Thermo-Optic Effects Chapter 7 Solid State Lasers By W. Keochner

Generation of heat in lasers (a) the energy difference of the photons between the pump band and the upper laser level is lost as heat to the host lattice; similarly, the energy difference between the lower laser level and the ground state is thermalized. The difference between the pump and laser photon energies, termed quantum defect heating, is the major source of heating in solid-state lasers. (b) In addition, nonradiative relaxation from the upper laser level to the ground state, and nonradiative relaxation from the pump band to the ground state will generate heat in the active medium. (c) In flashlamp-pumped systems, the broad spectral distribution of the pump source causes a certain amount of background absorption by the laser host material, particularly in the ultraviolet and infrared regions of the lamp spectrum. Absorption of lamp radiation by impurity atoms and color centers can further increase heating. Thermo-Optic Effects

7.1 Cylindrical Geometry The combination of volumetric heating of the laser material by the absorbed pump radiation and surface cooling required for heat extraction leads to a nonuniform temperature distribution in the rod. This results in a distortion of the laser beam due to a temperature- and stress-dependent variation of the index of refraction. The thermal effects which occur in the laser Material are thermal lensing and thermal stress- induced birefringence. Thermo-Optic Effects

r0r0 Coolant Flow Pump Light

The radial temperature distribution in a cylindrical rod is obtained from the one-dimensional heat conduction equation 7.1.1 Temperature Distribution We consider the case where the heat generated within the laser rod by pumplight absorption is removed by a coolant flowing along the cylindrical-rod surface. With the assumption of uniform internal heat generation and cooling along the cylindrical surface of an infinitely long rod, the heat flow is strictly radial, and end effects and the small variation of coolant temperature in the axial direction can be neglected. Thermo-Optic Effects Where K is the thermal conductivity, and heat is uniformly generated at a rate Q per unit volume

With the boundary condition for where is the temperature at the rod surface and is the radius of the rod, it follows that The temperature profile is parabolic, with the highest temperature at the center of the rod. The heat generated per unit volume can be expressed as where is the total heat dissipated by the rod and is the length of the rod. The temperature difference between the rod surface and the center is Thermo-Optic Effects

The transfer of heat between the rod and the flowing liquid creates a temperature difference between the rod surface and the coolant. A steady state will be reached when the internal dissipation P h is equal to the heat removed from the surface by the coolant where h is the surface heat transfer coefficient and T F is the coolant temperature. Combining with For the temperature at the center of the rod Thermo-Optic Effects

The boundary conditions for the heat transfer coefficient are a thermally insulated laser rod (h = 0), or unrestricted heat flow from the rod surface to a heat sink (h = ). For cases of practical interest the heat transfer coefficient is typically around h = 0.52Wcm2 C1. Thermo-Optic Effects the maximum temperature at the center is 114 C The temperature gradient between the center of the crystal and the surface is 57 C

7.1.2 Thermal Stresses Thermo-Optic Effects The temperature gradients generate mechanical stresses in the laser rod since the hotter inside area is constrained from expansion by the cooler outer zone. The stresses in a cylindrical rod, caused by a temperature distribution T(r), can be calculated from the Stress equations. Radial Stress Tangential Stress Axial Stress where the factor contains the material parameters is Youngs modulus, is Poissons ratio, and is the thermal coefficient of expansion.

We notice that the stress distributions also have a parabolic dependence on r The stress components represent compression of the material when they are negative and tension when they are positive. The center of the rod is under compression. The radial component of the stress goes to zero at the rod surface, but the tangential and axial components are in tension on the rod surface Thermo-Optic Effects

the stresses as a function of radius inside the Nd :YAG rod: Tensile strength of Nd :YAG is 1800 to 2100 kg/cm2 As the power dissipation is increased, the tension on the rod surface increases and may exceed the tensile strength of the rod, thereby causing fracture.

Thermo-Optic Effects It is of interest to determine at what power level this will occur The total surface stress max is the vector sum of and z : the tension on the surface of a laser rod depends on the physical constants of the laser material and on the power dissipated per unit length of the material, but does not depend on the cross section of the rod.

