1 Introduction

The high-quality linearly polarized laser source has been widely used in nonlinear optics [1], differential interference imaging [2], holographic microscopy [3] and so on. The linear polarization of laser output of isotropic laser crystal requires the addition of optical elements [4], but this will increase the resonator loss. Despite this, people are still actively exploring the method of outputting polarized laser in isotropic crystal lasers. For example, researchers have successfully realized the emission of various linearly polarized states in YAG/Tm: YAG/YAG crystals [5]. Tang [6] and colleagues, through the investigation of the singularity of polarized beam profiles, explored the mechanism by which isotropic laser directly produce linearly polarized laser output: coherent synthesis of linearly polarized laser output from intracavity polarization eigenmodes. Huang [7] and Wang [8] further explained the mechanism of polarization eigenmode degeneracy by measuring frequency differences in different polarization states. Typically, the frequency difference between two polarization eigenmodes in the resonator results from thermal or stress-induced effects. In the report [9], thermal effects are noted to impact both losses and the output of polarized eigenmodes. Changing the pump coupling ratios can alter the mode-matching coefficients (MMCs), leading to variations in heat. Additional investigation is required to determine the impact of these changes on the polarization mode locking of the Yb:YAG linearly polarized laser.

Trivalent ytterbium ions in solid-state media have high quantum efficiency, long fluorescence lifetime, and wide absorption and emission range. Yb: YAG crystals is widely used in the generation of ultra-short pulses [10] and have recently been shown to provide novel polarization-multiplexed sources for dual-comb spectroscopy [11, 12]. For femtosecond pulse applications, using different cavity parameters can have an impact on the pulse characteristics of Yb: YAG crystals [13]. For laser applications, Yb: YAG crystals show different wavelength and polarization characteristics in dual-wavelength lasers with different pumping power [14]. Hence, a more crucial focus lies in the in-depth examination of the linear polarization mechanism in the direct output of Yb:YAG lasers under varying pump conditions.

In this paper, Linearly polarized laser was emitted in the Yb:YAG laser at three different pump coupling ratios. The linearly polarized output was explained by the coherent superposition of two orthogonal polarization eigenmodes, utilizing the changes in the beam profiles and the frequency difference of the intrinsic modes. The experimental results showed that the mode matching coefficient affected the power loss within a certain range, but the resulting thermal change did not affect the polarization locking mechanism. Furthermore, M2 was measured for various polarization states, revealing that linearly polarized output states exhibited superior performance compared to partially polarized states. Simultaneously, spectral measurements revealed that the output wavelengths for different MMCs and polarization states were all around 1050 nm.

2 Experimental

The experimental setup is shown in Fig. 1, where a simple Fabry–Perot cavity of about 14 mm is constructed by a planar pump mirror (PM) and a planar output coupler (OC). The coating parameters of the planar pump lens are high transmittance to 940 nm pump beam and high reflectance to 1030 nm and 1050 nm. The planar output coupler has coating parameters of 5.6 and 4.2% transmittance in the 1030 nm and 1050 nm bands, respectively. The laser crystal is enveloped in indium foil and positioned on a copper heat sink cooled with water at a temperature of 287.9 K. The pump source is a 940 nm fiber-coupled laser diode pump with a fiber core diameter of 200 µm and a numerical aperture of 0.22. The pump beam is focused using a switchable telescope lens system (TLS) with imaging ratios of 2:1, 1:1, and 1:2, corresponding to waist radii of 50 µm, 100 µm, and 200 µm. In this experiment, a power meter (Gentec-EO, Inc.) and a Glan-Laser prism (transmittance range 350–2300 nm) were used as analyzers to detect the polarization state of the output laser, and the beam profiles are recorded with a CCD camera (DataRay, Inc.USA). After passing through the beam splitter prism (BS) and HR mirror (reflectivity range 800–1100 nm) a radio frequency (RF) spectrum analyzer (SIGLENT, Inc. SSA5083A, a bandwidth of 9 kHz to 13.6 GHz) and a spectrometer (Ocean Insight, Inc.HR4000, with a resolution of 0.47 nm) to monitor the changes in beat frequency and laser wavelength corresponding to the changes in the polarization state of the output laser.

