Beam quality improvement by gain guiding effect in end-pumped Nd:YVO<sub>4</sub> laser amplifiers

Zhen Xiang; Dan Wang; Sunqiang Pan; Yantao Dong; Zhigang Zhao; Tong Li; Jianhong Ge; Chong Liu; Jun Chen

doi:10.1364/OE.19.021060

1. Introduction

“Gain guiding” effect widely exists in all kinds of laser oscillators and amplifiers, such as diode-pumped solid state lasers (DPSSL) [1–3] and amplifiers [4–6], fiber optic lasers (FOL) [7], free electron lasers (FEL) [8] and microchip lasers [9]. When a laser beam goes through an active medium with spatially non-uniform gain distribution, the transverse profile of laser intensity is reshaped. This is the gain guiding effect. For most of the end-pumped laser amplifiers, the gain coefficient is higher in the center of the active medium than in the margin. Thus the lowest order laser mode owning a better overlap with the gain profile has higher gain than all the other modes. Meanwhile, in the wings of the pump region, higher thermal-induced stress and nonlinear refractive index profile produce the larger thermal phase aberration, which increases diffraction loss of higher order modes. The high gain for the lowest order mode and the high loss for higher order mode both lead to a high beam quality laser output.

For laser oscillators, high quality TEM₀₀ mode Gaussian laser output by gain guiding effect was demonstrated theoretically and experimentally by Salin.F [1]. In 2011, Yan etc. emphasized that, for most actual laser resonators especially pumped at high power, the combined guiding mechanism including gain guiding effect and index guiding effect was determinant for the transverse mode formation. The traditional geometric design criterion for TEM₀₀ mode resonator, which only considers the spatial diffraction effect, is not suitable for the case of high power pumping [3]. However, for laser amplifiers, previous studies generally focused on the excess spontaneous emission, which played a role in the phenomena of beam pointing fluctuation [6, 10]. Recently, Yan etc. experimentally demonstrated the beam quality enhancement by gain guiding effect in a Nd:YVO₄ laser amplifier [5]. Nonetheless, the influences of each parameter of the amplifier on the beam quality enhancement were not fully investigated in their work.

In this work, we develop a theoretical model for an end-pumped laser amplifier, in which gain guiding, gain saturation and thermal effect are all taken into account. In our model, the wave equation is solved by a practical algorithm, based on faster Fourier transform (FFT). Although this algorithm has something in common with that in the laser oscillators [3], there are still many differences between them. For a laser amplifier, the input laser beam has a great influence on the performance of the amplifier. The intensity profile of the input beam can impact the gain distribution in the active laser medium due to gain saturation effect. And the power of the input beam can change the thermal loading in the amplifier, which results in the changes of the temperature and refractive index distribution. The simulation results are discussed to show the influence of each design parameter on the performance of an amplifier. An end-pumped Nd:YVO₄ laser amplifier setup is built up and investigated experimentally. The experimental results agree well with the prediction of our theoretical model.

The paper is organized as follows. In chapter 2, the theoretical model is presented for the laser beam going through an active medium with non-uniform gain and refractive index profile. In chapter 3, the influences of several key parameters of the amplifier on beam quality improvement are discussed, such as the input beam quality, the beam filling factor between input beam and pump beam, the ratio between input power and pump power and the length of laser crystal. In chapter 4, an experimental setup of end-pumped Nd:YVO₄ laser amplifier is built up and investigated. The experimental results are compared with the theoretical simulations. The conclusions are drawn in the last chapter.

