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Pulse parameters of a synchronously pumped dye laser with arbitrary relaxation time
Volume 71, number 3,4
OPTICS COMMUNICATIONS
15 May 1989
PULSE PARAMETERS OF A SYNCHRONOUSLY P U M P E D DYE LASER W I T H ARBI'IRARY RELAXATION T I M E U. STAMM and F. WEIDNER 1 Friedrich-Schiller-Universit?it .lena, Sektion Physik, DDR-6900, Jena, G.D.R.
Received 18 June 1988; revised manuscript received 12 December 1988
Approximate analytical expressions describing position, peak intensity and duration of a steady-state pulse circulating in a synchronouslypumped dye laser near threshold are derived. Since these relations are valid for arbitrary energyrelaxation time of the lasing dye they include also the case of infrared emitting dyes. The results are compared with exact numerical simulations of the pulse evolution.
1. Introduction
The synchronously pumped dye laser (SPDL) is a reliable source of picosecond pulses tunable over the bandwidth of the laser dye. The synthesis of new photochemically stable infrared laser dyes [ 1 ] led to the extension of the accessible spectral region beyond 1 ttm [2-5 ]. These IR dyes are characterized by a very short energy relaxation time and a low quantum yield. In our paper [6 ] we have presented results of a numerical treatment of the steady-state regime of synchronously pumped IR dye lasers. We found that for typical pump lasers, such as cw mode-locked Nd:YAG, and usual cavity configurations, a change of cavity parameters influences the steady-state pulse characteristics particularly strongly for shorter fluorescence lifetimes. This is in agreement with the experiments reported in ref. [ 5 ]. Due to the numerical method employed, the so-called stepping model [ 7,8,12 ], results could only be obtained for positive values of the timing mismatch. Since, on the other hand, a steady-state treatment cannot reflect fluctuations of the pulse parameters due to spontaneous emission which is known to play a crucial role in SPDL [ 8,11 ], in our recent paper [ 9 ] we have given results of a numerical simulation of the pulse evo-
lution process in synchronously pumped IR dye lasers. We investigated the effect of spontaneous emission, timing mismatch and wavelength detuning from the centre frequency of the laser line on pulse characteristics and stability. In the present communication we develop some simple analytical expressions describing essential parameters of an SPDL pulse near threshold for arbitrary energy relaxation time of the laser dye. We determine the time at which threshold is exceeded as a function of the fluorescence lifetime, and give a condition for the position of the laser pulse in time. Using some approximations we derive expressions for the peak intensity and the pulse width in the single pulse operation of SPDL For a set of parameters the analytical results are compared with those of an exact numerical treatment of the pulse evolution process as presented in
ref. [91.
2. Basic equations
We assume a ring cavity configuration containing the active dye jet and a dispersive element for frequency tuning. Under steady-state conditions we obtain for the photon flux density of the laser pulse NL (t) the round trip equation [ ! 0 ]:
Jenoptik Jena GmbH, Forschungszentrum, DDR-6900 Jena,
G.D.R. 0 030-4018/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
165
Volume 71, number 3,4
OPTICS COMMUNICATIONS
NL(t+h)
where ~ ( x ) is the error function defined by
{
= R NL(t) exp[g(t) ] -- A----~d-t (NL(t) exp[g(t) ] ) + ~
2
15 May 1989
[
• (x) = 2 2 / ~ i exp ( - y 2 / 2 ) dy. o
d 2
~
(NL(t) e x p [ g ( t ) ] )
-- 4NL(t) exp[g(t)]
NL(t) exp[g(t)]
, (1)
where R is the reflectivity of the output mirror, Ato the bandwidth of the dispersive element, and h = (Ldye -- Lpump ) / u the timing mismatch caused by the detuning between the optical lengths of dye laser (Ldy~) and pump laser (Lpump). The gain g(t) in the active medium is described by the equation [ 6 ]
dg/dt=a21Np- a21NL(eg - 1 ) --g/T21 •
(2)
In the above equations t is the local time in a frame moving with the group velocity v of the dye laser pulse. The pump pulse is assumed to have a gaussian envelope
Np( t ) = ( eo/ v/~ao3rp) exp[
-
( t/1;p) 2 ]
with the duration zp and the normalized pump photon density %; ao3 and azl are the absorption and emission cross sections at pump and laser wavelength, respectively. The relaxation term g/T2~ becomes essential for infrared lasing dyes [5]. To simplify the analysis we consider the laser frequency to be near the centre of the laser transition. Therefore, phase modulations due to the saturation of the gain medium are negligible.
