dynamics of an upconversion er:yag laser with reabsorption losses

1630 J. Opt. Soc. Am. B/Vol. 21, No. 9 /September 2004 O. Toma and S. Georgescu

Dynamics of an upconversion Er:YAG laser withreabsorption losses

Octavian Toma and Serban Georgescu

National Institute for Laser, Plasma and Radiation Physics, Solid-State Quantum Electronics Laboratory,Bucharest R-76900, P.O. Box MG-36, Romania

Received November 17, 2003; revised manuscript received April 23, 2004; accepted April 30, 2004

The dynamical behavior of a green-emitting upconversion-pumped Er(0.5 at.%):YAG laser with reabsorptionlosses by an excited-state absorption process is studied. Stability analysis is used to identify the various re-gimes of laser emission and the bifurcations leading to them. The results are illustrated by Runge–Kuttaintegration of the rate equations. An analytical expression of the laser threshold is also presented. Calcu-lated values of the laser threshold and experimental results available in the literature are discussed. Themodel can be applied to the prediction and enhancement of the performances of the green Er31 upconversionlasers. © 2004 Optical Society of America

OCIS codes: 140.3430, 140.3500, 140.3580, 140.7300

1. INTRODUCTIONSelf-pulsing in lasers has been intensively studied in thepast few years, especially in connection with the erbium-doped fiber laser, whose self-pulsing behavior was ex-plained by the presence of ion pairs distributed along thefiber, acting as a saturable absorber.1–3

It was theoretically demonstrated4 that reabsorption oflaser radiation by excited-state absorption (ESA) can leadto self-pulsing in solid-state lasers, by a mechanism verysimilar to saturable absorption Q switching. The demon-stration was made with a simplified model, including athree-level laser scheme and an ESA process that de-pleted the pumped level, reabsorbing laser radiation.

Self-pulsing in an upconversion laser was first reportedby McFarlane.5 The laser material he used wasEr(5 at.%):YLiF4 , which could emit simultaneously at551 nm ( 4S3/2 → 4I15/2 transition) and 671 nm ( 4F9/2→ 4I15/2 transition). Laser oscillation in a pulsed modeat 551 nm, for a cw dye laser pumping at 791 nm, was ob-tained. The experimental study has revealed that thereare two pumping pathways for the laser emission at 551nm, one based on cooperative processes and the otherbased on ESA processes. The self-pulsing was explainedby the presence of a saturable ESA process (4I13/2→ 2H29/2) reabsorbing laser radiation. The simulta-neous emission at both wavelengths was also modulated,and this fact was explained by correlation of the ESApump pathway with the presence of the saturable ESA.Earlier, Johnson and Guggenheim,6 studying the directlypumped laser material Er:BaY2F8 , with various Er con-centrations, found the same ESA process responsible forself-pulsing. The high emission threshold of Er:YAG as agreen upconversion laser was also attributed to this para-sitic ESA.7 An estimation of the negative influence ofthis process on the Er:YLiF4 laser threshold was given byPollnau et al.8 by using numerical simulations.

Our aim is to study the effect of reabsorption of laserradiation by ESA on the dynamics of a real upconversion-

0740-3224/2004/091630-08$15.00 ©

pumped laser and to describe quantitatively the influenceof parasitic ESA on the emission threshold of this laser.For the elaboration of the model, we need spectroscopicdata as well as knowledge of the processes contributing atthe achievement of population inversion.

The system we are modeling is a 0.5-at.% Er:YAG laserpumped in IR on the transition 4I15/2 → 4I9/2 and emittingin green at 561 nm, on the 4S3/2 → 4I15/2 transition.Green laser emission at room temperature in this mate-rial at the same Er31 concentration was obtained byBrede et al.7 The Er:YAG material was chosen for ourmodel because all of its spectroscopic parameters neces-sary for simulations are known. Besides, for Er:YAG, theESA processes that populate the initial laser level 4S3/2have been elucidated.9 To simplify the model by exclud-ing from the rate equations the nonlinear terms due to co-operative upconversion processes, we chose a low Er con-centration for which cooperative upconversion processescan be neglected.

In Section 2 we present the model of the Er:YAG laserbased on the rate equations. The analytical expressionsof the steady-state solutions of the rate equations are pre-sented in Section 3. In Section 4, with a linear stabilityanalysis, the bifurcation values of the control parametersare determined. The nature of the various bifurcationsleading to changes in the laser working regime is also dis-cussed. The temporal behavior of the laser in various re-gimes is illustrated by numerical integration of the rateequations. In Section 5 the cw emission threshold is cal-culated, and the parameters that influence it are dis-cussed. Section 6 includes a discussion of experimentalresults available in the literature; experimental resultsare interpreted with our model.

