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On the accurate numerical simulation of pulses from dye lasers
Volume
28, number
3
OPTICS COMMUNICATIONS
March 1979
ONTHEACCURATENUMERICALSIMULATIONOFPULSESFROMDYELASERS E.J. MORGAN I%~~sics Dcparimr>nt,
Received
8 November
York UniversitjJ, Downsview,
Ontario,
Canada M3J 11’3
1978
A Fourier synthesis for the pumping lamp intensity and the inclusion of shock wave effects enable a set of rate equations to provide an accurate simulation of an output laser pulse. A comparison is made with experimental data from a coaxial flash pumped dye laser using Rh 6G dissolved in methanol.
A set of coupled rate equations for the photon number and the population of the upper level of the lasing transition has been the basis of numerical studies of pulsed dye lasers [ 1~ lo]. Such coupled equations have been solved by fourth order Runge-Kutta [e.g. 1,2,4] or by predictor-corrector computer routines [e.g. 81. Initially such equations assumed a pumpingpulse provided by a ruby or doubled Nd laser (Qswitched) [ 1.21. With such short pumping pulses no consideration of triplet state effects was required. However, the use of longer pump pulses from fast rise time flash lamps required the addition of triplet intersystem crossing terms, triplet absorption terms, and triplet state decay terms [3-61 to the equations. Later, Meyer et al. [l I] showed that such terms were unimportant for certain dyes (e.g. Rh 6G in ethanol) for flashlamp pulses of a few microseconds duration. Indeed, for co-axial flashlamps the passage of a shock wave disturbance through the dye medium is more harmful than triplet state effects [8,12,13]. However, no set of rate equations has included the effect of such a wave. Moreover, the goal of previous numerical computations has been to obtain an insight into the importance of various parameters on the laser pulse, rather than to produce an accurate simulation of an experimentally observed laser pulse. This work was undertaken to select terms and parameters in the rate equations which would model the experimentally observed pulse accurately. The numerical calculations were designed to reproduce the temporal shape, the total energy, and the instantaneous
photon flux inside the cavity of the experimental laser. This study was made in conjunction with experimental measurements on intra-cavity absorption and intracavity polarization with a co-axial flashlamp pumped dye laser [ 141. In common with other attempts at modelling a pulsed dye laser by means of rate equations [e.g. 151, the equations used in this study have the form: inversion dn/dt = w(t) - Bnq - n/r,, photons
,
dqldt = Bnq - q/rc + k, n/rD ,
where n = inversion, t = time, w(t) = pump rate, B = stimulated emission coefficient, 4 = photon number, 7D = lifetime of excited singlet state of dye, rC = laser cavity lifetime, k, = fraction of spontaneously emitted photons directed paraxially. Additional terms, or similar additional equations may be used to model a particular experiment (e.g. triplet terms [3-61, or other ‘modes’ [7,14] ). As written, these equations contain common assumptions: _ negligible optical effects from thermal gradients, - uniform deposition of pump radiation, - negligible spatial effects from cavity standing waves, - same temporal behaviour of pump pulse radiation at all pump wavelengths. The experimental part of the study involved a coaxial flashlamp pumped dye laser (Rh 6G dissolved in methanol), so the terms in the rate equations had to conform to that laser. Once a pulsed dye laser has 369
Volume
28, number
3
OPTICS COMMUNICATIONS
surged past threshold its photon flux tends to follow the pump intensity so particular care was taken to accurately reproduce the flashlamp pumping pulse. The shape of this pumping pulse was obtained with a silicon photo-diode assembly [ 161 and an osciloscope (risetime = 7 ns) which were used to record the lamp intensity obtained with a solution of methanol in the dye cell. No spectral resolution was attempted. A Fourier analysis of the recorded lamp pulse shape led to a function normalised to a peak value of unity [I 71 II j(f) =.f‘(to) + cu(t ~- fu) + 7
6, sin(knr/tt)
,
where lo = initial time (usually r0 = 0), I = time measured fr-om start of lamp pulse, tr = final time used in Fourier analysis (chosen to be longer than observed laser pulse duration), (Y= slope = (f(tr) - f’(to))/(tf to) , b, = Fourier coefficient, n = number ot terms in truncated series. This type of synthesis provides an analytic function that closely reproduces the shape of the observed tlashlamp intensity (cf. fig. la). The required pump function, w(t). is then given by [2,4,15],
where w,,, 1x = peak value for pump rate which is evaluated from
N=J
rv(i)dt=w,,,,,j. 0
f(r)dt=F*w,,,ax, 0
