Contribution of stress-dependent variation of refractive index to thermal lensing in Nd:glass laser rods

Jasbir S. Uppal and Jagdish C. Monga Bhabha Atomic Research Centre, Laser Division, Bombay 400 085, India. Received 27 April 1985. 0003-6935/85/223690-03$02.00/0. © 1985 Optical Society of America.

Thermal effects of the laser host material in optically pumped lasers are an important parameter for reliable oper­ation of repetitively pulsed solid-state lasers. These effects

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stem from the nonuniform heat dissipation in the laser rod, which is responsible for the optical distortions of the laser beam through the temperature and stress-dependent varia­tions of refractive index.1-3 Foster et al.4 and Koechner5

have studied the various effects contributing to the thermal lensing in Nd:YAG laser rods and have concluded that the major contribution toward thermal lensing arises due to the temperature-dependent variation of refractive index, where­as the stress-dependent term contributes only to the extent of ~20%. In a manner similar to Nd:YAG, it may be believed that in the case of Nd:glass as well, the contribution of a stress-dependent term is negligible, and there might be a tendency to ignore its contribution in estimating the ther­mally induced focal length of such lasers. The purpose of this Letter is to show that in the case of Nd:glass laser rods, the contribution of a stress-dependent variation of refractive index toward induced thermal lensing is not only significant but dominates that due to the temperature-dependent varia­tion of the refractive index.

In the case of flashlamp pumped Nd:glass lasers, the frac­tion of pump energy dissipated as heat inside the rod in­creases the temperature of the rod. Since the rod is cooled from the surface, the radial heat flow changes the initial temperature distribution to a parabolic temperature profile6

which decays with a certain time constant. The resulting refractive-index variation at any time during the cooling period can be represented by

where r is the radius at some point in the rod, and b is a constant that depends on the physical properties and tem­perature gradient within the laser rod. For the case where b is much larger than the length of the rod, the induced focal length is given by5

where n0 is the refractive index, and L is the length of the laser rod. If we include the contribution due to the endface curvature, the focal length fr,φ can be written as7

Table I. Physical and Optical Constants of Nd:YAG and Nd:Glass

where a is the radius of the laser rod, T(0) and T(a) refer to the temperature rise at the center and edge of the laser rod, respectively, a is the linear coefficient of thermal expansion, and Cr and Cφ are the functions of elastooptical coefficients for the radial and tangential components of polarized light, respectively. The first two terms in the parentheses of Eq. (3) refer to the contributions toward induced focal length due to the temperature and stress-dependent variations of the refractive index, respectively. The third term is due to the endface curvature caused by elongation of the rod.

Foster and Osterink4 and Koechner5 have derived the expressions for the functions of the elastooptic coefficients Cr and Cφ for Nd:YAG. Since Nd:glass is isotropic and has the same photoelastic matrix as that of Nd:YAG (cubic of class m3m) with 2p44 = p11 - P12, the expressions of Cr and Cφ for Nd:YAG are also applicable for Nd:glass material8:

where ν is Poisson's ratio, and P11, P12, and P44 are the elastooptic coefficients.

Using the physical constants from Table I, we find that in the case of Nd:YAG the contribution of the stress-dependent variation of the refractive index thermal lensing is only 22% of that of the temperature-dependent term, whereas for Nd:glass (LSG91H), this contribution is as much as 2.7 times that of the temperature-dependent term for the radial com­ponent of polarization. This is because, Nd:glass being an amorphous material, the values of elastooptic coefficients are much higher than the corresponding values for the crytal-line Nd:YAG material. This results in a larger variation in a stress-dependent refractive index for Nd:glass. Thermal lensing due to stress-dependent variations in Nd:glass is, therefore, ~4.8 times stronger than in Nd:YAG. On the other hand, the thermal coefficient of refractive index for Nd:glass is less than half of that of Nd:YAG, which reduces the thermal lensing due to temperature-dependent variation for Nd:glass to ~40% of the value for Nd:YAG. For the tangential component of polarization, the stress-dependent term is about two times that of the temperature-dependent term in Nd:glass, whereas in Nd:YAG, the stress term is <5% of the temperature-dependent term. The contribution due to the endface curvature term for both Nd:YAG and Nd:glass is of the same order and remains less than ~10% for the typical dimensions of laser rods.

The authors are grateful to P. K. Gupta for many helpful discussions.

References 1. G. D. Baldwin and E. P. Riedel, "Measurements of Dynamic

Optical Distortion in Nd-Doped Glass Laser Rods," J. Appl. Phys. 38, 2726 (1967).

2. L. M. Osterink and J. D. Foster "Thermal Effects and Transverse Mode Control in a Nd:YAG Laser," Appl. Phys. Lett. 12, 128 (1968).

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3. B. K. Sinha and N. Gopi, "Study of Thermal Relaxation in a Repetitively Pumped Nd:Glass Laser Rod," IEEE J. Quantum Electron. QE-16,433 (1980).

4. J. D. Foster and L. M. Osterink, "Thermal Effects in a Nd:YAG Laser," J. Appl. Phys. 41, 3656 (1970).

5. W. Koefhner "Thermal Lensing in Nd:YAG Laser Rod," Appl. Opt. 9, 2548 (1970).

6. J. M. Soures, S. Kumpan, and J. Hoose, "High Power Nd:Glass Laser for Fusion Applications," Appl. Opt. 13, 2081 (1974).

7. W. Koechner, Solid State Laser Engineering (Springer-Verlag, New York, 1976), pp. 353.

8. J. F. Nye, Physical Properties of Crystals (Clarendon, New York, 1957), pp. 250-51.

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