega / {2{Q_n}}}$$
where Zi is the imaginary part of the plasma-dispersion function; ${\Re}$ is the relative excitation coefficient, and Qn is the quality factor of the nth mode. The saturation coefficient is expressed as
$${\theta _{nm}} = 2{\gamma _a}{\gamma _b}{({\hbar \wp } )^{ - 2}}{\mathop{\rm Im}\nolimits} \{{{\theta_{nnmm}} + {\theta_{nmmn}}} \}({1 - {\delta_{nm}}} )$$
where ${\delta _{nm}}$ is Dirac function.

Next, the dual-longitudinal-mode operation state of the dual-isotope inflation ring laser was analyzed with the Lamb theory.

2.2 Dual-isotope theory

For the dual-longitudinal-mode operation laser gyro, the introduction of the Ne dual-isotope system modulated the profile of the intensity-tuning curve by adjusting the inflation ratio of the Ne dual-isotope to maintain the dual-longitudinal-mode ring laser operating in the self-biasing state. Next, the case of the Ne dual isotope in the gain medium was analyzed.

The contribution of the Ne dual isotopes to polarization can be treated separately [16,17]. First- and third-order polarizations were considered, and the total polarization is expressed as follows:

$$P = {f_1}{P_1}^{(1 )} + {f_2}{P_2}^{(1 )} + {f_1}{P_1}^{(3 )} + {f_2}{P_2}^{(3 )}$$
where P1 and P2 are the polarizations of the two isotopes, and f1 and f2 are the proportionality coefficients of the two isotopes. Under these conditions, the net gain and saturation coefficients were obtained by summing over the dual isotopes.
$$\begin{aligned} {\alpha _n} &= 2\nu {\Re} Q\{{{f_1}{Z_i}[{\gamma + i({\omega - {\nu_n}} )} ]{{[{{Z_i}(\gamma )} ]}^{ - 1}}} \\ &\quad{+ {f_2}{{\tilde{Z}}_i}[{\gamma + i({\tilde{\omega } - {\nu_n}} )} ]{{[{{{\tilde{Z}}_i}(\gamma )} ]}^{ - 1}}} \}- {\nu / {2{{\tilde{Q}}_n}}} \end{aligned}$$
$$\begin{aligned} {\theta _{nm}} &= 2{\gamma _a}{\gamma _b}{({\hbar \wp } )^{ - 2}}[{{f_1}} {\mathop{\rm Im}\nolimits} \{{{\theta_{nnmm}} + {\theta_{nmmn}}} \}({1 - {\delta_{nm}}} )\\ &\quad{ + {f_2}{\mathop{\rm Im}\nolimits} \{{{\theta_{nnmm}} + {\theta_{nmmn}}} \}({1 - {\delta_{nm}}} )} ]\end{aligned}$$
where $\tilde{\omega }$ is the resonant frequency of Ne22 atoms; ${Z_i}$, and ${\tilde{Z}_i}$ are the imaginary part of the plasma dispersion function corresponding to Ne20 and Ne22. The relationship between the most probable velocities u of Ne20 and Ne22 is presented as follows:
$$u = \tilde{u}\sqrt {{{\textrm{22}} / {\textrm{20}}}}$$

For the dual-longitudinal mode and four frequencies operation ring laser, the self-consistent equations of the intensity are written as follows:

