Table Of ContentSome fundamental considerations on the dynamics of class B laser threshold crossing
G.P. Puccioni1, T. Wang2,3, and G.L. Lippi2,3
1 Istituto dei Sistemi Complessi,
CNR, Via Madonna del Piano 10,
I-50019 Sesto Fiorentino, Italy
2 Institut Non Lin´eaire de Nice,
Universit´e de Nice Sophia Antipolis, France
3 CNRS, UMR 7335
1361 Route des Lucioles, F-06560 Valbonne, France ∗
(Dated: January 11, 2016)
6 With the help of a simple rate equation model, we analyze the intrinsic dynamics of threshold
1
crossing for Class B lasers. A thorough discussion of the characteristics and the limitations of
0
this very commonly employed model, which provides excellent qualitative predictions on the laser
2
behaviour,isoffered. Approximatesolutionsforthepopulationinversionandforthefieldintensity,
n upto thepoint where thelatter reaches macroscopic levels, are found anddiscussed, together with
a theassociated characteristic times. Numericalverifications test theaccuracy of thesesolutions and
J confirmtheirvalidity. Adiscussionoftheimplicationsonthresholddynamicsispresented,together
8 with themotivation for focussing on this – nowadays most common – class of lasers.
]
s I. INTRODUCTION apply quite well to most current lasers, including semi-
c
i conductor ones, in spite of their very small size [2].
t
p Lasers have been around for over half a century, and Before delving into the discussion, however, it is im-
o althoughinitiallyconsideredasapuretechnicalcuriosity portant to remark on the choice of the Class B laser
.
s (orasolutionlookingforaproblem,astheywereoftenre- for this discussion. The latter represent the ensemble
c ferredtointhe1960’s),theyhavebecomeubiquitous. As of lasers whose description is based on two main physi-
i
s a consequence, lasers are nowadays studied from differ- cal variables: the electromagnetic (e.m.) field intensity
y
ent points of view, depending on the kind of application andthepopulationinversion. Thesimplestdeviceofthis
h
which is the ultimate goal of the approach. This paper kindwillthereforebeasinglelongitudinalandtransverse
p
gives a contribution to the description of the dynamics mode laser, possessing a ring cavity with unidirectional
[
of the threshold crossing of a wide class of lasers, the emission (i.e., the so-called unidirectional ring cavity).
1 so-called Class B lasers [1], whose properties, although The latter can be obtained by inserting a non-reciprocal
v
well-known, are often viewed from a practical, rather element (e.g., a Faradayisolator [3]) inside a ring cavity.
2
7 than fundamental point of view. Thus, in order to help ComparedtotheClassAlaser,physicallydescribedby
8 understand the basics of these devices, we are going to the sole e.m. field intensity (e.g., He-Ne lasers and gas
1 highlight their temporal response to the threshold cross- lasers in general, dye lasers ...), the Class B device pos-
0 ing with the help of the simplest possible model. sesses a dynamics which is determined by the interplay
1. Thresholdisnormallyviewedasastaticproperty,and of the physics of the e.m. field and that of the mate-
0 is used as a concept of principle, regardless of the way rial’s response (restricted to the difference in population
6 the laser makes the transition from the off-state (below between the upper and lower state participating in the
1 threshold) to the on-state(above threshold). In the sim- lasing transition) [4]. To this class belong notably semi-
v: plified description based on the sole accounting of the conductor lasers, solid state lasers and some molecular
i stimulated photons, these two states correspond also to lasers (e.g., the CO , which finds applications both in
X 2
thelackofemissionandtothepresenceoflightemission, medicine – surgery or heat treatment – and in technol-
r respectively. Ofcourse,therealityismorecomplex,since ogy – cutting and drilling).
