## Abstract

Laser-induced thermal lens in optical components causes wavefront distortion of the laser beam and may affect performance and stability of optical systems such as high-power lasers. The bulging of the heated area, the temperature dependence of the refractive index, and the photoelastic effects are responsible for phase shifts damaging beam quality. The theoretical background for laser-induced beam distortion is well understood and applies only for axially symmetric thermal loadings, with the assumptions that the stresses follow thin-disk or long-rod approximations. This, in fact, limits the overall applications of this model. In this work, we developed an unified theoretical model for the optical path change in optical materials regardless of its thickness. The modeling is based on the solution of the thermoelastic equation and has a real description of the surface deformation caused in the optical element. In the appropriated limits, as expected, the model retrieves the thin-disk and the long-rod type distributions. Furthermore, we provided time-dependent radial expressions for the temperature, surface displacement, and stresses. The theory presented in this paper provides simple analytical tools for designing laser systems, and complements previous work allowing one to access optical distortions of materials ranging from thin-disk to long-rod-like distributions.

©2012 Optical Society of America

Wavefront distortion arising from thermal lens in optical materials poses a major problem in high-power laser systems and has been subject to exhaustive investigation over the past few decades [1–9]. In such optical components, nonuniform heating caused by the laser beam absorption compromises the system’s performance affecting the optical quality of the beam [1,2]. This effect has been mostly investigated regarding phase shift induced in optical windows. Particularly for windows, the wavefront distortion originates from the refractive index gradient thermally induced in addition to the thermoelastic deformation of the windows surface.

The theoretical treatment for this effect has been investigated in great detail [3–6] and its bases relies on the fact that windows, or optical elements, are subject to axially symmetric thermal loading and thermoelastic properties are isotropic in the plane of the windows. In addition, the stresses are constrained to follow either the plane-stress or plane-strain approximations. The plane-stress approximation applies to thin-disk geometries [4], where the windows thickness is much smaller than the windows radius. The plane-strain model, on the other hand, uses the long-rod approximation that satisfies the condition of much larger thickness compared to radius. In the absence of mechanical loading, the optical windows are subjected to cylindrical symmetric radial and azimuthal stresses that are related in a simple manner to the temperature distribution within the material. The phase shifts responsible for the distortion of the beam arise from the change in path length as the heated area deforms, from the temperature dependence of the refractive index, and from thermoelastic effects associated with nonuniform heating pattern. We show here that those assumptions are, in fact, too restrictive and develop a unified analytical theory for the optical path change in optical materials regardless of its thickness. The modeling is based on solution of the thermoelastic equation and has a real description of the surface deformation by the optical element. In the appropriated limits, the model retrieves the thin-disk (plane-stress) and long-rod (plane-strain) type distributions. This generalized model complements previous work [3–6] in the sense that it allows us to access optical distortions of materials ranging from thin-disk to long-rod like distributions, and could have significant impact on designing laser systems and predicting windows degradation of optical materials. Moreover, this effect is the foundation of many photothermal methods for material characterization, and particularly important for the thermal lens spectrometry, where the optical path change is directly linked to the thermo-optical properties of solid and liquid materials [10–12].

For an axial symmetric beam propagating along the $z$ direction, the optical path is defined as

We consider here a low optical absorption material of thickness ${l}_{0}$ (path length) and refractive index ${n}^{s}(r,z,t)$ centered in $z=0$, as shown in Fig. 1. The material is surrounded by a nonabsorbing fluid medium of refractive index ${n}^{f}(r,z,t)$. The optical path length can be then written asIn a first approximation, the thermal effects on the refractive index can be easily formulated as linearly dependent over the temperature range of interest as

The thermal stress index variation involves considerations relating to the thermoelastic effect and applies only to the sample as

Finally, the contribution originated from the surface deformation yields

Terms of second order proportional to the product $\mathrm{\Delta}n\mathrm{\Delta}l$ are neglected here. The above expressions show us that the optical path change induced by the laser beam is a result of thermal, stress, and strain effects. This complex problem could be simplified under approximative assumptions. For instance, if the beam radius is smaller than the radial dimension of the sample, for a relative short exposure time $t$, the temperature at the edges of the sample can be assumed constant. Moreover, if air is used as fluid, which is basically the case for optical windows in laser systems, ${n}_{0}^{f}\approx 1$, and the approximation of no heat flow across the interface material/fluid is valid [15]. Thus, the fluid contribution ${S}_{\text{th}}^{f}(r,t)$ can be safely neglected—and will be omitted from this point.

