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The effect of terminal level lifetime on three-micron laser emission in Er-doped crystals
Optics Communications 92 (1992) 67-72 North-Holland
OPTICS COMMUNICATIONS
The effect of terminal level lifetime on three-miCron laser emission in Er-doped crystals V. Lupei, S. Georgescu and V. Florea Institute o f Atomic Physics, P.O. Box MG-6, Bucharest R- 76900, Romania
Received 18 September 1991; revised manuscript received 10 February 1992
The effect of reducing the lifetime of the terminal laser level for three-micron laser emission in erbium doped crystals by codoping with ions as Nd 3+, Tm 3+ or Ho 3+ is discussed using the stationary regime solution of laser rate equations. It is shown that this effect could be beneficial or detrimental, depending on the specific laser transition. Reduction of Tt may also change the spectral characteristics of the emission.
1. Introduction Several high-concentration erbium-doped crystals show efficient three-micron 4Il t/2-~4It 3/2 laser emission at room temperature, though the lifetime of the terminal level is much longer (sometimes up to about two orders of magnitude) than that of the initial one. This fact usually should produce a selfsaturation of the laser transition. In early work, to overcome this "bottleneck" effect, co-doping with other ions to deexcite the 4I~3/2 Er 3+ level by energy transfer was proposed [ 1 ]. Although lately it was demonstrated that the three-micron emission of erbium lasers is based essentially on the existence of an up-conversion process from 4I~3/2, active at high Er 3+ concentrations [ 2-4 ], the attempt to improve the performances of these lasers by co-doping with ion deactivators of the 4I~3/2 level of Er 3+ continued. The results were apparently contradictory: the co-doping of Er 3+ :YAG with Nd 3+, Ho 3+ or Tm 3+, that reduce the lifetime of Er 3+ 4I~3/2 level, had a detrimental effect on the laser emission [ 5,6 ] while in the case of Er 3+, Cr 3+ : YSGG the co-doping with Ho 3+ was beneficial [ 7 ]. In this paper we show that these contradictory results of co-doping are the natural consequence of a unitary treatment of the three-micron Er 3÷ laser processes, using the rate equation model [ 3,4 ]. In order to make the discussion clearer we shall restrict to the
stationary regime (that governs the cw and long pulse laser emission) for which the rate equations can be solved analytically.
2. Theory Three-micron laser emission of highly Er a+ concentrated crystals involves several energy levels [ 3 ]. For xenon lamp pumping the excitation collects in the 4Sa/2 level, thermalized with 2H2~1/2. The pumping could be also made with appropriate sources in any level placed below 4S3/2, i.e. 4F9/2, 419/2, 4I~1/2 and even in the terminal laser level 4I~3/2 [ 8,9]. The main processes involved in the three-micron emission of Er 3+ in crystals axe represented in fig. 1. Even at relatively low Er 3+ concentrations the 4S3/2 level is de-excited by cross-relaxation processes. Thus, for concentrations up to 5 at.% (Er 3+ in YAG) the twoion processes are prevailing (fig. la): (4S3/2--}419/2) + (4Ii5/2--~4Ii3/2) [3,10], while for higher concentrations three-ion processes, involving the donor and two acceptors, become dominant (fig. lb): (4S3/2-, 4II5/2)D+ [ (4]15/2-'-}4113/2)A1+ (4115/2-',419/2)A2 ] [ 10-
12 ]. Both processes, characterized by the cross-relaxation parameter I4z4,have a similar result, leading to an equal population of 4Ii 1/2 and 4113/2 levels as a whole; however a population inversion between pairs of crystal field components (xi, yi) of these lev-
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
67
Volume92, number 1,2,3
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l
m
m
m
l l
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15 August 1992
20 l l l
m m
m
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mm ~w~Jm 2 H 2 . f 2
m
m
4F9/2
m
m
~[11/2
I/Jm m
0
m
~mm a)
b)
m~m~115~
c)
d)
Fig. 1. The main energytransfermechanismsactive in 3 ~tmEra+ emissionin crystals: (a) two-ioncross-relaxationfrom 483/2;(b) threeion cross-relaxationfrom 453/2;(c) up-conversionfrom 4Im3/2;(d) up-conversionfrom 4It I/2. els is built. Two other energy transfer (up-conversion) processes are involved: ( i ) (4Ii3/2~4115/2)+ (4Ii3/2~419/2) (fig. IC) with up-conversion parameter WI de-activates the final laser level and, via the rapid multiphonon relaxation 419/2 ~ 4I 11/2, repumps the initial one;
(ii) (4111/2--~4115/2)+
(4I 1I/2-'* 4F7/2)
(fig. ld) characterized by WE, de-activates the initial laser level, but due to multiphonon relaxation 4F7/2--,453/2 its negative effect is partially compensated. All the Er 3+ levels from 453/2 down are implied in the three-micron emission, but one can exclude safely 419/2 due to its very fast relaxation on the 4111/2level. Thus the laser processes can be described by the following rate equation system: dN4/dt= - N4/ T4 - W4N [ N4 + W2N 2 +Rp4No, d N 3 / d t = - N 3 / T 3 +N4/T4 +Rp3No,
(453/2)
- N2/ T2 - W2N ~ - p ~ + R2No = 0 ,
(4F9/2)
- N 1 / T I + N2/T2 - 2 W I N 2 + WEN 2
dN2/dt = - N 2 / T 2 + Na/T3 - 2 W 2 N2 + W I N 2 + W4N'~N4 -aqb( otN2 - flBl ) + Rp2No,
68
(2)
where RI =Rpl +Rv4,
(4113/2)
d ~ / d t = q~v[a(aN2 -fiN1 ) - p ] •
+ p ~ + RI No = 0 , oIN2 -- BN1 = p / a ,
(4111/2)
d N l / d t = - N I / T I + N2/T2 - 2 W I N 2 + W4N[N4 + tr~( otN2 - flNl ) + Rpt No,
Here NI to N4 are the populations of levels 4113/2, 4111/2, 4F9/2 and 453/2 respectively, 7"1to T4 being their lifetimes and Rp~ to R ~ are the pumping rates in these levels (the pump in 419/2 would be included in Rp2 ) . The coefficients a and fl are the Boltzmann thermal population fractions for the crystal field sublevels xi(4I~1/2) and yi(4113/2) involved in the laser transition, tr is the emission cross-section, p represents the total losses, v is the speed of light in the active medium and the exponent x takes the values 1 or 2 for the two-ion or three-ion cross-relaxation processes respectively. Under normal pumping rates the population No of the ground level 4115/2 can be taken as equal to the total erbium concentration. For stationary regime (dNi/dt=O; d ~ / d t = 0 ) and medium or high erbium concentrations (when W4N~)>> 1/T4) the rate equations for the 4113/2, 4111/2 levels and photon flux can be written as
( 1)
R2----Rp2 +Rp3 +Rp4.