a
some (incoherent) light is emitted even below threshold,
While the more complex interaction presents a fun-
but the simplified picture is justified both in terms of
damental interest for the study of the dynamics of these
emission strength – orders of magnitude weaker below
lasers,atthesametimeclassBlasersrepresentthesingle
than above threshold –, in terms of coherence of the ra-
most important source of coherent emission for techno-
diation (incoherent below threshold – i.e., spontaneous
logical applications, accounting for well over 90% of the
emission–,highlycoherentabovethreshold)andinterms
World laser sales [5] (2010 data). This practical interest
of directionality of the radiation (i.e., the appearence of
is closely relatedto their physicalproperties since to the
the collimated “pencil” beam above threshold). All these
Class B belong laserswhose materialtime constant (i.e.,
considerations hold very well for the macroscopic lasers
the timescale on which the population inversion reacts)
developedinthefirstdecadesoflaserfabricationandstill
is slower than the e.m. field’s. This implies that the
spontaneous losses, due to relaxation of the upper state
without contributing a photon to the stimulated compo-
∗ [email protected] nent, are strongly reduced; in other words, these devices
2
TABLE I. Transformations between the two forms of model TABLEII. Some typicalrelaxation constants valid for some
(equations (1–2) vs. equations(3–4)). selected, sample class B lasers.
Direct physicalmodel Normalized model Type of laser K(s−1) γ(s−1)
n ↔ 2γGI CO2 107 104
N ↔ KD Nd:YAG 108 104
G
R ↔ γKP Semiconductor 1011 109
G
κ ↔ K
dD
=−γ[(1+I)D−P], (4)
intrinsically possess much lower losses and offer an effi- dt
ciencywhichcanbeordersofmagnitudelargerthanthat
where I stands for the e.m. field intensity, D for the
ofClassAlasers. Thankstothisintrinsicandunmistake-
population inversion, K for the intensity losses, and P
able advantage, technological solutions have been devel-
represents the pump (i.e., energy supplied to the laser).
opedoverthe yearsto offerclassB lasersemitting virtu-
The recast version of the rate equations immediately
ally on all wavelengths from mid-IR to near-UV. Thus,
highlights the existence of the two time scales: the e.m.
in addition to the fundamental interest in studying the field intensity evolves over a timescale τ ∼ 1 (equa-
more complex dynamics, we find the practical motiva- I K
tion (3)), while the population inversion’s timescale is
tion of understanding threshold crossing as it occurs in τ ∼ 1 (equation(4)). Thisdirectlyillustratesthephys-
those lasers which are used in almost all of everyday’s D γ
ical characteristics which identify class B lasers. Typical
applications.
values of the relaxation constants are offered in Table II
for some selected sample devices.
These equations straightforwardly possess the follow-
II. MODEL PROPERTIES
ingdoublesetofsteady-statesolutions[8](theoverstrike
denoting the steady state):
There exist numerous derivations of the basic model
for a laser, and the Class B model [6] can be obtained
I =0 I =P −1
from the Maxwell-Bloch[7] model performing the adia- , (5)
D =P ! D =1 !
baticeliminationoftheatomicpolarization(cf. e.g.,[8]).
However, it is easy to show that they all reduce to the
where the threshold value in the normalizedform of this
so-called rate equations, which can be written directly
model is P =1 (i.e., I =0).
th
from physical considerations [9]:
A linear stability analysis of the above-threshold so-
dn lution [8, 11] immediately provides stable solutions (for
=−κn+GnN, (1)
P ≥1) with eigenvalues of the form:
dt
dN
1
=R−γN −GnN, (2) λ= −γP ± γ2P2−4γK(P −1) , (6)
dt
2
where n represents the photon number, N the difference h p i
where the square root takes imaginary values as soon as
between the number of atoms [10] in the upper and in
the lower level of the lasing transition, κ represents the 1 γ
losses for the photon number, G the coupling constant P >∼1+ 4K . (7)
betweenphotonsandatomicexcitation, Rstandsforthe
Giventhatγ ≪K forallClassB lasers,the eigenvalues,
pumprate(i.e., amountofenergysuppliedperunit time
equation (6), are (almost) always complex above thresh-
to the laser) and γ the spontaneous relaxation rate of
old, and represent a (damped) oscillation with angular
the population from the upper state. In writing equa-
frequency
tions(1-2)wehavemadetheimplicitassumptionthatwe
are considering a perfect four-level laser, with infinitely
ω ≈ γK(P −1), (8)
fast relaxation from the lower state towards a separate,
fundamental state [9]. This assumption does not quali- whereobtainingthisappproximateexpressionwehavene-
tativelychangethe resultsofouranalysisandis justified glectedthetermγ2P2,verysmallcomparedto4γK(P−
by its simplicity. Quantitative changes are discussed by 1) for all practical values of P.