2. LASER MODELOwing to the multitude of resonances between energy lev-els of Er31 in YAG, there are various ways of pumping theEr:YAG laser by ESA processes. For a pump wavelength

2004 Optical Society of America

O. Toma and S. Georgescu Vol. 21, No. 9 /September 2004 /J. Opt. Soc. Am. B 1631

around 800 nm, the usual way of pumping this laser is atwo-step absorption process that uses 4I9/2 as the pumplevel. Several chains of ground-state absorption (GSA)and ESA processes can be used to populate the 4S3/2 levelin these conditions, their relative importance changingwith the pump wavelength.9 We include in our model theabsorption processes represented in Fig. 1: 4I15/2→ 4I9/2 , 4I9/2 → 2H29/2 , 4I13/2 → 2H211/2 , and 4I11/2→ (4F5/2 , 4F3/2).

The laser transition is 4S3/2 → 4I15/2 . Our model in-cludes a parasitic ESA process at the laser wavelength4I13/2 → 2H29/2 . All cooperative energy-transfer pro-cesses are neglected. The model can be simplified by ourtaking into account that strong multiphonon transitionsconnect successive energy levels of Er in YAG. As a con-sequence, although the parasitic ESA process ends on the2H29/2 level, this level and the levels between it and 4S3/2have very short lifetimes so that we can safely considerthat parasitic ESA feeds directly the level 4S3/2 .

The rate equations that describe the dynamics of thelaser are

dN0

dt5 sem~ f51N5 2 f08N0!w 1

N1

T12 s0fN0 ,

dN1

dt5 2

N1

T11

N2

T22 sbfN1 2 s1N1w,

dN2

dt5 2

N2

T21

N3

T32 scfN2 ,

dN3

dt5 2

N3

T31

N4

T41 s0fN0 2 safN3 ,

dN4

dt5 2

N4

T41

N5

T5,

dN5

dt5 2

N5

T52 sem~ f51N5 2 f08N0!w 1 safN3

1 sbfN1 1 scfN2 1 s1N1w,

Fig. 1. Energy-level scheme of Er:YAG showing the various pro-cesses considered in our model. The ESA processes [a], [b], and[c] are various candidates for the second step of the pumping pro-cess. The ESA process of cross section s1 is the parasitic ESAprocess. The laser transition 4S3/2 → 4I15/2 is also shown.

dw

dt5 v@ sem~ f51N5 2 f08N0! 2 r 2 s1N1#

3 w 1 kN5

T5, (1)

where Ni are the populations of the six energy levels4I15/2 , 4I13/2 , 4I11/2 , 4I9/2 , 4F9/2 , and 4S3/2 (thermalizedwith 2H211/2); Ti are their lifetimes; f is the pumping IRphoton flux; and w is the laser photon flux. sem repre-sents the stimulated emission cross section of the lasertransition; sa , sb , and sc are the cross sections of theESA processes [a], [b], and [c], respectively (presented inFig. 1); and s1 is the cross section corresponding to theparasitic ESA process. f51 and f08 are the Boltzmann fac-tors that give the populations of the Stark levels 1 (low-est) of 4S3/2 and 8 (highest) of 4I15/2 , which represent theinitial and final laser levels. v 5 clp /@l8 1 (n 2 1)l#, crepresents the speed of light in vacuum, n is the index ofrefraction of the laser medium, and l, lp , and l8 are, re-spectively, the length of the active medium, its pumpedlength, and the length of the laser resonator. k is a factorthat takes into account the contribution of the spontane-ous emission to the laser flux, and r represents the lossesin the laser resonator:

r 5 r0 2log~R1R2!

2lp. (2)

r0 represents the losses in the active medium, and R1 andR2 are the reflectivities of the rear and output mirrors, re-spectively.

Uniform distribution of the pump radiation in the ac-tive medium has been assumed; for longitudinal pump-ing, this assumption is valid for an active medium lengthl much shorter than the inverse of the absorption coeffi-cient of the pump radiation.

As can be easily seen, the rate equations are not lin-early independent. The populations of the six levelstaken into account are connected by an equation express-ing the invariance of the total number of laser active ionsof Er31:

N0 1 N1 1 N2 1 N3 1 N4 1 N5 5 Nt . (3)

However, we used the system of Eqs. (1) to simplify theexpression of the matrix of the linearized system, whichwill be used in the stability analysis. For the calculationof the steady-state solutions of Eqs. (1), the equation cor-responding to N0 will be replaced by Eq. (3).