Tthr depends on the ratio P = rp/T21 and the parameter 7= 0t21~ p / a 0 3 g t h r. For small Ythe relaxation term in eq. (2) leads to gain depletion before laser threshold is attained. The minimum 7mi, for which threshold is just reached, is calculated from the condition that the maximum gain coefficient provided by eq. (2) with 7=Train is equal to the threshold gain gthr. The corresponding threshold time is found to be Tmax =
[ln(Tmin/x//~P) ] 1 / 2
(4)
It corresponds to the maximum threshold time for the given value of P. Tm,~ is shown as a function of P in fig. 1. Inserting eq. (4) into eq. (3) we obtain Ymm as a function of the ratio P=rp/T21. The two limiting cases of this curve (cf. fig. 1 ) of slow and very fast active dyes are Ymin= 1 and ~min= N//~P, respectively. The dashed line in fig. 1 corresponds to the asymptotic function v/-nP. For values ofP>~ 3 the deviation of the asymptotic function x/~P from 7rain is less then 10%. We may denote dyes with P>~ 3 as being fast. In fig. 2 the time evolution of the small signal gain
t
'°!I
I ~14
1.4~ 1
3. Laser threshold
The threshold condition of equal gain and loss during one cavity round trip provides for the threshold gain gthr =In ( l / R ) . Solving eq. (2) with NL=0 we obtain for the threshold time Tth r =/thr/Tp normalized to the pump pulse duration the transcendental equation e x p ( - P T t h r ) [1 - ~ ( x / ~ ( p / 2 - Tth~) )] = (2/7) exp( - ~p2) , 166
(3)
0
1
2
3
4
5
6 P
Fig. 1. Minimum value of ~=O/21~p/O/03gthr and corresponding time Tmaxof maximum gain defined by g( Tmax) =gthr in dependence on P = rp/T21. The dashed line corresponds to the asymptotic limit for very "fast" dyes.
Volume 71, number 3,4
OPTICS COMMUNICATIONS
g tg,,, !
15 May 1989
g/gt~ 1,0-
g/gt hr
1.0
1.0
a81
o.81
0,8-
0.61
0.61
0.6-
0.41
0.4
0.21
0.2-
a21 i
-2.0
0
20
~r,
~'.o
-2.0
0
2.0
4.0
~TP
-2.0
0
2.0
4.0
tiff
Fig. 2. Small signal gain below threshold for 7=7mi. and P = 0 (a), P=0.1 (b) and P = 1.0 (c).
g(t) for several values of P with 7= 7rain is depicted. The maximum value gthr=ln( 1/R) attained at time Tmax is shifted to earlier times with faster energy relaxation of the dye. The behaviour of the curve 7mi,(P) reflects the well known fact that, other things being equal, dyes with short relaxation time have to be pumped harder to reach threshold. If one wants to compare the laser operation of dyes with various energy relaxation times it is not useful to take the same 7-values for all. A real comparison is only possible if the threshold exceeding factor fl defined by 7=flTmin is kept constant.
ing wing to the center of the pulse as a consequence of the action of the spontaneous emission as a continuous source of radiation. Close to threshold we expand the gain eg~ 1 + g a n d obtain from eq. (5) for the gain go at the time to the expression 2 go ~g, hr + r-----~" Aco
(6)
to/to
The time To = can be estimated by eq. (3) substituting Tthr by To and using y = a 2 t ep/Oto3go. Before this equation can be solved we have to derive an expression for the dependence of rL on the laser parameters.
4. Position of the SPDL pulse in time 5. Calculation of pulse parameters Let us denote the time at which the laser pulse significantly saturates the gain as to. By analogy with ref. [ 12 ] we fix to by the condition [t=to = 0. In the following calculations the round trip equation ( 1 ) is analysed for h = 0 and in lowest order in Ato yielding
dg/dt
NL(/)
NL(t)=N~ax[2t-t°zL-(t-t°~2]'\ ZL / d
=R(NL(t)exp [g(t) ]
Aw dt {NL(t) e x p [ g ( t ) ] }
Our analytical estimation of the pulse parameters is based on the following approximations: - The laser is just above threshold. Only a single pulse circulates within the cavity. The pulse shape near the center is approximated by a parabola
)
.