$${\dot{I}_\textrm{I}} = 2{I_\textrm{I}}({{\alpha_\textrm{I}} - {\beta_\textrm{I}}{I_\textrm{I}} - {\theta_{\textrm{I II}}}{I_{\textrm{II}}} - {\theta_{\textrm{I III}}}{I_{\textrm{III}}} - {\theta_{\textrm{I IV}}}{I_{\textrm{IV}}}} )$$
$${\dot{I}_{\textrm{II}}} = 2{I_{\textrm{II}}}({{\alpha_{\textrm{II}}} - {\theta_{\textrm{II I}}}{I_\textrm{I}} - {\beta_{\textrm{II}}}{I_{\textrm{II}}} - {\theta_{\textrm{II III}}}{I_{\textrm{III}}} - {\theta_{\textrm{III V}}}{I_{\textrm{IV}}}} )$$
$${\dot{I}_{\textrm{III}}} = 2{I_{\textrm{III}}}({{\alpha_{\textrm{III}}} - {\theta_{\textrm{III I}}}{I_\textrm{I}} - {\theta_{\textrm{III II}}}{I_{\textrm{II}}} - {\beta_{\textrm{III}}}{I_{\textrm{III}}} - {\theta_{\textrm{III IV}}}{I_{\textrm{IV}}}} )$$
$${\dot{I}_{\textrm{IV}}} = 2{I_{\textrm{IV}}}({{\alpha_{\textrm{IV}}} - {\theta_{\textrm{IV I}}}{I_\textrm{I}} - {\theta_{\textrm{IV II}}}{I_{\textrm{II}}} - {\theta_{\textrm{IV III}}}{I_{\textrm{III}}} - {\beta_{\textrm{IV}}}{I_{\textrm{IV}}}} )$$
Subscripts I, II, III, and IV denote the frequencies split from the dual-longitudinal mode of the ring laser.

The naming rules for the four frequencies are presented in Fig. 1. Frequencies I and II were split from longitudinal Mode 1, and Frequencies III and IV were split from longitudinal Mode 2. Frequencies I and III were assumed to travel clockwise, whereas Frequencies III and IV were assumed to travel counterclockwise.

 figure: Fig. 1.

Fig. 1. Naming rule of four frequencies split from dual-longitudinal mode.

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Under the quasi-steady state, ${\dot{I}_n} = 0$, Eqs. (16)–(19) can be rewritten as

$${\alpha _\textrm{I}}\textrm{ = }{\beta _\textrm{I}}{I_\textrm{I}}\textrm{ + }{\theta _{\textrm{I II}}}{I_{\textrm{II}}}\textrm{ + }{\theta _{\textrm{I III}}}{I_{\textrm{III}}}\textrm{ + }{\theta _{\textrm{I IV}}}{I_{\textrm{IV}}}$$
$${\alpha _{\textrm{II}}}\textrm{ = }{\theta _{\textrm{II I}}}{I_\textrm{I}}\textrm{ + }{\beta _{\textrm{II}}}{I_{\textrm{II}}}\textrm{ + }{\theta _{\textrm{II III}}}{I_{\textrm{III}}}\textrm{ + }{\theta _{\textrm{III V}}}{I_{\textrm{IV}}}$$
$${\alpha _{\textrm{III}}}\textrm{ = }{\theta _{\textrm{III I}}}{I_\textrm{I}}\textrm{ + }{\theta _{\textrm{III II}}}{I_{\textrm{II}}}\textrm{ + }{\beta _{\textrm{III}}}{I_{\textrm{III}}}\textrm{ + }{\theta _{\textrm{III IV}}}{I_{\textrm{IV}}}$$
$${\alpha _{\textrm{IV}}}\textrm{ = }{\theta _{\textrm{IV I}}}{I_\textrm{I}}\textrm{ + }{\theta _{\textrm{IV II}}}{I_{\textrm{II}}}\textrm{ + }{\theta _{\textrm{IV III}}}{I_{\textrm{III}}}\textrm{ + }{\beta _{\textrm{IV}}}{I_{\textrm{IV}}}$$
Equations (20)–(23) can be described in matrix form as
$$\boldsymbol{\alpha } = \boldsymbol{\theta I}$$
The intensity is given by
$$\boldsymbol{I} = {\boldsymbol{\theta }^{ - 1}}\boldsymbol{\alpha }$$
Substituting Eqs. (13) and (14) into Eq. (25), the intensity-tuning curves for each frequency are illustrated in Fig. 2.

Figure 2 presents the simulation curve results of the In-ν. The simulation parameters are γa = 15.5 MHz, γb = 41 MHz, γ=128 MHz [18,19], ${\Re}$= 1.2. The optical length of the ring cavity L is 0.47 m, and the frequency spacing of the adjacent longitudinal modes is 640 MHz. The ratio of the Ne isotopes was Ne20: Ne22 = 0.53:0.47, and the frequency splitting quantity at the center frequency position of the Ne dual-isotope was 875 MHz [20]. Generally, in a complete gain linewidth of He–Ne laser (−640– 640 MHz), the intensity variation pattern of the frequencies split from the same longitudinal mode is similar. The subtle differences, for example, between Intensities III and IV in the range of −260– −100 MHz in Fig. 2 are caused by the saturation effect that occurs between these two frequencies in this range, and we will discuss this kind of effect in detail in Section 2.

 figure: Fig. 2.

