Simulation of the optical properties of Er:ZBLAN glass

Simulation of the optical properties of Er:ZBLAN glass

Hiroyuki Inoue a,*, Kohei Soga b, Akio Makishima c

a Department of Materials Engineering, School of Engineering, The University of Tokyo, 7-3-1, Hongo,

Bunkyo-ku, Tokyo 113-8656, Japanb Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, 7-3-1,

Hongo, Bunkyo-ku, Tokyo 113-8656, Japanc Center for New Materials, Japan Advanced Institute of Science and Technology, 1-1, Asahidai, Tatsunokuchi, Nomi,

Ishikawa 923-1292, Japan

Received 19 September 2001

Abstract

Absorption and emission spectra of Er3þ-doped ZBLAN glass at room temperature were estimated from structural

models prepared by using molecular dynamics (MD) simulation based on crystal field theory and point-charge ap-

proximation. The calculated absorption and emission spectra substantially agreed with the observed ones. Further-

more, it was found that the emission spectra at low temperatures could be also reproduced to some extent. The relation

between the splitting of the energy level and the crystal field parameters calculated from the Er3þ ion in each structural

model was examined. According to this relation, the energy levels could be classified into several groups. The splitting of

the group represented by the 4I9=2 level was dominated by the short-range structure around the Er3þ ions. The transition

process for the measurement of the well-defined fluorescence line narrowing (FLN) spectrum was estimated from the

characteristics of the splitting of the energy levels. � 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Er3þ-doped fibers have the potential for use asfiber amplifiers, fiber lasers and broadband lightsources operating in the 1:5 lm band [1]. Variousglass systems have already been investigated as theEr3þ ion hosts in order to realize high efficiencyand broadband operation. The width of the 1.5 lmband of Er3þ-doped fluoride glass, such as ZrF4–BaF2–LaF3–AlF3–NaF (ZBLAN) glass, is widerand flatter than that of the Er3þ-doped silica glass.

Thus, Er3þ-doped ZBLAN fiber is attractive forwavelength-division multiplexing (WDM) trans-mission systems as an optical amplifier. Severalproblems in relation to the use of a fluoride fiberamplifier have been indicated. One problem relatesto the noise figure of the Er3þ-doped fluoride fiberamplifier. The 4I11=2 level pumping at 980 nm ismore effective for low-noise performance than the4I13=2 level pumping at 1480 nm. The lifetime of the4I11=2 level of the Er

3þ ion in ZBLAN glass is about8.8 ms [2]. The excited state absorption from the4I11=2 level to the 4F7=2 level easily occurs. There-fore, 980 nm pumping cannot form a populationinversion between the 4I13=2 and 4I15=2 levels effi-ciently. This problem can be resolved by severalapproaches, i.e., by 970 nm pumping [3] and

Journal of Non-Crystalline Solids 298 (2002) 270–286

www.elsevier.com/locate/jnoncrysol

* Corresponding author. Tel.: +81-3 5841 7113; fax: +81-3

5841 8653.

E-mail address: [email protected] (H. Inoue).

0022-3093/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0022 -3093 (01 )01052 -3

Er3þ þ Ce3þ co-doping [2]. Some concerns aboutthe reliability of fluoride fibers and their connec-tivity to silica fiber still remain to be solved fortheir practical use. ZBLAN glass can be dopedwith a higher concentration of the Er3þ ionswithout quenching in comparison with the silicaglass. Therefore, ZBLAN fiber seems to be apossible candidate not only as a substitute for thesilica amplifier but also for new applications, suchas miniaturized amplifiers and lasers.

The optical properties of rare earth ions inglasses depend on the shapes of the spectra ofabsorption and emission. The shapes of the spectraare determined by the splitting of the energy levelsof 4f electrons, transition rates between the energylevels and the site-to-site variation of the rare earthions in the glasses. The splitting of the energylevels is determined by the atomic arrangementaround the rare earth ions in the glasses. The re-lation between the splitting of the energy levels andthe structure around the rare earth ions is de-scribed by the crystal field parameters based oncrystal field theory. A fluorescence line narrowing(FLN) technique is used to measure the splitting ofthe energy levels and the site-to-site variation ofthe rare earth ions in the glasses. Unfortunately,it is impossible to obtain crystal field parametersonly from the observed energy splitting. Therefore,the symmetry of the site of rare earth ions has beenassumed for the estimation of crystal field pa-rameters and the analysis of the structure aroundrare earth ions [4–15]. On the contrary, it is pos-sible that the relation in the glasses is discussedwithout assuming the symmetry. The model of theglass structure doped with the rare earth ions wasprepared by use of the molecular dynamics (MD)technique. When the point charge approximationis applied to the model, the splitting of the energylevels and the transition rates between them can beestimated for each rare earth ion in the structuralmodel. Initially, this method was applied to theEu3þ ions in glasses [16–26]. Recently, structuralmodels with other rare earth ions, Er3þ and Yb3þ

ions, have been reported [27–29] and the opticalproperties estimated by this method have beenextended to Tm3þ ions [30]. The transition ratebetween the energy levels of rare earth ions can beestimated on the basis of the Judd–Ofelt theory

[31,32]. The shape of the absorption and emissionspectra can be estimated by use of this method.More research on rare earth ions and glass systemsis necessary to confirm this method. The efficiencyof the excitation and energy transfer between rareearth ions can be better discussed by using thismethod.