The system (2) gives an exact analytical solution for the population N1 of 4Ii3/2:
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
able for discussing the effect of reducing T~ by codoping.
N, =(P[ (RI +R2) NolWI] 112
=[(I+~2) II2-F.][(RI"FR2)No/WI] 1/2
(3)
with
3. Discussion
= (2T,)
--I
[ (RI +R2)No WI ] - - 1 / 2 .
(4)
Below, or immediately after laser threshold the flux terms in eqs. (2) can be neglected and N2 = J [ (Rl +2R2) No~WE] 1/2
----[ ( P + y 2) 1/2--7] [(R~ +2R2) No/W2] 1/2 ,
(5)
with
v= 1 - N I / [ (R~ +2R2) NoT1 ] ,
(6)
and
y=(2T2)-l[ (Rt +2RE)NoW2] -I/2 .
(7)
The population difference between the crystal field components x; and Yi below laser threshold is
AN=otN2 -- flNi =otJ[ (RI + 2R2)No/ W2] t/2
[ _ q, I~ (W2"~l/2 ( R, +.~ ]'/21. XL1 ~\~-~/ \kT+--~-2j d
(8)
The parameter
p - (fl/ ol ) ( W2/ Wi ) 1/2
(9)
can be used as a figure of merit for the given laser transition [ 13 ]; it depends on the host crystal (via the up-conversion parameters I411, WE) and on the particular laser transition (via the Boltzmann coefficients a, fl). At threshold the gain must compensate for the losses p and the population difference must satisfy the third equation in (2). The system (2) gives an exact analytical solution for the photon flux density in the resonator:
~ = N o [R2+(R _l_R2)(l_p2)l_ 1__~ p otaT2 --Nl
15 August 1992
(P~-~2
2flW2~_ l - p 2
+ a20. ]
--~-i N,.
W2p O/20.2 (10)
AS we can observe from eqs. ( 2 ) - (8), (10) the lifetime T~ (4I, 3/2 ) affects the population of the levels, the population difference AN and the photon flux density. Consequently, all these expression are suit-
According to eqs. ( 3 ) - ( 6 ) shortening T~ implies a reduction of both N~ and N2 populations. While the reduction of N~ seems very normal, the reduction of N2 is a consequence of the coupling between 4111/2 and 4113/2 levels by up-conversion. The reduction of the lifetime Tt of 4Ii3/2 level by adding impurity ions as Nd 3+, Tm a+ or Ho a+ could be drastic (in some cases with orders of magnitude for few percent). The effect of reduction of T~ on the coefficients ~ and J (the T~ dependent part of the populations) is given in table 1 for two systems: Er3+:YAG (Er 3+ concentration 50 at.%) and Er3+:YSGG (10 at.°/~). Although several sets for rates W~ and WE are given in the literature [ 4, 1416], depending on the determination method, we used here the values from refs. [4,14]. In both these works the rates Wmand WE are determined from the analysis of the 4Ii1/2 decay for different pumping densities in various Er~ + levels. Thus, while the excitation conditions in ref. [4] (pumping in 4F9/2) make the method more sensitive to the I412process, the 4Ill/2 level being populated firstly after the excitation pulse, the experimental conditions in ref. [ 14 ], although much 10ss sensitive to the I412process (excitation in 4S3/2) assure accurate values for WI. Therefore, we have chosen W I a s determined in ref. [14] and the ratio W2/WI as obtained in ref. [4]. As for Er 3+:YAG [14], the T~, T2 values for Er3+:YSGG depend on Er 3+ concentrations, probably owing to some uncontrolled impurities, and we have used in our example some intermediate values between the low a n d high Er-concentration data [ 13,17 ]. Thus, the fo!lowing sets of spectroscopic parameters were choosen: for Er 3+ : YAG (50 at.%) T~=4× 10 -3, T2--88X 10-6 s [14], W~= 1.3× 10 -~5 cm3s -I, 1412=3.7× li0 -15 cm3s -l, and for Era+:YSGG (10 at.%) W l = l . 1 5 × 1 0 -~5 cm3s -l, I412=3.75×10 -15 cm3S-1 [13], T1=4.4×10 -3 s, 7"2=1.4×10 -3 s. In b o t h cases pumping rates RINo=R2No= 2.5 X 10~3 cm -3 s- 1 (assuring operation over threshold) were assumed arbitrarily as an illustrative example to Show the effect of Ti on J and 69
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15 August 1992
Table 1 The effect of reducing the lifetime T~ in case of Er3+ :YAg and Er3+ :YSGG on the coefficients ~ and to that determine the populations N~ and N2. Crystal
Tz (s)
t~
to
O/t0
Er: YAG 50 at.% Er
4 × 10 -3 1.5X 10 -3 1 × l0 -4 2 × l0 -5
0.895 0.890 0.789 0.572
0.997 0.987 0.823 0.420
0.898 0.901 0.958 1.362
Er:YSGG 10 at.% Er
4 . 4 × l0 -3 1.5× 10 -3 1 × 10 -4 2 × 10 -5
0.990 0.984 0.873 0.657
0.995 0.986 0.813 0.402
0.995 0.998 1.074 1.634
¢0. The data from table 1 show that the reduction of 7"1 has a more marked effect o n N I , although its effect on N2 is not negligible. From eq. (8) the condition to have positive population inversion below threshold restricts the maximal value of the figure of merit p to
~ (RI + 2R2~ '/2 P < tp \R-~1~ - 2 ]
"
(11)
Near the threshold, when q ~ 0 and the expressions (3), (5) for the populations NI, N2, respectively, are still valid, and the condition on p becomes more restrictive:
AN>~p/a,
p<~~~-(R'+2R2.~l/Z(l-a-~2 ) \ RI -I-R2 ,]
(12,
