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Abstract
The behavior of directly linearly polarized emission was investigated in an isotropic Nd:YAG laser with a weakly anisotropic Fabry-Perot cavity. In order to explore the polarization mechanism, the intracavity eigenmodes were analyzed theoretically. A theoretical model was developed and shown to be suitable for describing the orientation-dependent beam profiles of the polarizer-specific features observed in experiments. This model indicates that the linear polarization emission results from the coherently combined state of intracavity eigenmodes as opposed to an intrinsic polarization state. The coherent combination is attributed to the frequency degeneracy and spontaneous phase locking of polarized eigenmodes. For the first time, the polarization coupling mechanism based on the coherent combination of eigenmodes in an isotropic solid-state laser was demonstrated. Moreover, this study also contributes an effective method to judge different types of linear polarization states.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Among the crystals used in laser production, ion-doped yttrium aluminum garnet (YAG) offers particularly attractive thermo-mechanical properties [1], allowing high-power operation with reduced probability of fracture. However, optical isotropy generally results in unpolarized laser oscillations in free-running YAG lasers [2]. By contrast, anisotropic laser crystals, e.g. vanadate, are capable of generating linearly polarized eigenmodes owing to polarization-dependent absorption and stimulation cross sections [3]. Conventionally, the polarization output in isotropic crystalline lasers is conducted by intracavity polarization-selected elements [4], and is associated with increased intracavity losses. Therefore, the reduced properties of lasers possessing a directly polarized output seriously limits their application in systems requiring nonlinear frequency conversion [5], polarization-dependent amplification [6], and interferometers [7].
Nevertheless, many studies have reported obtaining a linearly polarized emission in isotropic crystalline lasers via several carefully designed approaches. For example, the influence of pump polarization on laser output polarization in a polarization-isotropic resonator has been investigated [8]. In addition, across two separate studies Dong and colleagues realized the self-selective polarized laser oscillations aligned to the crystal axis direction in a Yb:YAG microchip laser [9,10]. Linear self-polarized outputs were also observed in Cr,Nd:YAG and Yb:YAG/Cr4+:YAG passively switched lasers [11,12]. Furthermore, crystalline orientation and polarized-pump-independent linearly polarized lasing with a high degree of polarization has been achieved simply by realigning the resonator [2]. Though the behaviors of directly generated linearly polarized radiation from isotropic crystalline lasers were observed in many cases, several characteristics require further elucidation. For instance, the generating mechanism of spontaneous polarization in isotropic crystalline lasers and the extreme sensitivity of the degree and direction of polarization relative to both the laser resonator and the incident pump [13–15].
This paper discusses an experimental and theoretical investigation of the mechanism of linear polarization output in an isotropic solid-state Nd:YAG laser with a Fabry-Perot cavity. A model describing the specific features of the orientation-dependent profile of the polarizer observed in the experiments is presented. We demonstrate that the spontaneous linearly polarized emission results from a combined state of intracavity eigenmodes; the two eigenmodes are linearly polarized and orthogonal to each other.
2. Experiment
The experimental configuration of the laser system is shown in Fig. 1. In order to decrease the interference caused by the polarized pump light [16], a fiber-coupled 808 nm laser diode without obvious polarization characteristics was used as the pump source (NA=0.22, 200 µm core diameter). The pump beam was collimated by a telescopic lens system (TLS) with an imaging ratio of 1:2. A simple Fabry-Perot cavity was used to manipulate the directly linearly polarized output, consisting of a $2 \times 6 \times 8$ mm3 0.9 at. % [111]-Nd:YAG crystal with antireflection (AR) and high reflection (HR) coatings on the front facet at 808 and 1064 nm, respectively, and a flat output coupler (OC) with 15% transmittance. The laser crystal was wrapped with indium foil and mounted on a copper heat sink with a thermo-electric cooler module for efficient heat dissipation. The laser resonator length was set to about 30-40 mm. The polarization states of the output laser were analyzed by a Glan–laser prism (Ultra Photonics, Inc.) in combination with a power meter (Thorlabs, Inc.). Meanwhile, beam profiles were monitored by a CCD camera (DataRay Inc.), after passing through two optical wedges (OW) and a focusing lens (F). The OWs and F were used to reduce light intensity and narrow profiles, respectively. Meanwhile, an alternative optical path was used to monitor the detailed variation of the polarized beam. Among them, a polarizing beam splitter (PBS) was contributed to split the laser beam into a pair of orthogonal polarized components; the respective powers of the two components were monitored by power meters.
