Study on thermal characteristics of 266 nm deep ultraviolet laser generated by BBO crystal - 05

04 Theoretical study of thermal properties

The above experiment shows that the BBO crystal (www.wisoptic.com) generates serious heat in the process of frequency quadrupling. It is known that the energy band gap of the BBO crystal is 6.56 eV, while the single photon energy of 266 nm and 532 nm lasers is 4.66 eV and 2.33 eV respectively. Theoretically, the crystal does not have single photon absorption of 266 nm and 532 nm lasers. Only because the small amount of impurities contained in the crystal will introduce a new energy level structure and lead to weaker linear absorption, the dominant factor causing heat generation in the crystal is the two-photon absorption of 266 nm laser. While considering the two-photon absorption of 266 nm laser by BBO crystal in the process of frequency quadrupling, it is assumed that the two-photon absorption will cause the formation of dynamic color center defects in the crystal. The equations controlling the intensity of ultraviolet light and the density of color centers are:

Where: I₂₆₆ (x, y, z) is the 266 nm laser power density distribution; β_TPA is the two-photon absorption coefficient of BBO crystal for 266 nm laser; ω₂₆₆ is the angular frequency of 266 nm laser photons; σ is the color center defect absorption cross section; η_b is the color center defect bleaching efficiency; N (t) is the time-dependent color center density. When t = 0, N (t) = 0, so we can get:

Under a certain UV power, a series of nonlinear absorption losses caused by two-photon absorption and the color center defects generated by it inside the BBO (www.wisoptic.com) crystal can be measured by the nonlinear absorption coefficient β_NLA, whose expression is shown in formula (4). When the BBO crystal generates heat due to nonlinear absorption, a relatively stable temperature field distribution will be formed inside the crystal under fixed external temperature control conditions, which obeys the steady-state heat conduction Poisson equation:

Where: T (x, y, z) is the temperature distribution inside the crystal; k is the thermal conductivity, assuming that the coefficient is isotropic inside the crystal, the average value of the thermal conductivity in the directions perpendicular to and parallel to the optical axis can be taken; q (x, y, z) represents the heat generated by the heat source per unit volume per unit time inside the crystal. While considering the nonlinear absorption of the crystal to the 266 nm laser, the small amount of linear absorption of the crystal to the 266 nm and 532 nm lasers is also taken into account. Its expression is:

Where: α₅₃₂ and α₂₆₆ are the linear absorption coefficients of BBO crystal for 532 nm and 266 nm lasers respectively; _I532 (x, y, z) is the power density distribution of 532 nm laser. Since the quadruple frequency crystal is shorter, the power density of green light and ultraviolet light in the longitudinal direction of the crystal does not change much and the temperature of the crystal side is controlled, so the temperature change in the longitudinal direction of the crystal can be ignored. Formula (5) can be expressed as:

Assuming that the cross-sectional area of the BBO crystal is (a×b) mm², the side of the crystal is kept at a constant temperature of T₀ by a heating device, and a rectangular coordinate system is constructed with any vertex of the crystal as the origin. During the heat conduction process of the BBO crystal, it follows the first type of boundary conditions, then:

When the incident 532 nm green light and the generated 266 nm deep ultraviolet laser propagate along the center of the crystal, according to the boundary conditions, a general solution of the crystal temperature field consisting of two eigenfunctions can be given:

Where: A_mn is the unknown coefficient. Substituting equation (9) into the heat conduction equation (7) and using the orthogonality of trigonometric functions, we can get the expression of the unknown coefficient:

Assuming that the green light and ultraviolet laser are both TEM₀₀ mode Gaussian beams, the expression of their light intensity distribution is:

Where P₅₃₂ and P₂₆₆ are the powers of green light and ultraviolet light respectively; ω₀ is the waist radius of green light in the BBO crystal (www.wisoptic.com). Since green light and ultraviolet light are Gaussian beams and propagate along the center of the crystal, the light intensity is strongest at the center of the crystal during the frequency quadrupling process, and the heat generated is also the most. Therefore, the temperature inside the crystal is distributed in a gradient and the phase mismatch conditions at each location are also different. At this time, the highest conversion efficiency can be obtained by satisfying the phase matching condition at the center of the crystal. Therefore, under the injection of green light of different powers obtained in the frequency quadrupling experiment, the temperature offset ΔT of the crystal heating device when the optimal power ultraviolet light output is achieved is the heat generated at the center of the crystal, and the temperature distribution at the center of the crystal satisfies:

In the calculation of this paper, the crystal size a, b are both 5 mm, the green beam waist radius ω₀ is 350 μm, the thermal conductivity k is 1.4 W/(m·K), the linear absorption coefficients α₂₆₆ and α₅₃₂ of the crystal for 266 nm laser and 532 nm laser are both 0.01 cm^-1, the color center defect absorption cross section σ = 8×10^-17 cm², and the color center density reaches a steady state at each value, that is, the time t is ∞. Using finite element analysis software according to equations (9) to (13), the nonlinear absorption coefficient β_NLA and the normalized color center density at different matching temperatures that affect the crystal heat generation, different ultraviolet laser powers, can be theoretically solved. The calculation results are shown in Figure 5.