Beam stability improvement of high-power Lissajous modes by an off-axis pumped YVO<sub>4</sub>/Nd:YVO<sub>4</sub> laser

P. H. Tuan; W. C. Tsai; K. T. Cheng

doi:10.1364/OPTCON.468176

1. Introduction

Structured light with versatile amplitude, phase, and polarization distributions has become a dominant issue in optics society because of its myriad applications ranging from freestyling particle patterning [1], super-resolution microscopy [2], high-precision material processing [3], to high-capacity optical communication [4]. Even though much efforts have been put to improve and mature the existing technologies for flexibly generating high degrees-of-freedom structured light for customized purposes, there remains some prerequisite challenges before most of the laboratory research can be implemented into daily-life use. For instance, direct generation of on-demand optical fields [5] by a compact device is necessary for developing an easily-handling structured light tool with minimum requirement for the alignment skills. On the other hand, for better enriching the functionality of the tailored light as well as enhancing the efficiency for some related applications such as higher throughput in laser machining and stronger tolerance against the environment disturbance in optical communication, effectively power scaling of structured light while maintaining good beam stability and purity serves as the first-priority issue for the state of the art [6]. To meet the highly desirable requirement of at-source generation of high-power structured light, solid-state lasers with the inherent advantage of easier compensation for photon losses have been widely exploited to offer potential solutions for flexible light tailoring by intracavity mode filtering [7] or pump-control gain shaping [8]. Incorporating the selective pumping with the degenerate cavity design, a variety of high degree-of-freedom structured beams from the pure Hermite-Laguerre-Gaussian (HLG) eigenstates to the generalized coherent states of Lissajous modes by mode superposition have been demonstrated [9,10]. Even though power scaling for structured beams generated by solid-state lasers can be easily achieved by directly increasing the input power, the pump-induced thermal effects inevitably deteriorate the mode structure at high pump levels, which greatly hinders further applications requiring good beam stability.

In the studies of extremely high-power fundamental Gaussian mode operation in diode-pumped solid-state lasers, it has been reported that the output power performance and beam quality can be largely improved by using composite gain crystals with a diffusion-bonded undoped segment as an effective heat spreader [11,12]. Nevertheless, so far there is little discussion about using the composite gain crystal to enhance the output beam stability of high-power and high-order structured light. In this work, the comparative study on the pump-induced beam deformation of Lissajous modes by a uniformly-doped Nd:YVO₄ crystal and a YVO₄/Nd:YVO₄ composite gain crystal is demonstrated. At first, the theoretical analyses on the temperature and thermal stress fields inside the gain crystals are performed under different pump levels to validate the fact that the undoped YVO₄ front segment of the composite crystal can truly exhibit superb thermal management to effectively mitigate the pump-induced thermal effects. Based on the analysis, Lissajous modes with different transverse frequency ratios at accidental degenerate conditions [13] are then generated in an off-axis pumped cavity with a uniform or composite gain crystal to examine the beam morphology variation during power scaling. Experimental observations reveal that all Lissajous modes by the YVO₄/Nd:YVO₄ crystal can still maintain fairly stable and well-defined structures under a high pump level compared to the results by uniform crystal. Moreover, the power performance for Lissajous modes by the composite crystal also show significant improvement with an overall slope efficiency increment about 8%. To offer more quantitative comparison for the cavity and beam structure stability, the thermal lensing effect as well as the stress-induced beam envelope rotation and elongation are further analyzed. From the improved results with smaller beam rotation angles, beam aspect ratio changes, and thermal lens diopters, the thermal loading mitigation by the composite gain crystal can be estimated to be 2.6 times better than that by the uniform crystal. Finally, Lissajous modes by the composite crystal are transformed into the trochoidal beams via the astigmatic mode conversion to realize high-power and high-order vortex beams with clear beam morphologies.

2. Analyses for temperature and thermal stress distributions of gain crystal

To estimate the thermal mitigation ability by the undoped segment of composite gain crystal, temperature and thermal stress fields of end-pumped Nd:YVO₄ and YVO₄/ Nd:YVO₄ crystals are theoretically evaluated at first. Following the model used in Refs. [11,14], the steady-state temperature field inside a rectangular gain crystal with dimensions of l_x, l_y, and l_z can be written as the Fourier series of eigen-bases ψ_n,m,s(r)=sin(β_n·x) sin(γ_m·y)cos(δ_s·z) as