Fig. 2. Intensity tuning curve of each frequency in dual-longitudinal mode operation ring laser.

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In addition, limited by the gain linewidth of the He–Ne laser, the intensity variation of the strong and weak modes exhibited a certain pattern. When Frequencies I and II are tuned from −640 to 0 MHz, Frequencies III and IV are moved from 0 to 640 MHz simultaneously, as demonstrated in Fig. 3(a). When Frequencies I and II were tuned from 0 to 640 MHz, Frequencies III and IV moved below the loss threshold, and the novel longitudinal mode moved into the gain curve and began to oscillate, as demonstrated in Fig. 3(b). Therefore, affected by the laser gain, the profile of the intensity tuning curve of Frequencies I and II is the same as that of Frequencies III and IV in the range of −640– 640 MHz. In other words, the weak longitudinal mode repeats the mode-scanning process of the strong longitudinal mode. Here, only the influence of the dual-longitudinal mode on the profile of the intensity-tuning curve was considered. In the process of longitudinal-mode frequency tuning, the longitudinal modes of the novel oscillation, other than the original dual-longitudinal mode, are not considered. Hence, the intensity-tuning curve of the ring laser can be obtained by translating the intensity-tuning curve of the strong or weak mode. Thus, the profile of the intensity-tuning curve can be obtained as illustrated in Fig. 4.

 figure: Fig. 3.

Fig. 3. Frequency tuning and dual-longitudinal mode state. (a) Frequency I and II are tuned from −640 MHz to 0 MHz, and Frequencies III and IV are shifted from 0 to 640 MHz simultaneously. (b) Old frequency pair goes out, and novel frequency pair starts to oscillate.

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 figure: Fig. 4.

Fig. 4. Theoretical intensity tuning curve of dual-longitudinal mode operation ring laser.

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Suppose the ratio of Ne isotopes is Ne20: Ne22 = 0.53:0.47, the inflation pressure of the gain medium is 400 Pa, the optical length of the ring cavity is 0.47 m, and the simulation result of the intensity tuning curve is illustrated in Fig. 4. The curve was Gaussian. The curve was symmetrical because of the mixing of the Ne isotope ratios. There is a deformation on both sides of the curve, and it is related to the saturation effects between the modes, which are analyzed next.

3. Mode saturation effect

For a dual-longitudinal-mode ring laser, the competition between the splitting frequencies from the dual-longitudinal mode is more complex [21,22]. Competition occurs between two frequencies belonging to the same longitudinal mode and between any two frequencies split from different longitudinal mode pairs. According to the Lamb theory, for dual-longitudinal-mode operating ring lasers, the first- and third-order mode saturation effects are mainly considered [23,24]. The first order is the self-saturation coefficient, which describes the gain saturation phenomenon of the splitting frequencies. The third order is the mutual saturation effect, which describes the inverted particle competition between different frequencies. According to Eq. (14). The calculation results for the saturation coefficients are presented in Figs. 5(a) and 5(b).

 figure: Fig. 5.

Fig. 5. Normalized saturation coefficients and schematic of modes spectrums corresponding to saturation effect. (a) Saturation coefficients for Frequency I. (b) Saturation coefficients for Frequency III. Saturation coefficients for Frequencies II and IV can be obtained by change of subscripts I ↔ II, III ↔ IV;(c), (d), (e) are spectrums of dual-longitudinal mode and dual-isotope frequency distribution; (f), (g), (h) are schematics of burning hole of inverted population corresponding to (c), (d), and (e), respectively.

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For a dual-longitudinal-mode-operating ring laser, the saturation effects between the frequencies split from the two modes can be divided into two cases. The first is the intermodal saturation effect, which occurs between two frequencies split from a single longitudinal mode. The second is the intermodal saturation effect between two frequencies split from different longitudinal modes. The red curves in Figs. 5(a) and 5(b) and the red spectrum in Fig. 5(c) denote the first intermodal saturation effect. The blue curves in Figs. 5(a) and 5(b) and the blue spectra in Figs. 5(d) and 5(e) denote the second intermodal saturation effect.