These relations show that for three-micron generation of Er 3+ in a given crystal the possible transitions (emission wavelengths) between various crystal field sublevels, characterized by the figure of merit p, are selected by the wavelength and rate of pumping (RI, R2), emission cross-section tr, losses p and the coefficients a, ~oand t~. As shown in table 1 the coefficients ~, q~and their ratio t~/~oare dependent on the lifetime Tl: for large T~ as usually in crystals doped only with Er 3+, the ratios ~/~oare small and only transitions with small figure of merit p are possible, i.e. transitions that connect the lowest Stark sublevels of 4111/2 with the upper sublevels of 4Ii3/2. A drastic reduction of Ti (as it happens by co-doping the crystals with de-activators as Nd a+, Ho a+ or Tm 3+ ) leads to a marked increase of the ratio t~/~o, extending the range of p and new laser transitions 70
become possible (even those between the upper sublevels of4111/5 and the lowest one of 411a/2). Some of the new transitions made possible by reducing TI might have larger cross-sections; it is also possible that the population inversion for the original transition is reduced due to the lowering of #. Therefore the reduction of 7"1 by co-doping may change the spectrum of laser emission for three-micron Er 3+ doped crystals. The effect of reducing the lifetime T~ on the threemicron laser emission becomes apparent also from the examination of the photon flux density expression (eq. (10)). Although the condition that must be fulfilled by p for a positive flux density is identical to condition (12), eq. (10) enables a separation of relative contributions of various terms. Only the first and the last term in eq. (10) could be positive function of figure of merit p. The first (pump) term is positive when
+2R,)
p
I'2
,
(13)
while the last term, dependent on TI, is positive for transitions with p > 1. As a consequence, a large value of TI is beneficial for laser transitions with p < l since it reduces a negative term in eq. ( l 0). However, for transitions with p > 1, that render the last term in eq. ( l 0) positive, a smaller TI is beneficial. This is more evident for p larger than Po when the only positive term in eq. (10) is the last one. These considerations explain the detrimental effect of reducing T~ by co-doping with Nd 3+ or Ho 3+ on the 2.936 ~tm laser emission of Er a+ :YAG [ 5,6 ]
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
and the beneficial effect on the 2.702 ~tm emission in Er a+, Cr3+:YSGG co-doped with Ho 3+ [7]. A similar reduction of laser parameters by co-doping with T m 3+ and Ho 3+ was observed in the case of LuAG, which in many respects (for instance the total crystal field splitting of levels) is similar to YAG [ 18 ]. However, no available data on WI and WE in this system exist, so that a similar analysis as for YAG cannot be made at this moment. The long pulse laser emission at 2.936 ~m in Era+:YAG corresponds to the transition xE-y7 and has a figure of merit p = 0.43. Under xenon lamp pumping R, ~ R 2 ~Rv4 and Po = 1.225; thus the first term in eq. (10) is positive, while the last one is negative. Evidently, a reduction of Tm would be detrimental for this transition. However, for other transitions with p > l the reduction of 7"1 is beneficial and new laser transitions of shorter wavelength are, in principle, possible. In this case the first term in eq. (10) may become very small or even negative; this could reduce the efficiency of pumping, leading to a strong increase in the laser threshold that may preclude the stationary regime, making possible only the short-pulse laser emission. We note, also, that the reduction of TI below the value of T2 transforms the three-micron Er 3+ laser into a conventional four-level laser. Figure 2 shows the computed dependence of the photon flux density O, eq. (10), on pump rate R2/Vo for two long-pulse laser transitions: 2.936 ~tm emission of Er3+:YAG ( p = 0 . 4 3 ) and 2.794 Ilm emission o f E r 3+ in YSGG ( p = 1.09), for various TI values; it is evident that reduction of T~ has a positive effect for p > 1 and negative for the case with p < 1. In the case of Er a +, Cr 3+ : Y S G G cw emission was obtained at 2.702 ~tm under krypton laser pumping (2=647.1 nm); this emission involves a superposition of the transitions (xl-y~) and (x2-Y2) having a figure of merit p = 1.92. For this crystal the pump is made into the strong 4A2--*4T 2 absorption band of Cr 3+ and the excitation is then transferred very efficiently to the 419/2 level of Er a+. The presence of chromium modifies also the mechanism of de-excitation of 4S3/2 of Er 3+ (that is populated by the upconversion process W2) by the chain of transfer processes Er 3+ (4S3/2) -, Cr 3+ (4T 2 ) ~ Er a+ (419/2 ), characterized by a rate W~; it can also lead to a more effective depopulation of 4It3/2 by possible additional cross-relaxations between Er 3+ and Cr 3+ [ 19 ].