several authors (cf. e.g., Siegman’s book [9]). Notice that the model we are studying is entirely de-
With the help of the transformations detailed in Ta- terministic and does not take into account in any way
ble I it is possible to recast the rate equations into a the presence of spontaneous emission. In other words,
normalizedform,moresuitable forourdiscussion,asfol- the modelaccounts only for the stimulated fraction(i.e.,
lows: perfectly coherent) of the emitted photons and entirely
dI ignoresthe spontaneousone(negligible once thresholdis
=−K(1−D)I, (3) attained – cf. section IV for numerical estimates). An
dt
3
improvementonthismodelisrepresentedbyasetofrate r.h.s. of equation (4)) and obtain an approximate form
equations where the average contribution of the spon- for D(t>t˜):
taneous emission is accounted for in the field intensity
D(t)=D(t˜)+D(δt), (14)
equation(cf. e.g.,[12]). Thisaddition,however,doesnot
qualitatively alter the physical description, while it adds =1+(P −1) 1−e−γδt , δt≡t−t˜(15)
a good degree of mathematical complexity (cf., e.g., the ≈1+(P −1)[1−(1−γδt)] , (16)
(cid:0) (cid:1)
imperfect bifurcation problem in laser physics [13]). We
=1+(P −1)γδt, (17)
thereforeusethe simplermodel(equations(3–4))paying
closeattentiontotheinterpretationofitspredictionsand which holds as long as δt≪ 1, a condition which is very
γ
to the use of the boundary conditions (cf. discussion of
wellsatisfiedinpractice(andwhichcanbeeasilychecked
the value of I˜in section III). a posteriori – cf. section IV).
Wecannowusethisapproximatesolutiontogetanap-
proximatesolutionfor the initial phases ofthe e.m. field
III. DYNAMICAL THRESHOLD CROSSING intensity growth by replacing D(t) from equation (17)
into the rate equation for I (equation (3)), which can be
easily recast as:
When considering the transition from below to above
threshold we explicitely deal with the condition P(t = d(logI)
0−) < 1 and therefore I(t = 0) = 0 and D(t = 0) ≡ =γK(P −1)δt. (18)
dt
D = P(t = 0−), according to the first set of steady-
0
statevalues,equation(5). AssumingaHeavisidefunction Direct integration provides
shape for the pump (P(t < 0) < 1,P(t > 0) > 1), the
t t
model reduces to d(logI)=γK(P −1) (t′−t˜)dt′ (19)
Zt˜ Zt˜
dI 1
=0, (9) = γK(P −1)(t−t˜)2, (20)
dt 2
dD
=−γ(D−P), (10) whichonly holdsuntilatime t ,to be determined. The
dt M
l.h.s. of equation (19) provides logI(t) up to a constant
withtheinitialconditionsspecifiedabove. Thus,thee.m. (logI(t˜)) which corresponds to a mathematical diver-
field intensity remains constant, at zero, while the pop- gence,sinceI(t˜)=0. Besidesbeingunphysical,thisisan
ulation inversion starts growing exponentially according artefactofthemodel,whichconsidersonlythedetermin-
to istic evolution of the coherent fraction of the e.m. field:
spontaneous emission is not included in this description.