(1)$$T({\mathbf r}) = \sum\limits_{n,m,s} {{A_{n,m,s}}{\psi _{n,m,s}}({\mathbf r})} + {T_o}\;$$

with

(2)$${A_{n,m,s}} = \frac{8}{V}\frac{{\int_V {q({\mathbf r}){\kern 1pt} {\kern 1pt} {\psi _{n,m,s}}({\mathbf r}){d^3}{\mathbf r}} }}{{{K_x}\beta _n^2 + {K_y}\gamma _m^2 + {K_z}\delta _z^2}},$$

where β_n= nπ/l_x, γ_m= mπ/l_y, δ_s= sπ/l_z, r = (x, y, z) denotes the spatial coordinates, T_o is the environmental temperature, V = l_x·l_y·l_z is the crystal volume, and K_x, K_y, K_z are thermal conductivity coefficients along transverse (x, y) and longitudinal (z) axes. The heat source q(r) from an off-axis pumped Gaussian beam centered at (Δx, Δy) can be given by q(r)=Q_o·[exp(−αz)/(1−exp(−αz))]·exp{−2[(x−Δx)²+(y−Δy)²]/w_p²}, where Q_o= 2ξP_inα/πw_p², ξ is the fractional thermal loading, P_in is the input power, α is the absorption coefficient, and w_p is the effective spot size of pump beam. To make an explicit comparison on the pumped-induced thermal effects, a 10-mm-long uniformly-doped a-cut Nd:YVO₄ crystal and a 12-mm-long composite a-cut YVO₄/Nd:YVO₄ crystal with an additional 2-mm undoped front segment are considered in the analysis. The doping concentration of Nd³⁺ ions and the transverse dimensions of both gain crystals are chosen the same to be 0.2 at.% and 3×3 mm², respectively. Figure 1(a) shows the temperature fields of uniform and composite crystals at P_in= 8 W calculated by Eqs. (1) and (2) with the following parameters: Δx=Δy = 0.5 mm, α=0.32 mm^-1, T_o= 293 K, K_x= 5.23 W/(m·K), K_y= K_z= 5.1 W/(m·K), ξ=0.24, and w_p= 100 °m. It can be obviously seen that with the undoped front segment serving as a heat spreader, the temperature profile of the composite crystal is dramatically different from that of the uniform crystal. Considering the temperature distribution along z axis at the pump beam center T(Δx, Δy, z) as a function of P_in, not only the maximum hot-spot temperature but the longitudinal thermal gradient of the composite crystal can be found to significantly decrease as shown by Fig. 1(b). At P_in= 20 W, the maximum temperature difference between the uniform and composite crystals can exceed 100 K. Subsequently, thermal stress fields at the plane of the maximum temperature are further evaluated by solving the governing differential equation of Airy stress function Φ(r) under plane-stress approximation [15]:

(3)$$\left( {\frac{{{\partial^4}}}{{\partial {x^4}}} + \frac{{2{\partial^4}}}{{\partial {x^2}\partial {y^2}}} + \frac{{{\partial^4}}}{{\partial {y^4}}}} \right)\Phi ({\mathbf r}) ={-} \frac{E}{{1 - \nu }}\left( {\alpha_T^y\frac{{{\partial^2}T}}{{\partial {x^2}}} + \alpha_T^x\frac{{{\partial^2}T}}{{\partial {y^2}}}} \right),$$

where E is the Young’s modulus, ν is the Poisson ratio, $\alpha _T^x$ and $\alpha _T^y$ are the thermal expansion coefficients along the orthotropic axes of YVO₄ crystal. Considering the symmetry of system as well as the characteristics of temperature fields, the approximated solution for the Airy stress function can be given by

(4)$$\begin{array}{l} \Phi ({\mathbf r}) = [{{c_1}\cosh ({{\beta_1}y} )+ {c_2}y\sinh ({{\beta_1}y} )} ]\sin ({{\beta_1}x} )+ [{{c_3}\cosh ({{\gamma_1}x} )+ {c_4}x\sinh ({{\gamma_1}x} )} ]\sin ({{\gamma_1}y} )\\ \quad \quad \quad \;\; + \sum\limits_{n,m} {{R_{n,m}}({z_o})\sin ({{\beta_n}x} )\sin ({{\gamma_m}y} )} \end{array}$$

with

(5)$${R_{n,m}}({z_o}) = \left( {\frac{E}{{1 - \nu }}} \right)\sum\limits_{s = 0}^{} {\frac{{{A_{n,m,s}}\cos ({{\delta_s}{z_o}} )({\alpha_T^y\beta_n^2 + \alpha_T^x\gamma_m^2} )}}{{{{({\beta_n^2 + \gamma_m^2} )}^2}}}} ,$$