According to the Doppler effect of the gas laser [25], there is the following relationship between the velocity component in z direction of the inverted particle Vz and the light frequency νn.

$${V_z} = c({{\raise0.7ex\hbox{${{\nu_n}}$} \!\mathord{\left./ {\vphantom {{{\nu_n}} {{v_0}}}} \right.}\!\lower0.7ex\hbox{${{v_0}}$}} - 1} )$$
where c is the speed of light, and ν0 is the center frequency of the inverted particles. It is specified that the positive direction of the motion of the inverted particle is clockwise. In other words, if the velocity component of an inverted particle is clockwise, Vz > 0, and the converse is Vz < 0.

When one of the longitudinal modes is at the gain center of Ne20 or Ne22, the saturation effect between the two frequencies split from the single longitudinal mode that is located at the gain center of Ne20 or Ne22 reaches a peak, as the red curve θI II and θIII IV illustrated in Figs. 5(a) and 5(b). Take θI II in Fig. 5(a) as an example, because the amount of the frequency splitting due to Sagnac effect is significantly smaller than the frequency spacing of the adjacent longitudinal mode, $\nu _{\textrm{I}} \approx \nu _{\textrm{II}}$. From Eq. (26), it is obtained: $V_z^{\textrm{I}} = V_z^{\textrm{II}} \approx 0$. Therefore, Frequencies νI and νII consume the inverted particles with the same velocity component in the isotope Ne20. This causes the burning holes corresponding to Frequencies νI and νII to overlap, and intense mode competition occurs, as illustrated in Fig. 5(f). In Fig. 5(a), the two peaks of the saturation coefficient θI II appear at the gain center frequency of Ne20 and Ne22, respectively. That is, the −438 MHz and 438 MHz frequency positions corresponding to both sides of the center of the dual-isotope synthesis gain curve. In addition, notice that the value of θI II is slightly different in the two peaks, which is affected by the Gaussian-like type of the intensity tuning curve. In the −438 MHz frequency position, II is slightly greater than III, but in the 438 MHz frequency position, II is slightly less than III, where II and III are the oscillation intensities of Frequencies I and II, respectively. In Fig. 5(b), the saturation coefficient θIII IV follows a similar pattern, and the analysis process is the same with θI II.

When two longitudinal modes are symmetric about the gain center of Ne20 or Ne22, the intermodal saturation effect between the two frequencies split from the different longitudinal modes will be greatly enhanced, as the blue curve θI IV and θIII II illustrated in Figs. 5(a) and 5(b). Assume that the two longitudinal modes are symmetric about the center of the Ne20 gain, as Fig. 5(d) demonstrates. For Frequency νI, the inverted particles of the Ne20 isotope that contribute to the stimulated radiation have a velocity relationship.

$$V_z^{\textrm{I}}= c({{\raise0.7ex\hbox{${{\nu_{\textrm{I}} }}$} \!\mathord{\left./ {\vphantom {{{\nu_{\textrm{I}} }} {{v_0}}}} \right.}\!\lower0.7ex\hbox{${{v_0}}$}} - 1} )$$
where ν0 is the center frequency of the Ne20 gain curve. The frequency interval of the dual-longitudinal mode is 640 MHz, and the two longitudinal modes are symmetric about the center of Ne20 gain; hence, νI = (ν0 − 320)MHz, $V_z^{\textrm{I}} ={-} c\frac{{320}}{{{\nu _0}}}$. For Frequency νIV, because the propagation direction of the oscillation light transmission is opposite the specified positive direction of the inverted particles’ motion, the inverted particles of the Ne20 isotope need to satisfy the following velocity relation.
$$V_z^{\textrm{IV}} ={-} c({{\raise0.7ex\hbox{${{\nu_{\textrm{IV}}}}$} \!\mathord{\left./ {\vphantom {{{\nu_{\textrm{IV}} }} {{v_0}}}} \right.}\!\lower0.7ex\hbox{${{v_0}}$}} - 1} )$$
νIV can be expressed as νIV = (ν0 + 320) MHz; hence, $V_z^{\textrm{IV}} ={-} c\frac{{320}}{{{\nu _0}}} = V_z^{\textrm{I}}$. The above analysis reveals that Frequencies νI and νIV interact by consuming inverted particles with the same velocity component as the Ne20 isotope. This caused the burning holes corresponding to the two frequencies to overlap and created fierce mode competition, as demonstrated in Fig. 5(g). Therefore, the saturation coefficient θI IV peaks at −756 MHz in Fig. 5(a). In the same way, the mode competition between Frequencies νIII and νII can be proved, and the saturation coefficient θIII II peaks at −117 MHz in Fig. 5(b). In addition, the saturation coefficient θI IV peaks at 117 MHz in Fig. 5(a), and θIII II peaks at 756 MHz in Fig. 5(b), corresponding to the mode state in Fig. 5(e).