D(t)=(P −D )(1−e−γt)+D , (11) Aself-consistentsolutioncanonlybeobtainedbyinclud-
0 0
ing the spontaneous photons in the description, but the
which holds until the instant t˜ at which the popula- complexityofthemodelincreasesconsiderably;theaver-
tion inversion reaches the other solution, equations (5): age properties of the lasing transition, however, are still
D(t˜) = 1. Starting from this instant, the first of the correctly given by the set of rate equations (3-4). Thus,
model equations (3) acquires a positive r.h.s. and the we can use the correct physical condition (i.e., the aver-
e.m. field intensity starts to grow. agevalueofthe spontaneousemissioninthelasingmode
The value of t˜ can be easily obtained from equa- atthreshold) to estimate the value of I(t˜), thus avoiding
tion (11): the unphysical divergence. Indicating with I this value
0
(i.e., the value of I at t=t˜), we obtain the approximate
D(t˜)=1=(P −D )(1−e−γt˜)+D , (12) expression for the e.m. field intensity growth:
0 0
t˜=−1 log P −1 , (13) I(t)=I0e21γK(P−1)(t−t˜)2, (21)
γ P −D
(cid:20) 0(cid:21)
which already provides us with a wealth of (determinis-
where we are assured that t˜> 0 by the fact that P − tic)informationaboutthe initialphasesofthe growthof
1 < P −D . Starting from this instant, the full model, the laser intensity:
0
equations (3-4), must be used in its nonlinear form and
• thegrowthisexponentialbutwithaquadratictime
no closed solution exists for the time evolution of the
dependence – since the solution (21) holds only at
physical variables. However, if we concentrate on the
shorttimes,the quadraticgrowthintime indicates
initial phases of the e.m. field intensity growth, we can
aslowerinitialgrowththanwhatwouldresultfrom
gathersomeinformationonthe timescale overwhichthe
a linear time dependence;
laser turns on.
Assuming that the laser intensity I is small in its ini- • the speed at which the laser intensity grows de-
tial phases (I ≪ 1), we can suppose its influence on the pends on the distance of the pump from threshold
evolution of D to be negligible (since (1+I) ≈ 1 in the (P−1)–thelargerthepump,thefasterthegrowth;
4
• the time constant for the intensity growth is not IV. NUMERICAL VERIFICATIONS
proportional to K−1, as one would, mistakenly
but intuitively, expect from the timescale evolu-
tion of the intensity (cf. equation (3)), but rather 7e-07
to the geometric mean of the two time constants
(γK)−1/2; 6e-07
• theactualtimeconstantfortheexponentialgrowth 5e-07
is τexp = γK(P2−1), i.e., the distance of the pump 4e-07
to threshqold induces a hyperbolic lengthening of t (s)3e-07 1.28
thetimescale,withtheusualdivergence,typicalof 1.6
critical slowing down [14], taking place as P →1. 2e-07D(t)11..24
1
While intuitively unexpected, the dependence of the 1e-07 0.8
0.6
timescale on the product of the two relaxationconstants 0 0 2e-07 4e-07t (s)6e-07 8e-07 1e-06
for the physical variables is logical. Indeed, it is not 0.2 0.4 0.6 0.8
D
sufficient for the e.m. field intensity to grow at a rate 0
K−1 since the population inversion must have the time
FIG. 1. Comparison between the value of the time value
to increase as well in order for the photon number to
at which D = 1 occurs obtained from equation (13) – solid
develop. Notice that the relaxation oscillations, equa-
line – and theequivalent values obtained from the numerical
tion (8), appearing around the above-threshold solution
integrationofthemodel(equations(3,4))–dots. Forthisand
(thus, far beyond the intensity ranges we are consider-
thefollowing figuresthefollowing parameter valuesareused:
ing here) have the same parameter dependence as the γ = 1×106s−1, K = 1×108s−1, P = 2. The inset shows
time delay∆t≡t−t˜,apartfromanumericalcoefficient. thetemporal evolution ofthepopulation inversion computed
This pointis significantsince itshowshow the time con- fromequation(11)foratimeequalto 1;theasymptoticvalue
γ
stants appearing in all parts of the transient evolution is at P =2.
are closely relatedto eachother by the intrinsic physical
interplay between the laser variables.