10
15 August 1992 ~
2'
0 0
1
2
3
RpN o (10Z2cm-~ s'1)
Fig. 2. Photon fluxdensity Oversus pumping rate R2Nofor 2.794 jm~emissionof Era+ :YSGG (curves l --4correspondto 7"1= 4400, 1500, 100and 20 ~ts,respectively)and for 2.936 Itm emissionof Era+:YAG (curves 1'--4' correspondto/'1 = 4000, 1500, 100 and 20 ~ts,respectively). Because of all these effects the pump term in (10) is modified, the condition for positive pump becoming now p2<2/f, where the subunitary parameter f - [ 1+ ( W'4Ncr) / ( W~Vo) ] -1 describes the relative effect of the processes W~ and W4 and can be modified by a proper choice of the Er and Cr concentrations so as to make possible the emission at 2.702 ~tm [ 19]. For this transition the last term in (10) becomes positive ( p 2 _ t = 2 . 6 8 ) and thus we can expect that the reduction of/'1 would have important effects on the laser emission: reduction of threshold, improvement of pump efficiency and a nonlinear dependence of the photon I flux density on pump, due to the (R2No)i/2 dependence of the last term in (10) via N~. These effects ar~ evident from the computer calculation of • for various values of T~, presented ~n fig. 3, and they were indeed observed experimentally for the 2.702 ~tm CW laser emission under krypton laser pumping in Er 3+, Cr 3+ :YAG crystals co-doped with Ho 3+ [7].
4. Conclusions
The reduction of the lifetime of the terminal level 71
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i
15 August 1992
p o w e r characteristics o f the laser e m i s s i o n .
i
References
5l,
Cs 2 0
l
2
3
Rp N O t102ecm-3s-'J
Fig. 3. Photon flux density • versus pumping rate R2Nofor Er3+, Cr 3+ :YSGG (2= 2.702 ~tm); 1: Ti =4400 ~ts, 2: Tl = 1500 ;is, 3: TI =400 ;is, 4: TI = 100 lis, and 5: T, =20 ~ts.
4Ii3/2 o f t h r e e - m i c r o n laser crystals b y co-doping with ions such as N d 3+, H o 3+ or T m 3+ that de-activate this level b y energy t r a n s f e r has a c o m p l e x effect o n the laser e m i s s i o n ; it is a f u n c t i o n o f the figure o f m e r i t p. A n analysis o f the s o l u t i o n o f the rate equat i o n system for the s t a t i o n a r y regime o f g e n e r a t i o n shows that for laser t r a n s i t i o n s with p < l the reduct i o n o f TI w o u l d b e d e t r i m e n t a l , while for p > 1 it is beneficial. These p r e d i c t i o n s are c o n f i r m e d b y the negative effect o f c o - d o p i n g o n the long pulse 2.936 ~tm e m i s s i o n i n Er 3+ : Y A G ( p = 0 . 4 3 ) a n d b y the positive effect o n the cw e m i s s i o n at 2.702 ktm i n Er 3+, Cr 3+ : Y S G G ( p = 1.92). It is also s h o w n that the r e d u c t i o n o f T1 b y c o - d o p i n g with ions as N d 3 +, T m 3+, H o 3+, etc., c o u l d change the w a v e l e n g t h a n d
72
[ 1 ] A.A. Kaminskii, T.I. Butaeva, A.D. Ivanov, I.V. Michalov, A.G. Petrosyan, G.I. Rogov and V.A. Fedorov, Sov. Tech. Phys. Lett. 2 (1976) 308. [2]V.I. Zhekov, V.A. Lobachev, T.M. Murina and A.M. Prokhorov, Soy. J. Quantum Electron. 14 (1984) 123. [ 3 ] S. Georgescu, V. Lupei, I. Ursu, V.I. Zhekov, V.A. Lobachev, T.M. Murina and A.M. Prokhorov, Rev. Roum. Phys. 31 (1986) 857. [4]V.I. Zhekov, T.M. Murina, A.M. Prokhorov, M.I. Studenikin, S. Georgescu, V. Lupei and I. Ursu, Sov. J. Quantum Electron. 16 (1986) 274. [5 ] R. Kurtz, L. Pathe and J. Machan, Tunable Solid State Lasers, Technical Digest, Washington 1989, p. 224. [6] W.Q. Shi, M. Bass and M. Birnbaum, J. Opt. Soc. Am. B 6 (1989) 23. [7] G. Huber, E.V. Duczynski and K. Petermann, IEEE J. Quantum Electron. QE-24 (1988) 920. [8 ] S.A. Pollack, D.B. Chang and N.L. Moise, J. Appl. Phys. 60 (1986) 4077. [ 9 ] V.I. Zhekov, V.A. Lobachev, T.M. Murina, A.V. Popov, A.M. Prokhorov and M.I. Studenikin, Kvant. Electr. 16 (1989) 1138. [ 10] A. Lupei, V. Lupei, S. Georgescu, I. Ursu, V.I. Zhekov, T.M. Murina and A.M. Prokhorov, Phys. Rev. B 41 (1990) 10923. [ll]V.I. Zhekov, T.M. Murina, A.M. Prokhorov and M.I. Studenikin, Pis'ma v JETP 52 (1990) 1251. [ 12] V. Lupei, J. Lumin. 48/49 ( 1991 ) 157. [13] M.A. Noginov, S.G. Semenkov, V.A. Smirnov and I.A. Scherbakov, Optika i Spectrosk. 69 (1990) 120. [ 14] S. Georgescu, V. Lupei, A. Lupei, V.I. Zhekov, T.M. Murina and M.I. Studenikin, Optics Commun. 81 ( 1991 ) 186. [ 15 ] M.A. Noginov, V.A. Smirnov and I.A. Scherbakov, Optical and Quantum Electronics 22 (1990) $61. [ 16 ] W.Q. Shi, M. Bass and M. Birnhaum, J. Opt. Soc. Am. B 7 (1990) 1456. [ 17 ] P.F. Moulton, J.G. Manni and G.A. Rines, IEEE J. Quantum Electron. QE-24 (1988) 960. [18] M.A. Andriasyan, N.V. Vardanyan and R.B. Kostanyan, Kvant. Elecktr. 9 (1982) 1269. [19] V. Lupei, S. Georgescu and V. Florea, IEEE J. Quantum Electron., to be published.