To account for the stimulated emission, one shouldcomplete the terms s1N1 that stand for the parasitic ab-sorption as s1(N1 2 N6), where N6 is the population of2H29/2 . However, the population N6 is negligible incomparison with N1 , owing to the great difference intheir lifetimes (T1 5 6400 ms, T6 5 150 ns), and theterm accounting for the stimulated emission could besafely neglected.

In a recent study9 the excitation spectrum of Er:YAG(monitored by the green luminescence 4S3/2 → 4I15/2 ,


pumping interval corresponding to transition 4I15/2→ 4I9/2) was used to identify the most efficient pumpwavelength for this laser. The wavelength that wasfound the most efficient was 799.5 nm, corresponding tothe highest peak in the excitation spectrum. At thispump wavelength, the most important chain of processescontributing to the population of 4S3/2 is 4I15/2 → 4I9/2(GSA), followed by efficient multiphonon transitions to4I13/2 and 4I13/2 → 2H211/2 (ESA process [b] in Fig. 1).9

For the investigation of dynamics and threshold calcula-tion, we shall assume this way of pumping the upconver-sion laser under discussion, setting sa 5 sc 5 0. ESAprocesses [a] and [c] were included in our model for dis-cussion of experimental results available in the literature.

3. STEADY-STATE SOLUTIONSThe rate-equation-system Eqs. (1) can be solved forsteady-state solutions by our setting all the derivatives atthe left-hand side equal to zero and using the invariancecondition Eq. (3) instead of the equation corresponding toN0 . These solutions may not be experimentally obtain-able (in certain conditions the laser may pulsate indefi-nitely), but their stability (which we shall investigate byusing a stability analysis) decides whether the lasersettles down to a steady state or has a more complicateddynamical behavior.

The steady-state solutions of the system of Eqs. (1) canbe expressed as functions of the steady-state value of N5 .This can be accomplished with Eqs. (4); the overlinedcharacters denote the steady-state values of the five popu-lations and of the laser photon flux:

N0 5

Nt 1r

s12 S 1 1

sem

s1f51 1

T2 1 T3 1 T4

T5D N5

1 2sem

s1f08 1 s0f~T2 1 T3!

,

N1 5 2sem

s1f08N0 1

sem

s1f51N5 2

r

s1,

N2 5 s0fN0 1N5

T5,

N3 5T3

T2N2 ,

N4 5T4

T5N5 ,

w 5 S 1

r

sem

s1sbfA 2

1

T5D N5 1

sbf

s1S sem

rB 2 1 D ,

(4)

where A and B are given by

A

5

~ f51 1 f08! 1 s0f~T2 1 T3!f51 1T2 1 T3 1 T4

T5f08

1 2sem

s1f08 1 s0f~T2 1 T3!

,

B 5

2f08S Nt 1r

s1D

1 2sem

s1f08 1 s0f~T2 1 T3!

. (5)

N5 is the solution of the following second-degree equation:

sem

rAS 1

T52

sem

s1sbfA DN5

2 1 H Fsem

s1S sbf 2

1

T1D

2s0f

f08GA 2 2

sem2

s1rsbfAB 1

sem

rT5B

1 s0ff51

f08J N5 1

r

s1T11 Fsem

s1S sbf 2

1

T1D

2s0f

f08GB 2

sem2

s1rsbfB2 5 0. (6)

In the calculation of steady-state solutions, we ne-glected the spontaneous-emission factor k. This factor issmall and can be safely neglected above threshold; it isimportant only for starting the laser oscillation.

It is obvious that there are two steady-state solutions ofthe rate-equation system, each corresponding to one ofthe two solutions of Eq. (6). The existence of two steady-state solutions is due to quadratical nonlinearities of therate equations. These steady-state solutions representpossible asymptotic behaviors for the rate-equation-system Eqs. (1). Their physical meaning (whether theycan describe cw emission of the laser) depends on the val-ues of the various parameters and will be discussed inSection 4.

4. STABILITY ANALYSISIn this section we study the stability of the steady-statesolutions of the rate-equation system. Their stability de-termines the dynamical behavior of the laser: The laserwill settle down to cw operation if one of the steady-statesolutions is stable, whereas if they are both unstable, thelaser will choose another working regime.