(5)
On the leading edge of the pulse wing we assume an exponential increase of the laser intensity The time constant rL governs the smooth increase of the laser intensity from the lead-
(7)
- The pump rate is considered as constant during the laser pulse. - A l l exponentials are expanded to lowest order. The solution of eq. (2) is then given by
dNL/dt~.NL/TL.
167
Volume 71, number 3,4
g(t) =go +
OPTICS COMMUNICATIONS
'[._~L(
t-- to
$'p ,~
TL
'~ ( v - P g o )
15 May 1989
~2) NL/At3 0.00125 -
000100
'
000075
•
_,
-g°a2'N'~axz"
+ 3\
TL / J
'
(8)
with V=a2~NL(tO)rp and P=rp/T2,. This expression is inserted in the simplified round trip equation (5). Considering eq. (5) at the pulse m a x i m u m t = to + ZL and the time derivative at to+ ZL we obtain the equations 2
v-Pgo={gooz2,N#"zp-
2__~(~_p~
A(OTp \ T L /
'
(9)
0.00050 •
L
000025 '
o
0.5
;.o
;-/~,.
Fig. 3. Final steady state pulse shape calculated by an exact numerical treatment of the pulse evolution (solid line) and intensity parabola of the approximate treatment (dashed line ) for the laser parameters ot21/Oto3 0.24, P= 3, A(o% = 1600, R = 0.95, /~= 1.2. =
4
(~__E ~2
v - Pg° = g° o~2 1 N ~"* zP -- Ao)~---~k ZL /
(IO) "
max O~2)N L /A~.J
Using eq. (6) we find for the pulse duration
"L"Lz~&.j
ZL/Tp ~ ( 2 / P )
°°°i
X{ [ (2Aoxrp/P 2) (v--Pgthr) + 1 ] 1 / 2 1} - ' , (11)
10-3
and the intensity
a21N~ ax Ato
/'"- ,oo40oJ°
3
200 i
- 2go AoJzp 2
looi 10-' 1,1
Since v-Pg, hr increases with the threshold exceeding in the laser it is obvious that the pulse duration decreases and the intensity increases with increasing threshold exceeding factor. We want to compare these simple analytical expressions with the results of an exact numerical simulation of the pulse evolution process in analogy to those described in re£ [ 9 ]. We use the following set of laser parameters: a2~/ao3=0.24, P = 3 , Aw% = 1600, and R=0.95. In fig. 3 the steady-state pulse shape derived by the numerical simulation of the pulse evolution at a threshold exceeding factor f l = l . 2 (solid line) and the intensity parabola (dashed line) of the approximate analysis are depicted. The results for the peak intensity and the pulse duration in dependence on fl are given in fig. 4. Again the solid line corresponds to the exact numerical 168
1,2
1.3
#
1,1
1,2
1,3
Fig. 4. Peak intensity (a) and duration (b) of the SPDL-pulse provided by the numerical (solid curve) and the approximate (dashed curve) treatment in dependence on the threshold exceeding factor. The other parameters are as in fig. 3. simulation, the dashed line to the approximate treatment. These results prove that the analytical expressions ( 11 ), (12) are suited for a rough estimation of the pulse parameters to be expected in a SPDLjust above threshold in the single pulse regime. The deviation of the analytical results from the numerical treatment in the region of small fl (fl< 1.15) are caused mainly by the approximation of constant pump intensity during the laser pulse. Since the pulses become longer in this region the decrease of the pump intensity in the trailing edge of the pump pulse has
Volume 71, number 3,4
OPTICS COMMUNICATIONS "t't/ Tp 04 0.3
TL/TP 10-3
S
02 0.1
oCZlNCm°'/~,c~
10--~ 1
2
3
4.