Fig.1
figure 1

Schematic diagram of the Yb:YAG laser

Full size image

3 Experimental results and discussion

The polarization extinction ratio (PER) is used to characterize the degree of polarization of linearly polarized laser, which is expressed by the ratio of optical power of two beams in orthogonal polarization directions, usually expressed in decibels(dB). With the above setup, after collimation of the He–Ne laser, the replacement of the different TLSs was completed without moving the resonator. Pump beam with a power of 7 W was injected at three different TLSs, and the normalized power output of Glan-Laser prism at different angles (defined as 0° in the horizontal direction and 90° in the vertical direction) were recorded. The polarization characteristics of the output beam when the laser resonator is at maximum output power (free running), as shown in Fig. 2a. By fine-tuning the resonator, two stable linearly polarized states with polarization ratios greater than 26 dB were measured with all three TLSs. The polarization directions of the two linearly polarized laser were approximately perpendicular and nearly the same with the three pump coupling ratios, as shown in Fig. 2b.

Fig.2
figure 2

Variation of normalized transmission power with detector angle for three TLSs a free running operation and b misaligning operation

Full size image

To further explore the output polarization mechanism of Yb:YAG crystal at different TLSs, the slope efficiency was measured at various TLSs. The slope efficiency of partially polarized and two orthogonal polarized laser states at three different TLSs, as shown in Fig. 3a. In the figure, it can be observed that of the same TLSs, the slope efficiency of partially polarized and two orthogonal polarized laser states remains relatively unchanged. However, the slope efficiency of these three states all change significantly with different TLSs. At the TLS of 1:2, the slope efficiency is the highest at 40%; followed by 1:1 at 35%; and the lowest is at 2:1, which is 15%. To explain this phenomenon, a coordinate system was established on the front face of the crystal (xy-plane) and along the axial direction perpendicular to it (z-axis). Theoretically, the spatial overlapping integral \({\eta }_{0}\) was introduced with different coupling conditions of pump beam and the oscillating beam. Its numerical value also denotes the Mode Matching Efficiency (MMC) between the pump beam and the TEM00 fundamental mode eam [15, 16].

$$\eta_{0} = \frac{{\left( {\iiint\limits_{{}} {{\text{s}}_{l} \left( {x,y,z} \right)r_{p} \left( {x,y,z} \right)dv}} \right)^{2} }}{{\iiint\limits_{{}} {{\text{s}}_{l}^{2} \left( {x,y,z} \right)r_{p} \left( {x,y,z} \right)dv}}}$$
(1)
Fig.3
figure 3

a Slope efficiencies of partially polarized states, two orthogonally polarized states at three TLSs. b Distribution of thermal fields in the Yb:YAG crystal

Full size image

In the equation, \({s}_{l}\left(x,y,z\right)\) represents the normalized resonator mode intensity distribution, and \({r}_{p}\left(x,y,z\right)\) corresponds to the normalized pump distribution within the gain medium.

$$r_{p} (x,y,z) = \frac{{\alpha e^{ - \alpha z} }}{{\pi \times \omega_{p}^{2} (z) \times \left[ {1 - e^{ - \alpha l} } \right]}}\Theta (\omega_{p}^{2} (z) - x^{2} - y^{2} )$$
(2)
$$s_{l} (x,y,z) = \frac{2}{{\pi \times \omega_{0} \times 5}}e^{{\frac{{ - 2(x^{2} + y^{2} )}}{{\omega_{0} }}}}$$
(3)
$$\omega_{p} (z) = \omega_{p0} + \frac{{M^{2} \lambda }}{\pi }\left| {z - } \right.\left. {z_{0} } \right|$$
(4)

where \(\alpha\) is the absorption coefficient of the crystal at the pumping wavelength,\(l\) is the crystal length. \(\Theta\) is the Heaviside step function, \({\omega }_{0}\) is the radius of the fundamental mode girdle after the ABCD matrix calculation, which is 150 \(\mu m\). \({\omega }_{p}\left(z\right)\) is the pump beam waist in the crystal. The front face of the crystal is taken as the \({z}_{0}\) position.M2 is the beam quality factor of the pump beam. \(\lambda\) is the pump wavelength 940 nm. Different MMCs will affect the output power of the laser [17]. After theoretical calculations, it is learned that the MMCs of the TLSs at 2: 1, 1: 1 and 1: 2 were 0.12, 0.31 and 0.53, respectively. Increasing the MMCs is related to higher slope efficiency, which was consistent with experimental findings. The linear polarization output was realized by fine tuning the OC of the resonator at each TLSs, which has a little influence on power and slope efficiency. The change in the waist radius of the pump beam was directly influenced by different TLSs, leading to variations in endogenous heat within the crystal. Using finite element analysis, we conducted thermal simulations of laser crystals at various TLSs, building upon previous study [18, 19]. In the thermal simulation diagrams of Fig. 3b, changes in crystal heat were clearly observed at different TLS. According to the thermal-induced birefringence effect, variations in crystal heat could influence polarization. However, linearly polarized output laser was still achieved with different TLSs.