The matrix of the linearized equations system corre-sponding to the system of Eqs. (1) is


J 5 32n30 2 n50 n01 0 0 0 n05 n06

0 2n01 2 n51 n12 0 0 0 n16

0 0 2n12 n23 0 0 0

n30 0 0 2n23 n34 0 0

0 0 0 0 2n34 n45 0

n50 n51 0 0 0 2n05 2 n45 2n06 2 n16

2vn50 n61 0 0 0 vn05 v~n06 1 n16 2 r!

4 , (7)

where

n01 51

T1,

n05 5 semf51w,

n06 5 sem~ f51N5 2 f08N0!,

n12 51

T2,

n16 5 2s1N1 ,

n23 51

T3,

n30 5 s0f,

n34 51

T4,

n45 51

T5,

n50 5 semf08w,

n51 5 sbf 1 s1w,

n61 5 2vs1w. (8)

The question of the stability of steady-state solutionsreduces to finding the eigenvalues of the matrix J, ex-pressed in each equilibrium point (i.e., a steady-state so-lution), and studying their signs as functions of the con-trol parameters.10 We could not find analytically theeigenvalues of the matrix J because of its great dimen-sions, so we calculated them numerically by using a QRalgorithm.11

The matrix J will always have an eigenvalue equal tozero because Eqs. (1) are not independent. This fact isnot significant for our analysis and does not correspond toa bifurcation. It is determined only by the artificial in-crease of the dimension of the problem, by not using thecondition of invariance of the number of active Er31 ions.

For the numerical simulations, we used the values ofthe spectroscopic parameters given in Table 1. We as-sumed a monolithic laser resonator; thus v 5 c/n. Thecontrol parameters are the resonator losses r and thepump flux f. The losses were varied between 2 3 1023

and 2.8 3 1022 cm21; for a resonator with l 5 l8 5 lp5 2 cm, r0 5 2 3 1023 cm21, and R1 5 100%, this in-

terval corresponds to a variation of output mirror reflec-tivity R2 between 100% and 90%.

A study of the sign of the discriminant and solutions ofEq. (6) in the parameter space reveals that, at low valuesof pump photon flux, for all values of the losses r, the dis-

Fig. 2. (a) Stability diagram showing the domains in parameterspace that correspond to various laser regimes. For our configu-ration, the critical value of the resonator losses is rc 5 6.23 1023 cm21. (b) Detail of (a).

Table 1. Spectroscopic Parameters of Er:YAG

Parameter Value Reference

T1 6400 ms our measurementsT2 100 ms our measurementsT3 0.05 ms our measurementsT4 1.5 ms our measurementsT5 16.7 ms our measurementss0 (799.5 nm) 2 3 10221 cm2 Ref. 9sb (799.5 nm) 2 3 10221 cm2 Ref. 9s1 1.2 3 10221 cm2 Ref. 12sem 1.8 3 10220 cm2 Ref. 13f08 0.0169 calculatedf51 0.54 calculated


criminant is negative; therefore the equation has complexsolutions, without physical meaning. For a pump photonflux greater than some value depending on r, the dis-criminant is a positive and increasing function of thepump photon flux, regardless of the value of r. In thisdomain the solutions of Eq. (6) are of opposite signs, re-gardless of the values of the control parameters.

Consequently, the stability analysis will be performedin a limited domain in parameter space, where the dis-criminant is positive or zero. The only steady-state solu-tion of the rate-equation system whose stability has to beanalyzed is the solution determined by the positive solu-tion of Eq. (6), the only solution that has physical mean-ing.

The results of the stability analysis are synthesized inFigs. 2(a) and 2(b). The parameter space of the rate-equation system is divided into three domains: (1) underthe laser threshold, (2) a domain of cw emission, and (3) adomain in which the laser exhibits self-pulsing behavior.The domain of self-pulsing behavior is surrounded by thecw domain, as shown in Fig. 2(b).

Regarding the resonator losses r, there is a criticalvalue that separates two domains in the parameter space.Over this critical value (for our configuration, rc 5 6.23 1023 cm21), the resonator losses are high and deter-mine the dumping of the transient relaxation oscillationsof the laser, leading to a cw emission regime. Figure 3represents the real part of the eigenvalues of matrix Jfunction of the pump photon flux f for r 5 7.13 1023 cm21. The only bifurcation appearing in thiscase is a steady-state bifurcation; the bifurcation pointcorresponds to the laser threshold. For values of fgreater than the bifurcation value, all real parts of the ei-genvalues are negative. Therefore the steady-state solu-tion is stable and is the only physically significant solu-tion of the rate-equation system. The time dependence ofthe laser photon flux [obtained by Runge–Kutta integra-tion of the system of Eqs. (1)] corresponding to f 5 5.53 1018 cm22 ms21 and r 5 7.1 3 1023 cm21 is repre-sented in Fig. 4. The laser exhibits relaxation oscilla-tions and then settles down to cw emission.