5
P Fig. 5. Results of the approximate treatment for the dependence of the pulse intensity and duration on P= Zp/T2~.The other parameters are the same as in fig. 3. to be taken into account leading to smaller pulse intensities and slower saturation o f the gain. For higher fl values (fl> 1.3) the second order terms in the expansion of e g and in 1/Ato become significant. In fig. 5 the dependence o f the pulse width and intensity on the ratio P = zp/T2~ is depicted for fl= 1.2 and otherwise the same laser parameters as used in figs. 3, 4. As already stated in ref. [6] we obtain the longest pulses for the "slow" dye ( P = 0 ) since in S P D L with such dyes the time o f positive net gain during one round trip is the longest. For shorter energy relaxation time the absolute value of the p u m p energy increases and the time o f positive net gain decreases leading to more intense and shorter pulses. The growing influence of energy relaxation diminishes the intensity for P > 4 which in turn lengthens the pulses. But because o f the high p u m p power
15 May 1989
needed to exceed threshold and the decreasing pulse energy this region is not o f practical significance. In conclusion we have developed simple analytical expressions for the width and the peak intensity of a laser pulse from a S P D L operating in steady-state near threshold. These relations are valid for arbitrary energy relaxation time o f the lasing dye. With increasing p u m p energy the pulse intensity increases and the pulse width decreases monotonically. A comparison with computer simulations indicates the given formulaes to be suited for an estimate o f these pulse characteristics in the region considered here.
References [1 ] B. Kopainsky, P. Qiu, W. Kaiser, B. Sens and K.H. Drexhage, Appl. Phys. B 29 (1982) 15. [2 ] A. Seilmeier, W. Kaiser, B. Sens and K.H. Drexhage, Optics Lett. 8 (1983) 205. [ 3 ] A. Seilmeier, Opt. Quant. Electron. 16 (1984) 89. [ 4 ] P. Beaud, B. Zysset, A.P. Schwarzenbach and H.P. Weber, Optics Lett. 11 (1986) 24. [ 5 ] H. Roskos, S. Opitz, A. Seilmeier and W. Kaiser, IEEE J. Quam Electron. QE-22 (1986) 697. [6] U. Stamm, F. Weidner and B. Wiihelmi, Optics Comm. 63 (1987) 179. [ 7 ] J.M. Catherall, G.H.C. New and P.N. Radmore, Optics Len. 7 (1982) 319. [8] J.M. Catherall and G.H.C. New, IEEE J. Quant. Electron. QE-22 (1986) 1593. [9] U. Stature and F. Weidner, Appl. Phys. B (1989), to be published. [ 10] J. Herrmann and U. Motschmann, Appl. Phys. B 27 (1982) 27. [ 11 ] U. Stature, Appl. PlaysB 45 (1988) 101. [ 12] Z.A. Yasa, Appl. Phys. B 30 ( 1983) 135.
169
OPTICS COMMUNICATIONS
15 May 1989
PULSE PARAMETERS OF A SYNCHRONOUSLY P U M P E D DYE LASER W I T H ARBI'IRARY RELAXATION T I M E U. STAMM and F. WEIDNER 1 Friedrich-Schiller-Universit?it .lena, Sektion Physik, DDR-6900, Jena, G.D.R.
Received 18 June 1988; revised manuscript received 12 December 1988
Approximate analytical expressions describing position, peak intensity and duration of a steady-state pulse circulating in a synchronouslypumped dye laser near threshold are derived. Since these relations are valid for arbitrary energyrelaxation time of the lasing dye they include also the case of infrared emitting dyes. The results are compared with exact numerical simulations of the pulse evolution.
1. Introduction
The synchronously pumped dye laser (SPDL) is a reliable source of picosecond pulses tunable over the bandwidth of the laser dye. The synthesis of new photochemically stable infrared laser dyes [ 1 ] led to the extension of the accessible spectral region beyond 1 ttm [2-5 ]. These IR dyes are characterized by a very short energy relaxation time and a low quantum yield. In our paper [6 ] we have presented results of a numerical treatment of the steady-state regime of synchronously pumped IR dye lasers. We found that for typical pump lasers, such as cw mode-locked Nd:YAG, and usual cavity configurations, a change of cavity parameters influences the steady-state pulse characteristics particularly strongly for shorter fluorescence lifetimes. This is in agreement with the experiments reported in ref. [ 5 ]. Due to the numerical method employed, the so-called stepping model [ 7,8,12 ], results could only be obtained for positive values of the timing mismatch. Since, on the other hand, a steady-state treatment cannot reflect fluctuations of the pulse parameters due to spontaneous emission which is known to play a crucial role in SPDL [ 8,11 ], in our recent paper [ 9 ] we have given results of a numerical simulation of the pulse evo-
lution process in synchronously pumped IR dye lasers. We investigated the effect of spontaneous emission, timing mismatch and wavelength detuning from the centre frequency of the laser line on pulse characteristics and stability. In the present communication we develop some simple analytical expressions describing essential parameters of an SPDL pulse near threshold for arbitrary energy relaxation time of the laser dye. We determine the time at which threshold is exceeded as a function of the fluorescence lifetime, and give a condition for the position of the laser pulse in time. Using some approximations we derive expressions for the peak intensity and the pulse width in the single pulse operation of SPDL For a set of parameters the analytical results are compared with those of an exact numerical treatment of the pulse evolution process as presented in
ref. [91.