N = total number of pumped transitions, F = form factor [15]. The pump function provides good agreement between observation and calculation for the rising part of the laser pulse. The value of N can be selected to give agreement between the observed and calculated energies. This removes the need to know the details of the pump radiation absorption process, or of ageing effects in the dye, or of thermal refractive index gradients in the dye medium. Since the goal of the simulation is to study intra-cavity processes rather than pump absorption processes this is not a worrisome restriction. Triplet state absorption and singlet-triplet intersystem-crossing were assumed to be negligible for Rh 6G in methanol [ 111. The cavity lifetime, an important parameter in the rate equations, was calculated 370
March 1979
from the mirror cavity lifetime and cavity lifetime due to higher order mode losses as discussed by Theissing et al. [I 51. The latter lifetime incorporates the effects of cavity losses due to off-axis gain paths that miss the mirrors and therefore reduce the inversion without contributing to the useful photon flux. Their calculations [I 51 indicate that this parameter is dominant within the cavity. The falling part of the laser pulse is determined by shock wave effects which curtail the duration of the laser pulse [8,12,13] by spoiling the optical quality of the dye medium. However, incorporation of this effect into the rate equations is not a self evident process. Experience with numerical computation on pulsed dye lasers indicates that any effect on the pump pulse is rapidly reflected in the calculated laser output, so it was assumed that the deterioration of the optical quality of the dye medium affected the penetration of the pump radiation into that dye medium. This allowed the shock wave effect to be incorporated into the rate equations as follows. Firstly, the available laser volume was decreased by a factor (1 ~ u,t’/No )” as the wave proceeded radially inward, where R, is the inner radius of the dye cell, uW is the shock wave speed, and f’ is the time measured from the arrival of the wave at RO. In effect it was assumed that the passage of the shock wave caused that part of the dye medium to be opaque to laser radiation. Secondly, it was assumed that the poor optical quality of the disturbed medium would inhibit the passage of the pump radiation, and the resultant scattered pump radiation would be absorbed in the disturbed outer region. A further factor of (1 - r~,t’/R~)~ was somewhat arbitrarily assigned to this effect. These combined effects curtail the efficiency of the pump radiation severely, reducing it by an overall factor of (1 - ~,t’/f?~)~ after the arrival time of the shock wave. Although this approach produced an improved agreement between the observed and calculated laser pulses, it was still not a good match. Moreover, the observed laser pulse rarely had a uniform distribution of radiation in cross section, but tended to display an annular distribution. In short, the assumed uniform deposition of the pump radiation was not realised in practise. This particular aspect of the dye laser operation leas been examined by Blit and Ganiel [ 181 who state the conditions required for uniform pump energy deposition. These conditions were not realised in the present laser so the
Volume
28, number
3
OPTICS COMMUNICATIONS
model for the shock wave effect was modified to consider an annular, active zone whose cross section diminished with time as { 1 - (Rf - Ri)/(Ri - Ri)}, whereRI =R, -UWt’, R2 = inner radius of annular zone, R. = outer radius of annular zone (assumed equal to dye cell radius), t’ = t ~ t,, t, = arrival time of shock wave (i.e. transit time through cell wall). Hence, for t > t, the pump pulse, w(t), was multiplied by this factor. In practise, better agreement between the observed and calculated pulses was obtained when this factor was squared. Although somewhat arbitrary this improvement probably reflects a need to include further non-uniformity of the radial distribution of pump radiation and the effects of scattering of pump radiation by the turbulent, disturbed region. It may also include the effect of a reduction of off-axis gain directions, as the usable diameter decreases. A fourth order Runge-Kutta integration routine was used to compute the inversion and photon values at regular time intervals. The numerical computation was divided into two parts; one part prior to the arrival of the shock wave (t =Gta) and the other after its arrival (t > fa) whereupon the pump pulse was multiplied by the factor {l - (RI - Ri)/(Ri - Ri)}*. The Fourier synthesis was used to calculate the lamp output at the beginning and end of each computation interval. Within the interval of linear interpolation between these two calculated values was used in order to economize on computation time. The energy was calculated from
hF
J 0
March 1979
Tey% 4(t)