A verification of the approximate solutions is easily
The limits of validity of the solution we have obtained
obtained by comparing the analytical predictions to the
forthetransientcanbeeasilyestablishedinthefollowing
numerical values resulting from the integration of the
way. The transient dynamics of class B lasers is charac-
model, equations (3,4), obtained with a first-order Eu-
terizedbyadelayinthelaserintensitygrowth,accompa-
ler scheme programmed in GNU Octave. The temporal
nied by an overshoot of its value beyond its asymptotic
evolutionof the population inversion(cf. inset of Fig. 1)
statewithdampedoscillations[12]. Adynamicalanalysis
displays a growth corresponding to that of a saturating
can be performed, looking, among others indicators, at
exponential, as predicted by equation (11). The cross-
theshapeofthetrajectoryinphasespace[15]. Strongde- ing time t˜(t˜: D(t˜) = D = 1) can be easily found from
viations for the growth of the population inversion from
this trajectory (and more precisely from the numerical
the approximate solution, equation (17), are expected,
file). We also remark that, as implicit in the previous
and numerically found, when the laser intensity exceeds
discussion, the population inversion D grows beyond its
its asymptotic value I = P −1. Thus, we can set the
asymptotic value in the process of laser threshold cross-
limit of validity at a fraction of this value aI (a < 1, ing (cf. discussion in section V).
arbitrary) to determine the maximum time value tM for Fig. 1 shows the comparison between the time t˜nec-
which the approximate solution for I(t) (equation (21))
essary for the population inversion to reach its above-
holds:
threshold steady state value (continuous line) as pre-
a(P −1)=I0e21γK(P−1)(tM−t˜)2, (22) dtaicinteedd ffrroomm ethqueatteimonpo(1ra3l),traanjdecttohreyn(udmotesr)i.caNl toitmuenoebx--
pectedly, since the approximation used in the derivation
which immediately gives an estimate for t :
M of equation (13) is very well verified, the agreement is
excellent.
t =t˜+ 2 log a(P −1) . (23) Once threshold is crossed, the numerical integration
M
sγK(P −1) (cid:18) I0 (cid:19) has to start from a good estimate of the “initial” value
of the laser field intensity. The deterministic rate equa-
Since we are trying to estimate the time necessary to tion model does not account for the background noisy
attain a fraction a of the steady state value for the laser dynamics which holds the intensity value constant (in
intensity, it does not make sense to consider the limit average) around the value of the spontaneous emission.
P → 1, thus the potential divergences present in the Ifonestartsthe integrationofequations(3-4)withade-
expression on the r.h.s. of equation (23) lie outside the terministic initial condition (e.g., D = 0.5, as in one of
0
realm of the interesting physical parameter ranges. the simulations run for Fig. 1), during the whole tran-
5
2
beyond);smallsemiconductorlasersarecharacterisedby
N ≈ 105, while smaller cavities exit the realm of small-
sized lasers to approach the nanoscale. A more detailed
1.5 discussion, supported by stochastic calculations, can be
es found in [17, 18]. Here we use values 105 ≤ N ≤ 1010,
abl 2 specified in the figures as appropriate.
vari 1 The evolution of the population inversion following t˜
ser 1.5 is displayed in Fig. 2. The continuous lines (red online)
a
L 1 shows the population inversion numerically integrated
0.5 from the model, equations (3-4), while the dashed line
0.5
(green online) represents the predictions of the approxi-
mate expression, equation (11). The graph convincingly
00 2e-07 4e-07 6e-07 8e-07 1e-06
00 2e-07 4e-07 6e-07 8e-07 shows that the analytical approximation holds well be-
∆t (s) yond t˜, even once the laser intensity I starts growing
away from 0. Indeed, the two curves are superposed for
FIG.2. Timeevolutionofthelaservariablesasafunctionof times exceeding t=8×10−7s (for the parameter values
∆t=t−t˜: Laser intensity(dashed line– black online), pop-
of the figure), and remain very close until I ≈ I (I = 1
ulation inversion (continuous line – red online), approximate 2 2
for the chosen parameters).