According to the previous study [7, 18], from the front face of the crystal, we established a Jones matrix mathematical model M for analyzing the change of frequency difference (beat frequency information) between the polarization eigenmodes in the resonator and resolving the beam profiles.

$${\rm M}(\Delta \psi ,\Delta t,\theta ) = \left( {\begin{array}{*{20}c} {e^{ - i\Delta \psi } } & 0 \\ 0 & 1 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {1 + \Delta t} & 0 \\ 0 & {1 - \Delta t} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta } \\ {\sin \theta } & {\cos \theta } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & {e^{i\Delta \psi } } \\ \end{array} } \right)$$
(5)

where \(\Delta \varphi\) denotes the phase delay between the x and y directions caused by the residual anisotropy of the yttrium–aluminum-garnet crystals (ionic doping inhomogeneity, residual stresses, thermal stresses, etc.), \(\Delta t\) represents the intracavity loss anisotropy following the resonator misalignment., and \(\theta\) denotes the angle between the loss axis and the phase anisotropy axis. Brunel et al. [19, 20]. demonstrated that by adjusting the relative size and orientation of the phase and anisotropic losses, it is possible to achieve eigenmode frequency locking when the frequency difference between eigenmodes decreases to the locking region. Under the condition of a constant angle \(\theta\), frequency locking and linearly polarized laser output could be achieved by fine-tuning the OC to change \(\Delta t\) when \(\Delta t\) > \(\Delta \varphi\). The experimental results indicate that with different TLSs, the high degree of linearly polarized output could still be achieved. This suggested that frequency locking can be achieved at each TLSs when the angle \(\theta\) remains constant.

To further elucidate that the linear polarization output at different TLSs was still a result of frequency locking, the frequency difference between eigenmodes with various TLSs were measured using an RF spectrum analyzer. The changes in beat signal corresponding to the alteration of polarization were recorded. To observe clear beat signals, the resolution bandwidth (RBW) and video bandwidth (VBW) of the RF spectrum analyzer were set to 300 kHz and 30 kHz. When the laser free running with different TLSs, beat signals was observed on the RF spectrum analyzer. The beat signal detection method based on it: when the optical axis of the analyzer was aligned with its polarization direction, one of the orthogonal polarization eigenmodes could completely pass through the Glan-Laser prism and the other eigenmode was completely blocked. This phenomenon caused the beat frequency signal to almost disappear. When the optical axis of the Glan-Laser prism was oriented at a 45° angle to the polarization directions of the two eigenmodes, the components of the two eigenmodes in this direction were equal, resulting in the maximum intensity of the beat frequency signal. The beat signal intensity was recorded at 10° intervals. The weakest point was defined as 0°, reaching the maximum value at a 45° rotation and minimizing at 90°, with the pattern repeating until 360°. In the experiment, this signal was verified to be a polarization beat frequency value by rotating the angle of the Glan-Laser prism with different TLSs. The specific experimental results were shown in Fig. 4. Confirming the nature of this signal as a beat frequency signal and the laser oscillate in orthogonally polarized output when free running. In addition, the OC was fine-tuned at each TLSs, thus changing the output polarization state of the laser, as shown in Fig. 5. With the same direction of rotation, the OC was turned by the same angle for all three TLSs. Three selected polarization states and the beat frequency were determined. The observation revealed that as polarization transitioned from a partially polarized state to a linearly polarized state, the beat frequency gradually decreased until it vanished. After comparing, it was observed that the laser had different beat frequencies in three polarization states of the different TLSs. The magnitude of these beat frequency was related to the thermal-optic coefficient. Varied thermal energy in the three TLSs resulted in distinct changes in beat frequency.

Fig.4
figure 4

The intensity variation of the beat frequency of the rotating Glan-Laser prism is shown during the free running of the laser with TLSs of 2: 1, 1: 1 and 1: 2

Full size image
Fig.5
figure 5

a Variation of beat frequency for three different TLSs. b, c, d, Correspondence between laser polarization and beat frequency with three different TLS