Finally, the self-saturation coefficients βI, βIII are greater than the mutual saturation coefficients in Figs. 5(a) and 5(b). The results demonstrate that the suppression effect of the gain saturation on the frequency itself is much greater than that on the other frequencies in the ring laser. In addition, because the frequency interval between the adjacent two longitudinal modes pair is much larger than the frequency splitting amount of the Sagnac effect, the saturation effect between the two frequencies traveling in the same direction but split from the different longitudinal modes is the weakest, such as the green curves θI III and θIII I presented in Figs. 5(a) and 5(b).

The above research provides a theoretical basis for the selection of the best frequency tuning value, which is beneficial for maintaining laser stability and gyro sensitivity. Based on the mutual saturation coefficient, the first factor to be considered was the avoidance of intense mode competition. Hence, the optimal frequency position should avoid the peak values of the mutual saturation coefficients to ensure that each mode is away from the mode-competition region. However, this is difficult to achieve. The frequency positions at which the mutual saturation coefficients reach their peaks are different for the different longitudinal modes, and there are several mode-competition regions in the gain linewidth of the He–Ne ring laser. According to the operational characteristics of the dual-longitudinal mode and self-biasing laser gyro, the strong mode provides an interference signal to measure the angular velocity, whereas the weak mode provides the lock-out state. Under this constraint, the strong mode should be ensured to be primarily immune to mode competition, and the weak mode should oscillate at the same time. Therefore, at present, it is considered that the optimal frequency position for the strong mode to operate is −320 MHz from the center frequency position. Certainly, many factors should be considered when selecting the optimal operating frequency for the new dual-longitudinal-mode laser gyro, such as the angular velocity range of the inertial carrier, backscattering and non-uniform loss of the resonator, energy-level lifetime of the gain gas, and sensitivity of the rotational angular momentum in the gyro signal. These factors should be investigated in future studies.

4. Experiments

Figure 6 illustrates the experimental system utilized to obtain the ring-laser intensity-tuning curve. The experimental ring resonator comprises two parts. One part is the half-resonator of the mode control, which includes two spherical mirrors controlled by a piezoelectric ceramic. The other is the half-resonator of the stimulation. The exposed aperture was sealed by a fused quartz sheet at the angle of the Brewster’s window. The ring cavity was excited utilizing a high-frequency oscillator. This type of excitation requires a low voltage and generates less heat, which is conducive for stabilizing the profile of the intensity-tuning curve. In addition, a Fabry–Pérot cavity was utilized to separate strong and weak modes from the laser output. The optical length of the Fabry–Pérot cavity was 7 cm and FSR (free spectral region) was approximately 1 GHz. This is sufficient to distinguish the frequency intervals of adjacent longitudinal modes. The radius of curvature of the spherical mirrors was 6 m, and the optical reflectivity of the partial reflector was 98%. By Fabry–Pérot cavity and photomultiplier, a clear and stable longitudinal-mode spectrogram can be obtained in real time along with the PZT (Piezoelectric Ceramics) stretching process. The coordinate data of these longitudinal-mode spectrograms were acquired with a digital oscilloscope at high speed. After data processing, the profile of the intensity-tuning curve of the ring laser was obtained.

 figure: Fig. 6.

Fig. 6. Experimental system of ring laser intensity tuning curve.