hc = average energy per photon, T, = time at which laser pulse terminates, q(t) = photon number, rM = cavity lifetime due to mirror transmission alone. To compare the results with an actual laser required a few measurements. The shape of the output pulse of the laser was obtained by the same means as described previously for the flashlamp intensity. Care was taken to ensure that the photo-diode assembly would respond linearly to the output laser intensity. The output laser pulse energy was measured close to the exit mirror with a thermopile detector (Laser Instrumentation, Chertsey, U.K.). Table 1 shows a comparison of the experimental and calculated values for this study. In some applications 1141, this method of solving the differential rate equations proved to be very susceptible to numerical instabilities. Reducing the step size to remove the instability proved rather fruitless. It was then necessary to use a modified predictorcorrector method [ 191. This method could be relied upon to overcome the instability problems provided the step size was reasonably small. As shown in fig. la the Fourier synthesis accurately reproduces the shape of the pumping pulse with some reduction in the ‘noise’, although the early part of the lamp pulse is not reproduced as well as the remainder. Were this to prove a problem a parabolic fit could be
Table 1 Data for experiment and calculation ___ ___~_ ~~~-~ ._._~_~._._~~ Experiment Stored energy (pumped transitions) Laser energy Pulse duration FWHM of pulse Rmirror (mirror lifetime) Higher modes lifetime [ 15 j Cavity lifetime Dye lifetime (excited) Stimulated emission coeff Shock wave arrival time Shock wave speed in dye sol’n
33.8 13.9 650 360 0.40
J mJ ns ns -
Calculation (0.92 x 10’8) 13.5 mJ 700 ns 360 ns (2.87 ns) 0.902 ns 0.686 ns 7.4 ns 1.8 x lO-6 s-’ 250 ns 1.5 X lo3 m/s
371
Volume
28, number
0
200
3
OPTICS COMMUNICATIONS
600 400 TIME (ns6c.l
800
Fig. 1. Normalised pulses. (a) Flashlamp. Solid line: Fourier synthesis calculation. Closed circles: experimental points. (b) Laser. Solid lines: rate equation calculations. Closed circles: csperimental points.
used for this early part, or the lamp pulse could be divided into segments and a separate Fourier synthesis used over each section. In any event the synthesis gives a better simulation of the actual lamp pulse than the modified exponential expressions used previously [e.g. 4,151. Fig. 1b shows a comparison of three calculated curves for the output laser pulse with a set of experimental points. In all cases the values have been normalized to the peak value of the relevant set of data. Thus the calculated curves are compared with the normalized experimental points. The output laser pulse shape (based on figure la) which ignores shock wave effects peaks somewhat later than the experimental pulse with consequent distortion near the peak, and then decays too slowly. Clearly it is a markedly poorer simulation of the experimental shape than either of the other two curves. Each of them is a reasonable but not perfect representation of the observed shape, although the calculation based on an annular excitation region is nearer the observed shape than that based on the full diameter excitation region. Both provide a close simulation for the rising part of the laser pulse up to its peak. That neither curve matches the decay372
March 1979
ing slope of the observed laser pulse can be attributed to uncertainties in the radial distribution of the pump radiation, in the arrival time and subsequent speed of the shock wave, and to some variation of the flashlamp pulse from shot to shot. This study shows that an experimental dye laser pulse can be simulated well by a set of rate equations in which: a) the pumping pulse shape is accurately reproduced by a Fourier synthesis of the observed flash-lamp pulse, b) the cavity lifetime is determined by both mirror losses and higher order mode losses, as discussed by Theissing et al. [ 151, c) the optical disturbance created by the shock wave generated by the lamp discharge is included as a factor {l - uw(t - ta)/Ro}’ or {I - {[Ro -u, (t - t,)] 2 - Ri}/(Ri - Ri)}” which affects the pumping pulse for t > c,, d) the total laser energy is reproduced by a suitable choice of the total pumped transitions. As mentioned, this numerical model has been used to study intra-cavity absorption processes. In such applications some changes to the basic rate equations are needed. A check on the physical correctness of such changes can be obtained by noting how well the observed output laser pulse shape and energy are simulated by the modified rate equations. This research was supported by the National Research Council of Canada and by York University. The author is grateful to C.H. Dugan for many helpful discussions and suggestions, and to K.A. Innanen and K. Papp for useful advice in dealing with computational instabilities.
