Journal of Non-Crystalline Solids 322 (2003) 319–323
www.elsevier.com/locate/jnoncrysol
A comparison between different methods of calculating
the radiative lifetime of the 4I13=2 level of Er3þ in various glasses
B.J. Chen a, G.C. Righini
a,*
, M. Bettinelli b, A. Speghini
b
a
b
Optoelectronics and Photonics Department, Nello Carrara, Institute of Applied Physics (IFAC-CNR),
via Panciatichi 64, I-50127 Firenze, Italy
Dipartimento Scientifico e Tecnologico and INSTM, Universita di Verona, Strada Le Grazie 15, I-37134 Verona, Italy
Abstract
A comparison between the radiative lifetimes of the 4 I13=2 level of Er3þ derived from the conventional Judd–Ofelt
(J–O) method and from a simple formula, that corresponds to Einstein relation for the emission probability of a twolevel system, has been carried out in some typical glasses, such as germanate, phosphate, silicate, and tellurite. The
uncertainties of the radiative lifetimes are evaluated using standard error analysis; it comes out that uncertainties are
much larger (some tens %) when using the J–O method than with the simple formula (error less than 1%). The same
simple method can be used to calculate the radiative lifetimes of other rare earth ions, such as Ho3þ , Tm3þ and Yb3þ .
2003 Elsevier B.V. All rights reserved.
PACS: 78.20.)e; 78.20.Bh; 78.30.Ly; 78.40.)q
1. Introduction
Recent activity on rare earth (RE) doped glasses has been aimed at the development of integrated optical lasers and amplifiers which exploit
the 4 I13=2 fi 4 I15=2 transition of Er3þ centred at 1.54
lm [1]. The search for better and better host glass
materials is still under way, and Judd–Ofelt (J–O)
theory [2,3] represents an important tool associating experimental and theoretical data in the description of the f–f optical transitions of RE ions.
With J–O theory, in particular, the radiative
transition probabilities of RE ions can be calcu-
lated from their absorption spectra; as such it is
often used, even if some problems have been discussed, related to its application to practical
modelling processes and to the correctness of J–O
representation for a particular RE ion [4,5]. In the
present paper, we use a simple method to calculate
the radiative lifetime of the 4 I13=2 level of Er3þ in
different glass hosts, and we compare its results
with those obtained from the J–O procedure. A
similar method has been already suggested for
other RE ions [6], but, to the best of our knowledge, not for the Er3þ ion.
2. Experimental
*
Corresponding author. Tel.: +39-055 423 5239; fax: +39055 423 5350.
E-mail address: [email protected] (G.C. Righini).
Small samples, of typical size 20 mm · 10
mm · 1 mm, of trivalent erbium doped phosphate,
0022-3093/03/$ - see front matter 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0022-3093(03)00221-7
320
B.J. Chen et al. / Journal of Non-Crystalline Solids 322 (2003) 319–323
Table 1
Chemical composition and refractive index for the Er3þ -doped glasses under investigation
Glass
Chemical composition (mol%)
Refractive index
B–Ge–Na silicate
67.35SiO2 –8.79GeO2 –6.10B2 O3 –
15.98Na2 O–0.52Er2 O3 –1.26Yb2 O3
49PbO–50GeO2 –1Er2 O3
99Ba(PO3 )2 –1Er(PO3 )3
99Cd(PO3 )2 –1Er(PO3 )3
99Ca(PO3 )2 –1Er(PO3 )3
97.5Sr(PO3 )2 –2.5Er(PO3 )3
19ZnO–80TeO2 –1Er2 O3
1.53 ± 0.01
Lead germanate
Barium phosphate
Cadmium phosphate
Calcium phosphate
Strontium phosphate
Zinc tellurite
silicate, lead-germanate and zinc-tellurite glasses
were prepared by conventional melt-quenching
techniques, by using the procedures described
elsewhere [7–9]. The chemical compositions and
the refractive indexes of these glasses are reported
in Table 1. The refractive index of the silicate glass
was measured using m-line spectroscopy [10], employing an SF-6 glass coupling prism. For the
other glasses, the refractive indexes were taken
from the literature [8,11,12].
Electronic absorption spectra were measured
at room temperature in the visible region using a
ouble beam spectrophotometer (spectral bandwidth of 0.5 nm) (Kontron Uvikon 941 Plus)
and in the near infrared region using a Fourier transform infrared (FTIR) spectrometer
(spectral bandwidth of 2 cm1 ) (Nicolet Magna
760).