2. Theory and model

2.1 Principles of the simulation for beam propagation and power scaling

In the earlier research [11], solving the Maxwell's equations is the primary method to get the amplified laser distribution including intensity and phase profile. Analytical methods are always used for solving the Maxwell's equations, in which the gain distribution in the active medium is assumed to be a Gaussian profile or a quadratic profile. But in practice, the gain profile in the active medium changes remarkably with the distribution of pumping beam. Meanwhile, it depends on the input signal laser intensity considering gain saturation effect. Therefore, the former hypothesis on the gain distribution is not accurate enough.Our starting point for describing a laser beam passing through an amplifier is the Maxwell's equations. In steady state condition and with the approximations of monochromatic, paraxial, and slowly varying envelope, the wave equation in the positive z direction is [12]

\nabla_{⊥}^{2} ψ - 2 i k \frac{\partial ψ}{\partial z} = - (i k g + 2 k k_{0} Δ n) ψ,

where

\nabla_{⊥}^{2} = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}

,

k = n_{0} ω / c

,

k_{0} = ω / c

, Δn represents the nonlinear part of the refractive index mainly determined by the thermo-optic effect, g is the gain coefficient in the active medium and ω is the circular frequency. The electric field component E is given by

E = Re (ψ) \exp [i (ω t - k z)] .

In general, ψ, Δn and g are the functions of spatial coordinates x, y, z. Other parameters and their values used in our simulation are listed in Table 1 .

Table 1. Parameters for the Theoretical Model of Gain-Guided Laser Amplifier

View Table

The formal solution of Eq. (1) can be represented as following,

ψ (x, y, z + Δ z) = \exp [(- \frac{i}{2 k} \nabla_{⊥}^{2} + \frac{g}{2} - i k_{0} Δ n) Δ z] ψ (x, y, z) .

The Eq. (3) can be replaced to second-order accuracy in Δz by the symmetrized spit operator [13]

ψ (x, y, z + Δ z) = \exp (- \frac{i Δ z}{4 k} \nabla_{⊥}^{2}) \exp [(\frac{g}{2} - i k_{0} Δ n) Δ z] \exp (- \frac{i Δ z}{4 k} \nabla_{⊥}^{2}) ψ (x, y, z) .

The Eq. (4) is the theoretical basis of the following algorithm for solving the Eq. (1). In the right side of Eq. (4), there are three operators before the function ψ(x,y,z). Two of them are the same, i.e.

\exp [- (i Δ z / 4 k) \nabla_{⊥}^{2}]

, which represent the propagating along a distance of Δz/2 in a homogeneous medium. While the other one, i.e.

\exp [(g / 2 - i k_{0} Δ n) Δ z]

, represents the amplitude and phase changes.

We show the iterative procedure in the following way (see Fig. 1 ). The active medium in Fig. 1 is split into many slices (S₁, S₂, …, S_n). Taking the slice S_k as an example, we define ψ(x,y,z) and ψ(x,y,z+Δz) are the fields at the input and output surface of slice S_k respectively. We need three steps to get ψ(x,y,z+Δz) from ψ(x,y,z), i.e., firstly propagating a distance of Δz/2, secondly adding the amplitude and phase changes and thirdly propagating a distance of Δz/2 again. The three steps exactly correspond to the three operators on the RS of Eq. (4).

Fig. 1 The scheme for the iterative procedure of the slice algorithm operating in solving the Maxwell's equations.

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Define

ψ_{1} (x, y, z + Δ z / 2) = \exp (- \frac{i Δ z}{4 k} \nabla_{⊥}^{2}) ψ (x, y, z),

which represents the input beam ψ propagating a distance of Δz/2 in a homogeneous medium with refractive index n₀. With the help of scalar angle-spectrum theory of diffraction based on FFT, we have the solution of Eq. (5) as

ψ_{1} (x, y, z + Δ z / 2) = F^{- 1} {F [ψ (x, y, z)] \exp (i k (Δ z / 2) \sqrt{1 - λ_{l}^{2} f_{x}^{2} - λ_{l}^{2} f_{y}^{2}})},

where

F

,

F^{- 1}

are the Fourier transform operator and inverse Fourier transform operator respectively, and f_{x, y} are the spatial angular frequencies in the Fourier domain.

Define

ψ_{2} (x, y, z + Δ z / 2) = \exp [Δ z (\frac{g}{2} - i k_{0} Δ n)] ψ_{1} (x, y, z + Δ z / 2),

which represents the amplitude and phase changes of the input beam after going through a slice with length of Δz. Owing to the fact that Δz in our calculation is very small, it is reasonable to take g and Δn as constant in the range of Δz. The g is determined by input and pump beam intensity profile at the position of z+Δz/2. The Δn is determined by the temperature distribution at z+Δz/2 caused by pump beam.