Stress Fracture Limit The mechanical properties of the laser host material determine the maximum surface stress that can be tolerated prior to fracture. If max is the maximum surface stress at which fracture occurs, then we can rewrite Where is a thermal shock parameter. A larger Rs indicates a higher permissible thermal loading before fracture occurs. Thermo-Optic Effects

7.1.3 Photoelastic Effects Thermo-Optic Effects A change of refractive index due to strain is given by a small change in shape, size, and orientation of the indicatrix. The change is specified by small changes in the coefficients where P i jkl is a fourth-rank tensor giving the photoelastic effect. k l is a second-rank strain tensor.

In a cylindrical coordinate system the photoelastic changes in the refractive index for the r and polarizations are given by the refractive-index changes are given by where and are functions of the elasto-optical coefficients of Nd :YAG Thermo-Optic Effects

7.1.4 Thermal lensing Thermo-Optic Effects The change of the refractive index can be separated into a temperature- and a stress-dependent variation. where n(r) is the radial variation of the refractive index, n0 is the refractive index at the center of the rod, and n(r ) T, n(r ) are the temperature- and stress-dependent changes of the refractive index, respectively. The temperature-dependent change of the refractive index the refractive index shows a quadratic variation with radius r

The focal length of a lens-like medium, where the refractive index is assumed to vary according to is given by This expression is an approximation where it was assumed that the focal length is very long in comparison to the rod length. The distance f is measured from the end of the rod to the focal point. The total variation of the refractive index is obtained by Thermo-Optic Effects

Then The end effects: The deviation from flatness of the rod ends is obtained from where is the length of the end section of the rod over which expansion occurs. With

Thermo-Optic Effects The focal length of the rod caused by an end-face curvature is obtained from the thick-lens formula of geometric optics where the radius of the end-face curvature is the focal length of the rod caused by a physical distortion of the flat ends The combined the temperature- and stress-dependent variation of the refractive index and the distortion of the end-face curvature of the rod lead to where A is the rod cross-sectional area and P h is the total heat dissipated in the rod.

Thermo-Optic Effects theoretical and measured thermally induced back focal lengths of various laser rods are plotted as a function of lamp input. (A) the radially and (B) tangentially polarized beam components, and (C, D) measurements of average focal length for different rods and pump cavities

7.1.5 Stress Birefringence it was shown that the principal axes of the induced birefringence are radially and tangentially and that the magnitude of the birefringence increases quadratically with radius r a linearly polarized beam passing through the laser rod will experience a substantial depolarization. is radial refractive index component is tangential refractive index component is the polarization vector for incident radiation Thermo-Optic Effects

Radiation incident at point P must be resolved into two components, one parallel to and the other parallel to. Since, there will be a phase difference between the two components and the light will emerge elliptically polarized. Birefringence effects in pumped laser rods can be studied by collimated light beam from an HeNe laser Thermo-Optic Effects Thermal stresses in a 7.5-cm long and 0.63-cm-diameter Nd : LaSOAP crystal. Input power (a) 115W, (b) 450W, (c) 590W, and (d) 880W

When a birefringent crystal is placed between a polarizer and analyzer that are parallel, the transmitted intensity is given by Thermo-Optic Effects where is the angle between the polarizer and one of the principal birefringence axes and is the polarization phase shift of the light emerging from the crystal. The index difference,, leads to a phase difference the difference in optical path length normalized to the wavelength

As can be seen from this figure, at maximum lamp input power of 12kW, the path-length difference is approximately six wavelengths. we can calculate the total transmitted intensity by integrating over the cross-sectional area of the rod: Thermo-Optic Effects

with the intracavity power which is polarized orthogonal to the polarizer will actually be ejected from the cavity and represents the depolarization loss of the resonator Thermo-Optic Effects

for a TEM00 mode for which it was assumed that the beam radius For the same lamp input power the losses for the TEM00 mode are less than for a highly multimode beam. This is expected since the energy in the TEM00 mode is concentrated nearer the center of the rod, where the induced birefringence is smaller. Thermo-Optic Effects

The interaction of a linearly polarized beam with a birefringent laser rod and a polarizer not only leads to a substantial loss in power, but also a severe distortion of the beam shape. Output beam pattern for a high-power cw Nd :YAG laser (a) without and (b) with a Brewster plate in the cavity. Thermo-Optic Effects

7.1.6 Compensation of Thermally Induced Optical Distortions Thermo-Optic Effects Complete compensation of the thermal aberrations produced by a laser rod is difficult because: (a) The focal length depends on the operating conditions of the laser and changes with pump power and repetition rate. (b) The thermal lens is bifocal due to the stress-dependent variation of the refractive index. (c) Nonuniform pumping leads to nonspherical aberrations.