2.03 ± 0.01 [8]
1.5849 ± 0.0002
1.6039 ± 0.0002
1.5479 ± 0.0002
1.5585 ± 0.0002
2.08 ± 0.01 [12]
[11]
[11]
[11]
[11]
3. Calculations
For the B–Ge–Na silicate glass the experimental
oscillator strengths of the f $ f transitions were
obtained from the absorption spectrum (not
shown) and used in the frame of the J–O parameterisation scheme [13,14]. The three Xk parameters (k ¼ 2, 4 and 6) were calculated by fitting with
a least-squares method the electric dipole (ED)
contributions of the experimental oscillator
strengths of the observed transitions to the calculated ones. The matrix elements given by Carnall
et al. [15] were employed in the calculation. Magnetic dipole (MD) contributions were subtracted
from the observed oscillator strengths using the
procedure described elsewhere [16]. The values of
the intensity parameters for the B–Ge–Na silicate
glass are reported in Table 2.
Table 2
4
I13=2 fi 4 I15=2 transition barycenter vb , peak absorption cross-section r (peak), J–O parameters Xk and calculated lifetimes of the 4 I13=2
level for the Er3þ ion in the glasses under investigation (sJO : J–O procedure; sR : from Eqs. (2) and (3); d: relative difference between sJO
and sR )
Glass
vb (cm1 )
r (peak)
(1020 cm2 )
X2 (pm2 )
X4 (pm2 )
X6 (pm2 )
sJO (ms)
sR (ms)
B–Ge–Na
silicate
Lead germanate
Barium
phosphate
Cadmium
phosphate
Calcium
phosphate
Strontium
phosphate
Zinc tellurite
6643 ± 25
0.564 ± 0.002
4.05 ± 0.13
0.70 ± 0.15
0.403 ± 0.071
13.4 ± 5.6
14.10 ± 0.09
5.2
6563 ± 17
0.799 ± 0.002
4.76 ± 0.11
1.20 ± 0.12
0.807 ± 0.041
3.88 ± 0.70
3.68 ± 0.02
5.1
6560 ± 20
0.604 ± 0.002
5.02 ± 0.15
1.27 ± 0.18
0.923 ± 0.082
8.3 ± 2.1
7.22 ± 0.05
13.0
6580 ± 20
0.617 ± 0.002
5.34 ± 0.17
1.22 ± 0.21
0.851 ± 0.093
8.3 ± 2.5
8.46 ± 0.06
1.9
6601 ± 17
0.577 ± 0.002
5.48 ± 0.13
1.30 ± 0.15
0.765 ± 0.080
9.8 ± 2.4
8.23 ± 0.06
16.0
6587 ± 19
0.608 ± 0.002
5.033 ± 0.065
1.292 ± 0.089
0.926 ± 0.032
8.5 ± 1.1
8.23 ± 0.05
3.2
6602 ± 18
0.848 ± 0.002
6.30 ± 0.19
1.55 ± 0.26
1.29 ± 0.10
2.55 ± 0.71
2.59 ± 0.02
1.6
d (%)
B.J. Chen et al. / Journal of Non-Crystalline Solids 322 (2003) 319–323
sJO
1
¼
;
AED þ AMD
ð1Þ
which takes into account both the ED and MD
contributions.
On the other hand, given a general two-level
system, denoted as 1 and 2 the ground and the
excited state respectively, the radiative lifetime, sR ,
of the excited state, 2 is given by [19]
sR ¼
1
;
A21
ð2Þ
where A21 is the spontaneous emission probability
between states 2 and 1 (Einstein A coefficient). The
A21 coefficient, in turn, is related to the wave
number dependent absorption cross-section of the
2 ‹ 1 transition, rðvÞ, by the well-known equation
[19]
Z
g1
2 2
ð3Þ
A21 ¼ 8pcn vb rðvÞ dv;
g2
where g1 and g2 are the degeneracies of state 1 and
2, c is the velocity of light, n is the refractive index
of the medium, vb is the barycenter (in wave
numbers) of the absorption band. The absorption
and emission transitions between 4 I13=2 and 4 I15=2
of Er3þ ion are allowed by both ED and MD
mechanisms, and therefore A21 contains both the
ED and MD contributions.
From the absorbance spectra of the Er3þ -doped
glasses under investigation the absorption crosssections as a function of wave number were obtained from the well-known relation [13]
rðvÞ ¼ 2:303
AbsðvÞ
;
cl
ð4Þ
0.4
B-Ge-Na Silicate
0.0
Absorption Cross Section (10-20 cm2)
For the zinc-tellurite, lead-germanate and
phospate glasses under investigation, the J–O parameters were taken from the literature [8,17,18]
and are reported in Table 2 for comparison. The J–
O intensity parameters were then used to calculate
the ED contribution, AED , of the spontaneous
emission probability of the 4 I13=2 fi 4 I15=2 emission
transition of the Er3þ ion. The MD contribution,
AMD , of the spontaneous emission probability for
the 4 I13=2 fi 4 I15=2 emission transition was calculated following the procedure described in [16],
and finally the radiative lifetime, sJO , for the 4 I13=2
level was calculated using the relation [13]
321
0.4
Barium phosphate
0.0
0.4
Strontium phosphate
0.0
0.4
Calcium phosphate
0.0
0.4
Cadmium phosphate
0.0
0.4
Lead germanate
0.0
0.4
Zinc tellurite
0.0
6200
6400
6600
6800
7000
7200
Wavenumber (cm-1)
Fig. 1. Absorption cross-sections for the 4 I13=2 ‹ 4 I15=2 transition of the Er3þ ion in different oxide glasses.