Define

ψ (x, y, z + Δ z) = \exp (- \frac{i Δ z}{4 k} \nabla_{⊥}^{2}) ψ_{2} (x, y, z + Δ z / 2) .

Employing the similar method (scalar angle-spectrum theory of diffraction), we have the field at the output surface of the slice S_k,

ψ (x, y, z + Δ z) = F^{- 1} {F [ψ_{2} (x, y, z + Δ z / 2)] \exp (i k (Δ z / 2) . \sqrt{1 - λ_{l}^{2} f_{x}^{2} - λ_{l}^{2} f_{y}^{2}})} .

So far, we obtain the approximate solution of beam distribution passing through the slice S_k. By means of iteration (see Fig. 1), the laser beam distribution after going through the entire active medium could be obtained with numerical method.

2.2 Gain saturation effect

The gain coefficient g in the active medium is a key parameter for gain guiding effect. With the increase of g, the gain guiding effect is reinforced. For an end-pumped four-level laser system, the small signal gain coefficient g₀ is proportional to the pump power deposition density, which is a function of the incident pump intensity distribution P_pump and the absorption coefficient α at the pump wavelength λ_p. Assuming the normalized pump intensity profile does not change with the length of the gain medium, we can simply express g₀ as [14]

g_{0} = \frac{α σ_{21} τ_{f} P_{p u m p}}{h v_{p}} \exp (- α z) .

For small input power, the gain coefficient g nearly equals to the small signal gain coefficient g₀. However, in the situation of high input power, the gain coefficient g could be changed distinctly by the input laser distribution I_l (x,y,z). Therefore, the gain saturation effect must be taken into account. The basic equation governing the growth rate of the input beam along a lossless amplifier is given as

g (x, y, z) = \frac{1}{I_{l} (x, y, z)} \frac{d I_{l} (x, y, z)}{d z} = \frac{g_{0} (x, y, z)}{1 + I_{l} (x, y, z) / I_{s a t}},

where I_sat is the saturation intensity, whose physical meaning is that the input beam intensity required to reduce the gain to one half of the small signal value, and

I_{s a t} = h v_{l} / σ_{21} τ_{f} .

According Eq. (11), since the gain coefficient g and input laser distribution I_l (x,y,z) are interrelated, it is difficult to obtain an analytical expression of the gain coefficient. However, we can use the iteration algorithm mentioned above to solve this problem approximately.

2.3 Thermo-optic effect

The general steady-state heat conduction equation for an anisotropic cubic solid in Cartesian coordinates is given by [15]

K_{x} \frac{\partial^{2} T (x, y, z)}{\partial x^{2}} + K_{y} \frac{\partial^{2} T (x, y, z)}{\partial y^{2}} + K_{z} \frac{\partial^{2} T (x, y, z)}{\partial z^{2}} = - Q (x, y, z),

where T and Q are the temperature and thermal loading intensity in the active medium respectively. In this context, the crystal is pumped by a super-Gaussian laser beam. We suppose the distribution of thermal loading is the super-Gaussian profile in the radial direction and an exponential decay along the axis of the crystal rod. The thermal loading intensity distribution in the active medium is given by

Q (x, y, z) = Q_{0} \exp [- 2 {(x^{2} + y^{2})}^{2} / w_{p}^{4}] \exp (- α z),

where Q₀ is the thermal loading intensity in the center of the input surface of the crystal.

Q_{0} = \frac{η (I_{p} - I_{e x t r a c t})}{\underset{c r y s t a l}{∭} \exp [- 2 {(x^{2} + y^{2})}^{2} / w_{p}^{4}] \exp (- α z) d x d y d z},

in which I_extract is the extracted power by the input laser beam, i.e., I_extract = I_out – I_in. η is the fraction of residual pump power converted into heat. We set η=0.43 by considering the fact that “dark neodymium ions” which absorb pump photons but do not contribute to the population inversion, and the absorbed pump power partly changing into fluorescence. This value is the same as the value of non laser emitting process according to Walter Koechner [16].