2. Basic equations
We assume a ring cavity configuration containing the active dye jet and a dispersive element for frequency tuning. Under steady-state conditions we obtain for the photon flux density of the laser pulse NL (t) the round trip equation [ ! 0 ]:
Jenoptik Jena GmbH, Forschungszentrum, DDR-6900 Jena,
G.D.R. 0 030-4018/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
165
Volume 71, number 3,4
OPTICS COMMUNICATIONS
NL(t+h)
where ~ ( x ) is the error function defined by
{
= R NL(t) exp[g(t) ] -- A----~d-t (NL(t) exp[g(t) ] ) + ~
2
15 May 1989
[
• (x) = 2 2 / ~ i exp ( - y 2 / 2 ) dy. o
d 2
~
(NL(t) e x p [ g ( t ) ] )
-- 4NL(t) exp[g(t)]
NL(t) exp[g(t)]
, (1)
where R is the reflectivity of the output mirror, Ato the bandwidth of the dispersive element, and h = (Ldye -- Lpump ) / u the timing mismatch caused by the detuning between the optical lengths of dye laser (Ldy~) and pump laser (Lpump). The gain g(t) in the active medium is described by the equation [ 6 ]
dg/dt=a21Np- a21NL(eg - 1 ) --g/T21 •
(2)
In the above equations t is the local time in a frame moving with the group velocity v of the dye laser pulse. The pump pulse is assumed to have a gaussian envelope
Np( t ) = ( eo/ v/~ao3rp) exp[
-
( t/1;p) 2 ]
with the duration zp and the normalized pump photon density %; ao3 and azl are the absorption and emission cross sections at pump and laser wavelength, respectively. The relaxation term g/T2~ becomes essential for infrared lasing dyes [5]. To simplify the analysis we consider the laser frequency to be near the centre of the laser transition. Therefore, phase modulations due to the saturation of the gain medium are negligible.
Tthr depends on the ratio P = rp/T21 and the parameter 7= 0t21~ p / a 0 3 g t h r. For small Ythe relaxation term in eq. (2) leads to gain depletion before laser threshold is attained. The minimum 7mi, for which threshold is just reached, is calculated from the condition that the maximum gain coefficient provided by eq. (2) with 7=Train is equal to the threshold gain gthr. The corresponding threshold time is found to be Tmax =
[ln(Tmin/x//~P) ] 1 / 2
(4)
It corresponds to the maximum threshold time for the given value of P. Tm,~ is shown as a function of P in fig. 1. Inserting eq. (4) into eq. (3) we obtain Ymm as a function of the ratio P=rp/T21. The two limiting cases of this curve (cf. fig. 1 ) of slow and very fast active dyes are Ymin= 1 and ~min= N//~P, respectively. The dashed line in fig. 1 corresponds to the asymptotic function v/-nP. For values ofP>~ 3 the deviation of the asymptotic function x/~P from 7rain is less then 10%. We may denote dyes with P>~ 3 as being fast. In fig. 2 the time evolution of the small signal gain
t
'°!I
I ~14
1.4~ 1
3. Laser threshold
The threshold condition of equal gain and loss during one cavity round trip provides for the threshold gain gthr =In ( l / R ) . Solving eq. (2) with NL=0 we obtain for the threshold time Tth r =/thr/Tp normalized to the pump pulse duration the transcendental equation e x p ( - P T t h r ) [1 - ~ ( x / ~ ( p / 2 - Tth~) )] = (2/7) exp( - ~p2) , 166
(3)
0
1
2
3
4
5
6 P
Fig. 1. Minimum value of ~=O/21~p/O/03gthr and corresponding time Tmaxof maximum gain defined by g( Tmax) =gthr in dependence on P = rp/T21. The dashed line corresponds to the asymptotic limit for very "fast" dyes.
Volume 71, number 3,4
OPTICS COMMUNICATIONS
g tg,,, !
