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Modelling unusually fast non-radiative energy transfer in crystals
December i 994
Ma @ ia s ELSEVIER
Optical Materials 4 ( 1994) 31-33
Modelling unusually fast non-radiative energy transfer in crystals S.R. R o t m a n a, A. E y a l a y . K a l i s k y b, A. B r e n i e r c, C. P e d r i n i c, G. B o u l o n ° a Ben-Gurion University of the Negev, Department of Electrical and Computer Engineering, P.O. Box 653, 84105 Beer-Sheva, Israel b LaserDepartment, NuclearResearch Centre-Negev, P.O. Box 9001, 84190Beer-Sheva, Israel c Laboratoire de Physico-Chimie des Matkriaux Luminkscents, Universit~ Claude Bernard Lyon 1, Unit~ de Recherche associ~e au CNRS no 442, B6t. 205, 69622 Villeurbanne, France
Abstract
We examine the time decay of various Tm +3 sites in Cr, Tm, Ho-doped YAG. Certain sites can be identified as being paired with Cr +3, while others are isolated. A correspondencebetween the spectral and temporal behavior of each site has been established.
I. Introduction
Garnets based on rare-earth ions such as Tm +3 or Ho +3, codoped by Cr + 3, are of great interest for use as solid-state laser materials [ 1 ]. In particular, yttrium aluminum garnet (YAG), well known as the host of the neodymium YAG laser, has been doped with all three ions for use in a sophisticated scheme to achieve 2 ~tm lasing from the holmium ion [2 ]. T h e C r +3 is flashlamp pumped in its broad absorption bands (the 4T 2 and 4T1 states), the chromium ion transfers energy to the 3H4 thulium state, the excited thulium ion cross relaxes with a ground-state neighboring ion to produce two thulium ions in the 3F 4 state, and, finally, after energy hopping between thulium states the thulium transfers energy to the holmium 517 state which lases. A recent paper [ 3 ] has reported that the laser crystal Cr, Tm, Ho: YAG exhibits the formation of several different types of sites for the chromium, thulium, and holmium ions, with subsequent unique spectral and temporal characteristics at each site. However, the analysis of the Tm +3 was qualitative; the thulium 3H 4 emission is particularly difficult to analyze, because both its method of excitation (from the chromium ions) and its method of deexcitation
(cross relaxation with neighboring ground-state ions ) are nonexponential in behavior as well. In Ref. [2 ], we expanded the standard FoersterDexter theory into a model which analyzes cases where the possibility of spatial correlation of donors and acceptors exists. Our preliminary analysis shows that we can spectrally and temporally identify Tm +3 ions which are participating in a chromium-thulium pair. In Ref. [4], we developed a method to automatically derive the parameters needed in the model. In this paper, we will present the time decay of the various thulium sites when different chromium sites are pumped.
2. Correlated model
The standard expression for the number of excited donors ND(t) at time t, as developed by Inokuti et al., is derived by solving the equation ND(t)=No(O)
0925-3467/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0925-3467 ( 94 ) 00023-J
×lim
(f v
exp(-t/~g)
exp[-n(r)t]
w(r) d
0
,
(1)
S.R. Rotman et al. / OpticalMaterials4 (1994) 31-33
32
where n(r) is the rate of transfer between a donor and an acceptor separated by a distance r, and NA is the number of acceptors, r ~ is the time constant for spontaneous emission of the donor. The probability distribution of absorbers around the emitters is described by the function w(r). For the standard model, the probability of any particular acceptor being located at a distance r from the donor is 47rr 2 V
w(r) d V = - -
V
dr,
n( r) = ( ro/r)'( 1/z D) ,
(3)
where ro is the critical transfer distance, which modifies the magnitude of the interaction. In Eq. (3), s = 6 for dipole-dipole interactions, s = 6 for dipole-dipole interactions, s = 8 for dipole-quadrupole interactions, etc. The excited donor concentration is given by
× exp
[
z tg
c Co
(tl r(1-3/s)\7]
Co= 3 / ( 4zrr3) ,
J
(4)
(5)
F is the gamma function. The excited concentration of acceptors NA (t) is given by
N A ( t ) = e x p ( - - t / z A) i
exp(t'/z~)
(
No(t') rD
The solution of Eq. ( 1 ) given these constraints is
ND(t)=ND(O) exp[--t/roD--A(c/co)(nt/r~) 1/2 -cV,(B-A)
~+ (Zl) e x p ( - z l )
-cV2( 1 - B ) cI)+(z2) e x p ( - z 2 ) ] ,
(9)
c is NA/Vand V~and V2 are the volumes of the spheres with radii rl and r2, respectively, zi (where the index i can be 1 or 2) is given by
-=~, ( r o / r y t / r g .
