C J Mercer, Y H Tsang and D J Binks
Photon Physics Group, Department of Physics and Astronomy,
University of Manchester, M13 9PL, UK. colin.mercer@postgrad.manchester.ac.uk
1. Introduction
Abstract . We present here a model of a pulsed diode pumped passively Q-switched laser. The laser rate equations are solved numerically for a range of parameter values including output coupler reflectivity, pump pulse duration and doping levels within both the saturable absorber and gain medium. In particular, the conditions that result in a single output pulse per pump pulse are investigated. Pulse duration, peak power and pulse energy are also calculated. The example case for a Nd:YAG crystal with a
Cr:YAG passive Q-switch crystal is considered. The model is verified by comparison to experimental results.
Diode-pumped passively Q-switched lasers can be robust and efficient light sources which do not require moving parts, high voltages or complex control mechanisms to achieve nanosecond-scale pulsed operation. For continuous wave pumping, however, passive Qswitching generally results in a continuous stream of pulses emitted at a fixed repetition rate.
For applications requiring single pulses or pulses synchronized to an external signal, control of pulse timing can be regained by modulating the pump diodes.
Numerous models of Q-switched lasers have been presented previously. Degnan [1] notably derived a description of optimally coupled Q-switched lasers. Zhang et al. in 1997 [2] introduced a model for continuously pumped, passively Q-switched lasers, which included excited state absorption in the saturable absorber. Following this Dong in 2003 [3] accurately simulated the output characteristics of a continuously pumped passively Q-switched laser by introducing a pump power term to the rate equations. The results of this model were compared to experiment and good agreement was obtained [4].
Here we present a model of pulse pumped, passively Q-switched lasers. We investigate the conditions under which a typical pump pulse results in a single output pulse. We also calculate the pulse duration, energy and peak power obtained under these conditions.
2. Model
2.1. Laser Rate Equations
The three coupled rate equations for a passively Q-Switched Laser are [3], d
 dt

 t r
( 2

Nl

2
 g
N g l s

2
 e
N e l s
 ln(
1
R
)

L )

S (1.a)

dN dt dN g dt
   c

N

N


W p
(1.b)
   g c

N g

N so
 s

N g
(1.c)
Where,
N g

N e

N s 0
(1.d)
Where

is the photon density and N is the population inversion density, N g
is the instantaneous population density of the saturable absorber (SA) absorbing state. N e
is the population density of the excited state of the SA and N s0
is the total population density of the
SA. l is the length of the gain medium and l s is the length of the SA. c is the speed of light, t r is the round trip transit time in the laser resonator. R is the reflectivity of the output coupler and L is the roundtrip dissipative optical loss.
 is the inversion reduction factor. σ is the stimulated emission cross sectional area of the gain medium. σ e
is the excited state absorption cross section, σ g is the ground state absorption cross section of the SA.

and
 s are the lifetimes of the upper laser level of the gain medium and the excited state lifetime of the saturable absorber respectively. S represents the contribution from spontaneous emission; its inclusion was found to improve numerical stability. W p
is the pump rate density
[Wcm -3 s -1 ] into the upper laser level.
W p s ( t )

P p
1
 exp( h

A

2
 l ) p l s ( t ) (1.e)
Where s(t) is a square wave function introduced to represent the pulse pumping. Pulse energy E, peak power P, and pulse width
 p of a Q-switched laser can be calculated via [5],
E

P
 h

Al ' ln( t r
1
R
)
 
0

( t ) dt (2.a) h

Al ' ln( t r
1
R
)
 max
(2.b)
 p

E / P (2.c)
Where, A is the active area of the beam in the laser medium. h

is the laser photon energy.
Φ max is the maximum photon density in the laser cavity [4].
2.2. Time to first and second pulse
The time to build up to the first Q-switched pulse from the start of pumping of the gain medium, T
1
, is different from that of the time between the first and second pulse (T
2
) if pumping is continuous. This is because the population inversion density starts from zero for the first pulse but thereafter remains different from zero.
The following four expressions can be readily derived from previous work [3]. T
1 is given by,

T
1
 
[ln( W p

)
 ln( W p
 
N i
)] (3.a)
Where

is the radiative lifetime of the gain medium. N i
is the population inversion density immediately before an output pulse, and is given by,
N i

N cw

( N cw

N f
) exp(


T
2 ) (3.b)
Where N f
is the population inversion density immediately after the output pulse. N cw
is the population inversion density for the same system if continuously pumped and is given by,
N cw

W p

(3.c)
T
2
is given by,
T
2
 
[ln( W p
 
N f
)
 ln( W p
 
N i
)] (3.d)
If the pump pulse has a length p

, to obtain a single output pulse per pump pulse then the following condition must be met,
T
1
  p

T
1

T
2
(3.e)
3. Results
3.1. Comparison with experimental and analytical results.
To verify the model we compared the results with experimental and numerical results reported previously and also with analytical results. Below is a typical output from the program for a Cr, Nd:YAG self Q-switched laser. The same parameter values as used in reference [4] have been used unless they have been explicitly stated as being varied.
10 x 10
16 Output for Nd:YAG
10 x 10
16 Output for Nd:YAG
5 5
10
5
0
0 x 10
17
15
5
0
0
0
0 x 10
17
15
10
0.5
0.5
1
1
1.5
x 10
-4
1.5
x 10
-4
10
5
0
15
0 x 10
17
0
15
0 x 10
17
10
5
0
0
0.5
0.5
1
1
1.5
x 10
-4
1.5
x 10
-4
0.5
Time(s)
1 1.5
x 10
-4
0.5
Time(s)
1 1.5
x 10
-4
Figure 1: Output for Cr,Nd:YAG: Photon density, N (population inversion density) and Loss.
See reference [4] for parameter values used.