solution (equation (15)) for the population inversion (dash-
double dotted line – green online). The vertical dash-dotted The analyticalpredictions ofsectionIII haveprovided
line (blue online) marks the value of tM: the approximate alsoanestimateofthemaximumtimevalueforwhichthe
and the exact solution are still an excellent match up until approximate analysis holds. Fig. 3 shows a comparison
this time value. Inset: shape of the field intensity growth between the estimated time, as a function of the sponta-
according to the approximate solution, equation (21). The neous emission fraction (i.e., hI i/I). The agreement
sp,p
agreementisqualitativebutshowsthattheshapeofthecurve
here is somewhat less good than the one previously ob-
iswell reproduced–noticethatthevalueofγK hasbeen re-
tained, due to the fact that we have retained only the
placed by γK/2 for this graph (cf. text for details). The
linear term (first-order correction) in the expressions for
initial value used for the the field intensity (representing the
average spontaneous emission) is I0=1×10−10. the population inversion,equation(17), to obtainanap-
proximate behaviour for the initial phases of the inten-
sity growth, as reproduced by equation (21). It is from
this latter equation which we have estimated the maxi-
sient where D(t) < D = 1, the laser intensity decays mumtime,equation(23),representedasatimedifference
away to ever smaller numbers. This is an artefact of the ∆t ≡ (t −t˜) in Fig. 3. Notice, however, that the
max M
model and should not be mistaken for a physical effect. order of magnitude is correctly obtained and that the
Continuing the simulation from unphysically low values largesterroris ofthe orderof20%: itoccurs,notsurpis-
of the laser intensity (e.g., much lower than the average ingly, for the lower values of the spontaneous emission,
spontaneousemissionlevel)wouldartificiallyincreasethe which lead to longer values of t .
M
latency time needed to reach macroscopic intensity val-
We also remark that the threshold set for determining
ues, and thus affect the maximum values reached by the
the value of t (a = 0.1) falls well within the range of
M
populationinversion,and,asa consequence,bythe laser
validityfortheapproximateexpressionofthepopulation
intensity at its peak (not discussed here – the full time
inversion given by equation (17): the time t is marked
M
evolution can be seen, for instance, in Ref. [15]). Thus,
inFig.2bytheverticaldot-dashedline(blueonline)–at
it is crucial to consider a reasonable estimate of the av-
this instant, the numerical and the analytical expression
erage spontaneous emission. Traditionally, the following
for D(t ) coincide.
M
physical considerations have been employed to concep-
Finally, we look at the shape of the initial growth of
tually define threshold: for the stimulated emission to
the field intensity, as displayed in the inset of Fig. 2 for
overcome the spontaneous emission and concentrate on
comparison with the dashed line (black online) in the
the lasing mode all (or most) of the energy, the number
same figure. The overall shape is quite well reproduced,
of photons in the lasing mode has to equal the num-
evensurprisinglywellforanapproximatesolutionwitha
ber N of modes available for the spontaneous emission
growth rate as large as that of a quadratic exponential,
(i.e., the number of electromagnetic cavity modes). In
butforthe valueofthe time constantτ which, forthe
exp
otherwords,whileinaveragethenumberofspontaneous
sake of graphical comparison, has been doubled. Given
photons is hn i = 1 for each mode (j = 1...N), the
s,j therathercrudeapproximationsusedtoobtaintheshape
(average)number of stimulated photons in mode p must
of the growing intensity, equation (21), then used for es-
be hn i = N for lasing action to occur [16]. Thus, if
st,p timating the time t , the qualitative agreement is quite
M
we consider a laser whose cavity possesses N modes, its
satisfactory. As a last remark, the value of the delay
relative average spontaneous intensity at threshold will
time τ used in the numerical comparison, larger than
be hIspi = 1. Without entering into details, macro- the oneexpcoming from the analytical estimate, brings its
I N
scopic lasers have values of 107 < N < 1012 (and even valueabitclosertotheactualresponsetime(Fig.3)and
6
8e-07 (one-dimensional)phasespace,insuchsystemstheevolu-
tionofthefieldintensitytakesamonotonicformandthe
7e-07 only interesting aspects cover the delay time associated
with the threshold crossing due to the time-dependence
of the control parameter [11, 21].