References [ 11 D. Roess, J. Appl. Phys. 37 (1966) 2004. [2] P. Sorokin, J. Lankard, E. Hammond and V. Moruzzi, IBM J. Res. 11 (1967) 130. 131 W. Schmidt and I:. Schlfer, Z. Naturforschg. 22a (1967) 1563. 141 P. Sorokin, J. Lankard, V. Moruzzi and E. Hammond, J. Chem. Phys. 48 (1968) 4726. [5] R. Keller, IEEE J.Q.E. QE-6 (1970) 411. [6] B. Snavely, Proc. IEEE. 57 (1969) 1374. (71 R. Keller, E. Zalewski and N. Peterson, .I. Opt. Sot. Am. 62 (1972) 319.
Volume
28, number
3
OPTICS COMMUNICATIONS
[8] A. Hirth, K. Vollrath and J.-P. Fouassier, Optics Comm. 9 (1973) 139. 191 W. Brtinner and H. Paul, Ann. der Phys. 5 (1975) 366. [IO] H. Schroder, H. Neusser and W. Schlag, Optics Comm. 14 (1975) 395. [I I] Y. Meyer, P. Flamant, P. Gacoin, C. Loth and R. Astier, Lasers in physical chemistry and biophysics, ed. J. Joussof-Dubien (Elsevier, 1975) p. 129. [ 121 T. Ewanizky, R. Wright and H. Theissing, Appl. Phys. Lett. 22 (1973) 520. [ 131 S. Blit, A. Fisher and U. Ganiel, Appl. Opt. 13 (1974) 335. [14] F. Morgan, C.H. Dugan and A. Lee, Optics Comm. 27 (1978) 451;
March 1979
C.H. Dugan, A. Lee and F.J. Morgan, Appl. Opt. 17 (1978) 1012. [ 15 ] H. Theissing, R. Wright and T. Ewaniszky, Report AD77A-792, available from N.T.I.S., Springfield, Virginia 22151, U.S.A. [16] G. McCall, Rev. Sci. Instr. 43 (1972) 865. [17] C. Lanzos, Applied analysis, Chapt. 5 (Prentice-Hall, 1956). [18] S. Blit and U. Camel, Opt. and Quant. Elect. 7 (1975) 87. [ 191 R.W. Hamming, Numerical methods for scientists and engineers(2nd Ed., McGraw-Hill, 1973).
373
28, number
3
OPTICS COMMUNICATIONS
March 1979
ONTHEACCURATENUMERICALSIMULATIONOFPULSESFROMDYELASERS E.J. MORGAN I%~~sics Dcparimr>nt,
Received
8 November
York UniversitjJ, Downsview,
Ontario,
Canada M3J 11’3
1978
A Fourier synthesis for the pumping lamp intensity and the inclusion of shock wave effects enable a set of rate equations to provide an accurate simulation of an output laser pulse. A comparison is made with experimental data from a coaxial flash pumped dye laser using Rh 6G dissolved in methanol.
A set of coupled rate equations for the photon number and the population of the upper level of the lasing transition has been the basis of numerical studies of pulsed dye lasers [ 1~ lo]. Such coupled equations have been solved by fourth order Runge-Kutta [e.g. 1,2,4] or by predictor-corrector computer routines [e.g. 81. Initially such equations assumed a pumpingpulse provided by a ruby or doubled Nd laser (Qswitched) [ 1.21. With such short pumping pulses no consideration of triplet state effects was required. However, the use of longer pump pulses from fast rise time flash lamps required the addition of triplet intersystem crossing terms, triplet absorption terms, and triplet state decay terms [3-61 to the equations. Later, Meyer et al. [l I] showed that such terms were unimportant for certain dyes (e.g. Rh 6G in ethanol) for flashlamp pulses of a few microseconds duration. Indeed, for co-axial flashlamps the passage of a shock wave disturbance through the dye medium is more harmful than triplet state effects [8,12,13]. However, no set of rate equations has included the effect of such a wave. Moreover, the goal of previous numerical computations has been to obtain an insight into the importance of various parameters on the laser pulse, rather than to produce an accurate simulation of an experimentally observed laser pulse. This work was undertaken to select terms and parameters in the rate equations which would model the experimentally observed pulse accurately. The numerical calculations were designed to reproduce the temporal shape, the total energy, and the instantaneous
photon flux inside the cavity of the experimental laser. This study was made in conjunction with experimental measurements on intra-cavity absorption and intracavity polarization with a co-axial flashlamp pumped dye laser [ 141. In common with other attempts at modelling a pulsed dye laser by means of rate equations [e.g. 151, the equations used in this study have the form: inversion dn/dt = w(t) - Bnq - n/r,, photons
,
dqldt = Bnq - q/rc + k, n/rD ,
