## Abstract

In this work, we are reporting a new approach to compute the overlap efficiency of end pumped solid-state laser systems. Unlike existing methods in which the overlap integral is computed with a linearize approximation near the threshold, in this method the inverse of the overlap integral is computed numerically in the above threshold regime for several values of circulating fields. Now by fitting a linear curve to this data the overlap efficiency is obtained. The effect of the beam quality factor is also taken into account. It is demonstrated that the linearized approximation near the threshold can give rise to 50% error in overlap efficiency. The method was used to estimate the overlap efficiency in different types of axially pumped lasers.

© Optical Society of America

## 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]

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

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

and

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

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

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

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

The normalised circulating power *P′*_{circ}
is
defined in terms of saturation intensity *I*_{sat}
as

$${P\prime}_{\mathit{circ}}={P}_{\mathit{circ}}/{I}_{\mathit{sat}}=\u3008{A}_{m}\u3009{I\prime}_{\mathit{circ}},$$

where *I′*_{circ}
is the normalized circulating
intensity, 〈*A*_{m}
〉 is the average
mode area inside the gain medium and is defined as

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

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

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

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

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

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

The output power from the laser will be given as

If we define the overlap efficiency *η*_{o}
as
follows

where *η*_{a}
is the absorption efficiency of pump
power inside the gain medium. The output power equation can be rewritten as

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

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

it is clear from Eq.20 that for any given resonator conditions the threshold
pump power is proportional to the factor
1/*α*_{a}*C*. 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

Considering a single transverse mode *TEM*
_{00},
*f*_{m}
(*x*, *y*,
*z*) can be given as

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

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

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]

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 0.5*mm*. thick gain medium is
*α*_{a}
=4.2*mm*
^{-1}.
The pump beam parameters along both axes were
*ω*_{px}
=100*µm*,
${M}_{x}^{2}$
=60,
*ω*_{py}
=10*µm*, and
${M}_{y}^{2}$
=2.5. Mode radii (1/*e*
^{2}) are
*ω*_{mx}
=40*µm*,
and *ω*_{my}
=50*µm*.
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 0.5*mm*.
thick axially pumped Nd: *YVO*4 microchip laser with absorption
coefficient
*α*_{a}
=4.2*mm*
^{-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
5*mm* with absorption coefficient
*α*_{a}
=0.6*mm*
^{-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.

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