Fig. 1. Schematic diagram of the operation and measurement of the linear polarization regime for the continuous-wave Nd:YAG laser.
3. Results and discussion
Using the resonator parameters listed above, we evaluated the degree of polarization of the laser primarily for reference. The degree of linear polarization was measured by recording the orientation-dependent power transmission of the Glan-laser prism. It was quantified via the polarization extinction ratio (PER), defined as the ratio of optical powers in the two polarization directions, often stated in decibels. Under the free running regime, no obvious polarization characteristics were recorded, as shown in Fig. 2(a). By refining the resonator alignment, a pair of stable linear polarization states with a PER of 25 dB were obtained, as shown in Fig. 2(b); these two polarized lasers were oriented in the horizontal and vertical directions, respectively (we defined the horizontal as 0° and vertical as 90°).
Fig. 2. Normalized transmission power as a function of polarizer angle for the (a) free running and (b) misaligning operation.
Under the free running operation and misaligning operation (vertical polarization) operation, a PBS was used to split the laser beam into two orthogonal polarized components. Then, the power curves of the vertical and horizontal components were monitored. During the free running regime, the powers of the two polarized components increased linearly, as illustrated in Fig. 3(a). Conversely, under misaligning operation, more details of the correlated variations of the two components were manifested, as displayed in Fig. 3(b). After the laser threshold, the power of the vertical component rose rapidly with increasing incident pump power, while the horizontal component remained almost invariant, until the PER reached a maximum value 25 dB (A point, correspond to about 3W incident pump power). After reaching A point, by contrast, the vertical component remained invariant whereas the horizontal one continued increasing. The overall performance of polarization variation is as follows: the PER of output laser was gradually increasing after the threshold, peaking at about 25 dB when the pump power is around 3W, then fell off dramatically to unpolarized state. That revealed that the linearly polarized emission with high PER is only obtained within a certain pump power range corresponding to a specific resonator alignment.
Fig. 3. Power curves of two orthogonal polarization components passing through a PBS under (a) free running operation, (b) misaligning operation.
Next, we analyzed the beam profiles of the spontaneous polarization output corresponding to the maximum PER (A point) and observed the orientation-dependent beam profiles. After passing through a Glan–laser prism, the laser beams were monitored by a CCD camera. Through incremental rotation of the prism to different orientations, beam profile variations were captured. Two kinds of beam profile changes were observed in the vicinity of the extinction position, and the two types of situations may be alternative with a certain of randomness in same experiment: in situation 1, when the Glan–laser prism was just at the extinction direction of the polarized laser, by increasing the gain of the CCD (8 dB), four discrete spots could be observed, as shown in Fig. 4(e). Slight deviation of the prism from the extinction position led to a single inclined elliptical spot in the position of left or right deviation, as shown in Figs. 4(d) and 4(f), respectively. Moreover, the major axes of these two elliptical spots were roughly perpendicular to each other. In situation 2, when the Glan–laser prism was just at the extinction direction of the polarized laser, by increasing the gain of CCD (8 dB), two discrete spots could be observed, as shown in Fig. 4(h). Slight deviation of the prism from the extinction position resulted in a single inclined elliptical spot in the position of left or right deviation, as shown in Figs. 4(g) and 4(i), respectively. Similarly, major axes of the two elliptical spots were also perpendicular.
Fig. 4. (a)-(c) Profiles after passing through a linear polarizer oriented in different angles of -5°, 0° and 5° under free running regime as reference. (d)-(i) Experimental results of profiles analyzing of polarized beams near the extinction position in (d)-(f) situation 1, and (g)-(i) situation 2. α represented the azimuth angle of polarizer. The extinction position of polarized beam was defined as 0°. Arrows are used to indicate the orientations of the polarizer.