where n = inversion, t = time, w(t) = pump rate, B = stimulated emission coefficient, 4 = photon number, 7D = lifetime of excited singlet state of dye, rC = laser cavity lifetime, k, = fraction of spontaneously emitted photons directed paraxially. Additional terms, or similar additional equations may be used to model a particular experiment (e.g. triplet terms [3-61, or other ‘modes’ [7,14] ). As written, these equations contain common assumptions: _ negligible optical effects from thermal gradients, - uniform deposition of pump radiation, - negligible spatial effects from cavity standing waves, - same temporal behaviour of pump pulse radiation at all pump wavelengths. The experimental part of the study involved a coaxial flashlamp pumped dye laser (Rh 6G dissolved in methanol), so the terms in the rate equations had to conform to that laser. Once a pulsed dye laser has 369
Volume
28, number
3
OPTICS COMMUNICATIONS
surged past threshold its photon flux tends to follow the pump intensity so particular care was taken to accurately reproduce the flashlamp pumping pulse. The shape of this pumping pulse was obtained with a silicon photo-diode assembly [ 161 and an osciloscope (risetime = 7 ns) which were used to record the lamp intensity obtained with a solution of methanol in the dye cell. No spectral resolution was attempted. A Fourier analysis of the recorded lamp pulse shape led to a function normalised to a peak value of unity [I 71 II j(f) =.f‘(to) + cu(t ~- fu) + 7
6, sin(knr/tt)
,
where lo = initial time (usually r0 = 0), I = time measured fr-om start of lamp pulse, tr = final time used in Fourier analysis (chosen to be longer than observed laser pulse duration), (Y= slope = (f(tr) - f’(to))/(tf to) , b, = Fourier coefficient, n = number ot terms in truncated series. This type of synthesis provides an analytic function that closely reproduces the shape of the observed tlashlamp intensity (cf. fig. la). The required pump function, w(t). is then given by [2,4,15],
where w,,, 1x = peak value for pump rate which is evaluated from
N=J
rv(i)dt=w,,,,,j. 0
f(r)dt=F*w,,,ax, 0
N = total number of pumped transitions, F = form factor [15]. The pump function provides good agreement between observation and calculation for the rising part of the laser pulse. The value of N can be selected to give agreement between the observed and calculated energies. This removes the need to know the details of the pump radiation absorption process, or of ageing effects in the dye, or of thermal refractive index gradients in the dye medium. Since the goal of the simulation is to study intra-cavity processes rather than pump absorption processes this is not a worrisome restriction. Triplet state absorption and singlet-triplet intersystem-crossing were assumed to be negligible for Rh 6G in methanol [ 111. The cavity lifetime, an important parameter in the rate equations, was calculated 370
March 1979
from the mirror cavity lifetime and cavity lifetime due to higher order mode losses as discussed by Theissing et al. [I 51. The latter lifetime incorporates the effects of cavity losses due to off-axis gain paths that miss the mirrors and therefore reduce the inversion without contributing to the useful photon flux. Their calculations [I 51 indicate that this parameter is dominant within the cavity. The falling part of the laser pulse is determined by shock wave effects which curtail the duration of the laser pulse [8,12,13] by spoiling the optical quality of the dye medium. However, incorporation of this effect into the rate equations is not a self evident process. Experience with numerical computation on pulsed dye lasers indicates that any effect on the pump pulse is rapidly reflected in the calculated laser output, so it was assumed that the deterioration of the optical quality of the dye medium affected the penetration of the pump radiation into that dye medium. This allowed the shock wave effect to be incorporated into the rate equations as follows. Firstly, the available laser volume was decreased by a factor (1 ~ u,t’/No )” as the wave proceeded radially inward, where R, is the inner radius of the dye cell, uW is the shock wave speed, and f’ is the time measured from the arrival of the wave at RO. In effect it was assumed that the passage of the shock wave caused that part of the dye medium to be opaque to laser radiation. Secondly, it was assumed that the poor optical quality of the disturbed medium would inhibit the passage of the pump radiation, and the resultant scattered pump radiation would be absorbed in the disturbed outer region. A further factor of (1 - r~,t’/R~)~ was somewhat arbitrarily assigned to this effect. These combined effects curtail the efficiency of the pump radiation severely, reducing it by an overall factor of (1 - ~,t’/f?