OPTICS COMMUNICATIONS
The effect of terminal level lifetime on three-miCron laser emission in Er-doped crystals V. Lupei, S. Georgescu and V. Florea Institute o f Atomic Physics, P.O. Box MG-6, Bucharest R- 76900, Romania
Received 18 September 1991; revised manuscript received 10 February 1992
The effect of reducing the lifetime of the terminal laser level for three-micron laser emission in erbium doped crystals by codoping with ions as Nd 3+, Tm 3+ or Ho 3+ is discussed using the stationary regime solution of laser rate equations. It is shown that this effect could be beneficial or detrimental, depending on the specific laser transition. Reduction of Tt may also change the spectral characteristics of the emission.
1. Introduction Several high-concentration erbium-doped crystals show efficient three-micron 4Il t/2-~4It 3/2 laser emission at room temperature, though the lifetime of the terminal level is much longer (sometimes up to about two orders of magnitude) than that of the initial one. This fact usually should produce a selfsaturation of the laser transition. In early work, to overcome this "bottleneck" effect, co-doping with other ions to deexcite the 4I~3/2 Er 3+ level by energy transfer was proposed [ 1 ]. Although lately it was demonstrated that the three-micron emission of erbium lasers is based essentially on the existence of an up-conversion process from 4I~3/2, active at high Er 3+ concentrations [ 2-4 ], the attempt to improve the performances of these lasers by co-doping with ion deactivators of the 4I~3/2 level of Er 3+ continued. The results were apparently contradictory: the co-doping of Er 3+ :YAG with Nd 3+, Ho 3+ or Tm 3+, that reduce the lifetime of Er 3+ 4I~3/2 level, had a detrimental effect on the laser emission [ 5,6 ] while in the case of Er 3+, Cr 3+ : YSGG the co-doping with Ho 3+ was beneficial [ 7 ]. In this paper we show that these contradictory results of co-doping are the natural consequence of a unitary treatment of the three-micron Er 3÷ laser processes, using the rate equation model [ 3,4 ]. In order to make the discussion clearer we shall restrict to the
stationary regime (that governs the cw and long pulse laser emission) for which the rate equations can be solved analytically.
2. Theory Three-micron laser emission of highly Er a+ concentrated crystals involves several energy levels [ 3 ]. For xenon lamp pumping the excitation collects in the 4Sa/2 level, thermalized with 2H2~1/2. The pumping could be also made with appropriate sources in any level placed below 4S3/2, i.e. 4F9/2, 419/2, 4I~1/2 and even in the terminal laser level 4I~3/2 [ 8,9]. The main processes involved in the three-micron emission of Er 3+ in crystals axe represented in fig. 1. Even at relatively low Er 3+ concentrations the 4S3/2 level is de-excited by cross-relaxation processes. Thus, for concentrations up to 5 at.% (Er 3+ in YAG) the twoion processes are prevailing (fig. la): (4S3/2--}419/2) + (4Ii5/2--~4Ii3/2) [3,10], while for higher concentrations three-ion processes, involving the donor and two acceptors, become dominant (fig. lb): (4S3/2-, 4II5/2)D+ [ (4]15/2-'-}4113/2)A1+ (4115/2-',419/2)A2 ] [ 10-
12 ]. Both processes, characterized by the cross-relaxation parameter I4z4,have a similar result, leading to an equal population of 4Ii 1/2 and 4113/2 levels as a whole; however a population inversion between pairs of crystal field components (xi, yi) of these lev-
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
67
Volume92, number 1,2,3
OPTICS COMMUNICATIONS l
l
m
m
m
l l
m
m
m
m
m m
m m
15 August 1992
20 l l l
m m
m
15
mm ~w~Jm 2 H 2 . f 2
m
m
4F9/2
m
m
~[11/2
I/Jm m
0
m
~mm a)
b)
m~m~115~
c)
d)