15 May 1989
g/gt~ 1,0-
g/gt hr
1.0
1.0
a81
o.81
0,8-
0.61
0.61
0.6-
0.41
0.4
0.21
0.2-
a21 i
-2.0
0
20
~r,
~'.o
-2.0
0
2.0
4.0
~TP
-2.0
0
2.0
4.0
tiff
Fig. 2. Small signal gain below threshold for 7=7mi. and P = 0 (a), P=0.1 (b) and P = 1.0 (c).
g(t) for several values of P with 7= 7rain is depicted. The maximum value gthr=ln( 1/R) attained at time Tmax is shifted to earlier times with faster energy relaxation of the dye. The behaviour of the curve 7mi,(P) reflects the well known fact that, other things being equal, dyes with short relaxation time have to be pumped harder to reach threshold. If one wants to compare the laser operation of dyes with various energy relaxation times it is not useful to take the same 7-values for all. A real comparison is only possible if the threshold exceeding factor fl defined by 7=flTmin is kept constant.
ing wing to the center of the pulse as a consequence of the action of the spontaneous emission as a continuous source of radiation. Close to threshold we expand the gain eg~ 1 + g a n d obtain from eq. (5) for the gain go at the time to the expression 2 go ~g, hr + r-----~" Aco
(6)
to/to
The time To = can be estimated by eq. (3) substituting Tthr by To and using y = a 2 t ep/Oto3go. Before this equation can be solved we have to derive an expression for the dependence of rL on the laser parameters.
4. Position of the SPDL pulse in time 5. Calculation of pulse parameters Let us denote the time at which the laser pulse significantly saturates the gain as to. By analogy with ref. [ 12 ] we fix to by the condition [t=to = 0. In the following calculations the round trip equation ( 1 ) is analysed for h = 0 and in lowest order in Ato yielding
dg/dt
NL(/)
NL(t)=N~ax[2t-t°zL-(t-t°~2]'\ ZL / d
=R(NL(t)exp [g(t) ]
Aw dt {NL(t) e x p [ g ( t ) ] }
Our analytical estimation of the pulse parameters is based on the following approximations: - The laser is just above threshold. Only a single pulse circulates within the cavity. The pulse shape near the center is approximated by a parabola
)
.
(5)
On the leading edge of the pulse wing we assume an exponential increase of the laser intensity The time constant rL governs the smooth increase of the laser intensity from the lead-
(7)
- The pump rate is considered as constant during the laser pulse. - A l l exponentials are expanded to lowest order. The solution of eq. (2) is then given by
dNL/dt~.NL/TL.
167
Volume 71, number 3,4
g(t) =go +
OPTICS COMMUNICATIONS
'[._~L(
t-- to
$'p ,~
TL
'~ ( v - P g o )
15 May 1989
~2) NL/At3 0.00125 -
000100
'
000075
•
_,
-g°a2'N'~axz"
+ 3\
TL / J
'
(8)
with V=a2~NL(tO)rp and P=rp/T2,. This expression is inserted in the simplified round trip equation (5). Considering eq. (5) at the pulse m a x i m u m t = to + ZL and the time derivative at to+ ZL we obtain the equations 2
v-Pgo={gooz2,N#"zp-
2__~(~_p~
A(OTp \ T L /
'
(9)
0.00050 •
L
000025 '
o
0.5
;.o
;-/~,.