~)~ after the arrival time of the shock wave. Although this approach produced an improved agreement between the observed and calculated laser pulses, it was still not a good match. Moreover, the observed laser pulse rarely had a uniform distribution of radiation in cross section, but tended to display an annular distribution. In short, the assumed uniform deposition of the pump radiation was not realised in practise. This particular aspect of the dye laser operation leas been examined by Blit and Ganiel [ 181 who state the conditions required for uniform pump energy deposition. These conditions were not realised in the present laser so the
Volume
28, number
3
OPTICS COMMUNICATIONS
model for the shock wave effect was modified to consider an annular, active zone whose cross section diminished with time as { 1 - (Rf - Ri)/(Ri - Ri)}, whereRI =R, -UWt’, R2 = inner radius of annular zone, R. = outer radius of annular zone (assumed equal to dye cell radius), t’ = t ~ t,, t, = arrival time of shock wave (i.e. transit time through cell wall). Hence, for t > t, the pump pulse, w(t), was multiplied by this factor. In practise, better agreement between the observed and calculated pulses was obtained when this factor was squared. Although somewhat arbitrary this improvement probably reflects a need to include further non-uniformity of the radial distribution of pump radiation and the effects of scattering of pump radiation by the turbulent, disturbed region. It may also include the effect of a reduction of off-axis gain directions, as the usable diameter decreases. A fourth order Runge-Kutta integration routine was used to compute the inversion and photon values at regular time intervals. The numerical computation was divided into two parts; one part prior to the arrival of the shock wave (t =Gta) and the other after its arrival (t > fa) whereupon the pump pulse was multiplied by the factor {l - (RI - Ri)/(Ri - Ri)}*. The Fourier synthesis was used to calculate the lamp output at the beginning and end of each computation interval. Within the interval of linear interpolation between these two calculated values was used in order to economize on computation time. The energy was calculated from
hF
J 0
March 1979
Tey% 4(t)
hc = average energy per photon, T, = time at which laser pulse terminates, q(t) = photon number, rM = cavity lifetime due to mirror transmission alone. To compare the results with an actual laser required a few measurements. The shape of the output pulse of the laser was obtained by the same means as described previously for the flashlamp intensity. Care was taken to ensure that the photo-diode assembly would respond linearly to the output laser intensity. The output laser pulse energy was measured close to the exit mirror with a thermopile detector (Laser Instrumentation, Chertsey, U.K.). Table 1 shows a comparison of the experimental and calculated values for this study. In some applications 1141, this method of solving the differential rate equations proved to be very susceptible to numerical instabilities. Reducing the step size to remove the instability proved rather fruitless. It was then necessary to use a modified predictorcorrector method [ 191. This method could be relied upon to overcome the instability problems provided the step size was reasonably small. As shown in fig. la the Fourier synthesis accurately reproduces the shape of the pumping pulse with some reduction in the ‘noise’, although the early part of the lamp pulse is not reproduced as well as the remainder. Were this to prove a problem a parabolic fit could be
Table 1 Data for experiment and calculation ___ ___~_ ~~~-~ ._._~_~._._~~ Experiment Stored energy (pumped transitions) Laser energy Pulse duration FWHM of pulse Rmirror (mirror lifetime) Higher modes lifetime [ 15 j Cavity lifetime Dye lifetime (excited) Stimulated emission coeff Shock wave arrival time Shock wave speed in dye sol’n
33.8 13.9 650 360 0.40
J mJ ns ns -
Calculation (0.92 x 10’8) 13.5 mJ 700 ns 360 ns (2.87 ns) 0.902 ns 0.686 ns 7.4 ns 1.8 x lO-6 s-’ 250 ns 1.5 X lo3 m/s
371
Volume
28, number
0
200
3
OPTICS COMMUNICATIONS
600 400 TIME (ns6c.l
800
Fig. 1. Normalised pulses. (a) Flashlamp. Solid line: Fourier synthesis calculation. Closed circles: experimental points. (b) Laser. Solid lines: rate equation calculations. Closed circles: csperimental points.