We consider the case of a continuous wave incident beam with a Gaussian ${\mathrm{TEM}}_{00}$ intensity profile. It is assumed that the attenuation of light intensity along the material thickness can be neglected—low optical absorption approximation. Thus, the temperature rise distribution with no axial heat flow to the surroundings is given by the solution of the following heat conduction differential equation [16]:

The thermoelastic equation for the stress and surface displacement caused by a laser-induced nonuniform temperature distribution, in the quasi-static approximation, can be expressed in cylindrical coordinates by introducing the scalar displacement potential $\mathrm{\Psi}$ and the Love function $\psi $ following by the Poisson’s equation [17],

and the Biharmonic equation, ${\alpha}_{T}$ is the linear thermal expansion coefficient, and $\nu $ the Poisson’s ratio.The components of the displacement vector (${u}_{z}$) and the stress (${\sigma}_{ii}$) are obtained from $\mathrm{\Psi}(r,z,t)$ and $\psi (r,z,t)$ by the relations

Recently [13,15,18], we have obtained semianalytical expressions for the temperature and thermoelastic potentials. In those solutions, we were assuming that the radial dimension of the sample was large enough for the temperature to be considered constant at the edges of the sample, that is, $T(\infty ,z,t)=0$. In fact, the solutions were validated by all numerical finite elemental analysis modeling with real boundary conditions [15,18]. Using this approximation, and assuming no azimuthal dependence for the temperature distribution, we can write the temperature rise as

${J}_{n}(\xi )$ is the Bessel function of the first kind, andThe above expressions allow us to write the displacement and stresses in a simple manner as

The symmetric and antisymmetric combinations of the radial and azimuthal aberrations to the optical path length lead to

Using Eqs. (14) and (27), the thermal expansion contribution can be written as

Finally, with Eqs. (8), (22), (33), and (35), we can write $S(r,t)$ for free stress-birefringent material, that is ${q}_{\parallel}\approx {q}_{\perp}$, as

To illustrate and support the analytical solutions, Eqs. (36) to (40), we perform simulations considering physical parameters of a calcium fluoride windows (${\mathrm{CaF}}_{2}$) [6]. Figure 2 shows the normalized temperature, $T(r,t)/T(0,t)$, and surface displacement, ${u}_{z}(r,t)/{u}_{z}(0,t)$, profiles for windows with different thicknesses at an exposure time of $t=0.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{s}$ and $\omega =50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$. The temperature profile has a sharp shape compared with the surface displacement for thick windows, which is caused by the mechanical inertia of the surface to expand upon temperature change. Both the temperature and the surface deformation profiles become similar only for very small thickness. This, in fact, validates the thin-disk, or the plane-stress, approximation. However, it is important to note that approximating the surface deformation by the temperature profile is reasonably acceptable only for very thin samples.

The limit cases, plane-stress and plane-strain, are better explored by analyzing the optical path change. Figure 3 shows the optical path $S(r,t)$ scaled by a factor ${l}_{0}{Q}_{0}$ for $r=\omega $ and different thicknesses. $S(r,t)$ was evaluated by Eq. (36) and the limit cases by Eqs. (37) and (38), respectively. The results illustrate well the total phase shift moving from a plane-stress approximation, for small ${l}_{0}$, to the plane-strain as ${l}_{0}$ increases.

Figure 4 shows the time and radial evolution of the optical path $S(r,t)$ for a windows of thickness ${l}_{0}=2.0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$. The radial dependence of the optical path length induces aberration in the wavefront propagation. The corresponding phase shift is given by

where $\lambda $ is the laser wavelength. In fact, Eq. (41) provides an expression for analyzing the thermal lens effect caused by a laser-induced optical element. This effect is important in high-power laser systems and has numerous applications in photothermal effect based techniques [10–12,15] for material characterization, such as in the thermal lens spectrometry. As for high-power laser systems, a criterion normally used to characterize the laser beam distortion is the Strehl ratio—the ratio of the intensity at the point of maximum intensity in the observation plane with and without phase aberration. For instance, in an untruncated Gaussian illumination with beam radius $\omega $, it is given by [19]In conclusion, we have developed a unified analytical theory for the wavefront distortion caused by laser-induced thermal lens. The theory applies not only to plane-stress and plane-strain type distributions, but also for stress and strain contributions within these limit cases. Radial and time-dependent expressions for the temperature, surface displacement, and stresses have been obtained. This generalized model could have significant impact on designing laser systems and predicting laser-induced windows degradation in optical materials. Furthermore, the optical path description presented in this work has direct application in thermal lens spectrometry, correlating optical path change to thermo-optical properties of solid materials.

## ACKNOWLEDGMENTS

The authors are thankful to the Brazilian agencies CAPES, CNPq, and Fundação Araucária for the financial support of this work.

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