where z_o locates at the plane with the highest temperature. In terms of the Airy stress function, the normal stress and shear stress fields can be given by σ_xx=∂²Φ/∂y², σ_yy=∂²Φ/∂x², and σ_xy= −∂²Φ/∂x∂y, respectively. The unknown coefficients c₁, c₂, c₃, and c₄ can be determined by the boundary conditions as σ_xx=σ_yy=σ_xy= 0 at all edges. It is worthy to note that the term of basis expansion in Eq. (4) to satisfy the thermal gradient profile dominates the pump-induced stress field especially for the case under tight-focusing pumping. By using Eqs. (3) to (5) with E = 1.33×10² GPa, ν=0.33,$\alpha _T^x$=11.37 ×10⁻⁶ K^-1 and $\alpha _T^y$=4.43 ×10⁻⁶ K^-1[15], normal stress fields σ_xx(x, y) and σ_yy(x, y) as well as the dependences of maximum normal stresses on P_in for the uniform and composite crystals can be evaluated as shown in Fig. 1(c). Because of the off-axis pumping and the birefringent thermal coefficients of a-cut YVO₄ crystals, the compressive normal stress fields can be seen to reveal obvious anisotropic morphologies. With the great reduction of thermal loading, the maximum normal stress of the composite crystal can be found to be less than 50% of the value for the uniform crystal. Since the anisotropic stress fields have been studied to be the main cause for the principal axis rotation of gain crystal indicatrix [16], effectively diminishing the thermal stress is beneficial for alleviating the laser beam deformation during power scaling.

Fig. 1. (a) Calculated temperature fields with P_in= 8 W and Δx=Δy = 0.5 mm for the uniform and composite gain crystals. (b) Temperature distributions along z axis at the pump beam center as a function of pump power. (c) The compressive normal stress fields and the corresponding maximum thermal stresses as a function of pump power.

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3. Experimental demonstration for beam stability improvement

To validate that the thermal effect mitigation by the composite crystal can actually lead to prominent beam stability improvement for high-order structured modes, an off-axis pumped Nd:YVO₄ laser was designed to generate Lissajous beams at different pump levels as shown in Fig. 2. A concave mirror with a radius of curvature ρ=30 mm as well as a plane mirror with the effective transmittance of 5% at 1064 nm were respectively used as the front mirror and output coupler to form the spherical cavity. The pump source was a 20-Watt 808-nm fiber-coupled diode laser with a fiber core diameter of 200 °m and a numerical aperture of 0.22. A coupling lens set with unity magnification and an effective focal length of 50 mm was utilized to reimage the pump beam into the gain crystal. All experimental parameters for the cavity configuration are the same as those used in Ref. [17]. An a-cut uniformly-doped Nd:YVO₄ crystal and an a-cut YVO₄/Nd:YVO₄ composite crystal with dimensions and doping conditions as those in theoretical analyses were exploited as the gain media for comparison. To realize Lissajous modes with sharper morphologies by high-order Hermite-Gaussian mode superposition, off-axis displacements along two transverse directions were both set to be 0.5 mm.

Fig. 2. The experimental setup to generate high-order Lissajous beams.

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Under 2D off-axis pumping, Lissajous structured modes can be achieved by finetuning the cavity length L_cav to satisfy the accidental degenerate condition given by $L_{{P / Q}}^{\textrm{(}p,q\textrm{)}}$= L_P_/Q+ξ^(p,q)d [17], where L_P_/Q=ρsin²(Pπ/Q) is the degenerate cavity length to lead to longitudinal-transverse coupling with the frequency ratio of P/Q, ξ^(p,q) is the astigmatic factor given by ξ^(p,q)= (p−q)/2(p + q), and d is the effective optical path difference related to the crystal birefringence. For a more stable cavity condition with abundant accidental degenerate levels, we choose P/Q = 1/4 in the experiment. Figure 3(a) shows the average output power as a function of pump power P_in for Lissajous beams with different transverse frequency ratios (p, q) by the uniform and composite crystals. For the cases by the composite crystal, it can be seen that the overall power performance is obviously improved with the slope efficiency to be 8% higher than the cases by the uniform crystal. At P_in= 20 W, the maximum output power for Lissajous modes by composite crystal can reach up to 9.4 W with an optical-to-optical conversion efficiency to be 47%. The deformed beam morphologies for the Lissajous modes with increasing P_in are displayed in Figs. 3(b) and 3(c) for the cases by uniform and composite crystals, respectively. The generated Lissajous beams can be seen to gradually rotate, elongate, and become blurred with the increasing P_in for all cases. Besides, the effective localized region of Lissajous curve intensity will broaden as P_in increases because of the enlarging mode size from the thermal lensing. However, comparing the experimental results, it can be apparently found that the beam structure degradation is significantly alleviated by the composite crystal to remain relative stable and pure mode patterns even at P_in= 20 W.

Fig. 3. (a) Average output power versus pump power for Lissajous beams with different frequency ratios (p, q) by the uniform and composite crystals. The corresponding beam morphologies with increasing P_in for cases by (b) uniform and (c) composite crystals.

