The YAG crystal belongs to the cubic crystal system and is generally considered to be optically isotropic^[1]. Therefore, when the rare-earth-doped YAG laser does not use an anisotropic optical element in the cavity, the output laser has no specific polarization orientation. Because of the weak birefringence, which is generally caused by weak stress, these lasers are orthogonal and dual-polarized^[2]. However, since 1987, Nd:YAG^[3,4], Tm:YAG^[5], Er:YAG^[6], and Yb:YAG^[7] have achieved polarization control laser output. Kravtsov et al. studied the influence of the polarization of the pump light in the isotropic cavity on the degree of polarization of the laser output^[8]. By rotating the angle of the laser crystal, Dong et al. realized the self-selective linearly polarized laser output consistent with the crystal axis direction in the Yb:YAG microchip laser and also in the Cr:Nd:YAG and Yb:YAG/Cr4+Q-switched laser^[9–11]. Dong et al. used the anisotropy of the crystal axis direction to realize the linearly polarized laser output of the Yb:YAG microchip^[12]. Otsuka and Chu achieved a polarization output consistent with the pump polarization in Nd:YAG ceramics^[13,14]. In 2014, by fine-tuning the laser cavity, Li realized a linearly polarized laser with a high degree of polarization independent of the crystal orientation and pump polarization in the YAG/Tm:YAG/YAG laser crystal^[5].

The explanation of the linearly polarized radiation behavior directly produced by the isotropic solid-state laser generally uses the analysis of the local symmetry of the dopant ion distribution in the matrix crystal^[12,15-18]. This analysis only yields a slight difference in gain or loss. It is not sufficient for explaining the selective output of the laser polarization mode. In our previous work, the output polarized laser was proved to be a coherent superposition of two eigen polarization modes by the analysis of the polarized spot profile^[19,20]. Although the experimental results are in good agreement with the explanation, the specific frequency difference changes are not measured and analyzed, which is important evidence. In order to fully verify the correctness of the analysis, this paper measured the frequency difference between the polarization modes under different polarization ratios. The route of the frequency locking between two polarization modes was clearly observed, which proved that the polarized laser output of the isotropic Nd:YAG laser is the frequency locking and coherent superposition of two polarization modes.

The coherent combination of two polarized lights means that their frequencies are the same^[21]. Brunel has proved that in a near isotropy cavity, by adjusting the relative amount and direction of weak phase and loss anisotropies, the frequency difference between two eigen polarization modes will be reduced^[22]. When the angle between the axes of these two anisotropies is 45° and the loss anisotropy is larger than the phase anisotropy, the frequency difference will fall into the locked region, which is called frequency degeneration^[23–26]. It provides the possibility to coherently combine the two eigenmodes into the linear polarization.

The Jones matrix of a round trip^[20,27] corresponding to the above mentioned cavity is given by M(Δφ,ΔT)=(e−iΔφ001)(1−ΔTΔT1)(100eiΔφ),(1)where the Δφ is the round-trip phase difference caused by weak birefringence, and ΔT is the loss anisotropy. The angle between the axes of the phase and the loss anisotropies is 45°.

The eigenvalues of the matrix are calculated to be λ±=cosΔφ±δ,(2)where δ=ΔT2−(sinΔφ)2.

The corresponding eigenstates of polarization are E±=11+ΔT2[1(±δ−isinΔφ)/ΔT].(3)

When δ<0, the eigenvalues are conjugate complex numbers. Then, the frequency difference of the two eigenstates must be Δν=2πarccos(cosΔφ1−ΔT2)cL,(4)where c is velocity of light, and L is the optical length of the cavity.

When δ≥0, the eigenvalues are real numbers. In this case, the frequency of the eigenstates is the same.

The relationship between Δν and ΔT in the case of fixed Δφ is calculated and shown in Fig. 1.

As shown in Fig. 1, when the phase anisotropy Δφ is fixed, the frequency difference begins to decrease as the loss anisotropy ΔT increases. When the loss anisotropy ΔT is greater than or equal to the phase anisotropy Δφ, the frequency difference is zero. The continued increase of loss anisotropy ΔT will not change the state of frequency degeneracy. The frequency difference with the phase anisotropy and loss anisotropy is important evidence for verifying the frequency locking of the polarized eigenmodes.

The experimental device is shown in Fig. 2. A fiber-coupled semiconductor laser diode (LD) with a center wavelength of 808 nm produced by LIMO is used as a pumping source. The numerical aperture of the fiber is 0.22, and the fiber core diameter is 200 µm. The pump light is collimated and incident on the end face of the gain medium by a pair of plano-convex lenses, and the coupling ratio of the shaping system is 1:2. The gain medium used in the experiment is a Nd:YAG crystal with the Nd3+ doping concentration of 1% and the size of 3mm×3mm×10mm. One coated surface of the Nd:YAG crystal S1 has high transmittance to the pump light at 808 nm as well as high reflectivity to 1064 nm (R>99.93%), and the other coated surface S2 has a high transmittance to 808 nm and 1064 nm. In order to achieve good crystal temperature control, the crystal is wrapped with indium foil and placed on a copper block and cooled by a thermoelectric cooler (TEC) module. The temperature is controlled at about 286.5 K. The output coupling mirror M1 is coated with a transmittance of 10% at 1064 nm wavelength. It is placed on a movable platform to adjust the cavity length. The length of the laser cavity is controlled between 20 and 150 mm. The pump light is filtered from the output laser through the M2 filter, which is highly reflective to 808 nm and highly transparent to the 1064 nm wavelength. The Glan prism is used as a polarization analyzer (transmission range 350–2300 nm). The laser power with different polarization characteristics is measured by the power meter. The beam profile is measured by a charge coupled device (CCD). The power spectrum of the output power is measured by a spectrum analyzer, which consists of a silicon-based photodetector (Thorlabs DET025A/M) and spectrum analyzers (Keysight N9000A).