Full size image

At the same time, when the state of linearly polarized output was achieved by the laser, the special change in beam profile was observed. As shown in Fig. 6a, using a CCD infrared camera to detect the beam profiles of different TLSs, beam profiles similar to Gaussian distributions were observed. After introducing the Glan-Laser prism and rotating it to the extinction position, oriented perpendicular to the polarization direction, two profiles were observed. In Fig. 6a, when the Glan-Laser prism was rotated + 5° or − 5° from the extinction position. The beam profiles had transformed into a singularly skewed elliptical profiles with either a leftward or rightward orientation. Additionally, the long axis direction was nearly vertical. In partially polarized states, the rotation of the Glan-Laser prism did not alter the beam profiles. Theoretically, the eigenvector calculated by Eq. (5) were used to describe the optical vector corresponding to polarization eigenmodes. The intensity of the two eigenvector fields correspond to a Gaussian-like distribution. Because of the instability caused by thermal effects, there were slight differences between the two polarization eigenmodes inside the cavity. The simulation involved the partial overlap of the laser from these two eigenmodes for computation. After passing through the Glan-Laser prism, linearly polarized laser exhibited a split in the beam profiles at the extinction position. A slight rotation of the Glan-Laser prism in two opposite directions resulted in a set of orthogonally oriented elliptical beam profiles. However, there was no change of partially polarized laser. Through the above analysis of the spot, it could be observed that the nature of the linearly polarized output achieved in the Yb:YAG laser was different from the linearly polarized output achieved by polarization eigenmodes selection. For laser linearly polarized laser through polarization eigenmodes selection, the beam profiles exhibited only differences in brightness as the Glan-Laser prism rotated at different angles. Near the extinction position, there were no changes in the beam profiles as described above. At three different TLSs, the linearly polarized extinction beam profiles remained roughly the same. After rotating the Glan-Laser prism, a set of orthogonally oriented elliptical beam profiles appeared. The thermal changes resulting from the replacement of TLSs had a minimal impact on the overlap of polarization eigenmodes. In terms of beam profiles variations, it could be observed that the linearly polarized laser with the three different TLSs was a result of the coherent superposition of the two eigenmodes. The thermal changes did not have a significant impact on this coherent superposition (Fig. 7).

Fig.6
figure 6

a Experimental results of profiles analyzing of polarized beams near the extinction position and theoretical image. b Profiles after passing through a linear polarizer oriented in different angles of − 5°, 0° and 5° with free running regime as reference

Full size image
Fig.7
figure 7

The wavelengths of laser output in partially polarized and linearly polarized states with three TLSs

Full size image

In order to better elucidate that the beam profile peculiarities were not caused by the generation of higher-order transverse modes with different TLS. To explore laser performance in different polarization states, measurements of M2 were conducted with various conditions. Both partially polarized and linearly polarized lasers exhibited M2 less than 1.5 with different TLSs. However, the M2 of linearly polarized lasers surpassed that of partially polarized states, attributed to the coherent superposition of two eigenmodes within the resonator. The specific M2 were listed in Table 1.

Table 1 M2 of polarized lasers and partially polarized state lasers at different TLS
Full size table

In the experiment, due to the broad emission spectrum of Yb3+, the output wavelengths of the OC around 1030 and 1050 nm were very close. When fine-tuning the resonator with various TLSs, the alterations in transmission were minimal. The spectra of linearly polarized and partially polarized states with different TLSs were measured, and it was observed that the spectral center remained around 1049 nm. This indicated that the replacement of TLS and fine-tuning of OC did not cause any influence on the output wavelength. The experimental results were shown in Fig. 7

4 Conclusion

In this paper, we achieved a linearly polarized laser output with an extinction ratio greater than 26 dB in a Yb:YAG laser by varying the losses at different TLSs. It's worth noting that the changes in the output beating frequency and the profiles of the linearly polarized laser confirmed that the generation of linearly polarized laser results from the coherent combination of two orthogonally polarized eigenmodes. The changes in crystal heat do not affect the linearly polarized output. Due to fine-tuning of the OC to control losses, the free-running output and the linearly polarized laser output exhibit nearly identical slope efficiency. Measurement of beat frequency values during free-running confirms that at different MMCs, the polarization eigenmodes frequency locking can be realized when the loss anisotropy is greater than the phase anisotropy in the resonator. The analysis of the beam profiles of the linearly polarized output elucidated that with different TLSs, the linearly polarized output is coherently generated by two orthogonally polarized eigenmodes. Simultaneously, we measured the M2 values with different polarization states, all yielding values less than 1.5. The M2 for linearly polarized laser should be superior to that of partially polarized states. With variations in TLSs and polarization states, the output wavelength remains approximately the same. In a word, at different TLSs, different MMCs will affect the output power. However, it did not affect the polarization output of the polarization mode frequency-locked linear polarization. The heat changes are not intense enough to disrupt polarization locking's degeneracy conditions.