The temperature gradient induces the refractive index changes Δn in the active medium. For a paraxial beam propagating in the z direction, neglecting the axial strain and strain-induced birefringence, we could get

Δ n = \frac{d n}{d T} (T - T_{0}) .

2.4 Methods to evaluate the beam quality

In previous researches, due to the function of mode selection of laser resonators, stable output distribution could be obtained, and beam quality could be judged by near or far field distribution [17]. However, this way is not suitable for a high power pumped laser amplifier because the intense thermal lens with aberration remarkably changes the mode of input beam. The near and far field distribution of output laser is visibly different. We evaluate beam quality by calculating the beam quality factor M², which is invariable no matter in the near or far field. Based on the beam propagation equation in free-space (see Eq. (6)), the hyperbolic-fit method is employed to calculate the M² factor of input and output beams. The hyperbolic-fit equation of laser beam width is (2w)² = Az² + Bz + C. The corresponding expression of M² factor is [18]

M^{2} = \frac{π}{λ} \sqrt{A C - \frac{B^{2}}{4}},

where A, B, C is the fitting coefficient and w is the beam radius. In this paper, all beam radii are calculated by means of second-order moment as follows,

w_{x}^{2} = \frac{4 \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} x^{2} E (x, y, z) E^{*} (x, y, z) d x d y}{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E (x, y, z) E^{*} (x, y, z) d x d y},

where we suppose the beam centre of gravity is at its symmetric center. For a symmetrical Cartesian system, we have w = w_y = w_x.

3. Simulation results and analyses

The configuration of a typical end-pumped laser amplifier is shown in Fig. 2 . The composite YVO₄/Nd:YVO₄ crystal is end-pumped by fiber-coupled diodes. The pump beam has a super-Gaussian profile. We suppose that it does not diverge over the length of gain medium, i.e., the pump beam radius w_p is constant.

Fig. 2 The configuration of a typical end-pumped laser amplifier.

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Simulations are performed based on the theory in chapter 2 to show the dependence of output beam quality on related parameters in the gain-guided laser amplifier. If not specially specified in the following calculations and discussions, the following values are fixed w_p=500μm, w_l=400μm, I_p=25W, I_in=10W, L=16mm, $M_{in}^{2}$ = 2.2. It should be noticed that w_l here is not the beam waist radius but the beam radius at the input surface of the crystal. $M_{in}^{2}$ is the M² factor of the input beam.

To show the relationship between output beam quality and input beam quality, three different input beams with spherical aberration are composed, which are the Gaussian beam with spherical aberration (GA), the hollow beam with spherical aberration (HA), and the quasi-Gaussian beam with spherical aberration (QGA). Their distributions and beam quality factors are shown in Fig. 3 . These input beams are formed by means of amplitude distribution multiplying the phase with spherical aberration, i.e.,

ψ (x, y, 0) = A (x, y, 0) \exp [i φ (x, y, 0)],

where A and φ represent the amplitude and phase distribution respectively. For most of the high power oscillators and amplifiers, the deterioration of beam quality is caused by the spherical aberration in phase. The phase distribution φ has the expression as

φ (x, y, 0) = c_{0} {(x^{2} + y^{2})}^{2} + φ_{0},

where φ₀ is a constant and c₀ is the spherical aberration coefficient. In this way, we have the input signal beams for our simulations, which are similar to those in actual master-oscillator power-amplifier (MOPA) systems. If we consider only the amplitude distribution of the three input beams (i.e. φ=φ₀ with c₀ is zero), the GA, HA and QGA beams have little difference on the beam quality factors, which are 1.0, 1.15 and 1.1 respectively. By changing the value of c₀, we could obtain different beam quality factors. In this way, Fig. 3 shows three different beams with the same beam quality factor of 2.2 but different intensity profiles.

Fig. 3 The power distributions and fitting curves of three composed input beams. (a) the Gaussian beam with spherical aberration (GA), (b) the hollow beam with spherical aberration (HA), (c) the quasi-Gaussian beam with spherical aberration (QGA).