The free-running output power without the Glan prism and the power corresponding to the angle of the Glan prism are shown in Fig. 3. The maximum output power is 0.323 W at the pump power of 1.37 W. By rotating the angle of the Glan prism, the polarization of the output power is found to be near unpolarized.

The frequency difference variations with the angle of the analyzer in the case of the free-running state are shown in Fig. 4. As shown in Fig. 4(a), in the power spectrum, there exist some signal peaks whose signal intensity varies under the analyzer of 10°, 20°, and 30° in the case of the free-running state. As shown in Fig. 4(b), the intensity varies from zero to maximum with the detecting angle range of 45°. This can be explained by the beating of the two eigen polarization modes. There are two eigen polarization modes in the cavity. In the case of the free-running state, the frequency difference between them will not change with the detecting angle, but the beating frequency intensity will change. When the analyzer direction coincides with a certain eigen polarization mode, since the component of the other eigen polarization mode is almost nothing, the beating intensity of these two modes should be near zero. When the angle between the analyzer direction and the eigen polarization mode is 45°, the components of the two eigen polarization modes in the analyzer direction are equal, and then the intensity of the beating signal should be the largest. Based on the above analysis, the direction of the eigen polarization mode can be measured in the case of the free-running state. The change of the frequency beating with the analyzer angle is tested at different positions of the Nd:YAG. At different positions, the direction of the eigen polarization mode is different. We selected a position whose direction of the eigen polarization mode is 45° away from the horizontal direction to further carry out the experiment.

By finely adjusting the cavity mirrors horizontally, a near linearly polarized laser can be obtained, which is shown in Fig. 5. The maximum output power is 0.312 W at a pump power of 1.37 W. By rotating the angle of the Glan prism, the polarization of the output power is found to be near linear. Through the maximum power P₁ and the minimum power P₀, the polarization extinction ratio (PER) of the polarized light can be defined as PER = 10 × lg(P₁/P₀) (dB) and is usually expressed in decibels. As high as 31 dB of the PER can be achieved in the experiment.

The beating frequency of the two eigen modes was measured during the adjusting. The variation was clearly observed. As shown in Fig. 6, the frequency difference decreases and suddenly disappears during the transition from the free-running state to the frequency locking state. As shown in Fig. 7, the PERs corresponding to the frequency differences are recorded in detail during the experiment. The experimental results show that as the PER increases, the frequency difference decreases and suddenly disappears, which indicates frequency locking.

The relationship between the PER and the beating frequency is shown in Fig. 8. The frequency difference in the free-running state is 10.2 MHz. By adjusting the output coupler (OC), the frequency difference begins to decrease, from 10.2 MHz to 9.6 MHz, and the PER is 4.1 dB. Continuing to adjust the OC, the beating frequency is reduced to 9.3 MHz, and the PER becomes 8.9 dB. When the PER reaches 20.1 dB, the beating frequency is reduced to 9.0 MHz. After that, finely tuning the OC, the beating frequency suddenly disappears. The extinction ratio at this time reaches 31 dB. It can be seen that with the fine adjustment of the OC, the frequency difference between the two eigen polarization modes begins to decrease in a small range, and the PER of the output laser rises sharply. When the frequency difference is reduced to about 9 MHz, the tiny OC adjustment can realize the complete disappearance of the frequency difference, and the PER reaches a maximum of 31 dB.

We further studied the influences of different OC transmittances on the polarization frequency using the OC transmittances of 5%, 10%, 15%, 20%, and 30%, respectively. The optical axis is consistent with the last time after replacing the output mirror. As shown in Fig. 9(a), with the change of the OC transmittance, experiments show that the realization of the linearly polarized laser does not depend on the coupling rate of the OC. In the next step, the influence of different cavity lengths on the polarization state of the output laser was studied. The adjustment range of the cavity length L is 20–150 mm, and the step size is 10 mm. The optical axis is consistent with the last time after changing the cavity length. As shown in Fig. 9(b), with the adjustment of the cavity length, the experiment shows that the realization of the linearly polarized laser does not depend on the cavity length.

The beam profile of spontaneous polarization output in the free-running state and the maximum PER is analyzed. After the output laser passes through the analyzer, the beam profile under different rotation angles of the analyzer is recorded by the CCD. The prism is gradually rotated to different directions relative to the polarization direction, and the variations of the profile are observed and shown in Fig. 10.

In Case 1, it is a laser with unpolarized output, the obtained spot has nothing to do with the direction of the Glan prism, and it is a circle. Case 2 is the laser output of the maximum PER. When the Glan prism is exactly in the extinction direction of the polarized laser, four spots can be observed in the figure. When the prism is slightly deviated from the extinction position, there is only one oblique irregular ellipse at the left- and right-skewed position. Moreover, the major axes of a pair of elliptical spots approximately intersect.

As shown in Fig. 11, using the same calculation of reference^[20], by changing the ellipticity parameters and position parameters, various dislocations of the two eigenmodes are simulated. The experimental results are well consistent with the theoretical simulation.

The frequency difference of the eigen polarization modes was determined by the power spectrum analysis during the generation of a linearly polarized output laser with an isotropic Nd:YAG crystal. The frequency difference of the eigenmodes, which is caused by the weak anisotropy inside the crystal, was clearly observed to be reduced with the tiny adjustment of the cavity. The locking route can be clearly shown on the power spectrum. The frequency locking of the eigen polarization modes was experimentally proved to be the physical mechanism of the polarized output with isotropic laser media.