Experimental and theoretical study of the weak and asymmetrical thermal lens effect of Nd:YLF crystal for σ and π polarizations

Abstract

The thermal lens effect of Nd:YLF crystal for different polarized beams is experimentally and theoretically studied in this paper. In the experiment, the different thermal lens effects of Nd:YLF crystal along a and c axes for π- and σ-polarized probe beams are observed, and the values of the focal lengths are measured. The theoretical analysis is made to explain the extremely weak thermal lens effect along the c axis, as well as that for the σ-polarized beam. And it is corroborated that the complementation among the thermo-optical coefficient, the thermal end bulging and the photoelastic effects contributes to the weak and asymmetry thermal lens effect of Nd:YLF crystal.

Appendix

Thermal bulging of the end surface of the end pumped crystal is studied here. Unlike the side pumped crystal, where the pump power and the heat load distribution could be regard as uniform, the pump power and the heat load in the end pumped crystal is exponentially decayed along the z axis of the crystal, as well as the temperature distribution. For the non-uniform distribution of temperature along the z axis, the thermal bulging of the end surface of the crystal cannot be easily expressed by the function below [10]

$$\Delta l(0) - \Delta l(r) = \alpha r_{0} \left[ {T(0) - T(r)} \right] = \alpha r_{0} \frac{{P_{\text{heat}} r^{2} }}{4Al\kappa }.$$

(12)

Here, Δl(r) and T(r) are the relative deformation distribution and temperature distribution in the cross section of the side pumped crystal rod, respectively. And with the thick lens focal length equation, the focal length caused by the bulging of the end surface of the crystal is

$$f = \frac{1}{{2d^{2} l/dr^{2} (n - 1)}} = \frac{A\kappa }{{P_{\text{heat}} }}\left[ {\frac{{\alpha r_{0} (n - 1)}}{l}} \right]^{ - 1} .$$

(13)

To the end pumped crystal rod, the deformation on the end surface could be expressed as below [5, 15]

$$\Delta l(r) = (n - 1)(1 + \nu )C_{\alpha } (r)\alpha \int_{0}^{l} {T(r,z)dz}$$

(14)

where C _α (r) is a scaling factor for the expansion of the infinite element, it varies in the range of (0,1). What is different from the citation [5] here is that we consider the difference of the C _α in the radial direction due to the variation of the temperature distribution in the radial direction. And the end bulging of the end surface is

$$\Delta l(0) - \Delta l(r) = (n - 1)(1 + \nu )D_{\alpha } (r)\alpha \int_{0}^{l} {\Delta T(r,z)dz}$$

(15)

where D _α (r) is another scaling factor which we named the relative scaling factor and it has a relationship with C _α (r) as below

$$D_{\alpha } (r) = \frac{{C_{\alpha } (0)\int_{0}^{l} {T(0,z)dz} - C_{\alpha } (r)\int_{0}^{l} {T(r,z)dz} }}{{\int_{0}^{l} {\Delta T(r,z)dz} }}.$$

(16)

And $\Delta T(r, z)$ is the temperature difference between the rod section center and the location with the radius of r,

$$\Delta T(r,z) = T(0,z) - T(r,z).$$

(17)

And the value of T(r, z) and $\Delta l(r)$ could be numerically calculated with the infinite element simulation by ANSYS software. The parameters used in the simulation are listed in Table 1. And the simulation results of the integrated temperature distribution and the deformation distribution along z direction in the crystal are shown in Figs. 8 and 9.

Fig. 8

Temperature distribution integrated along z axis

Fig. 9

Simulated displacement of the finite elements in Nd:YLF crystal at 50 W pump power

Then, we could calculate the values of C _α (r) and D _α (r) with Eqs. (14)–(16). And the calculated values are plotted in Fig. 10. From the figure, we could discover that the value of C _α (r) and D _α (r) are both increased with the increase of the radius r. The value of C _α (r) increases from 0.31 to 0.42 with an average value of just around 0.35, which is given in citation [5]. D _α (r) could be calculated out with the values of C _α (r), and it varies from 0.023 to 0.058. Here, we leaves out the value of D _α (0) = 0 for that it is meaningless. And from the curve of D _α (r), we could see that the value of D _α (r) is not monotonically increasing, but reaches the top at r = 1.6–1.8 mm, and makes a little decrease at the rod edge of r = 2.0 mm. The value of D _α (2) in need in our calculation is 0.055. The value of D _α (r) is determined by the calculated value of Δl(r) and $\mathop \smallint \limits_{ 0}^{\text{l}} \Delta {\text{T(r,z)dz}}$. The smaller value of D _α (2) is because of that the value of Δl(2) is relatively smaller than that of Δl(r) (r < 2). And the smaller Δl(2) may be caused by the constraint by the shear stress on the interface of the crystal and the heat sink.

Fig. 10

Variation of the values of C _α (r) and D _α (r) with the increase in radius r