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Figure 4 shows the M² factor of output beam as a function of the M² factor of input beam for w_p = 400μm. The three sets of dots represent the situations for three different input beams respectively. The beam quality improvements are obvious in Fig. 4 for all these beams, though their input beam distributions are quite different. The best output beam quality is obtained (M² = 1.08) with the input beam M² factor of 1.5. We also observe that if the input beam has a very good beam quality, for example M² = 1.1, the quality of output beam is deteriorated instead of getting better after going through the amplifier, which was also illustrated by Minassian etc. in experiments [19]. This is mainly caused by two aspects. Firstly, the thermal distortion induced by spherical aberration in the amplifier is larger than that of the input beam in this situation, which could remarkably modify the phase distribution of the input beam, and lead to the degradation of beam quality. Secondly, the physical mechanism of gain guiding effect is that it magnifies the fundamental mode but suppresses higher order modes, which results in better beam quality laser output. The input beam with high beam quality means that it has only a small percentage of higher order modes. This results in the limited function of gain guiding effect for beam quality improvement. Therefore, in this situation, the gain guiding effect could not enhance the beam quality, whereas the thermal distortion deteriorates it.

Fig. 4 The relationship between M² factor of input beam and M² factor of output beam.

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Figure 5 shows the M² factor of output beam and the output power with different pump radii in the situation of w_l = 300, 400, 500μm respectively. For each curve in the Fig. 5(a), there is an optimum pump radius w_p, which corresponds to the best output beam quality. The optimum w_p always has a value of ~0.9 times of w_l. This means an effective transverse gain aperture for mode selection. If w_p is too small, such as w_p = 100μm, output beam quality is deteriorated rapidly due to severe thermal effect induced by the concentrating pump. If w_p is too large, such as w_p = 900μm, M² factor of the output beam is approximately equal to that of the input beam. This is because the large pump area decreases the gain intensity, which weakens the gain guiding effect according to Eq. (11). For the output power from the amplifier, things are little different. Larger pump radius always leads to lower output power. One can see in Fig. 5(b) that an increase of the pump radius leads to an almost linear decrease of the output power for all these three situations.

Fig. 5 The dependency of the M² factor of output beam and output power on the pump radii in the situation of w_l = 300, 400, 500μm respectively.

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We have mentioned above that there are two main mechanisms in the amplifier that influence the output beam quality. One is the fundamental mode selection mechanism based on gain guiding effect. The other is the thermal spherical aberration induced wavefront deterioration. The former improves beam quality and the latter deteriorates it. Both of them are related to many parameters in the amplifiers, such as pump power, pump beam radius, beam filling factor (defined as a ratio of the input beam radius to the pump beam radius), gain saturation effect and so on. To obtain better output beam quality, we should strengthen the gain guiding effect and weaken the thermal effect. But the two mechanisms are interrelated. For example, if we increase the pump power to strengthen the gain guiding effect, the thermal effect always becomes severer too. Therefore, we have to make a compromise between the two mechanisms. Figure 6 shows the M² factor of output beam as a function of input power with the pump power of 25W and 40W respectively. One can see that the output beam quality in the case of I_p=25W is better than that in the case of I_p = 40W even though the gain guiding effect is obvious stronger in the latter than in the former. In the situation of 40W pump power, the thermal effect is nearly doubled with the same pump radius. The thermal effect becomes the main mechanism that influences the beam quality at this time. This result is confirmed by Schulz etc. in the experiment of end-pumped laser amplifier [20]. Meanwhile, we find in Fig. 6 that the beam quality goes better with the increasing input power. This is hard to believe because the gain guiding effect is weakened due to the gain saturation according to Eq. (11). But one can understand it easily considering thermal effect. Higher input power means more extracted power from the amplifier and less thermal loading. This is very helpful to weaken the phase distortion and improve the beam quality. This prediction agrees well with experimental results shown in chapter 4.

Fig. 6 The dependency of the M² factor of output beam on the input power in the situation of I_p = 25, 40W respectively.