(10)
~ + is a degenerate hypergeometric function of two variables [ 4 ]. Methods to automatically fit A, q, and r2 to the data have been discussed in Ref. [4 ].
In Ref. [3], several Cr + 3 ion sites were identified. In particular, excitation wavelengths of 687.3 and 687.2 nm are preferentially exciting unpaired and paired (with a thulium ion) chromium ions. Similarly, in Ref. [3], the emission of T m ÷3 at 809 and 806.6 nm are from unpaired and paired (with a chrom i u m ion) thulium ions. These are the wavelengths used for Cr +3 pumping and T m +3 emission. In Figs. 1-4 are shown the decay from T m +s ions; the experimental procedure is described in Ref. 2. Fig.
d [ N D ( t ' ) ] ) dt' dt' "
0
0 ~ -0.5 " ~
(6) In Ref. [ 2], we suggest an extension of this model by allowing w(r) to vary as a function of the degree of correlation, i.e.
A4z~r2 w(r) d V = ~ d r ,
(8)
3. Results
where c is the concentration o f acceptors and Co, the critical concentration, is
X
Ar 3 + B( r32- r 3) =r 3.
(2)
where w(r) is normalized over the total volume V. The multipole interaction is given by
ND(t) = N D ( 0 )
where A is a free parameter controlling the degree of correlation and B is constrained by
Experimental data Theoretical Uncorrelated fit
z -I.5
0
B41rr2 -
=
V
dr,
4~zr2 dr, V
F1
--
,
-5
o
~-
;
;
4'
;
;
¢
%
TIME (ms)
r2
(7)
Fig. 1. 809 nm (unpaired) Tm +3 ionic emission after pumping (unpaired) Cr +s ions at 687.3 nm.
S.R. Rotman et al. / Optical Materials 4 (1994) 31-33 O'
0 Experimental
~ ,
-0.5
33
..... .
.
.
.
data
Theoretical
Correlated
Theoretical
Uncorreloted
E xperimentol data Theoretical C o r r e l a t e d fit Theoretical Uncorreloled fit
fit -Q5
fit
-I
>-
J
..... 1 ....
E-
co
o
-2.5 -&5
.
0
I
2
3 TfME
4 (ms)
5
6
-3
7
8 TIME
Fig. 2. 806.6 nm (paired) Tm +3 ionic emission after pumping (paired) Cr +3 ions, at 687.2 nm. Table 1 Parameters for the fits shown in Figs. 1-4 for use in the model given in Ref. [2]. ~ and rb~ are equal to 13 ms and 100 gs, respectively, rD is equal to V~ [ 4/3 ) ~ No ] t/3. Fig.
ro/rD
1 2 3 4
0.63 0.51 0.51 0.57
A
rt/ro
-
-
6.55 4.13 2.5
0.35 0.39 0.45
(ms)
Fig. 3. 809 nm (unpaired) Tm +3 ionic emission after pumping (paired) Cr +3 ions at 687.2 nm. 0
~. ,
Experimental dQIo Theoreticot Corre)oted f i t Theoretical Uncorrelated fit
-- ..... . . . . .