These phenomena arising from the orientation dependency of the beam profiles of the polarizer are unusual. Supposing that the linear polarization output above is an eigenstate, then the beam profiles should not demonstrate orientation dependency linked to the rotation of the polarizer; only a uniform variation in the intensity of the beam profiles should occur. Therefore, the images in Fig. 4 imply that we may not be observing eigen-polarization radiation of the type associated with anisotropic crystalline lasers (e.g. a-cut Nd:YVO4). For isotropic crystalline lasers, it has been demonstrated that they generally sustain the oscillation of dual-polarization eigenstates with an eigenfrequency difference, supported by the low value of the Lamb mode-coupling constant [17,18]. Then, the incoherent combination of these eigenstates eventually results in an unpolarized output [19–22], as depicted in Fig. 5(a). If there was a coherent combination of two orthogonal polarization eigenmodes, under a certain phase difference (see Fig. 5(b)), small drifts in the intracavity isotropy may lead to significant variations of the polarization states. In addition, when the spatial position and intensity distribution of the two eigenmodes are identical, signifying that they overlap perfectly, there would be no variation in the beam profiles due to the process of rotating the polarizer. Inevitably, however, there are tiny differences between the two intracavity eigenmodes caused by natural factors. In other words, two eigenmodes in a laser cavity could not overlap perfectly in reality. Nonperfect overlapping leads to the dislocation of the two optical vector fields. Due to constructive and destructive interference at different places, an unusual performance of the profiles analyzing the polarized beam was inferred.
Fig. 5. The output polarization dependence on the of intracavity eigenmodes in isotropic crystalline lasers, under (a) incoherent combination of intracavity eigenmodes, (b) coherent combination of intracavity eigenmodes.
The coherent combination of two polarized beams signifies that they are synchronous with respect to frequency [23,24]. By adjusting the relative amounts and orientations of the phase and loss anisotropies, Brunel and co-workers proved theoretically and experimentally that eigenfrequency degeneracy could be realized as soon as the eigenfrequency difference decreases to the extent that it falls within the locking region [25–27]. According to the principle describing the coherent combination of polarized beams [28], a pair of orthogonally polarized components with the same frequency could be combined to produce various polarized states for different phase differences, including linear polarization. This physical mechanism could provide a possible route to achieve the coherent combination of orthogonally polarized intracavity eigenmodes.
To this end, we intended to build a mathematical model to simulate the profiles for analyzing polarized beams, focusing on profile behaviors for processes in which the two intracavity eigenmodes are unable to overlap perfectly. Referring to Brunel et al. [29], we setup a model based on two main parameters: phase anisotropy $\Delta \mathrm{\varphi }$ and loss anisotropy $\Delta t$. $\Delta \mathrm{\varphi }$ represents the phase retardance between the x- and y-directions caused by the residual anisotropy (inherent or thermally caused) of the laser crystal, and $\Delta t$ represents the intracavity loss anisotropy following the resonator misalignment. Then a round-trip Jones matrix containing the polarization states of the intracavity eigenmodes was established, which assumed the self-consistency of the electromagnetic field in the resonator. Starting from the middle of the cavity, the matrix ${\textbf M}$ is expressed as
Fig. 6. (a)-(b) Schematic diagrams of two orthogonally polarized eigenmodes. Arrows are used to indicate the polarization distributions.
By changing the ellipticity parameters and position parameters, various dislocated situations of the two eigenmodes were simulated. Two types of situations were identified that conformed with the experimental observation of the analysis profiles, as shown in Fig. 7. In situation 1, ${x_1} = {x_2},\; {y_1} = {y_2},\; {\sigma _{x1}} = {\sigma _{y2}} \ne {\sigma _{y1}} = {\sigma _{x2}}$, the geometric center of two eigenmodes are coincident, but their sizes are different. When α was just at 0° (extinction position), four circular spots were observed; when α deviated slightly to the left or right from 0°, two spots on a diagonal were merged gradually, and others on another diagonal were weakened simultaneously to the extent that they almost disappear. Thus, eventually only a single inclined elliptical spot was observed on both the sides of the extinction position, respectively, and the major axes of the two elliptical spots were roughly perpendicular to each other. In situation 2, ${x_1} \ne {x_2},\; {y_1} \ne {y_2},\; {\sigma _{x1}} = {\sigma _{y2}} \ne {\sigma _{y1}} = {\sigma _{x2}}$, the geometric center of two eigenmodes are not coincident, and their sizes are also different. When $\alpha $ was just at 0°, two circular spots horizontally distributed were observed; when $\alpha $ deviated slightly to the left or right from 0°, one was elongated gradually, and another one was weakened simultaneously to the extent that it almost disappears. As a result, only a single inclined elliptical spot was observed on the near sides of the extinction position. Similarly, the major axes of two elliptical spots were also perpendicular to each other.