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Figure 7 depicts the M² factor of output beam and output power versus the length of crystal from 2mm to 18mm. The M² factor of output beam decreases with the increase of crystal length till L = 14mm and then keeps nearly unchanged (M² = 1.2). Furthermore, the output power increases with the increase of crystal length till L = 16mm.

Fig. 7 The M² factor of output beam and output power versus the length of crystal.

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This chapter discusses in detail, the simulation results of laser amplification and the design considerations for beam quality improvement and the power scaling in an end-pumped laser amplifier. Many design parameters in this amplification process have to be optimized, where various tradeoffs are involved. For example, one should make a compromise between the output beam quality and output power firstly. The degradation in beam quality, caused by thermally induced spherical aberrations, will reduce the brightness more than the potential increase in brightness due to power amplification. Secondly, with end-pumping scheme, the overlap between the signal beam and pump beam (i.e., the beam filling factor) is critical. To improve the beam quality, the signal beam radius is usually chosen to be a little larger than the pump beam radius. If the beam filling factor is too small, the gain in the wings is lost by the spontaneous emission. However, if it is too large, the thermal spherical aberrations in the wings of the pump distribution can significantly reduce the beam quality. Similarly, the pump beam size needs to be optimized. If it is too small, stress fracture of the crystal may occur and thermal lens may become strong. If it is too large, the extraction efficiency in the crystal will be reduced due to a lower inversion density. Thirdly, the ratio between input power and pump power should be carefully concerned. This is related to the two main problems in the amplifier, i.e., the gain saturation effect and the thermal loading in the active medium. Although higher input power weakens the gain guiding effect due to gain saturation, it means higher exaction efficiency and lower thermal loading, which leads to a better output beam quality and higher output power. The value of the power ratio should be optimized experimentally. To sum up, the improvement of thermal management and the enhancement of gain guiding effect are the effective ways to obtain the output beam with high power and good beam quality simultaneously.

4. Experimental results

The experimental setup is the same as that shown in Fig. 2. A composite a-cut Nd:YVO₄ crystal is used as the gain medium, which is an efficient four-level laser crystal to provide high gain when being pumped at 808nm and lasing at 1064nm. The composite crystal consisted of a 2mm thermal bonding end-cap on both ends and 16mm 0.3 at.% doped region. The undoped end cap was used to weaken the thermal effects [5]. Both end surfaces of the crystal are anti-reflection (AR) coated at 808nm and 1064nm. A fiber coupled laser diode (LD) delivering maximum output power of 50W is used to pump the laser crystal. Using a simple telescope imaging system as the coupling lens system, we could obtain the different pump spot sizes. The fiber had a 0.22 numerical aperture and a 400μm diameter. The laser crystal is pumped through a dichroic 45° mirror which is AR-coated for the pump wavelength and high-reflection coated for the laser wavelength. The temperature of the LD is controlled by a thermoelectric cooling module (TECM), and the center wavelength of the LD could be temperature-tuned by adjusting the temperature of the LD with TECM. The effective cooling of laser crystal is realized by mounting the crystal with indium foil in a water cooled copper holder.

The signal beam is from a Nd:YVO₄ oscillator. We can control the signal beam quality by changing the resonator length and the pump power. To show the beam quality improvement due to the gain guiding effect in the amplifier, we use a signal beam with M² factor of 2.2. It is coupled by a positive lens into the amplifier. The signal beam has a nearly Gaussian profile in the far field after the coupling lens. But it has different intensity patterns at different positions in the focal region. This is typical for a Gaussian beam with strong spherical aberrations. In this way, we can have signal beams with the same M² factor but different intensity patterns. The beam intensity profiles of signal beam and output beam from the amplifier are shown in Fig. 8 . The intensity profiles in Fig. 8(a) and Fig. 8(c) are from the same signal beam but at different positions near the focal region. The profiles in Fig. 8(b) and Fig. 8(d) are from the corresponding output beam respectively. The beam quality is improved to be M² = 1.4 for both of the two situations.

Fig. 8 The beam profiles and beam quality factors before and after the amplification. (a), (c) are from the same signal beam but at different positions near the focal region; (b), (d) are from the corresponding output beam respectively.

