## Abstract

In the LD end pumped rectangular laser gain medium, the thermal induced refractive index is not only non-uniformly transversely, but also non-uniformly and distributed along the pumping beam propagation, the effect of thermal lens is a distributed not a lumped lens effect as previously considered. In this paper, the effect of a distributed thermal lens is analyzed.

© 2013 OSA

## 1. Introduction

LD end-pumped solid-state lasers are outstanding by their high efficiency, compact package and beam quality [1]. LD end pumping rectangular gain is distributed uniformly along the slow axis and non-uniformly along the fast axis, but shows the gradual change along the propagation direction. Therefore, the thermal lens effect of the laser crystal is gradually changing along the laser propagation direction. However, currently most of theoretical studies on LD pumping thermal lens effect are based on the uniform distribution of temperature along propagation direction [2] and circularly symmetrical two-dimensional distribution of thermal lens [3], so that thermal focal length are calculated by treating the medium as single lens. Furthermore, a compound lens model was also proposed by simply using the method of geometrical optics but neglecting the separation distance between each elemental component lens in some references, in which the optical path between the neighboring elemental lenses is not taken into consideration, obviously, there exist the optical path between the neighboring elemental lenses. In this paper, the optical path between the neighboring elemental lenses is taken into consideration which is more close to the practical case.

The temperature gradient is produced by LD end pumping in the rectangular laser medium in the directions of the fast axis and beam propagation. The temperature gradient and thermal stress make the original refractive index changes in the medium. The refractive index is distributed uniformly in the slow axis direction of the rectangular laser medium which is parallel to the cooling surfaces and non-uniformly in the fast axis which is perpendicular to the cooling surfaces. Therefore, the thermal lens effect in LD end pumped rectangular laser medium is a one-dimensional cylindrical thermal lens effect.

In this paper, the distributed cylindrical thermal lens effect of a rectangular laser crystal Yb:YAG with doping concentration of 2.5%, which has the absorption coefficient of 2.8/cm at the pumping wavelength 940nm, has been analyzed. The assumption of 70% of the absorbed pumping power transformed into heat is made. The pumping beam at the end surface of rectangular medium has a size of 10mm in width and from 0.2 to 0.6mm in height by using different focal length of beam forming optics. The thermal refractive index distribution in LD end pumped rectangular Yb:YAG crystal is derived by using the thermal distribution in reference [4]. The relationship between the equivalent thermal focal length in LD end pumped rectangular Yb:YAG crystal and the pumping power density and pumping beam size is studied.

The single and double end pumping Yb:YAG crystal structure are shown in Fig. 1 and Fig. 2. The thermal stress theory used in this paper is subject to the basic hypothesis of materials elasticity mechanics and thin plate theory [5]. Based on these hypothesis, there are only the normal stress ${\text{\sigma}}_{y}$ and ${\text{\sigma}}_{\text{z}}$ in the y and z direction respectively. With superposition method, the thermal stress in the slab are obtained [6]

Consequently, the refractive index is expressed as [4]

From Fig. 3 and Fig. 4,it can be seen that the refractive index distribution along z direction, which is the direction of pumping beam propagation, is gradually decaying under the condition of the single end pumping and symmetrical along z direction under the condition of the double end pumping, the refractive index distribution has a symmetric bell shape along x direction (fast axis) because of the cooling effect of the large surfaces of the rectangular medium, and the refractive index distribution is uniform along y direction (slow axis). When the laser beam propagates into the rectangular medium along z direction, the symmetric bell shape of the refractive index along x direction will cause the laser beam bending toward to the center, and the uniform refractive index distribution along y direction will not cause any bending of the laser beam, therefore, the end pumped rectangular medium can only have the focusing effect on y direction, the thermal lens effect in LD end pumped rectangular laser medium is a cylindrical lens. Because the refractive index is not uniform along z direction which is the direction of the pumping beam propagation, the focal length of the thermal cylindrical lens gradually change along z direction.

## 2. The equivalent group of the thermal cylindrical lens in LD end pumped rectangular laser medium

As described in the above section, the focal length of the thermal cylindrical lens gradually change along z direction, for the simplicity of analysis, the thermal cylindrical lens effect along z direction, the pumping beam propagation direction, is considered to be divided into a group of the elemental thin thermal cylindrical lens with the identical thickness of △Z, the summation of all thickness of the elemental thin thermal cylindrical lens will be the length of rectangular laser medium. For each elemental thin thermal cylindrical lens, the focal length will be different from the others because of the different refractive index at the different z position.