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The optical length of the experimental ring resonator is 0.47 m. He: Ne = 15:1, Ne20: Ne22= 0.53:0.47, the inflation pressure was 400 Pa. Figure 7 presents a photograph of the experimental resonator filled with gain gas. These filling parameters are beneficial for the symmetry of the intensity tuning curve.

 figure: Fig. 7.

Fig. 7. Half resonator of stimulation is filled with Ne isotope gain medium.

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The scheme for accurately detecting the profile of the intensity-tuning curve is presented in Fig. 8. The voltage applied to the PZT pushed the spherical mirror. Correspondingly, the frequencies of the longitudinal oscillation modes were scanned. The intensity amplitudes of the oscillation modes were modulated with the profile of the intensity tuning curve. Hence, the profile of the intensity tuning curve was obtained by tracking the amplitude of the strong mode and adding the amplitude of the weak mode.

 figure: Fig. 8.

Fig. 8. Profile of the intensity tuning curve is obtained by F–P cavity. (a) Longitudinal modes spectrum of state ①; (b) Longitudinal modes spectrum of state ②; (c) Intensity tuning curve is obtained.

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 figure: Fig. 9.

Fig. 9. Experimental result of intensity tuning curve in dual-longitudinal mode operation ring laser.

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The experimental results of the intensity tuning curve in the dual-longitudinal-mode operating ring laser are presented in Fig. 9. The profile of the intensity-tuning curve is similar to that of a Gaussian curve. The change in the slope on both sides of the curve is caused by the mode saturation effect. In Fig. 9, when the tuning frequency is in the range of 320 to 640 MHz, the profile of the intensity tuning curve exhibits a deformation, which is mainly caused by the change in cavity loss, resulting in the expansion of the PZT. Overall, the experimental results are consistent with the theoretical analysis. The results in Fig. 9 are only qualitative. In the future, based on an accurate calibration of the Fabry–Pérot cavity, a quantitative analysis of the experimental curve of intensity tuning will be our future research direction.

5. Conclusion

The intensity coupling characteristics of the dual-longitudinal-mode ring laser were investigated employing Lamb theory. In particular, the saturation effects between the dual-longitudinal mode and the four frequencies in the same polarization state were analyzed utilizing the Ne dual-isotope in the gain medium. The analysis proved that the saturation effects between the co-propagating waves were weaker than those between the inverse waves split from a single longitudinal mode. The suppression effect of gain saturation on the frequency itself was much greater than that on other frequencies in the ring laser. The saturation effects between the inverse traveling waves reach an extreme value when the four frequencies are symmetric about the gain center of Ne20 or Ne22. Finally, the theoretical intensity curve coincided with the experimental results. This study provides a theoretical basis for realizing a special dual-longitudinal-mode operation and self-biasing laser gyro.

Appendix: General saturation coefficients

In the theory of the multimode laser operation, the expression of the general saturation coefficients θnμρσ are given by

$${\theta _{n\mu \rho \sigma }} = {({{\wp / {2\hbar }}} )^2}\omega {\Re} {[{2{Q_n}{z_i}(\gamma )} ]^{ - 1}}\sum\limits_{t = 1}^4 {({{{{N_{tw}}} / {\bar{N}}}} )} {T_{tw}}$$
$$\bar{N} = ({{1 / L}} )\int_0^L {N({s,t} )ds} $$
$${N_l} = ({{1 / L}} )\int_0^L {N(s,t){e^{{{ - i\mathrm{2\pi }ls} / L}}}} ds$$
where the acceptable values of subscripts n, μ, ρ, σ must observe Eq. (7), a perturbation tree was introduced to simplify the calculation of third-order polarization [26]. Subscript t in Ntw and Ttw denotes the branch count of the perturbation tree. Subscript w is a parameter related to the direction of the oscillating mode, and its definition is listed in Table 1. $\bar{N}$ is the average inverted population difference, Ntw is the excitation integral for the traveling-wave ring laser, defined in Table 2.

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Table 1. Definition of Parameter w

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Table 2. Definition of excitation integral Ntw for dual-longitudinal mode operation ring laser

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where CW and CCW are the modes propagating in the clockwise and counterclockwise directions, respectively. None of these terms contain rapidly varying integrands, and hence vanish.