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Figure 9 shows the M² factor of output beam with different input beam radii in theory and in experiment respectively. It can be seen from the curve that w_l slightly larger than w_p makes the best results for beam quality enhancement in a gain-guided laser amplifier. This has been predicted in chapter 3. And it is shown again the beam filling factor is very important for an end-pumped laser amplifier.

Fig. 9 The M² factor of output beam with different input beam radius in experiments and in theory.

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Figure 10 shows the M² factor of output beam with different input power for the pump power of 25W and 40W in theory and in experiment respectively. The curves in the two pictures show the agreement between the experiments and theory, which also confirms our model and analyses in chapter 2 and chapter 3. We suppose that the small distinction between theoretical and experimental results is mainly due to the absorption coefficient. It is difficult to get the exact value of the absorption coefficient in experiments, which is related to the operating temperature and the drive current of pumping LD.

Fig. 10 The M² factor of output beam with the input beam power in experiment and theory for I_p = 25, 40W respectively.

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In general, the LD operating temperature influences the pump wavelength, which leads to variation of active medium absorption coefficient. This is an effective instrument for us to adjust the performance of the amplifier, such as the gain and thermal distribution. We measured the optical spectrum of the pump diodes in our experiments. With the diodes output power of 25W, the central wavelength is 807nm at 30°C. It can be temperature-tuned with a coefficient of 0.3nm/°C. Figure 11 shows the output beam quality from the amplifier with different LD temperature. The absorption coefficient is not only determined by the pump diodes wavelength but also by Nd³⁺ ion doping concentration in the active medium. In our experiment, with 0.3 at.% doping concentration, the output beam quality goes better with higher LD operating temperature.

Fig. 11 The M² factor of output beam with LD operating temperature in experiment.

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5. Conclusions

Considering thermal effect, gain guiding effect and gain saturation effect comprehensively, we develop a theoretical model for an end-pumped laser amplifier. Numerical simulations are performed to show the output beam quality and output power in a Nd:YVO₄ laser amplifier with different input beam quality, different signal and pump beam radius, different input and pump power. Some useful conclusions are drawn for the design consideration of an effective amplifier. Several parameters, such as the beam filling factor, the ratio between input and output power, the absorption coefficient of active medium, have to be optimized. Moreover, an experiment is performed for an end-pumped Nd:YVO₄ laser amplifier. The experiment results approve the theoretical analyses and predictions. The theoretical analyses for a dual-end pumped and multiple-stages amplification will be carried out in the next step.

Acknowledgments

Project supported by the National Natural Science Foundation of China (Grant No. 60908013).

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Parameters	Symbol	Unit	Value
Pump beam wavelength	λ_p	nm	808
Signal beam wavelength	λ_l	nm	1064
Linear refractive index of the active medium	n₀		2.18
Stimulated emission cross section	σ₂₁	m²	15.6×10⁻²³
Upper state lifetime	τ_p	s	90×10⁻⁶
Thermal conductivity coefficient in x direction	K_x	W/mK	5.1
Thermal conductivity coefficient in y direction	K_y	W/mK	5.1
Thermal conductivity coefficient in z direction	K_z	W/mK	5.23
Pump beam frequency	v_p	Hz	3.71×10¹⁴
Signal beam frequency	v_l	Hz	2.82×10¹⁴
Planck constant	h	J×s	6.63×10⁻³⁴
Surroundings temperature	T₀	°C	20
Absorption coefficient	α	cm⁻¹	6
Input beam intensity	I_in	W
Output beam intensity	I_out	W
Pump beam intensity	I_p	W

Beam quality improvement by gain guiding effect in end-pumped Nd:YVO₄ laser amplifiers

Abstract

1. Introduction

2. Theory and model

2.1 Principles of the simulation for beam propagation and power scaling

2.2 Gain saturation effect

2.3 Thermo-optic effect

2.4 Methods to evaluate the beam quality

3. Simulation results and analyses

4. Experimental results

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (11)

Tables (1)

Equations (20)

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