Fig. 1. The main energytransfermechanismsactive in 3 ~tmEra+ emissionin crystals: (a) two-ioncross-relaxationfrom 483/2;(b) threeion cross-relaxationfrom 453/2;(c) up-conversionfrom 4Im3/2;(d) up-conversionfrom 4It I/2. els is built. Two other energy transfer (up-conversion) processes are involved: ( i ) (4Ii3/2~4115/2)+ (4Ii3/2~419/2) (fig. IC) with up-conversion parameter WI de-activates the final laser level and, via the rapid multiphonon relaxation 419/2 ~ 4I 11/2, repumps the initial one;
(ii) (4111/2--~4115/2)+
(4I 1I/2-'* 4F7/2)
(fig. ld) characterized by WE, de-activates the initial laser level, but due to multiphonon relaxation 4F7/2--,453/2 its negative effect is partially compensated. All the Er 3+ levels from 453/2 down are implied in the three-micron emission, but one can exclude safely 419/2 due to its very fast relaxation on the 4111/2level. Thus the laser processes can be described by the following rate equation system: dN4/dt= - N4/ T4 - W4N [ N4 + W2N 2 +Rp4No, d N 3 / d t = - N 3 / T 3 +N4/T4 +Rp3No,
(453/2)
- N2/ T2 - W2N ~ - p ~ + R2No = 0 ,
(4F9/2)
- N 1 / T I + N2/T2 - 2 W I N 2 + WEN 2
dN2/dt = - N 2 / T 2 + Na/T3 - 2 W 2 N2 + W I N 2 + W4N'~N4 -aqb( otN2 - flBl ) + Rp2No,
68
(2)
where RI =Rpl +Rv4,
(4113/2)
d ~ / d t = q~v[a(aN2 -fiN1 ) - p ] •
+ p ~ + RI No = 0 , oIN2 -- BN1 = p / a ,
(4111/2)
d N l / d t = - N I / T I + N2/T2 - 2 W I N 2 + W4N[N4 + tr~( otN2 - flNl ) + Rpt No,
Here NI to N4 are the populations of levels 4113/2, 4111/2, 4F9/2 and 453/2 respectively, 7"1to T4 being their lifetimes and Rp~ to R ~ are the pumping rates in these levels (the pump in 419/2 would be included in Rp2 ) . The coefficients a and fl are the Boltzmann thermal population fractions for the crystal field sublevels xi(4I~1/2) and yi(4113/2) involved in the laser transition, tr is the emission cross-section, p represents the total losses, v is the speed of light in the active medium and the exponent x takes the values 1 or 2 for the two-ion or three-ion cross-relaxation processes respectively. Under normal pumping rates the population No of the ground level 4115/2 can be taken as equal to the total erbium concentration. For stationary regime (dNi/dt=O; d ~ / d t = 0 ) and medium or high erbium concentrations (when W4N~)>> 1/T4) the rate equations for the 4113/2, 4111/2 levels and photon flux can be written as
( 1)
R2----Rp2 +Rp3 +Rp4.
The system (2) gives an exact analytical solution for the population N1 of 4Ii3/2:
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OPTICS COMMUNICATIONS
able for discussing the effect of reducing T~ by codoping.
N, =(P[ (RI +R2) NolWI] 112
=[(I+~2) II2-F.][(RI"FR2)No/WI] 1/2
(3)
with
3. Discussion
= (2T,)
--I
[ (RI +R2)No WI ] - - 1 / 2 .
(4)
Below, or immediately after laser threshold the flux terms in eqs. (2) can be neglected and N2 = J [ (Rl +2R2) No~WE] 1/2
----[ ( P + y 2) 1/2--7] [(R~ +2R2) No/W2] 1/2 ,
(5)
with
v= 1 - N I / [ (R~ +2R2) NoT1 ] ,
(6)
and
y=(2T2)-l[ (Rt +2RE)NoW2] -I/2 .
(7)
The population difference between the crystal field components x; and Yi below laser threshold is
AN=otN2 -- flNi =otJ[ (RI + 2R2)No/ W2] t/2
[ _ q, I~ (W2"~l/2 ( R, +.~ ]'/21. XL1 ~\~-~/ \kT+--~-2j d
(8)
The parameter
p - (fl/ ol ) ( W2/ Wi ) 1/2
(9)
can be used as a figure of merit for the given laser transition [ 13 ]; it depends on the host crystal (via the up-conversion parameters I411, WE) and on the particular laser transition (via the Boltzmann coefficients a, fl). At threshold the gain must compensate for the losses p and the population difference must satisfy the third equation in (2). The system (2) gives an exact analytical solution for the photon flux density in the resonator:
~ = N o [R2+(R _l_R2)(l_p2)l_ 1__~ p otaT2 --Nl
15 August 1992
(P~-~2
2flW2~_ l - p 2
+ a20. ]
--~-i N,.