05
r2/rD 0.99 0.98 0.94
-I.5
-2
1 represents the decay of an unpaired Tm ÷3 ion when the Cr + 3 ions are pumped. As we would expect, the standard Foerster-Dexter model reasonably well fits the data. Fig. 2 shows the decay of the paired Tm +3 ions when the paired C r +3 ions are pumped. In this case, as we would expect, a strong positional correlation (represented by a high value o f A) is measured. (See Table 1 for the actual parameters calculated. ) Fig. 3 is the unpaired Tm ÷3 ion emission when the paired Cr ÷3 is pumped; Fig. 4 is the paired Tm +3 ion emission when the unpaired Cr +3 is pumped. Interestingly enough, both emissions exhibit a strong correlation effect. In the case of Fig. 3, this may be due to an energy migration over the Tm +3 sites to an unpaired Tm +3 after the fast transfer of energy from a paired Cr +3 to a paired Tm ÷3. In the case o f Fig. 4, this may indicate a migration over the Cr +3 sublattice. Further measurements at different wavelengths are necessary. 4. Conclusions A simple extension o f the Inokuti-Hirayama model shows that energy transfer occurs particularly quickly between paired Cr +3 and Tm+3; the unpaired Cr +3
TIME
(ms)
Fig. 4. 806.6 nm (paired) Tm +3 ionic emission after pumping (unpaired) Cr +3 ions at 687.3 nm.
and Tm +3 ions exhibit the standard energy transfer properties. Further study of energy migration over the Cr +3 and Tm +3 sublattices is necessary.
Acknowledgement This work was partially funded by the Israel National Council for Research and Developement, Jerusalem, Israel.
References [ 1 ] V.A. Smirnov and I.A. Shcherbakov, IEEE J. Quantum Electron. QE-24 ( 1988 ) 949. [2] S.R. Rotman, Y. Kalisky, A. Brenier, C. Pedrini, G. Boulon and F.X. Hartmann, J. Appl. Phys. 72 (1992) 224, and the references therein. [ 3 ] W. Nie, Y. Kalisky, C. Pedrini, A. Monteil and G. Boulon, Opt. Quant. Elec. 22 (1990) S123. [4] A. Eyal and S.R. Rotman, Chem. Phys. Lett. 206 (1993) 113.
Ma @ ia s ELSEVIER
Optical Materials 4 ( 1994) 31-33
Modelling unusually fast non-radiative energy transfer in crystals S.R. R o t m a n a, A. E y a l a y . K a l i s k y b, A. B r e n i e r c, C. P e d r i n i c, G. B o u l o n ° a Ben-Gurion University of the Negev, Department of Electrical and Computer Engineering, P.O. Box 653, 84105 Beer-Sheva, Israel b LaserDepartment, NuclearResearch Centre-Negev, P.O. Box 9001, 84190Beer-Sheva, Israel c Laboratoire de Physico-Chimie des Matkriaux Luminkscents, Universit~ Claude Bernard Lyon 1, Unit~ de Recherche associ~e au CNRS no 442, B6t. 205, 69622 Villeurbanne, France
Abstract
We examine the time decay of various Tm +3 sites in Cr, Tm, Ho-doped YAG. Certain sites can be identified as being paired with Cr +3, while others are isolated. A correspondencebetween the spectral and temporal behavior of each site has been established.
I. Introduction
Garnets based on rare-earth ions such as Tm +3 or Ho +3, codoped by Cr + 3, are of great interest for use as solid-state laser materials [ 1 ]. In particular, yttrium aluminum garnet (YAG), well known as the host of the neodymium YAG laser, has been doped with all three ions for use in a sophisticated scheme to achieve 2 ~tm lasing from the holmium ion [2 ]. T h e C r +3 is flashlamp pumped in its broad absorption bands (the 4T 2 and 4T1 states), the chromium ion transfers energy to the 3H4 thulium state, the excited thulium ion cross relaxes with a ground-state neighboring ion to produce two thulium ions in the 3F 4 state, and, finally, after energy hopping between thulium states the thulium transfers energy to the holmium 517 state which lases. A recent paper [ 3 ] has reported that the laser crystal Cr, Tm, Ho: YAG exhibits the formation of several different types of sites for the chromium, thulium, and holmium ions, with subsequent unique spectral and temporal characteristics at each site. However, the analysis of the Tm +3 was qualitative; the thulium 3H 4 emission is particularly difficult to analyze, because both its method of excitation (from the chromium ions) and its method of deexcitation
(cross relaxation with neighboring ground-state ions ) are nonexponential in behavior as well. In Ref. [2 ], we expanded the standard FoersterDexter theory into a model which analyzes cases where the possibility of spatial correlation of donors and acceptors exists. Our preliminary analysis shows that we can spectrally and temporally identify Tm +3 ions which are participating in a chromium-thulium pair. In Ref. [4], we developed a method to automatically derive the parameters needed in the model. In this paper, we will present the time decay of the various thulium sites when different chromium sites are pumped.