Fig. 3. Final steady state pulse shape calculated by an exact numerical treatment of the pulse evolution (solid line) and intensity parabola of the approximate treatment (dashed line ) for the laser parameters ot21/Oto3 0.24, P= 3, A(o% = 1600, R = 0.95, /~= 1.2. =
4
(~__E ~2
v - Pg° = g° o~2 1 N ~"* zP -- Ao)~---~k ZL /
(IO) "
max O~2)N L /A~.J
Using eq. (6) we find for the pulse duration
"L"Lz~&.j
ZL/Tp ~ ( 2 / P )
°°°i
X{ [ (2Aoxrp/P 2) (v--Pgthr) + 1 ] 1 / 2 1} - ' , (11)
10-3
and the intensity
a21N~ ax Ato
/'"- ,oo40oJ°
3
200 i
- 2go AoJzp 2
looi 10-' 1,1
Since v-Pg, hr increases with the threshold exceeding in the laser it is obvious that the pulse duration decreases and the intensity increases with increasing threshold exceeding factor. We want to compare these simple analytical expressions with the results of an exact numerical simulation of the pulse evolution process in analogy to those described in re£ [ 9 ]. We use the following set of laser parameters: a2~/ao3=0.24, P = 3 , Aw% = 1600, and R=0.95. In fig. 3 the steady-state pulse shape derived by the numerical simulation of the pulse evolution at a threshold exceeding factor f l = l . 2 (solid line) and the intensity parabola (dashed line) of the approximate analysis are depicted. The results for the peak intensity and the pulse duration in dependence on fl are given in fig. 4. Again the solid line corresponds to the exact numerical 168
1,2
1.3
#
1,1
1,2
1,3
Fig. 4. Peak intensity (a) and duration (b) of the SPDL-pulse provided by the numerical (solid curve) and the approximate (dashed curve) treatment in dependence on the threshold exceeding factor. The other parameters are as in fig. 3. simulation, the dashed line to the approximate treatment. These results prove that the analytical expressions ( 11 ), (12) are suited for a rough estimation of the pulse parameters to be expected in a SPDLjust above threshold in the single pulse regime. The deviation of the analytical results from the numerical treatment in the region of small fl (fl< 1.15) are caused mainly by the approximation of constant pump intensity during the laser pulse. Since the pulses become longer in this region the decrease of the pump intensity in the trailing edge of the pump pulse has
Volume 71, number 3,4
OPTICS COMMUNICATIONS "t't/ Tp 04 0.3
TL/TP 10-3
S
02 0.1
oCZlNCm°'/~,c~
10--~ 1
2
3
4.
5
P Fig. 5. Results of the approximate treatment for the dependence of the pulse intensity and duration on P= Zp/T2~.The other parameters are the same as in fig. 3. to be taken into account leading to smaller pulse intensities and slower saturation o f the gain. For higher fl values (fl> 1.3) the second order terms in the expansion of e g and in 1/Ato become significant. In fig. 5 the dependence o f the pulse width and intensity on the ratio P = zp/T2~ is depicted for fl= 1.2 and otherwise the same laser parameters as used in figs. 3, 4. As already stated in ref. [6] we obtain the longest pulses for the "slow" dye ( P = 0 ) since in S P D L with such dyes the time o f positive net gain during one round trip is the longest. For shorter energy relaxation time the absolute value of the p u m p energy increases and the time o f positive net gain decreases leading to more intense and shorter pulses. The growing influence of energy relaxation diminishes the intensity for P > 4 which in turn lengthens the pulses. But because o f the high p u m p power
15 May 1989
needed to exceed threshold and the decreasing pulse energy this region is not o f practical significance. In conclusion we have developed simple analytical expressions for the width and the peak intensity of a laser pulse from a S P D L operating in steady-state near threshold. These relations are valid for arbitrary energy relaxation time o f the lasing dye. With increasing p u m p energy the pulse intensity increases and the pulse width decreases monotonically. A comparison with computer simulations indicates the given formulaes to be suited for an estimate o f these pulse characteristics in the region considered here.
References [1 ] B. Kopainsky, P. Qiu, W. Kaiser, B. Sens and K.H. Drexhage, Appl. Phys. B 29 (1982) 15. [2 ] A. Seilmeier, W. Kaiser, B. Sens and K.H. Drexhage, Optics Lett. 8 (1983) 205. [ 3 ] A. Seilmeier, Opt. Quant. Electron. 16 (1984) 89. [ 4 ] P. Beaud, B. Zysset, A.P. Schwarzenbach and H.P. Weber, Optics Lett. 11 (1986) 24. [ 5 ] H. Roskos, S. Opitz, A. Seilmeier and W. Kaiser, IEEE J. Quam Electron. QE-22 (1986) 697. [6] U. Stamm, F. Weidner and B. Wiihelmi, Optics Comm. 63 (1987) 179. [ 7 ] J.M. Catherall, G.H.C. New and P.N. Radmore, Optics Len. 7 (1982) 319. [8] J.M. Catherall and G.H.C. New, IEEE J. Quant. Electron. QE-22 (1986) 1593. [9] U. Stature and F. Weidner, Appl. Phys. B (1989), to be published. [ 10] J. Herrmann and U. Motschmann, Appl. Phys. B 27 (1982) 27. [ 11 ] U. Stature, Appl. PlaysB 45 (1988) 101. [ 12] Z.A. Yasa, Appl. Phys. B 30 ( 1983) 135.
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