W2p O/20.2 (10)
AS we can observe from eqs. ( 2 ) - (8), (10) the lifetime T~ (4I, 3/2 ) affects the population of the levels, the population difference AN and the photon flux density. Consequently, all these expression are suit-
According to eqs. ( 3 ) - ( 6 ) shortening T~ implies a reduction of both N~ and N2 populations. While the reduction of N~ seems very normal, the reduction of N2 is a consequence of the coupling between 4111/2 and 4113/2 levels by up-conversion. The reduction of the lifetime Tt of 4Ii3/2 level by adding impurity ions as Nd 3+, Tm a+ or Ho a+ could be drastic (in some cases with orders of magnitude for few percent). The effect of reduction of T~ on the coefficients ~ and J (the T~ dependent part of the populations) is given in table 1 for two systems: Er3+:YAG (Er 3+ concentration 50 at.%) and Er3+:YSGG (10 at.°/~). Although several sets for rates W~ and WE are given in the literature [ 4, 1416], depending on the determination method, we used here the values from refs. [4,14]. In both these works the rates Wmand WE are determined from the analysis of the 4Ii1/2 decay for different pumping densities in various Er~ + levels. Thus, while the excitation conditions in ref. [4] (pumping in 4F9/2) make the method more sensitive to the I412process, the 4Ill/2 level being populated firstly after the excitation pulse, the experimental conditions in ref. [ 14 ], although much 10ss sensitive to the I412process (excitation in 4S3/2) assure accurate values for WI. Therefore, we have chosen W I a s determined in ref. [14] and the ratio W2/WI as obtained in ref. [4]. As for Er 3+:YAG [14], the T~, T2 values for Er3+:YSGG depend on Er 3+ concentrations, probably owing to some uncontrolled impurities, and we have used in our example some intermediate values between the low a n d high Er-concentration data [ 13,17 ]. Thus, the fo!lowing sets of spectroscopic parameters were choosen: for Er 3+ : YAG (50 at.%) T~=4× 10 -3, T2--88X 10-6 s [14], W~= 1.3× 10 -~5 cm3s -I, 1412=3.7× li0 -15 cm3s -l, and for Era+:YSGG (10 at.%) W l = l . 1 5 × 1 0 -~5 cm3s -l, I412=3.75×10 -15 cm3S-1 [13], T1=4.4×10 -3 s, 7"2=1.4×10 -3 s. In b o t h cases pumping rates RINo=R2No= 2.5 X 10~3 cm -3 s- 1 (assuring operation over threshold) were assumed arbitrarily as an illustrative example to Show the effect of Ti on J and 69
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OPTICS C O M M U N I C A T I O N S
15 August 1992
Table 1 The effect of reducing the lifetime T~ in case of Er3+ :YAg and Er3+ :YSGG on the coefficients ~ and to that determine the populations N~ and N2. Crystal
Tz (s)
t~
to
O/t0
Er: YAG 50 at.% Er
4 × 10 -3 1.5X 10 -3 1 × l0 -4 2 × l0 -5
0.895 0.890 0.789 0.572
0.997 0.987 0.823 0.420
0.898 0.901 0.958 1.362
Er:YSGG 10 at.% Er
4 . 4 × l0 -3 1.5× 10 -3 1 × 10 -4 2 × 10 -5
0.990 0.984 0.873 0.657
0.995 0.986 0.813 0.402
0.995 0.998 1.074 1.634
¢0. The data from table 1 show that the reduction of 7"1 has a more marked effect o n N I , although its effect on N2 is not negligible. From eq. (8) the condition to have positive population inversion below threshold restricts the maximal value of the figure of merit p to
~ (RI + 2R2~ '/2 P < tp \R-~1~ - 2 ]
"
(11)
Near the threshold, when q ~ 0 and the expressions (3), (5) for the populations NI, N2, respectively, are still valid, and the condition on p becomes more restrictive:
AN>~p/a,
p<~~~-(R'+2R2.~l/Z(l-a-~2 ) \ RI -I-R2 ,]
(12,
These relations show that for three-micron generation of Er 3+ in a given crystal the possible transitions (emission wavelengths) between various crystal field sublevels, characterized by the figure of merit p, are selected by the wavelength and rate of pumping (RI, R2), emission cross-section tr, losses p and the coefficients a, ~oand t~. As shown in table 1 the coefficients ~, q~and their ratio t~/~oare dependent on the lifetime Tl: for large T~ as usually in crystals doped only with Er 3+, the ratios ~/~oare small and only transitions with small figure of merit p are possible, i.e. transitions that connect the lowest Stark sublevels of 4111/2 with the upper sublevels of 4Ii3/2. A drastic reduction of Ti (as it happens by co-doping the crystals with de-activators as Nd a+, Ho a+ or Tm 3+ ) leads to a marked increase of the ratio t~/~o, extending the range of p and new laser transitions 70
become possible (even those between the upper sublevels of4111/5 and the lowest one of 411a/2). Some of the new transitions made possible by reducing TI might have larger cross-sections; it is also possible that the population inversion for the original transition is reduced due to the lowering of #. Therefore the reduction of 7"1 by co-doping may change the spectrum of laser emission for three-micron Er 3+ doped crystals. The effect of reducing the lifetime T~ on the threemicron laser emission becomes apparent also from the examination of the photon flux density expression (eq. (10)). Although the condition that must be fulfilled by p for a positive flux density is identical to condition (12), eq. (10) enables a separation of relative contributions of various terms. Only the first and the last term in eq. (10) could be positive function of figure of merit p. The first (pump) term is positive when
+2R,)
p
I'2
,
(13)
while the last term, dependent on TI, is positive for transitions with p > 1. As a consequence, a large value of TI is beneficial for laser transitions with p < l since it reduces a negative term in eq. ( l 0). However, for transitions with p > 1, that render the last term in eq. ( l 0) positive, a smaller TI is beneficial. This is more evident for p larger than Po when the only positive term in eq. (10) is the last one. These considerations explain the detrimental effect of reducing T~ by co-doping with Nd 3+ or Ho 3+ on the 2.936 ~tm laser emission of Er a+ :YAG [ 5,6 ]