Thus, the total effect of the thermal cylindrical lens of LD end pumped rectangular laser medium can be described by the compound effect of a group of elemental thermal cylindrical lens. The method will be discussed in details below.

The LD pumped YAG medium is the lens-like medium whose refractive index n(x,y) varying near the z axis (the pumping beam propagation direction) is described by [8,9]

The above equation solution of the refractive index distribution is the numeric solution. For the LD end pumped rectangular laser medium, the refractive index in terms of the coefficient$\text{\gamma}\left(\text{z}\right)$ can be simply described by

For each elemental thermal cylindrical lens, the transformation matrix of the thermal lens takes the form of [10]

The each elemental thermal cylindrical lens has the equivalent geometrical thickness and the different focal length along pumping beam propagation direction, the focal length of the elemental cylindrical thermal lenses is gradually variable along the pump beam propagation direction, as shown in Fig. 5 for single end pumped and Fig. 6 for double end pumped.

Because the small thickness of the elemental cylindrical thermal lens and gradual variation of the thermal refractive index along the pumping beam propagation direction, it can be approximately thought that the refractive index of each elemental cylindrical thermal lens is constant along z direction which is the value of the refractive index at z position and only variable along x direction. Therefore, the LD end pumping rectangular laser medium can be made to be equivalent to a group of elemental cylindrical thermal lens with different focal length, and each elemental cylindrical thermal lens can be described by Eq. (8). Based on the ABCD law, the ray matrix relating the output plane to the input plane becomes

Based on the Eq. (9), the total thermal lens effect induced by the gradually variable refractive index distribution in the LD end pumping rectangular laser medium can be described as an equivalent thermal lens focal length [9]

Based on the above theory analysis and the data in Table 1, the total thermal focal length as a function of pumping beam radius and pumping intensity at the end surface of the rectangular medium can be discussed in details.

In following analysis, the total thermal cylindrical lens effects are studied in two categories of the pulse pumping source and the continuous pumping source. When the pulse pumping source is used, the laser medium is only heated during the pumping pulse duration; the thermal induced refractive index distribution is derived from the highest temperature distribution. When the continuous pumping source is used, the laser medium is heated continuously; the thermal induced refractive index distribution is derived from the steady state temperature distribution. For the high repetitive rate pumping source, it can be regarded as the case of continuously pumping at the average power.

In the following analysis, the temperature distribution is first calculated, and the stress distribution is obtained based on the temperature distribution, the refractive index distribution is the combination of the temperature gradient induced refractive index variation and the stress induced refractive index variation, the refractive index distribution can give every matrices M_{m}, each of which is equivalent to one different lens subject to its location in the gain medium, the product of matrices finally provide the value of 1/C_{m}, which gives the focal length of the total cylindrical thermal lens.

For the single pulsed pumping, the thermal induced refractive index distribution is derived from the highest temperature distribution of the laser medium; the total cylindrical thermal focal length can be given by Eq. (10). Figure 7(a) and 7(b) show the total thermal focal length as a function of the pumping beam radius with different pumping peak power intensity

In Fig. 7(a) and 7(b), the power source is a pulse LD laser, the pumping peak power were kept from 300watts to 1300watts to maintain the pumping peak power density of 20kw/cm^{2} with the pumping size on the end surface of the rectangular crystal from 0.15mm to 0.65mm high and 10mm wide; the pumping peak power were kept from 375watts to 1625watts to maintain the pumping peak power density of 25kw/cm^{2} with the pumping size on the end surface of the rectangular crystal from 0.15mm to 0.65mm high and 10mm wide; the pumping peak power were kept from 450watts to 1950watts to maintain the pumping peak power density of 30kw/cm^{2} with the pumping size on the end surface of the rectangular crystal from 0.15mm to 0.65mm high and 10mm wide. The pumping laser pulse width is kept at 300us; the heat induced refractive index distribution is derived at the highest temperature distribution of the laser mediums.

