A new approach to compute Overlap efficiency in axially pumped Solid State Lasers

Rakesh Kapoor; P. K. Mukhopadhyay; Jogy George

doi:10.1364/OE.5.000125

1. Introduction

In the last decade, the development of efficient, high power diodes and new laser materials has generated considerable interest in diode pumped solid state lasers. Axially pumped lasers offer higher efficiency and better beam quality than side pumped lasers. The qualities of axially pumped lasers make them more suitable for applications in laser medicine, high-density optical storage, color display, and optical testing. For optimum design of an axially pumped solid state laser, one of the most important parameter that determines the laser efficiency and the output power is the spatial overlap between the pump beam and the laser mode. The influence of the pump and laser mode size on the laser threshold and slope efficiency has been investigated by several authors [1, 2, 3, 4, 5]. Most of these workers use an average pump spotsize. Although this approach simplifies the problem of obtaining analytical expressions for optimisation, the effect of some parameters, such as the divergence of pump beam and waist location inside the crystal can not be taken into account. In some reports the overlap integral is computed numerically [6, 7] and the pump beam divergence and waist location have been taken into account. But in all these works, the overlap integrals are based on a linearized approximation near the threshold [5] and this approximation is only valid if the circulating intensity in the resonator is negligible in comparison to the saturating intensity. In practical conditions, this assumption is not valid above the threshold regimes of pump power.

In this work, we are reporting a new approach to calculate the overlap efficiency in axially pumped cw or quasi cw solid state lasers. In this approach we applied the linearized approximation in the above threshold regime. We first computed the overlap integral numerically for different values of circulating intensity. Then inverse of the overlap integral is plotted as a function of circulating field. Now by fitting a linear curve to this data, two coefficients are obtained and these coefficients are further used to estimate the overlap efficiency. We applied this method to calculate the mode overlap for both circular and elliptic pump beams. We have tried to find out how the pump beam with different beam quality-factor M ² can affect the efficiency of an axially pumped Solid State Laser. We have also tried to find out the effect of pump waist size on the overlap efficiency for a constant M ² value.

2. Theoretical Analysis

According to rate equation analysis, if I_p (z) is the incident pump intensity at any plane inside the gain medium and the reflectivity of the two end mirrors is equal to unity, then in steady state, if we neglect the standing wave effects, the saturated gain coefficient g(z) at that plane can be given as[8]

g (z) = \frac{η_{q} α_{a} I_{p} (z) ⁄ I_{sat}}{(1 + 2 I_{circ} ⁄ I_{sat})},

Where α_a is the absorption coefficient of the gain medium at the pump intensity and I_sat is the saturation intensity of the gain medium, η_q is the product of quantum efficiency and quantum defect of the system. In most of the four level solid state lasers the quantum efficiency is approximately equal to one therefore η_q is given by

η_{q} \approx \frac{ω_{m}}{ω_{p}},

where ω_m is the laser frequency and ω_p is the pump beam frequency. Eq.1 is valid only for the plane wave approximation, in real lasers the pump intensity as well as the mode intensity inside the gain volume, is a function of position. Therefore, the gain coefficient inside the gain volume is also a function of position. If we neglect the saturation of the pump power absorption, then the spatial distribution of pump intensity inside an axially pumped gain volume can be defined as

I_{p} (x, y, z) = \frac{P_{p}}{A_{p} (z)} f_{p} (x, y, z) Exp (- α_{a} z),

and

A_{p} (x) = \iint f_{p} (x, y, z) d x d y,

where P_p is the incident pump power, f_p (x, y, z) is the spatial profile function at a plane z inside the gain medium and A_p (z) is the pump area at the same plane. If P_circ is the total circulating power inside the resonator, the I_circ (x, y, z) at any point inside the gain medium can similarly be defined as

I_{circ} (x, y, z) = \frac{P_{circ}}{A_{m} (z)} f_{m} (x, y, z),

where f_m (x, y, z) is the spatial mode profile function at a plane z inside the gain medium and A_m (z) is the mode area at the same plane. If g(x, y, z) is the value of the saturated gain coefficient at a particular position (x, y, z) inside the gain volume, then the change in circulating power in one round trip is defined as

Δ P_{circ} = ∭ d I (x, y, z) d x d y = 2 \iint \int_{0}^{l} g (x, y, z) I_{circ} (x, y, z) d z d x d y,

where l is the length of the gain medium. Now for the small gain approximation the total saturated round trip gain G can be defined as

G = \frac{2 \int \int \int g (x, y, z) I_{circ} (x, y, z) d V}{P_{circ}}

Therefore, the total gain inside a resonator can be computed as follows

G = 2 η_{q} α_{a} {P'}_{p} \int \int \int \frac{Exp (- α_{a} z) f_{p} (x, y, z) f_{m} (x, y, z)}{A_{p} (z) A_{m} (z) (1 + \frac{2 {p'}_{circ} f_{m} (x, y, z)}{A_{m} (z)})} d V,

The normalised circulating power P′_circ is defined in terms of saturation intensity I_sat as