For values of r lower than the critical value, the behav-ior of the laser can be more complex. In Figs. 5(a) and5(b) is represented the dependence on f of the real parts

Fig. 3. Real part of the eigenvalues of matrix J for r 5 7.13 1023 cm21 . rc . The constant eigenvalues l5 5 0, l65 20.667 ms21, and l7 5 220 ms21 have been omitted. Thesteady-state bifurcation point at fth 5 4.359 3 1018 cm22 ms21.

of eigenvalues of matrix J, for r 5 4.5 3 1023 cm21.The plot shows three bifurcation points. The first bifur-cation point represents a steady-state bifurcation: Asingle real eigenvalue equals zero. For values of fgreater than this bifurcation value (but lower than thenext bifurcation value), all real parts of the eigenvaluesare negative, denoting the stability of the steady-state so-lution. Thus between the first two bifurcation values off, the laser regime is cw; the first bifurcation correspondsto the laser-emission threshold. The Runge–Kutta solu-tion of the rate-equation system for values of parametersin this domain exhibits qualitatively the same relaxationoscillations and tendency to a steady state as in Fig. 4.

The second bifurcation point represents a Hopf bifurca-tion: Two complex-conjugated eigenvalues have purelyimaginary values. After the bifurcation point, the realparts of the two eigenvalues become positive, denoting theinstability of the steady-state solution of the rate equa-tions. That is, in this domain of the parameter space thelaser emits no longer in the cw regime; it exhibits a self-pulsing behavior. The time dependence of the laser pho-ton flux for values of control parameters f 5 4.43 1018 cm22 ms21 and r 5 4.5 3 1023 cm21, which liein this domain, is represented in Fig. 6. The laser outputis a train of pulses of 952 ns FWHM and a 30.3-kHz rep-etition rate. With increasing pump flux f at a fixed valueof r, the width of the laser pulses decreases, while theirrepetition rate increases, denoting a tendency to cw as fapproaches its third bifurcation value. If the pump fluxis fixed and the resonator losses are decreased, both therepetition rate and the width of the laser pulses decrease.

The third bifurcation point represents another Hopf bi-furcation. The real parts of the same two eigenvalues be-come now negative, denoting that the steady-state solu-tion is again stable. For values of f greater than thisbifurcation value, the laser is in a cw regime. The timedependence of the laser photon flux exhibits qualitativelythe same evolution to a steady state through relaxationoscillations as in Fig. 4.

The first two bifurcation points in Fig. 5 are close toeach other and become closer with decreasing r, as can beseen in the bifurcation diagram in Fig. 2, where the twolines that limit the cw regime approach one another as rdecreases. That is, with decreasing r, the zone of the cw

Fig. 4. Relaxation oscillations obtained by numerical integra-tion of rate-equation-system Eqs. (1), for f 5 5.53 1018 cm22 ms21 and r 5 7.1 3 1023 cm21.


regime that immediately follows the laser threshold be-comes thinner, and it is difficult to obtain a cw emission ofthe laser in this zone. The Hopf bifurcation values of fdepart from one another with decreasing r; this can alsobe observed in the bifurcation diagram, in which the self-pulsing zone in the parameter space enlarges as r de-creases. That is, at low values of resonator losses, pulsedlaser emission can be obtained in a wide range of valuesof the pump photon flux f. This would allow one to ob-tain laser pulses of various widths and at various repeti-tion rates, suitable for multiple applications.

5. THRESHOLD CALCULATIONThe laser-emission threshold can be determined fromrate-equations (1) written for steady-state operation. Asthe stability analysis has shown, the laser-operation re-gime is always cw immediately after threshold, so we mayuse the steady-state equations to determine the laserthreshold. Making w 5 0 (negligible laser photon flux)and using the invariance condition for the total number oflaser active ions, we can obtain the threshold pump pho-ton flux as the solution of the second-degree equation

S f51 2r

semNt

T2 1 T3 1 T4 1 T5

T5Df 2 2 S s1

semsbT5

11

sbT5

r

semNt

T1 1 T2 1 T3

T1Df

21

s0sbT1T5S f08 1

r

semNtD 5 0. (9)

Fig. 5. (a) Real part of the eigenvalues of matrix J for r 5 4.53 1023 cm21 , rc . There is visible a Hopf bifurcation point atf3 5 4.843 3 1018 cm22 ms21. We omit l5 5 0, l65 20.667 ms21, and l7 5 220 ms21 for the same reason asabove. (b) Detail of (a), showing another two bifurcations: asteady-state bifurcation point at f1 5 4.116 3 1018 cm22 ms21

and a Hopf bifurcation point at f2 5 4.117 3 1018 cm22 ms21.