For the continuous pumping, the thermal induced refractive index distribution is derived from the steady state temperature distribution of the laser medium; the total cylindrical thermal focal length can be given by Eq. (10). Figure 7(c) and 7(d) show the total thermal focal length as a function of the pumping beam radius with different pumping power intensity

In Fig. 7(c) and 7(d), the power source is a continuous LD laser, the pumping power were kept from 300watts to 1300watts to maintain the pumping power density of 20kw/cm^{2} with the pumping size on the end surface of the rectangular crystal from 0.15mm to 0.65mm high and 10mm wide; the pumping peak power were kept from 375watts to 1625watts to maintain the pumping power density of 25kw/cm^{2} with the pumping size on the end surface of the rectangular crystal from 0.15mm to 0.65mm high and 10mm wide; the pumping power were kept from 450watts to 1950watts to maintain the pumping power density of 30kw/cm^{2} with the pumping size on the end surface of the rectangular crystal from 0.15mm to 0.65mm high and 10mm wide. The heat induced refractive index distribution is derived at the steady state temperature distribution.

Figure 7(e) and 7(f) show that the total cylindrical thermal focal length is a function of pumping intensity with the different pumping beam radius when the pumping source is a pulse laser diode.

In Fig. 7(e) and 7(f), the pumping source is a LD pulse laser, the peak power were kept from 50watts to 350watts to have the pumping peak power density from 5kw/cm^{2} to 35kw/cm^{2} with the pumping size of 0.10mm high and 10mm wide on the end surface of the rectangular crystal; The pumping peak power were kept from 100watts to 700watts to have the pumping peak power density from 5kw/cm^{2} to 35kw/cm^{2} with the pumping size of 0.20mm high and 10mm wide on the end surface of the rectangular crystal; The pumping peak power were kept from 150watts to 1050watts to have the pumping peak power density from 5kw/cm^{2} to 35kw/cm^{2} with the pumping size of 0.30mm high and 10mm wide on the end surface of the rectangular crystal. The pumping laser pulse width is kept at 300us; the heat induced refractive index distribution is derived at the highest temperature distribution.

Figure 7(g) and 7 (h) show that the total cylindrical thermal focal length is a function of pumping intensity with the different pumping beam radius when the pumping source is a continuous laser diode.

In Fig. 7(g) and 7(h), the pumping source is a LD continuous laser, the power were kept from 100watts to 350watts to have the pumping power density from 10kw/cm^{2} to 35kw/cm^{2} with the pumping size of 0.10mm high and 10mm wide on the end surface of the rectangular crystal; The pumping power were kept from 200watts to 700watts to have the pumping power density from 10kw/cm^{2} to 35kw/cm^{2} with the pumping size of 0.20mm high and 10mm wide on the end surface of the rectangular crystal; The pumping power were kept from 300watts to 1050watts to have the pumping power density from 10kw/cm^{2} to 35kw/cm^{2} with the pumping size of 0.30mm high and 10mm wide on the end surface of the rectangular crystal. The heat induced refractive index distribution is derived at the steady state temperature distribution.

From Fig. 7, it can be seen that the heat induced thermal focal length by a pulse pumping power is much smaller than that by a continuous pumping power. The higher pumping power intensity can produce the decreased thermal focal length. When the different pumping beam radius is kept at the same pumping power intensity, the bigger pumping beam radius will produce the decreased thermal focal length because the higher pumping power will produce the higher temperature and the higher temperature gradient.

## 3. Conclusion

For the LD end pumped rectangular laser medium, the heat induced refractive index distribution is uniform along the direction parallel to the large cooling surface of the rectangular medium, and is not uniform and bell shaped along the direction vertical to the two large cooling surface of the rectangular medium, and is variable along the direction of the pumping beam propagation. Thus, LD end pumped rectangular laser medium can be considered as a group of the elemental thermal cylindrical lens, each elemental thermal cylindrical lens with the different focal length made the contribution to the total thermal lens effect.

Based on the above analysis, it can be found that the total cylindrical thermal focal length decrease with the increase of pumping power intensity, and for the same pumping power intensity, the bigger pumping beam radius the shorter the thermal focal length. Because the increase of the pumping power intensity and the pumping beam radius at the pumping intensity will increase the temperature and the temperature gradient of the laser medium under the same cooling conditions.

It can also be found that the total cylindrical thermal focal length of single end pumped laser is almost two times longer than that of double end pumped laser. The thermal effects of the continuously pumping laser diodes are more significant than that of the pulsed pumping laser diodes in the laser medium.

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