{P'}_{p} = P_{p} / I_{sat}

where I′_circ is the normalized circulating intensity, 〈A_m 〉 is the average mode area inside the gain medium and is defined as

〈 A_{m} 〉 = \frac{1}{l} \int_{0}^{l} \int \int f_{m} (x, y, z) dx dy dz,

In steady state, the total gain inside a resonator is equal to total loss therefore

G = 2 α_{0} z + \ln (\frac{1}{R_{1} R_{2}}) \equiv L + T_{1} + T_{2},

where R ₁ and R ₂ are the reflectivities of the two mirrors, and T ₁ and T ₂ are the corresponding transmissions. α ₀ is the absorption coefficient at the lasing wavelength. All the intracavity losses are clubbed in term L. If T ₂ is the output coupler transmission, the output power from a laser is given as

P_{out} = T_{2} P_{circ},

With the help of Eq.8, Eq.11 and Eq.12, one can compute the laser output power. Eq.8 looks quite complicated to compute, as the knowledge of the pump distribution function and mode distribution function inside the gain volume is necessary. In most of the practical conditions the output power of a laser changes linearly with input power, this is only possible if the total saturated round trip gain is approximated as follows

G = 2 η_{q} α_{a} {P'}_{p} [\frac{C}{(1 + 2 D {I'}_{circ})}],

where C and D are constants. By comparing Eq.13 with Eq.8, the value of the integral in Eq.8 will be given as

\int \int \int \frac{Exp (- α_{a} z) f_{p} (x, y, z) f_{m} (x, y, z)}{A_{p} (z) A_{m} (z) [1 + \frac{2 〈 A_{m} 〉 {I'}_{circ} f_{m} (x, y, z)}{A_{m} (z)}]} d V = \frac{C}{1 + 2 D {I'}_{circ}},

Here the constant C will have inverse of length dimensions and D is a dimensionless quantity. Now from Eq.11 and Eq.13, we get the following relation

2 η_{q} α_{a} {P'}_{p} [\frac{C}{(1 + 2 D {P'}_{circ} / 〈 A_{m} 〉)}] = (T + L),

where T=T ₁+T ₂. The circulating power, that must built up inside the resonator in order to saturate the gain factor down to where it can just be equal to total cavity losses, is given by

P_{circ} = \frac{I_{sat} 〈 A_{m} 〉}{2 D} [\frac{2 η_{q} α_{a} {P'}_{P} C}{T + L} - 1],

The output power from the laser will be given as

P_{out} = \frac{T_{2} η_{q} α_{a} P_{p} C 〈 A_{m} 〉}{D (T + L)} - \frac{T_{2} I_{sat} 〈 A_{m} 〉}{2 D},

If we define the overlap efficiency η_o as follows

η_{o} = \frac{α_{a} C 〈 A_{m} 〉}{η_{a} D},

where η_a is the absorption efficiency of pump power inside the gain medium. The output power equation can be rewritten as

P_{out} = \frac{T_{2}}{(T + L)} η_{q} η_{a} η_{o} P_{p} - \frac{T_{2} I_{sat} 〈 A_{m} 〉}{2 D},

with the help of Eq.19, one can easily find the threshold power. It will correspond to the pump power for which the output power is equal to zero. Therefore, the threshold power P_th can be given as

P_{th} = \frac{I_{sat} (L + T)}{2 η_{q} α_{a} C},

the expression for the slope efficiency m can also be obtained from Eq.19.

m = \frac{η_{o} η_{a} η_{q} T_{2}}{(L + T)},

it is clear from Eq.20 that for any given resonator conditions the threshold pump power is proportional to the factor 1/α_aC. If we compare this factor with the match function which describes the spatial overlap of pump beam and resonator modes [7], then the value of the match function F can be given as

F = 2 α_{a} C 〈 A_{m} 〉,

Considering a single transverse mode TEM ₀₀, f_m (x, y, z) can be given as

f_{m} (x, y, z) = Exp [- \frac{2 x^{2}}{ω_{mx}^{2} (z)} - \frac{2 y^{2}}{ω_{my}^{2} (z)}],

where ω_mx and ω_my are the radii of the beam along the x-axis and y-axis respectively. The pump profile function f_p (x, y, z) for a diode laser can be written as

f_{p} (x, y, z) = Exp [- \frac{2 x^{2}}{ω_{px}^{2} (z)} - \frac{2 y^{2}}{ω_{py}^{2} (z)}],

where ω_px and ω_py are the radii of the beam along the x-axis and y-axis respectively. The expression for the beam radius can be given as

ω_{p} (z) = ω_{p 0} {[1 + {(M^{2} z ⋋ ⁄ π ω_{p 0}^{2})}^{2}]}^{\frac{1}{2}},

where λ is the wavelength in the medium, ω_p ₀ is the radius at the waist, and M ² is the beam quality factor. The value of M ² and ω_p ₀ can be different for the x-axis and y-axis. For a particular beam, the beam quality factor M ² is a constant and is related to the waist radius ω ₀ and far field divergence angle θ of a beam as [9]