6e-07
(s)max theThCelalassrgBerlpahsears,eisnpsatceeada,ssroecnidaetersdtwhiethdtyhneamphicyssicnsono-f
∆t 5e-07
trivial, since it allows for a non-monotonic evolution of
the laser intensity, in addition to the appearance of an
4e-07 intrinsic time delay, superposed to the one induced by
the bifurcation [11, 22]. This intrinsic delay stems from
the fact that: 1. the field intensity cannot grow until
3e-07
1e-10 1e-09 1e-08 1e-07 1e-06 1e-05
the population inversionhas reachedits threshold value,
Spontaneous emission fraction
and2. asufficientamountofinversionisneededtoallow
for the growth of the photon number. This introduces a
FIG.3. Comparison between thevalueofthetimevaluefor
which I(tM) = aI (plotted as ∆tmax = tM −t˜) as a func- causalelement whichrequires the population to growby
tionofthevalueofthespontaneousemission (I0). Solid line: a sufficient amount for the field intensity to approachits
approximate expression – equation (23) –; dots: numerical above-thresholdvalue;itisalsothecausefortheappear-
integration of the model (equations (3,4)). Cf. text for de- anceofatimescaleproportionaltothegeometricmeanof
tails. The discrimination level for the intensity has been set the two relaxation constants (γ and K). In the absence
at a=0.1 (i.e., 10% of the asymptotic value I, according to of this causal component, one should have expected the
equation (5) – right hand group of solutions).
fieldintensitytogrowataratecontrolledbyK,oncethe
population inversion has reached its threshold value.
Summarizing the results of this paper, we have ob-
to the relaxation oscillation period, estimated from the
tained approximate expressions for the times at which
linear stability analysis, equation (8).
the population inversion reaches its threshold and the
field intensity attains macroscopic values, together with
approximatesolutionsfor both variables within the time
V. DISCUSSION AND CONCLUSIONS
intervals just defined. The agreement is quite satisfac-
tory in all cases (and even excellent in some), in spite of
Theusualpictureoflaserthresholdisbasedonastatic the extreme simplicity of the analysis. These considera-
representation,wherethe fieldbecomes coherentassoon tionsallowforadeeperinsightintothethresholdcrossing
as the pump rate exceeds its threshold value. This pic- properties of Class B lasers.
turerestsonthevalidityoftheassumptionofaninfinitely
largesystem(i.e.,the thermodynamicallimit[19]forthe
laser), which is very well satisfied by a large class of ex- ACKNOWLEDGMENTS
isting devices: in practice all solid state lasers (even mi-
crodisks)andalltraditionalgaslasers,highpowerlasers, GLLis gratefultoallthe students thathavetakenthe
etc. [20]. Refinements become necessary when studying graduate course in Laser Dynamics for the stimulating
different laser classes. Recent work has shown that the discussions and questions and to all the collegues with
well-established characterization of coherence properly whom, in the past decades, he has had the opportunity
holds only for Class A devices [18] and in these systems to discuss issues related to laser transients.
adynamicalperspectiveinthecrossingofthresholddoes
not reserve particular surprises. Due to the restricted
[1] Tredicce J R, Arecchi F T , Lippi G L, and Puccioni G limited technological importance.
P 1985 J. Opt. Soc. Am. B2 173–183. [5] http://spie.org/x38563.xml
[2] The borders between below threshold and above thresh- [6] Arecchi F T and Bonifacio R 1965 IEEE J. Quantum
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