Table 1 compares our numerical results with the numerical and experimental results previously presented by Dong [4]; comparison is also made with the results of analytical solutions [6].
Table 1: Comparison of model results against analytical solutions and Dong
2004 [4] experimental and modelling results.
Power(W) Energy(μJ)
Pulse
Width(ns)
First
Pulse(μs)
Second
Pulse(μs)
Dong(2004)
Model 780
Dong(2004)
Experimental 680
Analytical
Solutions
Model
725
643
3.5
3.3
2.8
2.7
4.5
5
3.9
4.2
34
35.3
35.8
64
66.2
66.8
The model presented here calculated the results by integrating the area for the photon density calculated and using equations 2.a, b and c and the height of the peak for the modelled pulse. The results from our program lie between those of Dong 2004 [4] and the analytical results. Dong 2004 appears closer to the experimental results reported in his paper. However, it must be considered that in that experiment the Q-switch pulses were reordered with a Silicon photodiode detector with a 1.5ns rise time. As this rise time of the detector is of the same order of the Q-switch pulse time, the accuracy of the measurement taking this into account puts the Matlab model’s results in good agreement and within the error of the experimental measurement.
3.2. Time to first Pulse
The main parameters determining the time to first and second pulse are the pump rate, reflectivity and the doping of the SA. Altering all of these will effect the time to first pulse, but the more lengthened the time to first pulse is the less efficient the laser is as more energy is lost through the spontaneous emission during build up time of the laser. In figure 2(a) the effect of doping on the first and second pulse times is shown.
Time for First and Second Pulses to Occur
400
300
200
100
0
0.5
1
Doping(x10
17 cm
-3
)
1.5
Second
Pulse
First
Pulse
Time to First Pulse as a Function of
Reflectivity of the Output Coupler
100
80
60
40
20
0
50 150 250
Time (μs)
350
Figure 2: (a) Length of time to the first and then second Q-switched pulses as a function of doping level in the saturable absorber. (b): Change in the time for the first pulse to occur against the reflectivity of the output coupler.
It can be seen that as the doping of the SA increases the time to first and second pulse increases. All the other parameters are kept constant. The gap between first and second pulse also increases. As shown in figure 2(b), reducing the reflectivity also results in a longer time to first and second pulses.. This is expected because the cavity losses are increased as the reflectivity is lowered. These results show that the system efficiency is reduced in

ensuring only a single output pulse is produced per pump pulse because to do so requires a relatively high loss cavity.
3.3. Pulse width and Energy
Pulse width and pulse energy as a function of reflectivity for different doping levels in the SA is shown in figures 3(a) and 3(b), respectively.
Pulse Width for Doping 0.6-
700
1.1x10
17 cm
-3
9
Energy Output for Doping Levels
0.6-1.1x10
17
cm
-3
600
500
400
300
200
0.6
0.7
0.8
0.9
1
1.1
8
7
6
5
4
3
0.6
0.7
0.8
0.9
1
1.1
100
2
1
0 0
30 50 70 90
Reflectivity(%)
30 50 70 90
Reflectivity(%)
Figure 3: (a) Pulse width as a function of reflectivity. (b) Pulse energy as a function of reflectivity.
The results show that as the reflectivity approaches low values the pulse width begins to increase sharply. The point at which it begins to do this depends on the doping of the SA. For a higher doping of the SA typically a higher pulse width is obtained for any given reflectivity.
Figure 3(b) shows the change in pulse energy as a function of reflectivity of the output coupler for different doping levels. For higher doping levels, a higher energy is obtained. This is due to the longer time taken to saturate the SA, and hence a longer build-up time for the gain medium. It can be seen that although a higher doping at any value of reflectivity will result in higher output energy, the optimum reflectivity to maximize the output energy at any doping level, will change towards a slightly lower reflectivity at a higher doping levels. Note that if a fixed pulse width is required when increasing the doping, a reduced reflectivity is needed to maintain the desired pulse duration thereby reducing the extraction efficiency.
There is thus a trade off between efficiency and energy output if a particular pulse width is required.
4. Conclusion
This work shows that it is possible to gain control over the repetition rate of a passively Qswitched laser with the use of QCW diode pumping. The time to first and second pulses of a passively Q-switched laser can be controlled as they are functions of the pump power, output coupler reflectivity and the doping level of the saturable absorber. Altering these parameters will alter the time to first and second pulse; however, to lengthen the time is at a cost of efficiency. Efficiency will be reduced as the pump rate is lowered, or the doping in the saturable absorber is increased, or the output coupler reflectivity is decreased. Once the

pump pulse of a QCW diode is longer than the time to first pulse, but shorter than the time for second pulse to occur then only one output pulse pump pulse can be produced.
References
[1] Degnan J. Theory of the optimally coupled Q-switched laser. IEEE journal of quantum electronics 1989; 25(2):214-220.
[2] Xingyu Zhang W. Optimization of Cr-doped saturable-absorber Q-switched lasers. IEEE journal of quantum electronics 1997; 33(12):2286-2294.
[3] Jun Dong E. Numerical modelling of CW-pumped repetitively passively Q-switched
Yb:YAG lasers with Cr:YAG as saturable absorber. Optics communications 2003;
226(1):337-344.
[4] Jun Dong P. Experiments and numerical simulation of a diode-laser-pumped Cr, Nd:YAG self-Q-switched laser. Journal of the Optical Society of America B, Optical physics 2004;
21(12):2130-2136.
[5] Degnan J. Optimization of passively Q-switched lasers. IEEE journal of quantum electronics 1995; 31(11):1890-1901.
[6] Koechner W. Solid State Laser Engineering, fifth ed.1999. Berlin, Springer. 1999.