In Eq. (9) we have put into evidence two groups of pa-rameters that will play an important role in the expres-sion of the laser threshold; we denote them

j 5r

semNt,

z 5s1

semsbT5. (10)

The discriminant of Eq. (9) can be written in the formof a second-degree polynomial in j:

D 51

sbT1T52 F ~T1 1 T2 1 T3!2

sbT1

24~T2 1 T3 1 T4 1 T5!

s0Gj2 1

1

sbT1T5F2~T1

1 T2 1 T3!z 24f08

s0T5~T2 1 T3 1 T4 1 T5!

14f51

s0Gj 1 z2 1

4f08f51

s0sbT1T5. (11)

For values of the spectroscopic parameters correspond-ing to Er:YAG, the roots of the polynomial D are bothnegative, and the coefficient of its quadratic term is posi-tive. Consequently, D will be positive for all positive val-ues of j (i.e., of resonator losses r), and Eq. (9) will havetwo real solutions for every positive value of j. The signsof these two solutions will be determined by the sign ofthe coefficient of the quadratic term in Eq. (9) because theother two coefficients are always negative. It is easy tosee that the coefficient of the quadratic term is positive ifj satisfies the condition

j < f51

T5

T2 1 T3 1 T4 1 T5. (12)

Thus, for j satisfying expression (12), Eq. (9) has solu-tions of opposite signs. The positive solution gives thevalue of the threshold pump flux. If j does not satisfy ex-pression (12), the coefficient of the quadratic term in Eq.(9) is negative, and Eq. (9) has two negative solutions.That is, for values of resonator losses r greater than a

Fig. 6. Laser pulses obtained by numerical integration of rate-equation-system Eq. (1), for f 5 4.4 3 1018 cm22 ms21 and r5 4.5 3 1023 cm21.


critical value [given by expression (12)], the laser emis-sion is impossible for all values of pump photon flux.

For values of j satisfying expression (12), the laserthreshold is the positive solution of Eq. (9):

fth 5

S z 11

sbT5

T1 1 T2 1 T3

T1j D 1 AD

2S f51 2T2 1 T3 1 T4 1 T5

T5j D . (13)

From Eq. (13) it can be seen that the laser thresholddepends on r and s1 only through the parameters j and z.As can be seen from Eqs. (11) and (13), the thresholdpump flux is an increasing function of these two param-eters. The increase of fth with r is influenced by themagnitude of stimulated emission cross section sem andthe doping level Nt of the active medium. Greater valuesof these two parameters (in the limits of our low-concentration model) determine a laser threshold lesssensitive at the increase of resonator losses. Similarly,the increase of the laser threshold with s1 is influencedby the stimulated emission cross section sem and the life-time T5 of the initial laser level. An increase of these twospectroscopic parameters would determine a laser thresh-old less sensitive at the increase of the parasitic absorp-tion cross section. The competition between the secondstep of the pumping process (cross section sb) and theparasitic ESA, which share the initial energy level, re-sults in the presence of sb at the denominator of z. Thus,for greater values of sb , the second step of the pumpingprocess decreases the population of the 4I13/2 energy level,diminishing the effect of parasitic ESA on the laser emis-sion.

The theoretical limit to which the laser threshold canbe decreased by decreasing the resonator losses is givenby

fth~r 5 0 ! 51

2 f51F z 1 S z2 1

4f08f51

s0sbT1T5D 1/2G . (14)

In the absence of the parasitic ESA (z 5 0), the mini-mum threshold reduces to the pump photon flux neces-sary for obtaining the population inversion:

fth~r 5 0;s1 5 0 ! 5 S f08

f51s0sbT1T5D 1/2

. (15)

For the Er:YAG material considered here, z2 is muchgreater than the second term under the square root in Eq.(14), so that the dependence of fth(r 5 0) on the parasiticESA cross section is almost linear. This fact also denotesthat the contribution of parasitic ESA to the laser thresh-old is much greater than the pump photon flux requiredfor achieving the population inversion in this three-levellaser. For illustration, in the situation considered here,fth(r 5 0;s1 5 0) 5 2.9 3 1017 cm22 ms21, whereasfth(r 5 0) 5 4.1 3 1018 cm22 ms21. For longitudinalpumping, if we consider a pump beam waist of 50 mm, thevalues of the threshold pump flux correspond respectivelyto the threshold powers of 1.4 and 20.1 W.