ω_{0} θ = \frac{M^{2} ⋋}{π},

With the knowledge of the pump-beam quality factor, pump waist radii and mode waist radii along with their respective waist positions inside the resonator, the integral given in Eq.14 can be solved numerically for various values of normalised circulating intensity I′_circ . In the linearized approximation near the threshold, both sides of Eq.14 are expanded into a series. The values of the constant C and D are obtained only from the first two terms of the series. While in the proposed method no such approximation is made. The value of the inverse of the integral can be plotted as a function of I′_circ and by fitting a linear curve the values of the constant C and D can be obtained. There after the value of overlap efficiency and threshold power for a given setup can be computed with the help of Eq.18 and Eq.20. Although by neglecting the higher order terms of the series in the linearized approximation near the threshold, the computed value of the threshold pump-power [5, 7] will not be affected much, but the same is not true for overlap efficiency. The overlap efficiency computed with such approximations will be quite erroneous.

3. Simulations

To verify these facts an elliptic pump spot and an elliptic TEM ₀₀ mode profile were considered. The absorption coefficient of a 0.5mm. thick gain medium is α_a =4.2mm ^-1. The pump beam parameters along both axes were ω_px =100µm, $M_{x}^{2}$ =60, ω_py =10µm, and $M_{y}^{2}$ =2.5. Mode radii (1/e ²) are ω_mx =40µm, and ω_my =50µm. With these parameters, we computed the variation of the match function and overlap efficiency with the position of the pump-beam focal plane in the gain medium. The computations were done with both the linearized approximations near the threshold and the proposed methods. Fig.1(a) and Fig.1(b) show the variation of the match function and overlap efficiency with focal plane position. The threshold pump-power of a laser is inversely proportional to the match function. Therefore, it can be seen that the values of the match function obtained with both methods are almost the same. However the values of overlap efficiency computed with the linearized approximation near the threshold are about 25% less than the values obtained with our method. This corresponds to about 50% error. We also plotted in Fig.2 the variation of the inverse of the integral in Eq.14 with respect to I′_circ for different pump beam position. It can be seen that in all the cases the curves fit to the straight lines with a regression coefficient not less than 0.9986. This validates our assumption of the linear approximation for the overlap integral in Eq.13. Now we applied this method to a 0.5mm. thick axially pumped Nd: YVO4 microchip laser with absorption coefficient α_a =4.2mm ^-1 and n=2.165. The variation of the overlap efficiency with respect to the mode radius was computed. The laser mode was taken as a circular TEM ₀₀ mode. The pump-beam was considered to be elliptic with two different spots sizes $M_{x}^{2}$ =40, and $M_{y}^{2}$ =2.0. The results are shown in Fig.3. We have also computed the variation of the overlap efficiency with respect to pump-beam spot size in an axially pumped Nd: YAG laser. A crystal of thickness 5mm with absorption coefficient α_a =0.6mm ^-1 and n=1.82 was considered. The computations were done for a circular pump beam with different values of beam quality factor M ² and different values of laser-mode waist radius ω_m ₀. The results are shown in Fig.4. Values of overlap efficiency with M ²=1 correspond nearly to the constant pump-beam spot along the gain medium.

4. Conclusion

We have demonstrated with computer simulations that the overlap efficiency computed with the methods based on linearized approximation near the threshold can give a value 25% less than the one computed in above threshold regimes. Our method is based on the computation of the total overlap integral for several values of circulating field intensity. The method also takes care of pump beam divergence and pump beam waist position. Therefore, the overlap efficiency computed with this method is more accurate. Although the analysis is used for a four level system, but it is equally valid for a three level system in linear pump power absorption regime.

Figure 1. Variation of (a) match function and (b) overlap efficiency with pump-beam waist position with respect to entrance plane of gain medium. The pump beam profile is taken as an elliptic profile. The match function and overlap efficiency is computed with both, the linearize approximation near threshold and the proposed method.

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Figure 2. Variation of inverse of overlap integral with normalised circulating field intensity in different pump beam conditions. A linear fit to these plots gives the value of constants C and D. Solid lines are the linear fit to the plotted data. z is the distance of pump beam waist from the entrance plane. The pump beam spot in the gain medium increases with z.

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Figure 3. Variation of match function with mode waist radius ω_m ₀ for two different values of pump waist sizes. The pump beam profile is taken as an elliptic profile and ω_p ₀=[ω_px , ω_py ].

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Figure 4. Variation of overlap efficiency η ₀ with pump waist radius for four different values of beam quality factor M ². (a) The mode waist radius ω_m ₀=100µm, (b) The mode waist radius ω_m ₀=300µm, (c) The mode waist radius ω_m ₀=500µm.

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References

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A new approach to compute Overlap efficiency in axially pumped Solid State Lasers

Abstract

1. Introduction

2. Theoretical Analysis

3. Simulations

4. Conclusion

References

Cited By

Figures (4)

Equations (27)

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