Where Ttw is the velocity integration term appearing in the third-order polarization expression. The expressions for Ttw are expressed in Eqs. (32)–(34), and the definitions of the complex frequency ${\zeta _{tk}}$ are listed in Table 3.

$${T_{t1}} = [{{1 / {{\zeta_{t2}}}}} ]{{[{Z({{\zeta_{t1}}} )+ Z({{\zeta_{t3}}} )} ]} / {({{\zeta_{t1}} + {\zeta_{t3}}} )}}$$
$${T_{t2}} ={-} [{{1 / {{\zeta_{t2}}}}} ]{{[{Z({{\zeta_{t1}}} )- Z({{\zeta_{t3}}} )} ]} / {({{\zeta_{t1}} - {\zeta_{t3}}} )}}$$
$${T_{t3}} = \frac{1}{2}\frac{1}{{{\zeta _{t1}} - {{{\zeta _{t2}}} / 2}}}\left( {\frac{{Z({{\zeta_{t3}}} )- Z({{\zeta_{t1}}} )}}{{{\zeta_{t3}} - {\zeta_{t1}}}} - \frac{{Z({{\zeta_{t3}}} )- Z\left( {\frac{1}{2}{\zeta_{t2}}} \right)}}{{{\zeta_{t3}} - \frac{1}{2}{\zeta_{t2}}}}} \right)$$

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Table 3. Definition of complex frequencies ${\zeta _{tk}}$ appeared in velocity integration Ttw

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Funding

National Natural Science Foundation of China (62075180, 61605156, 61803299); Natural Science Basic Research Program of Shaanxi Province (2016JQ6073); Shaanxi Province Qin Chuangyuan “scientists + engineers” team construction (2023KXJ-150).

Disclosures

The authors declare no conflict of interest.

Data availability

The data supporting this study are available from the corresponding author upon request.

References

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Data availability

The data supporting this study are available from the corresponding author upon request.

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Figures (9)
Fig. 1.
Fig. 1. Naming rule of four frequencies split from dual-longitudinal mode.

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Fig. 2.
Fig. 2. Intensity tuning curve of each frequency in dual-longitudinal mode operation ring laser.

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Fig. 3.
Fig. 3. Frequency tuning and dual-longitudinal mode state. (a) Frequency I and II are tuned from −640 MHz to 0 MHz, and Frequencies III and IV are shifted from 0 to 640 MHz simultaneously. (b) Old frequency pair goes out, and novel frequency pair starts to oscillate.

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Fig. 4.
Fig. 4. Theoretical intensity tuning curve of dual-longitudinal mode operation ring laser.

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Fig. 5.
Fig. 5. Normalized saturation coefficients and schematic of modes spectrums corresponding to saturation effect. (a) Saturation coefficients for Frequency I. (b) Saturation coefficients for Frequency III. Saturation coefficients for Frequencies II and IV can be obtained by change of subscripts I ↔ II, III ↔ IV;(c), (d), (e) are spectrums of dual-longitudinal mode and dual-isotope frequency distribution; (f), (g), (h) are schematics of burning hole of inverted population corresponding to (c), (d), and (e), respectively.

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Fig. 6.
Fig. 6. Experimental system of ring laser intensity tuning curve.

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Fig. 7.
Fig. 7. Half resonator of stimulation is filled with Ne isotope gain medium.

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Fig. 8.
Fig. 8. Profile of the intensity tuning curve is obtained by F–P cavity. (a) Longitudinal modes spectrum of state ①; (b) Longitudinal modes spectrum of state ②; (c) Intensity tuning curve is obtained.

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Fig. 9.
Fig. 9. Experimental result of intensity tuning curve in dual-longitudinal mode operation ring laser.