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To provide more quantitative measures on the improved cavity stability by the composite crystal, the enlarging thermal lensing effect with the increasing pump power is further estimated by analyzing the variation of degenerate cavity length L_P/Q. Based on the cavity ABCD law, the longitudinal-transverse frequency ratio can be given by Δf_T/Δf_L= P/Q = cos^-1[(A + D)/2]/2π. As a result, the effective focal length of an internal thermal lens located fairly close to the front mirror can be linked to the degenerate cavity length as f_th= 2ρL_P/Q/{ρ[1−cos(2πP/Q)]−2L_P/Q}. From the measured cavity length $\textrm{L}_{{1 / 4}}^{\textrm{(2,2)}}$ at each pump power, the corresponding f_th can be evaluated as shown in Fig. 4(a). To explicitly compare the mitigation ability for the thermal lensing effect, the diopter of thermal lens which is proportional to the input power as D_th= 1/f_th= C·P_in is depicted in Fig. 4(b). It is obvious that D_th for the cases by the composite crystal are greatly reduced compared with the cases by the uniform crystal especially at high pump levels. The diopter coefficient of thermal lens for the composite crystal can be evaluated to be C_comp= 1.2×10⁻⁵ mm^-1W^-1 which is 2.6 times smaller than C_uni= 3.2×10⁻⁵ mm^-1W^-1 for the uniform crystal.

Fig. 4. (a) The estimation of effective focal length for thermal lens by measuring the variation of degenerate cavity length at each pump power. (b) Dependences of thermal lens diopters as a function of pump power for the uniform and composite crystals.

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On the other hand, for quantitatively comparing the degrees of Lissajous mode deformation, the beam similarity S defined as the correlation integral between the normalized wave intensity I_Pin at each pump level with respect to that at 2 W, i.e. S=∫I_Pin(r)· I_2W(r)d²r, is calculated as an indicator as shown in Fig. 5(a). With better thermal effect mitigation, the beam similarity for Lissajous modes by the composite crystal can be maintained to be higher than 90% even when P_in= 20 W. Next, the beam structure rotation and elongation induced by the anisotropic thermal stress are further analyzed according to experimental mode patterns shown in Figs. 3(b) and 3(c). Figures 5(b) and 5(c) reveal the aspect ratio b/a for beam envelope as well as the beam rotation angle θ with respect to the horizontal axis as depicted by the inset of Fig. 3(b). Once again, it can be clearly seen that the gradually increasing rotation angle and aspect ratio with the enlarging P_in are dramatically reduced by using the composite crystal.

Fig. 5. Dependences of the defined (a) beam similarity, (b) aspect ratio, and (c) rotation angle on the increasing pump power for the Lissajous modes by the uniform and composite crystals.

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Finally, the corresponding trochoidal vortex beams by performing astigmatic transformation on Lissajous modes during power scaling are examined to provide useful references for the research on generating high-power and high-order structured vortex fields. Figure 6 shows the transformed mode patterns corresponding to Lissajous modes of (p, q) = (3, 1) and (6, -2) at different pump levels for the uniform and composite crystals. With more stable and purer Lissajous mode structures, the transformed trochoidal beams by the composite crystal can be seen to preserve cleaner mode morphologies with fairly good circular symmetry.

Fig. 6. Transverse patterns of trochoidal beams corresponding to Lissajous modes of (p, q) = (3,1) and (6, -2) at different pump power P_in.

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4. Conclusion

In conclusion, the beam structure deformation of Lissajous modes generated in an off-axis pumped degenerate cavity during power scaling is systematically compared by using a uniformly-doped Nd:YVO₄ crystal and a YVO₄/Nd:YVO₄ composite gain crystal. Consistent with the theoretical prediction, Lissajous modes by the composite gain crystal with the undoped front segment as an effective heat spreader have been confirmed to preserve more stable and purer beam structures than the cases by the uniform crystal because of the better thermal mitigation ability. The quantitative analyses on the experimental results verify that not only the thermal lensing diopter has been greatly reduced to be less than 50% of the value for the uniform crystal but also the pumped-induced beam rotation and elongation have been prominently alleviated by the composite crystal. Moreover, the output power performance for Lissajous modes generated by the composite crystal also exhibit significant improvement with the overall slope efficiency to be 8% higher than the case by the uniform crystal. This demonstration can provide a potential solution for the at-source generation of high-order and high-power structured light with superior beam structure and cavity stability.

Funding

Ministry of Science and Technology, Taiwan (MOST-108-2112-M-194-005-MY3).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Beam stability improvement of high-power Lissajous modes by an off-axis pumped YVO₄/Nd:YVO₄ laser

Abstract

1. Introduction

2. Analyses for temperature and thermal stress distributions of gain crystal

3. Experimental demonstration for beam stability improvement

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (5)

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