Fig. 7. (a)–(f) Theoretically simulated results of analysis profiles near the extinction position for (a)–(c) situation 1 and (d)–(f) situation 2. α represents the azimuth angle of the polarizer. The extinction position was defined as 0°. Arrows are used to indicate the orientations of the simulated polarizer.
The experimental observations and theoretical simulations involving the profile-based analysis of the polarized beam exhibited close agreement. The direct polarization emission was not a polarized eigenstate but, essentially, a composite state from the coherent combination of eigenmodes.
The analysis profiles of the polarized beam shown in Fig. 7, demonstrate that the laser system oscillates stably in dual-polarization eigenstates with frequency degeneracy. The two types of situation corresponding to changing profiles may originate from the spatial randomness of the weak anisotropy in the gain media, which caused by the instability of thermal effects. The spatial randomness is reflected in the relative value of these parameters: (${x_1},{y_1}$) and (${x_2},{y_2}$) and (${\sigma _{x1}},{\sigma _{y1}}$) and (${\sigma _{x2}},{\sigma _{y2}}$). Apparently, the weak phase anisotropy introduced by residual birefringence in the laser crystal results in eigenfrequency splitting. Moreover, the intracavity loss anisotropy due to the adjustment of the resonator mirrors partially offset the effect of phase anisotropy. Therefore, the eigenfrequency difference decreased to fall within the locking region. In addition, the fact that the dual-polarization eigenmodes share the same frequency facilitates coherent combination, and the final polarization states are dependent on the initial phase difference between them. Based on the observation that the linearly self-polarized regime does not change over time, it can be deduced that the phase difference was eventually stabilized in $\Delta \phi = 0/\mathrm{\pi }$ without active phase locking. When pump power deviates from the locking point A (3W), the changing of thermal distribution in the crystal lead to the variation of phase anisotropy and frequency difference. The regime of frequency degeneracy would be destroyed and the condition of coherent combination no longer be met under this kind of resonator alignment. As a consequence, the degree of polarization decreased gradually to unpolarized state due to incoherent combination. Therefore, the polarization states of the output laser depend strongly on the stability of the frequency degeneracy and phase locking, which could also justifiably explain the sensibility and instability of the polarization. In other words, to stabilize the polarization states in isotropic laser devices, some measures (e.g. realigning resonator, maintaining thermal stability) should be implemented to ensure that the eigenfrequency difference falls into the locking region, with phase locking strictly at 0/π. In addition, any perturbations that could disrupt the locking state, such as fluctuation of pump power, thermal instability, resonator misalignment and so on, should be avoided.
4. Conclusion
In conclusion, through experimental and theoretical investigations, we successfully explained the mechanism of the directly linearly polarized output in an isotropic Nd:YAG laser. Under the condition of frequency degeneracy and phase locking, the coherent combination of the two orthogonally polarized eigenmodes finally results in the linearly polarized emission. The mechanism could also be an important guideline to resolve the sensibility and instability of polarization in similar laser devices. Furthermore, beam profile-based analysis of the polarized beam was shown to be an effective method for judging different types of linear polarization states, including both eigenstates and combined states. Thus, we believe that similar polarization phenomena and analytical results of polarized spots can be observed in other isotropic laser crystals (eg. c-cut Nd:YVO4), and this will be the focus of subsequent studies. Meanwhile, the mechanism underlying the self-locking phase between the two intracavity eigenmodes merits further elucidation in our future studies.
Funding
National Natural Science Foundation of China (51761135115, 61575199, 61875199, 61975208); Chinese Academy of Sciences (XDB20000000); Natural Science Foundation of Fujian Province (2019J02015).
Acknowledgments
The authors would like to thank the National Natural Science Foundation of China (61575199, 61875199, 61975208, and 51761135115), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB20000000), and the Science Foundation of Fujian province (2019J02015) for their support of this research.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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