Analytical solutions for anisotropic time-dependent heat equations with Robin boundary condition for cubic-shaped solid-state laser crystals

Thermal lens in diode-pumped square-section rods of Nd-doped phosphate glass and 0.5%Nd:5%Lu:CaF2 crystal: simulations and experiments

We performed simulations and experiments of wavefront distortions induced by propagating through diode-pumped square-section amplifying laser rods of Nd-doped phosphate glass and 0.5%Nd:5%Lu:CaF₂. We observed that depending on the material, wavefront distortions' profile can vary from a circular lens-like distortion to a complex astigmatic distortion. We showed that this difference comes from the relative sign of piezo-optic tensor coefficients.

Tunable thermally induced gradient index for extended depth of field

Extended depth of field (EDoF) imaging remains desirable in the consumer market due to its reduced cost through omission of refocusing hardware. This paper describes the study of a heated plastic plate and its effect on EDoF imaging. It is shown that the depth of field of an imaging device can be extended by heating either the periphery or core of a homogeneous element-solving the heat equation for a parallel plate generates a radial gradient index structure, which can be modeled in Zemax ray-tracing software. For an F/2 system with the dimensions of a typical mobile-device camera, the heating arrangement shifts the peak value of the modulation transfer function (MTF) at Nyquist/2 from object distance 1350 to 750 mm. This study suggests the possibility of a fixed-focus camera with distinct modes: a "high-quality" mode operational over the standard depth of field when the plate is not heated; a "macro" mode triggered by heating the plate periphery; and a "wide-focus" mode triggered by heating the lens core. Importantly, the reduction in MTF generally associated with EDoF solutions is not present when the plate is unheated. The plate, having no optical power when unheated, can be added to existing off-the-shelf lens designs. Such a heating arrangement could provide a very-low-cost refocusing alternative where moving parts are not desirable.

Measurement of thermal effects of diode-pumped solid-state laser by using digital holography

Thermal lensing is one of the most important factors that can affect the performance of high-power solid-state lasers, such as limiting the power scaling capability and deteriorating output beam quality. In this paper, a novel and accurate measurement of digital holography is proposed to determine the thermal lensing of diode-pumped solid-state lasers with high resolution. The digitally recorded hologram can reveal the phase change when light travels through the laser gain medium. From the phase map, we can obtain the index variations induced by temperature differences inside the laser crystal when it is pumped by laser diodes, as well as determine the focal length of the integrated thermal lensing focus length. There was much work on measuring the static laser medium thermal lens because there is no laser output from the cavity in the setup. Our experiment setup was able to achieve online measurement with laser output at the same time. The measuring result can provide an accurate guide for compensating the thermal lensing in laser design to achieve high-power output and good beam quality. Moreover, detailed index variations in the direction of the laser crystal cross-section can be numerically reconstructed, by which the thermal effects, pump uniformity, crystal uniformity, etc., can be revealed from the holography result.

Heat-coupled Gaussian continuous-wave double-pass optical parametric oscillator: thermally induced phase mismatching for periodically poled MgO:LiNbO3 crystal

In this paper, a model describing the thermal effects on the optical parametric oscillator (OPO) of Gaussian continuous waves in double-pass cavities is presented. Eight equations, including forward and backward nonlinear waves, heat, and thermally induced phase mismatching equations, were coupled and solved simultaneously to investigate the effect of heat generation on the OPO's efficiency. The model was applied for a periodically poled MgO:LiNbO₃ crystal with an excellent agreement with experimental data. The numerical calculations have been carried out by a homemade code written in Intel FORTRAN, which used a finite difference method.

Time-resolved thermal lens spectroscopy with a single-pulsed laser excitation beam: an analytical model for dual-beam mode-mismatched experiments

Pulsed laser beam excitations are more commonly used in thermal lens spectroscopy (TLS) than continuous-wave (CW) ones, because CW excitations limit the measurement to linear absorption processes [J. Opt. A5, 256 (2003)]. In this work, we present a new and full analytical model for a single-pulsed laser excitation dual-beam mode-mismatched TLS for low absorption solid-state and liquid samples. Our model has been based on a new solution of time-dependent heat equation for a finite-radius cylindrical sample exposed to a single-pulsed excitation laser beam. For low absorbent samples, unlike previous models, all aberration terms associated in the thermal lens were taken into account in Fresnel integration. Besides, the model provides a full analytical mathematical expression for the temperature rise, normalized signal intensity, and Z-scan photothermal lens signal. The model was confirmed with experimental data of distilled deionized water with excellent agreement. Therefore, the model allows us to extract thermo-optical properties of samples in an analytical and more accurate way.