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Tables (3)

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Table 1. Definition of Parameter w

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Table 2. Definition of excitation integral Ntw for dual-longitudinal mode operation ring laser

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Table 3. Definition of complex frequencies ζ t k appeared in velocity integration Ttw

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Equations (34)

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E ˙ n + 1 2 ( ν n / Q n ) E n = − 1 2 ( ν n / ε 0 ) Im ( P n )
ν n + ϕ ˙ n = Ω n − 1 2 ( ν n / ε 0 ) E n − 1 Re ( P n )
P n ( 1 ) ( t ) = − ℘   N ¯ ( ℏ K u ) − 1 E n ( t ) Z [ γ + i ( ω − ν n ) ]
Z ( ζ ) = i K π − 1 / 2 ∫ − ∞ ∞ exp ⁡ [ − ( V / u ) 2 ] ( ζ + i k V ) − 1 d V
ζ = γ + i ( ω − ν n )
P n ( 3 ) = 2 ( ε 0 / ν n ) ∑ ∑ ∑ E μ E ρ E σ θ n μ ρ σ e i ψ n μ ρ σ
( a ) n = μ , ρ = σ = m , ( b ) n = σ , μ = ρ = m .
I ˙ n = 2 I n ( α n − ∑ m θ n m I m )
I n = 1 2 ( ℘ E n / ℏ ) 2 ( γ a γ b ) − 1
α n = Z i [ γ + i ( ω − ν n ) ] ν n ℜ [ 2 Q Z i ( γ ) ] − 1 − ω / 2 Q n
θ n m = 2 γ a γ b ( ℏ ℘ ) − 2 Im { θ n n m m + θ n m m n } ( 1 − δ n m )
P = f 1 P 1 ( 1 ) + f 2 P 2 ( 1 ) + f 1 P 1 ( 3 ) + f 2 P 2 ( 3 )
α n = 2 ν ℜ Q { f 1 Z i [ γ + i ( ω − ν n ) ] [ Z i ( γ ) ] − 1 + f 2 Z ~ i [ γ + i ( ω ~ − ν n ) ] [ Z ~ i ( γ ) ] − 1 } − ν / 2 Q ~ n
θ n m = 2 γ a γ b ( ℏ ℘ ) − 2 [ f 1 Im { θ n n m m + θ n m m n } ( 1 − δ n m ) + f 2 Im { θ n n m m + θ n m m n } ( 1 − δ n m ) ]
u = u ~ 22 / 20
I ˙ I = 2 I I ( α I − β I I I − θ I II I II − θ I III I III − θ I IV I IV )
I ˙ II = 2 I II ( α II − θ II I I I − β II I II − θ II III I III − θ III V I IV )
I ˙ III = 2 I III ( α III − θ III I I I − θ III II I II − β III I III − θ III IV I IV )
I ˙ IV = 2 I IV ( α IV − θ IV I I I − θ IV II I II − θ IV III I III − β IV I IV )
α I  =  β I I I  +  θ I II I II  +  θ I III I III  +  θ I IV I IV
α II  =  θ II I I I  +  β II I II  +  θ II III I III  +  θ III V I IV
α III  =  θ III I I I  +  θ III II I II  +  β III I III  +  θ III IV I IV
α IV  =  θ IV I I I  +  θ IV II I II  +  θ IV III I III  +  β IV I IV
α = θ I
I = θ − 1 α
V z = c ( ν n / ν n v 0 v 0 − 1 )
V z I = c ( ν I / ν I v 0 v 0 − 1 )
V z IV = − c ( ν IV / ν IV v 0 v 0 − 1 )
θ n μ ρ σ = ( ℘ / 2 ℏ ) 2 ω ℜ [ 2 Q n z i ( γ ) ] − 1 ∑ t = 1 4 ( N t w / N ¯ ) T t w
N ¯ = ( 1 / L ) ∫ 0 L N ( s , t ) d s
N l = ( 1 / L ) ∫ 0 L N ( s , t ) e − i 2 π l s / L d s
T t 1 = [ 1 / ζ t 2 ] [ Z ( ζ t 1 ) + Z ( ζ t 3 ) ] / ( ζ t 1 + ζ t 3 )
T t 2 = − [ 1 / ζ t 2 ] [ Z ( ζ t 1 ) − Z ( ζ t 3 ) ] / ( ζ t 1 − ζ t 3 )
T t 3 = 1 2 1 ζ t 1 − ζ t 2 / 2 ( Z ( ζ t 3 ) − Z ( ζ t 1 ) ζ t 3 − ζ t 1 − Z ( ζ t 3 ) − Z ( 1 2 ζ t 2 ) ζ t 3 − 1 2 ζ t 2 )
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