6. DISCUSSION OF EXPERIMENTAL DATAIn a series of papers,7,13,14 laser action in low-concentrated (0.5 and 1 at.%) Er:YAG at room tempera-ture was reported. The Er:YAG laser was investigatedwith various methods of pulsed pumping: direct pump-ing at 486 nm (dye laser, 50-ns pump pulses) and upcon-version pumping at 810 nm (Ti:Al2O3 laser, 50-ms pumppulses). For direct pumping, the threshold of the Er:YAGlaser was lower than that corresponding to theEr(1 at.%):YLiF4 . However, for upconversion pumpingat 810 nm, laser emission in Er:YAG required a pump fluxthat seemed to slightly damage the laser crystal: Theoutput energy of Er:YAG was orders of magnitude lowerthan that of Er:YLiF4 and decreased after a few pumpshots. For the improvement of the GSA cross section, akrypton-ion laser at 647 nm was also used as an addi-tional pumping source that matches the GSA into the4F9/2 level in many crystals. With this way of pumpingthe Er:YAG laser, the output energy decreased after a fewpump pulses and then stabilized; the output energy wasstill much lower than that of Er:YLiF4 .

We analyzed the dynamics of the Er:YAG laser pumpedat 810 nm. Spectroscopic data were obtained by ourmeasurements: s0 (647 nm) ' 10220 cm2, s0 (810 nm)' 10222 cm2, and sc (810 nm) ' 4 3 10221 cm2. Thestability analysis of the steady-state solution of Eqs. (1)(for sa 5 sb 5 0) resulted in a stability diagram similarto Fig. 2. Similarly, there is always a cw emission do-main at low pump powers. This fact enabled us to usethe steady-state rate equations for the calculation of thelaser threshold. We found a similar expression for thezero-losses limit of the laser threshold:

fth8~r 5 0 ! 51

2 f51F z8 1 S z82 1

4f08f51

s0scT2T5D 1/2G .

(16)

Here z8 5 s1T1 /( semscT2T5). The presence of the ratioT1 /T2 in the expression of z8 should be noted; this can beexplained by the fact that at threshold the populations oflevels 1 and 2 now satisfy N1 /N2 5 T1 /T2 . The intrin-sic threshold for the achievement of population inversionhas an expression similar to Eq. (15):

fth8~r 5 0;s1 5 0 ! 5 S f08

f51s0scT2T5D 1/2

. (17)

Substitution of numerical values into Eqs. (16) and (17)yields fth8(r 5 0) 5 1.2 3 1020 cm22 ms21 and fth8(r5 0;s1 5 0) 5 6.8 3 1018 cm22 ms21, respectively.These two values are, respectively, 29 and 25 timesgreater than those obtained in the case of 799.5-nmpumping. For longitudinal pumping, if we consider apump beam waist of 50 mm, the values of the thresholdpump flux correspond respectively to the threshold pow-ers of 580 and 35 W.

For an upconversion pumping at 810 nm aided by thekrypton-ion laser at 647 nm, Eqs. (16) and (17) remainvalid. Substitution of numerical values into Eqs. (16)and (17) yields fth8(r 5 0) 5 1.2 3 1020 cm22 ms21 andfth8(r 5 0;s1 5 0) 5 6.8 3 1017 cm22 ms21, respec-tively. It can be observed that the zero-losses-limitthreshold practically does not change, whereas the inver-


sion threshold is decreased ten times. These facts can beeasily explained by the fact that s0 appears only in theexpression of the inversion threshold and does not influ-ence the parameter z8, which is responsible for the lossescaused by parasitic ESA. Thus, according to our calcula-tions, the krypton laser does not influence the laserthreshold of Er:YAG pumped by ESA at 810 nm. Themain shortcoming of this way of pumping is the use of4I11/2 as the initial level for the second step of the pump-ing scheme. This level has a lifetime much shorter thanthat of 4I13/2 ; this fact influences negatively the competi-tion between the second step of the pumping process andthe parasitic ESA, which is manifested in the parameterz8.

In conclusion, we can explain the weak performances ofthe Er:YAG laser in the studies cited above by the factthat the pumping process based on ESA from 4I11/2 levelis not favorable for the Er:YAG laser. This way of pump-ing gives a high laser threshold. Thus, for one to obtainlaser emission in Er:YAG, a great pump power that couldcause damage to the laser crystal was necessary.