Thermally induced phase mismatching in a repetitively Gaussian pulsed pumping KTP crystal: a spatiotemporal treatment

Thermally induced phase mismatching (TIPM) has been proven to be an influential issue in nonlinear phenomena. It occurs when refractive indices of crystal are changed due to temperature rise. In this work, the authors report on a modeling of spatiotemporal dependence of TIPM in a repetitively pulsed pumping KTP crystal. Gaussian profiles for both spatial and temporal dependences of pump beam were used to generate second-harmonic waves in a type II configuration. This modeling is of importance in predicting the nonlinear conversion efficiency of crystals when heat is loaded in the system. To this end, at first, an approach to solve the heat equation in a repetitively pulsed pumping system with consideration of the temperature dependence of thermal conductivity and realistic cooling mechanisms such as conduction, convection, and radiation, is presented. The TIPM is then calculated through the use of experimental thermal dispersion relations of KTP crystal. The results show how accumulative behaviors of temperature and TIPM (with its reverse sign) happen when the number of pulses is increased. Fluctuations accompanying temperature and TIPM were observed which were attributed to the off-time between successive pulses. Moreover, in this work, a numerical procedure for solving a repetitively pulsed pumped crystal is introduced. This procedure enables us to solve the problem with home-used computing machines.

Complete anisotropic time-dependent heat equation in KTP crystal under repetitively pulsed Gaussian beams: a numerical approach

In this work, a thorough and detailed solution for the time-dependent heat equation for a cylindrical nonlinear potassium titanyl phosphate (KTP) crystal under a repetitively pulsed pumping source is developed. The convection and radiation boundary conditions, which are usually ignored in the literature, have been taken into account, and their importance on the temperature distribution has been discussed in detail. Moreover, the temperature dependence of thermal conductivity of KTP was considered in the calculations, and its impact is discussed. It is shown that the radiation term has a negligible effect and can be dropped safely, while the temperature dependence of thermal conductivity is more influential, such that ignorance of it brings some errors into the modeling. The time evolution of the temperature while the crystal is pumping with a train of successive Gaussian pulses until reaching equilibrium is shown. To accomplish numerical calculations, we developed a homemade code written with the finite difference time domain method in Intel Fortran (ifort) and ran it with the Linux operating system.

Pulsed Bessel-Gauss beams: a depleted wave model for type II second-harmonic generation

In this work, a three-dimensional and time-dependent nonlinear wave model to describe the generation of pulsed Bessel-Gauss second-harmonic waves (SHWs) is presented. Three coupled equations, two for ordinary and extraordinary fundamental waves and one for extraordinary SHWs, describing type II second-harmonic generation (SHG) in a KTiOPO₄ (KTP) crystal were solved by considering the depletion of fundamental waves (FWs). The results examined the validity of nondepleted wave approximation against the energy of pulses, beam spot size, and interaction length. It was shown that for pulses with spot sizes of ω_f=80  μm and energy of 0.8j, the nonlinear interaction was accomplished over a distance of ∼5  mm. Therefore, for KTP crystals with lengths longer than 5 mm, the nondepleted wave approximation can no longer be valid. To be valid, the crystal must be shorter than the interaction length, i.e., 5 mm.

Heat coupled Gaussian continuous-wave double-pass type-II second harmonic generation: inclusion of thermally induced phase mismatching and thermal lensing

A model describing the thermal effects in type II second harmonic generation (SHG) of Gaussian continuous-wave (CW) in a double-pass cavity is presented. The thermally induced phase mismatching (TIPM) along with thermal lensing was included in the classical SHG formalism through the interposing the heat and TIPM equations. To this end, eight equations were coupled together and solved simultaneously to reveal how the SHG is affected in time when heat is generated in the crystal. The model showed an excellent agreement with experimental data [Opt. Laser Tech.34, 333-336 (2002)]. Furthermore, a numerical procedure, which was developed in this work, is introduced for simultaneously solving the SHG, heat, and TIPM equations with home-used computing machines.

Temperature nonuniformity occurring during the cooling process of a KDP crystal and its effects on second-harmonic generation

The temperature nonuniformity occurring during the cooling process of a KDP crystal is studied, along with its effects on the second-harmonic generation (SHG) of a high-average-power laser. A comprehensive model is proposed incorporating principles of thermodynamics, mechanics, and optics, and it is applied to investigate the temperature nonuniformity and its effects. The temperature rise caused by linear absorption is calculated, while the temperature nonuniformity occurring during the cooling process is analyzed using the finite-element method (FEM). The stress induced by the nonuniformity is then studied using the FEM, and the trend of its change is determined. Moreover, the changes in refractive index caused by the stress are calculated, the results of which are used to determine the variations in the induced phase mismatch. The SHG efficiency considering the phase mismatch is eventually obtained by solving the coupling wave equations. The results demonstrate that the temperature nonuniformity has negative effects on the SHG efficiency.

Simulation of temperature and thermally induced stress of human tooth under CO2 pulsed laser beams using finite element method

The authors report the simulation of temperature distribution and thermally induced stresses of human tooth under CO2 pulsed laser beam. A detailed tooth structure comprising enamel, dentin, and pulp with realistic shapes and thicknesses were considered, and a numerical method of finite element was adopted to solve time-dependent bio-heat and stress equations. The realistic boundary conditions of constant temperature for those parts embedded in the gingiva and heat flux condition for those parts out of the gingiva were applied. The results which were achieved as a function of energy density (J/cm(2)) showed when laser beam is irradiated downward (from the top of the tooth), the temperature and thermal stresses decrease quickly as a function of depth that is a result of strong absorption of CO2 beams by enamel. This effect is so influential that one can use CO2 beams to remove micrometer layers while underlying tissues, especially the pulp, are safe from thermal effects.