used for this early part, or the lamp pulse could be divided into segments and a separate Fourier synthesis used over each section. In any event the synthesis gives a better simulation of the actual lamp pulse than the modified exponential expressions used previously [e.g. 4,151. Fig. 1b shows a comparison of three calculated curves for the output laser pulse with a set of experimental points. In all cases the values have been normalized to the peak value of the relevant set of data. Thus the calculated curves are compared with the normalized experimental points. The output laser pulse shape (based on figure la) which ignores shock wave effects peaks somewhat later than the experimental pulse with consequent distortion near the peak, and then decays too slowly. Clearly it is a markedly poorer simulation of the experimental shape than either of the other two curves. Each of them is a reasonable but not perfect representation of the observed shape, although the calculation based on an annular excitation region is nearer the observed shape than that based on the full diameter excitation region. Both provide a close simulation for the rising part of the laser pulse up to its peak. That neither curve matches the decay372
March 1979
ing slope of the observed laser pulse can be attributed to uncertainties in the radial distribution of the pump radiation, in the arrival time and subsequent speed of the shock wave, and to some variation of the flashlamp pulse from shot to shot. This study shows that an experimental dye laser pulse can be simulated well by a set of rate equations in which: a) the pumping pulse shape is accurately reproduced by a Fourier synthesis of the observed flash-lamp pulse, b) the cavity lifetime is determined by both mirror losses and higher order mode losses, as discussed by Theissing et al. [ 151, c) the optical disturbance created by the shock wave generated by the lamp discharge is included as a factor {l - uw(t - ta)/Ro}’ or {I - {[Ro -u, (t - t,)] 2 - Ri}/(Ri - Ri)}” which affects the pumping pulse for t > c,, d) the total laser energy is reproduced by a suitable choice of the total pumped transitions. As mentioned, this numerical model has been used to study intra-cavity absorption processes. In such applications some changes to the basic rate equations are needed. A check on the physical correctness of such changes can be obtained by noting how well the observed output laser pulse shape and energy are simulated by the modified rate equations. This research was supported by the National Research Council of Canada and by York University. The author is grateful to C.H. Dugan for many helpful discussions and suggestions, and to K.A. Innanen and K. Papp for useful advice in dealing with computational instabilities.
References [ 11 D. Roess, J. Appl. Phys. 37 (1966) 2004. [2] P. Sorokin, J. Lankard, E. Hammond and V. Moruzzi, IBM J. Res. 11 (1967) 130. 131 W. Schmidt and I:. Schlfer, Z. Naturforschg. 22a (1967) 1563. 141 P. Sorokin, J. Lankard, V. Moruzzi and E. Hammond, J. Chem. Phys. 48 (1968) 4726. [5] R. Keller, IEEE J.Q.E. QE-6 (1970) 411. [6] B. Snavely, Proc. IEEE. 57 (1969) 1374. (71 R. Keller, E. Zalewski and N. Peterson, .I. Opt. Sot. Am. 62 (1972) 319.
Volume
28, number
3
OPTICS COMMUNICATIONS
[8] A. Hirth, K. Vollrath and J.-P. Fouassier, Optics Comm. 9 (1973) 139. 191 W. Brtinner and H. Paul, Ann. der Phys. 5 (1975) 366. [IO] H. Schroder, H. Neusser and W. Schlag, Optics Comm. 14 (1975) 395. [I I] Y. Meyer, P. Flamant, P. Gacoin, C. Loth and R. Astier, Lasers in physical chemistry and biophysics, ed. J. Joussof-Dubien (Elsevier, 1975) p. 129. [ 121 T. Ewanizky, R. Wright and H. Theissing, Appl. Phys. Lett. 22 (1973) 520. [ 131 S. Blit, A. Fisher and U. Ganiel, Appl. Opt. 13 (1974) 335. [14] F. Morgan, C.H. Dugan and A. Lee, Optics Comm. 27 (1978) 451;
March 1979
C.H. Dugan, A. Lee and F.J. Morgan, Appl. Opt. 17 (1978) 1012. [ 15 ] H. Theissing, R. Wright and T. Ewaniszky, Report AD77A-792, available from N.T.I.S., Springfield, Virginia 22151, U.S.A. [16] G. McCall, Rev. Sci. Instr. 43 (1972) 865. [17] C. Lanzos, Applied analysis, Chapt. 5 (Prentice-Hall, 1956). [18] S. Blit and U. Camel, Opt. and Quant. Elect. 7 (1975) 87. [ 191 R.W. Hamming, Numerical methods for scientists and engineers(2nd Ed., McGraw-Hill, 1973).
373