In many pump configurations pump radiation is more intense at the center than at the periphery of the rod. The focal length of a given area in the rod is inversely proportional to the intensity of the absorbed pumped radiation. the focal length at the center of the rod is shorter than at the edges. the thermally induced refractive index profile contains terms that are higher than quadratic. A negative lens will remove the quadratic term; however higher- order effects cannot be compensated. Thermo-Optic Effects

The most common approaches are the insertion of a negative lens in the resonator Bifocusing (if thermal lensing compensation is combined with birefringence compensation) is eliminated and depolarization losses are minimized in resonators containing polarized beams. The objective of birefringence compensation is to achieve equal phase retardation at each point of the rods cross section for radially and tangentially polarized radiation. This can be accomplished by rotating the polarizations, either between two identical laser rods or in the same rod on successive passes, such that the radial and tangential components of the polarizations are exchanged.

Thermo-Optic Effects For example, birefringence compensation in an oscillator containing two identical laser heads can be achieved by inserting a 90 quartz rotator between the laser rods. The rotator produces a 90 rotation of every component of the electric field of the laser beam. The part of a mode that is radially polarized in the first rod, is tangentially polarized in the second rod. Since each part of the beam passes through nearly identical regions of the two rods, the retardation induced by one rod is reversed by the other.

7.2 Slab and Disk Geometries Thermo-Optic Effects The upper and lower surfaces are maintained at a constant temperature by water-cooling, and the sides are uncooled. thermal gradients are negligible in the x- and z-directions and the thermal analysis is reduced to a one-dimensional case, y axis.

Thermo-Optic Effects The maximum temperature that occurs between the surface and the center of the slab (y = d/2) is given by where Q is the heat deposition, d is the thickness, and K is the thermal conductivity of the slab. The temperature rise causes stress in the slab according to The surfaces are in tension and the center is under compression

Thermo-Optic Effects the maximum temperature difference allowed between the surface and the center before thermal fracture occurs With (thermal-shock parameter) for Nd : glass, one obtains For slabs of finite width w, the power per unit length at the stress fracture limit is given by where is the aspect ratio of a finite slab. It is interesting to compare the surface stress of a rod and slab for the same thermal power absorbed per unit length:

for x and y polarized light, respectively. The parameter is the contribution from thermal focusing, that is the parameters and are related to stress-induced focusing Thermo-Optic Effects The temperature and stress profile leads to a birefringent cylindrical lens. The focal lengths of the birefringent lens are where and are the stress optic coefficients for stress applied parallel and perpendicular to the polarization axis.

Thermo-Optic Effects 7.3 End-pumped Configurations In contrast to transversally pumped systems, heat deposition in end-pumped lasers is very inhomogeneous. The very localized heat deposition leads to highly nonuniform and complex temperature and stress profiles. Besides the temperature and stress-dependent variations of the refractive index, the contribution of end bulging to the formation of a thermal lens can be substantial in end-pumped lasers.

Thermo-Optic Effects Inhomogeneous local heating and nonuniform temperature distribution in the laser crystal lead to a degradation of the beam quality due to the highly aberrated nature of the thermal lens. An end-pumped laser rod has a temperature profile across the pumped region which is a function of the distribution of pump radiation.

An Nd :YAG crystal with, a 15W pump beam from a diode array was assumed to be focused onto an Nd :YAG rod of 4.75mm radius. The pump beam, which enters the laser crystal from the left along the z- axis, has a Gaussian intensity distribution and a spot-size radius of 0.5mm in the x-direction. It was assumed that 32% of the incident pump radiation is converted to heat. Thermo-Optic Effects

A Gaussian pump beam incident on the crystal has been assumed where 0 is the absorption coefficient and w p is the (1/e 2 ) Gaussian radius of the pump beam. With P h the fraction of the pump power that results in heating, the effective focal length for the entire rod can be expressed by where K is the thermal conductivity of the laser material and dn/dT is the change of refractive index with temperature.

Thermo-Optic Effects An end-pumped Nd :YAG rod with a length of 20 mm and a radius of 4.8mm was pumped with a fiber-coupled laser-diode array. The output from the fiber bundle was imaged onto the crystal surface into a pump spot with radius w p = 340m.

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Thermo-Optic Effects Chapter 7 Solid State Lasers By W. Keochner