7. CONCLUSIONSWe presented a rate-equation model of an upconversiongreen Er(0.5 at.%):YAG laser emitting on the transition4S3/2 → 4I15/2 . The laser exhibits reabsorption losses bythe parasitic ESA process 4I13/2 → 2H29/2 . We put intoevidence the various laser-emission regimes and foundthe values of the control parameters (pump photon flux fand resonator losses r) corresponding to various laser re-gimes. The dynamical behavior of the laser was illus-trated by numerical integration of the rate-equation sys-tem.

We also found an analytical expression of the laserthreshold, in the presence of parasitic ESA. A discussionof the influence of various parameters on the laser thresh-old was presented, based on significant groups of param-eters that appear in the expression of the laser threshold.We showed that, for this laser, parasitic ESA makes themain contribution to the laser threshold.

The results of our model helped us to explain the ex-perimental results available in the literature. A discus-sion of the various pump processes used in the experi-ments was presented. We proposed a way of pumpingthe green Er:YAG laser that could, according to our calcu-lations, reduce the threshold and enhance performancesof the green Er:YAG laser.

ACKNOWLEDGMENTSThis study was supported by the CERES (FundamentalResearch of Social, Economic, and Cultural Interest)Project 128/2001.

O. Toma, the corresponding author, can be reached bye-mail at [email protected].

REFERENCES1. F. Sanchez, P. Le Boudec, P.-L. Francois, and G. Stephan,

‘‘Effects of ion pairs on the dynamics of erbium-doped fiberlasers,’’ Phys. Rev. A 48, 2220–2229 (1993).

2. E. Lacot, F. Stoeckel, and M. Chenevier, ‘‘Dynamics of anerbium-doped fiber laser,’’ Phys. Rev. A 49, 3997–4007(1994).

3. S. Colin, E. Contesse, P. Le Boudec, G. Stephan, and F.Sanchez, ‘‘Evidence of a saturable-absorption effect inheavily erbium-doped fibers,’’ Opt. Lett. 21, 1987–1989(1996).

4. F. Sanchez and A. Kellou, ‘‘Laser dynamics with excited-state absorption,’’ J. Opt. Soc. Am. B 14, 209–213 (1997).

5. R. A. McFarlane, ‘‘Dual wavelength visible upconversion la-ser,’’ Appl. Phys. Lett. 54, 2301–2302 (1989).

6. L. F. Johnson and H. J. Guggenheim, ‘‘New laser lines inthe visible from Er31 ions in BaY2F8 ,’’ Appl. Phys. Lett. 20,474–477 (1972).

7. R. Brede, E. Heumann, J. Koetke, T. Danger, G. Huber, andB. Chai, ‘‘Green up-conversion laser emission in Er-dopedcrystals at room temperature,’’ Appl. Phys. Lett. 63, 2030–2031 (1993).

8. M. Pollnau, W. Luthy, and H. P. Weber, ‘‘Population mecha-nisms of the green Er31:LiYF4 laser,’’ J. Appl. Phys. 77,6128–6134 (1995).

9. S. Georgescu, O. Toma, C. Florea, and C. Naud, ‘‘ESA pro-cesses responsible for infrared-pumped, green and violet lu-minescence in low-concentrated Er: YAG,’’ J. Lumin. 101,87–99 (2003).

10. J. D. Crawford, ‘‘Introduction to bifurcation theory,’’ Rev.Mod. Phys. 63, 991–1037 (1991).

11. B. N. Parlett, ‘‘The QR algorithm,’’ Comput. Sci. Eng. 2,38–42 (2000).

12. P. LeBoulanger, J.-L. Doualan, S. Girard, J. Margerie, andR. Moncorge, ‘‘Excited-state absorption spectroscopy ofEr31-doped Y3Al5O12 , YVO4 , and phosphate glass,’’ Phys.Rev. B 60, 11380–11390 (1999).

13. T. Danger, J. Koetke, R. Brede, E. Heumann, G. Huber, andB. H. T. Chai, ‘‘Spectroscopy and green upconversion laseremission of Er31-doped crystals at room temperature,’’ J.Appl. Phys. 76, 1413–1422 (1994).

14. R. Brede, T. Danger, E. Heumann, G. Huber, and B. Chai,‘‘Room-temperature green laser emission of Er:LiYF4 ,’’Appl. Phys. Lett. 63, 729–730 (1993).

dynamics of an upconversion er:yag laser with reabsorption losses

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