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
and the beneficial effect on the 2.702 ~tm emission in Er a+, Cr3+:YSGG co-doped with Ho 3+ [7]. A similar reduction of laser parameters by co-doping with T m 3+ and Ho 3+ was observed in the case of LuAG, which in many respects (for instance the total crystal field splitting of levels) is similar to YAG [ 18 ]. However, no available data on WI and WE in this system exist, so that a similar analysis as for YAG cannot be made at this moment. The long pulse laser emission at 2.936 ~m in Era+:YAG corresponds to the transition xE-y7 and has a figure of merit p = 0.43. Under xenon lamp pumping R, ~ R 2 ~Rv4 and Po = 1.225; thus the first term in eq. (10) is positive, while the last one is negative. Evidently, a reduction of Tm would be detrimental for this transition. However, for other transitions with p > l the reduction of 7"1 is beneficial and new laser transitions of shorter wavelength are, in principle, possible. In this case the first term in eq. (10) may become very small or even negative; this could reduce the efficiency of pumping, leading to a strong increase in the laser threshold that may preclude the stationary regime, making possible only the short-pulse laser emission. We note, also, that the reduction of TI below the value of T2 transforms the three-micron Er 3+ laser into a conventional four-level laser. Figure 2 shows the computed dependence of the photon flux density O, eq. (10), on pump rate R2/Vo for two long-pulse laser transitions: 2.936 ~tm emission of Er3+:YAG ( p = 0 . 4 3 ) and 2.794 Ilm emission o f E r 3+ in YSGG ( p = 1.09), for various TI values; it is evident that reduction of T~ has a positive effect for p > 1 and negative for the case with p < 1. In the case of Er a +, Cr 3+ : Y S G G cw emission was obtained at 2.702 ~tm under krypton laser pumping (2=647.1 nm); this emission involves a superposition of the transitions (xl-y~) and (x2-Y2) having a figure of merit p = 1.92. For this crystal the pump is made into the strong 4A2--*4T 2 absorption band of Cr 3+ and the excitation is then transferred very efficiently to the 419/2 level of Er a+. The presence of chromium modifies also the mechanism of de-excitation of 4S3/2 of Er 3+ (that is populated by the upconversion process W2) by the chain of transfer processes Er 3+ (4S3/2) -, Cr 3+ (4T 2 ) ~ Er a+ (419/2 ), characterized by a rate W~; it can also lead to a more effective depopulation of 4It3/2 by possible additional cross-relaxations between Er 3+ and Cr 3+ [ 19 ].
10
15 August 1992 ~
2'
0 0
1
2
3
RpN o (10Z2cm-~ s'1)
Fig. 2. Photon fluxdensity Oversus pumping rate R2Nofor 2.794 jm~emissionof Era+ :YSGG (curves l --4correspondto 7"1= 4400, 1500, 100and 20 ~ts,respectively)and for 2.936 Itm emissionof Era+:YAG (curves 1'--4' correspondto/'1 = 4000, 1500, 100 and 20 ~ts,respectively). Because of all these effects the pump term in (10) is modified, the condition for positive pump becoming now p2<2/f, where the subunitary parameter f - [ 1+ ( W'4Ncr) / ( W~Vo) ] -1 describes the relative effect of the processes W~ and W4 and can be modified by a proper choice of the Er and Cr concentrations so as to make possible the emission at 2.702 ~tm [ 19]. For this transition the last term in (10) becomes positive ( p 2 _ t = 2 . 6 8 ) and thus we can expect that the reduction of/'1 would have important effects on the laser emission: reduction of threshold, improvement of pump efficiency and a nonlinear dependence of the photon I flux density on pump, due to the (R2No)i/2 dependence of the last term in (10) via N~. These effects ar~ evident from the computer calculation of • for various values of T~, presented ~n fig. 3, and they were indeed observed experimentally for the 2.702 ~tm CW laser emission under krypton laser pumping in Er 3+, Cr 3+ :YAG crystals co-doped with Ho 3+ [7].
4. Conclusions
The reduction of the lifetime of the terminal level 71
Volume 92, number 1,2,3
10
OPTICS COMMUNICATIONS
i
15 August 1992
p o w e r characteristics o f the laser e m i s s i o n .
i
References
5l,
Cs 2 0
l
2
3
Rp N O t102ecm-3s-'J
Fig. 3. Photon flux density • versus pumping rate R2Nofor Er3+, Cr 3+ :YSGG (2= 2.702 ~tm); 1: Ti =4400 ~ts, 2: Tl = 1500 ;is, 3: TI =400 ;is, 4: TI = 100 lis, and 5: T, =20 ~ts.
4Ii3/2 o f t h r e e - m i c r o n laser crystals b y co-doping with ions such as N d 3+, H o 3+ or T m 3+ that de-activate this level b y energy t r a n s f e r has a c o m p l e x effect o n the laser e m i s s i o n ; it is a f u n c t i o n o f the figure o f m e r i t p. A n analysis o f the s o l u t i o n o f the rate equat i o n system for the s t a t i o n a r y regime o f g e n e r a t i o n shows that for laser t r a n s i t i o n s with p < l the reduct i o n o f TI w o u l d b e d e t r i m e n t a l , while for p > 1 it is beneficial. These p r e d i c t i o n s are c o n f i r m e d b y the negative effect o f c o - d o p i n g o n the long pulse 2.936 ~tm e m i s s i o n i n Er 3+ : Y A G ( p = 0 . 4 3 ) a n d b y the positive effect o n the cw e m i s s i o n at 2.702 ktm i n Er 3+, Cr 3+ : Y S G G ( p = 1.92). It is also s h o w n that the r e d u c t i o n o f T1 b y c o - d o p i n g with ions as N d 3 +, T m 3+, H o 3+, etc., c o u l d change the w a v e l e n g t h a n d
72
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