where AbsðvÞ is the wave number dependent absorbance, l and c are the thickness and the ion
concentration (in ions/cm3 ), respectively. In Fig. 1
the absorption cross-sections in the range 6200–
7200 cm1 for the glasses under investigation are
shown. The observed band centered at about 6600
cm1 is due to the 4 I13=2 ‹ 4 I15=2 absorption transition of the Er3þ ion. We see how large is the
dependence of the absorption cross-section on the
glass host. These absorption cross-sections are
typical of the glasses investigated and in agreement
with recent results reported in the literature
[20,21].
4. Results
The sJO s of the 4 I13=2 level of the Er3þ ion for the
glasses under investigation, calculated using relation (1), are reported in Table 2. The uncertainties
on the calculated radiative lifetimes, sJO , were
322
B.J. Chen et al. / Journal of Non-Crystalline Solids 322 (2003) 319–323
obtained using standard error analysis [22] and are
also reported in Table 2. The uncertainties are in
some cases surprisingly high (even 20–30% of sJO )
but they can be considered as an upper limit of the
experimental errors, because they are calculated
using the simplifying assumption that the uncertainties of the Xk parameters are uncorrelated.
The radiative lifetimes, sR , for the same glasses
were instead calculated by using Eqs. (2) and (3);
the sR are reported in Table 2 as well, together
with the transition barycenters for the different
samples. The uncertainties on sR were also calculated by standard error analysis using the standard
deviations of the integrated absorption crosssection.
5. Discussion
Table 2 shows that the relative differences of the
radiative lifetimes sJO and sR are equal or less than
5% for most of the oxide glasses under investigation, and are much larger only for barium and
calcium phosphate glasses (13% and 16%, respectively).
Let us now analyse the uncertainties on sJO and
sR . Usually, the r.m.s. error for the J–O procedure
is 8% [7–9,17,18] and that is the reason why some
authors have considered the J–O theory as a semiquantitative theory. However it can give a useful
contribution in estimating transition probabilities
with an accuracy generally not worse than 10%.
Due to the theoretical complexity, an accurate
theory for the calculation of f–f transition intensities of trivalent lanthanide ions is still absent.
From Eqs. (1)–(3), considering the propagation
of the errors of different experimental quantities in
an equation [22], calculations show that the
precision on the sR is mainly dependent on the
standard deviation of the integrated absorption
cross-section and of the barycenter of the 4 I13=2 ‹
4
I15=2 absorption band. In fact, the measurement
of the refractive index usually is quite accurate: as
an example, according to the data reported in
Table 1, in the worst case the error on n is
±0.7%. On the other hand, the precision on the
sJO is due to a complicated function involving the
sum of the standard deviations on the absorption
cross-sections and on the barycenters of all the
bands included in the J–O procedure, and the
relative uncertainties on the J–O parameters could
be as high as 10% (see Table 2). As a consequence,
the data in Table 2 indicate that the uncertainties
on the sR are at least an order of magnitude
smaller than those on sJO .
6. Conclusions
The radiative lifetime of the 4 I13=2 level of Er3þ
ion in various oxide glasses was calculated both by
the standard J–O method (sJO ) and by using a
simple procedure based on the Einstein relation
for the spontaneous emission probability between
two states (sR ). The latter procedure implies
the calculation of the oscillator strength of the
4
I13=2 ‹ 4 I15=2 absorption band, and avoids the
data fitting process which is necessary in the J–O
method.
The comparison between the results obtained
with the two methods confirms that the simpler
procedure is feasible and gives results in agreement
with those obtained from the more laborious J–O
procedure. Indeed, the values of sR could be considered also more reliable than sJO , as the assessed
uncertainties on the sR are at least an order of
magnitude smaller than those on sJO .
Moreover, we mention that the simple procedure used in the present paper, of immediate
physical meaning, can be usefully employed at
least in the first stage of design and characterization of novel glasses or crystalline materials containing REs. Besides the 4 I13=2 ‹ 4 I15=2 transition in
Er3þ , this method can also be used for calculating
the radiative lifetimes for other transitions from
the first excited state to the ground state, such as
5
I7 fi 5 I8 for Ho3þ , 3 F4 fi 3 H6 for Tm3þ or
2
F5=2 fi 2 F7=2 for Yb3þ .
Acknowledgements
B.J.C. thanks ICTP, Trieste, for his fellowship
in the frame of TRIL Program. M.B. and A.S.
thank Erica Viviani (Universita di Verona) for
expert technical assistance. The technical assis-
B.J. Chen et al. / Journal of Non-Crystalline Solids 322 (2003) 319–323
tance of Roberto Calzolai (IFAC-CNR) is gratefully acknowledged.
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