Posted: March 24th, 2023

Analysis of the laser cavity and Q-switched operation

The efficiency of the laser is governed by the available pump power and the overlap between the
pump beam and the laser cavity mode (as well as the cavity losses). In general, the pumped gain
volume is designed to be smaller in diameter than the laser cavity mode. This ensures that the gain
available is as high as possible, and that the lowest order transverse laser mode (TEM
00
) operates
preferentially.

The laser cavity is a simple hemispherical design, with a flat crystal surface forming one mirror
and a spherically-curved concave output coupler as the other. The radius of curvature of the output
coupler is 7.5 cm. This cavity is “stable” (ie can support a stable laser mode) if the output coupler
is placed within approximately 7.5 cm of the high reflector.
The cavity stability is determined by matching the curvature of the gaussian beam of the
wavefront to that of the end mirrors. (Thus the beam is reflected with a constant phase front
across its profile.) This is calculated in terms of the cavity g parameters.
g
1
= 1 - L/R
1
and
g
2
= 1 - L/R
2
where L is the optical path length of the cavity and R
1

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and R
2
are the mirror radii of curvature.
R
1
and R
2
are positive if the mirrors are concave toward the laser cavity. The laser crystal is 8
mm long and has a refractive index of 1.78. The Q-switch is 3 mm long with a refractive index
of approximately 1.6.

Real solutions for the cavity modes only exist when
0≤g
1
g
2
≤1.
These parameters can be used to estimate the cavity mode spotsizes w
1
and w
2
(diameters) at the
end mirrors using:
w
L g
1
2

 ( )
1


2
g g g
1 1 2
w
L g
2
2

 ( )
1


1
g g g
2 1 2

and
.

24/07/2018. 5
When the laser is Q-switched, it produces short, high-power pulses, at a repetition frequency
governed by the Q-switch “switching” rate. The pulse width from the laser is determined by the
pump power (the number of times above threshold that the laser is being pumped) and the cavity
lifetime 
c
. The pulsewidth 
p
may be approximated by:

 
r r
r r
( )
ln( )1

 
p
c

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where r is the initial inversion ratio (the number of times above threshold that the laser is pumped)
and (r) is the energy extraction efficiency as a function of inversion ratio. The energy extraction
efficiency is approximately 100 % for r >2. Thus 
p
is of the order of 
c .
The cavity lifetime is
given by:
where T
rt

is the cavity round trip time and 2
0
c
rt

T
p






1


2
ln
0
1 2
R R

p is the distributed cavity loss in one round trip. The
mirror reflectivities are R
1
and R
2
.

Question: Using the Melles Griot graphs supplied (see Resource Notes) determine the spotsize of
the pump beam after it is focused by the 0.29 pitch GRIN lens. Assume a distance of 0.5 mm
between the GRIN lens and the fibre. For best mode matching, the optimum laser alignment has
the pump focused on the back face of the laser crystal

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Question: Calculate the laser cavity mode spotsizes at each end mirror. Assume the measured
distance between the two end mirrors is 6.5 cm. Show that this forms a stable cavity.

Question: Assuming a low distributed cavity loss and an energy extraction efficiency of 100 %,
calculate the cavity lifetime and hence estimate the Q-switched pulsewidth.

Question: What could you do to reduce the pulsewidth?

References:
A. Siegman, Lasers, University Science Books, Mill Valley Ca, 1986.
C. Davis, Lasers and Electro-optics, Cambridge University Press, Cambridge, 1996.
Melles Griot catalogue on Gradient Index Lenses.
J. Quellette, The Diode Pumped Laser Revolution, AIP Industrial Physicist pp7-9 1996.
24/07/2018. 6
Diode current to power conversion data

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