2. Correlated model
The standard expression for the number of excited donors ND(t) at time t, as developed by Inokuti et al., is derived by solving the equation ND(t)=No(O)
0925-3467/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0925-3467 ( 94 ) 00023-J
×lim
(f v
exp(-t/~g)
exp[-n(r)t]
w(r) d
0
,
(1)
S.R. Rotman et al. / OpticalMaterials4 (1994) 31-33
32
where n(r) is the rate of transfer between a donor and an acceptor separated by a distance r, and NA is the number of acceptors, r ~ is the time constant for spontaneous emission of the donor. The probability distribution of absorbers around the emitters is described by the function w(r). For the standard model, the probability of any particular acceptor being located at a distance r from the donor is 47rr 2 V
w(r) d V = - -
V
dr,
n( r) = ( ro/r)'( 1/z D) ,
(3)
where ro is the critical transfer distance, which modifies the magnitude of the interaction. In Eq. (3), s = 6 for dipole-dipole interactions, s = 6 for dipole-dipole interactions, s = 8 for dipole-quadrupole interactions, etc. The excited donor concentration is given by
× exp
[
z tg
c Co
(tl r(1-3/s)\7]
Co= 3 / ( 4zrr3) ,
J
(4)
(5)
F is the gamma function. The excited concentration of acceptors NA (t) is given by
N A ( t ) = e x p ( - - t / z A) i
exp(t'/z~)
(
No(t') rD
The solution of Eq. ( 1 ) given these constraints is
ND(t)=ND(O) exp[--t/roD--A(c/co)(nt/r~) 1/2 -cV,(B-A)
~+ (Zl) e x p ( - z l )
-cV2( 1 - B ) cI)+(z2) e x p ( - z 2 ) ] ,
(9)
c is NA/Vand V~and V2 are the volumes of the spheres with radii rl and r2, respectively, zi (where the index i can be 1 or 2) is given by
-=~, ( r o / r y t / r g .
(10)
~ + is a degenerate hypergeometric function of two variables [ 4 ]. Methods to automatically fit A, q, and r2 to the data have been discussed in Ref. [4 ].
In Ref. [3], several Cr + 3 ion sites were identified. In particular, excitation wavelengths of 687.3 and 687.2 nm are preferentially exciting unpaired and paired (with a thulium ion) chromium ions. Similarly, in Ref. [3], the emission of T m ÷3 at 809 and 806.6 nm are from unpaired and paired (with a chrom i u m ion) thulium ions. These are the wavelengths used for Cr +3 pumping and T m +3 emission. In Figs. 1-4 are shown the decay from T m +s ions; the experimental procedure is described in Ref. 2. Fig.
d [ N D ( t ' ) ] ) dt' dt' "
0
0 ~ -0.5 " ~
(6) In Ref. [ 2], we suggest an extension of this model by allowing w(r) to vary as a function of the degree of correlation, i.e.
A4z~r2 w(r) d V = ~ d r ,
(8)
3. Results
where c is the concentration o f acceptors and Co, the critical concentration, is
X
Ar 3 + B( r32- r 3) =r 3.
(2)
where w(r) is normalized over the total volume V. The multipole interaction is given by
ND(t) = N D ( 0 )
where A is a free parameter controlling the degree of correlation and B is constrained by
Experimental data Theoretical Uncorrelated fit
z -I.5
0
B41rr2 -
=
V
dr,
4~zr2 dr, V
F1
--
,
-5
o
~-
;
;
4'
;
;
¢
%
TIME (ms)
r2
(7)
Fig. 1. 809 nm (unpaired) Tm +3 ionic emission after pumping (unpaired) Cr +s ions at 687.3 nm.
S.R. Rotman et al. / Optical Materials 4 (1994) 31-33 O'
0 Experimental
~ ,
-0.5
33
..... .
.
.
.
data
Theoretical
Correlated
Theoretical
Uncorreloted
E xperimentol data Theoretical C o r r e l a t e d fit Theoretical Uncorreloled fit
fit -Q5
fit
-I
>-
J
..... 1 ....
E-
co
o
-2.5 -&5
.
0
I
2
3 TfME
4 (ms)
5
6
-3
7
8 TIME
Fig. 2. 806.6 nm (paired) Tm +3 ionic emission after pumping (paired) Cr +3 ions, at 687.2 nm. Table 1 Parameters for the fits shown in Figs. 1-4 for use in the model given in Ref. [2]. ~ and rb~ are equal to 13 ms and 100 gs, respectively, rD is equal to V~ [ 4/3 ) ~ No ] t/3. Fig.
ro/rD
1 2 3 4
0.63 0.51 0.51 0.57
A
rt/ro
-
-
6.55 4.13 2.5
0.35 0.39 0.45
(ms)
Fig. 3. 809 nm (unpaired) Tm +3 ionic emission after pumping (paired) Cr +3 ions at 687.2 nm. 0
~. ,
Experimental dQIo Theoreticot Corre)oted f i t Theoretical Uncorrelated fit
-- ..... . . . . .
05
r2/rD 0.99 0.98 0.94
-I.5
-2
1 represents the decay of an unpaired Tm ÷3 ion when the Cr + 3 ions are pumped. As we would expect, the standard Foerster-Dexter model reasonably well fits the data. Fig. 2 shows the decay of the paired Tm +3 ions when the paired C r +3 ions are pumped. In this case, as we would expect, a strong positional correlation (represented by a high value o f A) is measured. (See Table 1 for the actual parameters calculated. ) Fig. 3 is the unpaired Tm ÷3 ion emission when the paired Cr ÷3 is pumped; Fig. 4 is the paired Tm +3 ion emission when the unpaired Cr +3 is pumped. Interestingly enough, both emissions exhibit a strong correlation effect. In the case of Fig. 3, this may be due to an energy migration over the Tm +3 sites to an unpaired Tm +3 after the fast transfer of energy from a paired Cr +3 to a paired Tm ÷3. In the case o f Fig. 4, this may indicate a migration over the Cr +3 sublattice. Further measurements at different wavelengths are necessary. 4. Conclusions A simple extension o f the Inokuti-Hirayama model shows that energy transfer occurs particularly quickly between paired Cr +3 and Tm+3; the unpaired Cr +3
TIME
(ms)
Fig. 4. 806.6 nm (paired) Tm +3 ionic emission after pumping (unpaired) Cr +3 ions at 687.3 nm.
and Tm +3 ions exhibit the standard energy transfer properties. Further study of energy migration over the Cr +3 and Tm +3 sublattices is necessary.
Acknowledgement This work was partially funded by the Israel National Council for Research and Developement, Jerusalem, Israel.
References [ 1 ] V.A. Smirnov and I.A. Shcherbakov, IEEE J. Quantum Electron. QE-24 ( 1988 ) 949. [2] S.R. Rotman, Y. Kalisky, A. Brenier, C. Pedrini, G. Boulon and F.X. Hartmann, J. Appl. Phys. 72 (1992) 224, and the references therein. [ 3 ] W. Nie, Y. Kalisky, C. Pedrini, A. Monteil and G. Boulon, Opt. Quant. Elec. 22 (1990) S123. [4] A. Eyal and S.R. Rotman, Chem. Phys. Lett. 206 (1993) 113.