In this paper, we show the absorption andemission spectra of the Er3þ ions in ZBLAN glassat room temperature simulated from structuralmodels prepared by MD simulation. Then, theemission spectra at low temperatures are estimatedand compared with the observed FLN spectra.Finally, the relation between the splitting of theenergy levels and the structure around Er3þ ions isdiscussed.

2. Experimental conditions

2.1. Glass preparation

Glass with a composition of 52ZrF4 � 20BaF2 �3:5LaF3 � 3AlF3 � 20NaF � 0:5InF3 � 1ErF3 was pre-pared. The powders were mixed and melted in agold crucible at 900�C for 15 min. The melt wascast into a preheated aluminum mold. All of theseprocesses were conducted in a glove box under anatmosphere of dry nitrogen gas. The glass ob-tained was cut into a 10� 25� 5 mm3 shape andpolished.

The absorption spectrum was measured witha self-recording spectrophotometer in the wave-length range 200–2600 nm. Fluorescence spec-trum was measured by 255 nm excitation with afluorescence spectrophotometer at room temper-ature.

2.2. Structural models for the Er3þ-doped ZBLANglass

The structure models of the Er3þ-dopedZBLAN glass prepared by MD simulation wereused. The distance of the Er–F pair in the crystalsis about 0.01 �AA longer than that of the Tm–F pair[30]. This difference is within the range of the dis-tribution of the first coordination of the Tm–Fpair in the structural models for the Tm3þ-doped

H. Inoue et al. / Journal of Non-Crystalline Solids 298 (2002) 270–286 271

ZBLAN glass. Here, the structural models for theTm3þ-doped ZBLAN glass were used as thestructural models for the Er3þ-doped ZBLANglass. The summary of the preparation proceduresof the structural models is as follows: Three hun-dred ninety three ions (Zr4þ : 53; Ba2þ : 20; La3þ :3; Al3þ : 3; Naþ : 20; Re3þ : 1; F� : 293; Re3þ ¼the rare earth ion) were placed randomly in a cu-bic cell with periodic boundary conditions. A cellparameter of 1.759 nm was determined from theexperimental density of the glass. Simulations werecarried out at a constant volume. The potentials ofthe Born–Mayer type were used with formal ioniccharges, and the parameters used are listed inTable 1. The Coulomb force was evaluated byEwald summation. To obtain the variation of thesites of rare earth ions in the glass structure, MDsimulation was performed for 300 different sets ofrandom initial coordinates. The temperature of thesimulation was lowered from 3000 to 300 K with atime step of 1 fs for 10 000 time steps (10 ps). After5000 time steps (5 ps) at 300 K, the coordinates ofthe last step were used for further calculation.Computations were made with a HITAC SR2201computer in the Information Technology Centerat the University of Tokyo.

2.3. Calculation of the splitting of 4f energy levelsand the transition rate between them

A detailed method of the calculations can befound in the literature [33–35]. The Hamiltonian

describing the electrostatic field of rare earth ioncan be written as follows:

HCF ¼Xkq

A�kq

Xi

rki Ckq rið Þ; ð1Þ

where CkqðriÞ is an irreducible spherical tensoroperator of rank k, operating on the ith electronwhose radius is ri, and Akq is a crystal field pa-rameter. On the basis of the point charge ap-proximation, one can write

Akq ¼ � e2

4pe0

Xj

qjRkþ1j

CkqðRjÞ; ð2Þ

where qj is the charge on the jth ion in the struc-tural models, and Rj is its distance from the rareearth ion. Only operators of rank 2, 4 and 6 areneeded to determine relative energies within the fn

manifold. Once the ion positions and charges areknown, the only undetermined variables are thevalues of the powers of the electron radii rki . Thevalues were obtained by use of a DV-Xa calcula-tion [36] for a trivalent Er ion to which the 5d and5g orbits were added. It is known that the modi-fication of the values is necessary for the repro-duction of the observation [37]

hrki ¼ akhrkiDV-Xa; ð3Þwhere ak is a phenomenological parameter for themodification. The sum in Eq. (2) was evaluatedwith the coordinates taken from the MD simula-tion in a sphere with a radius of 20 nm from eachrare earth ion. The values of the energy-level pa-

Table 1

The potential and the parameters used in MD simulation

Na Ba Re La Al Zr F

Z +1 +2 +3 +3 +3 +4 )1

Aij(10�16 J) Na Ba Tm La Al Zr F

Na 1.00 4.84 3.25 5.83 1.29 2.90 1.04

Ba 8.34 13.09 23.76 5.71 13.64 5.07

Re 3.17 4.48 4.14 7.82 3.39

La 6.51 7.01 13.25 6.11

Al 1.93 3.91 1.30

Zr 2.49 3.02

F 0.84

Born–Mayer potential

Uij ¼e2

4pe0

ZiZj

rijþ Aij exp

�� rij

q

�; q ¼ 0:03 nm:

272 H. Inoue et al. / Journal of Non-Crystalline Solids 298 (2002) 270–286

rameters for the SLJ terms determined by Canallet al. [38] for the Er3þ ion in the LaF3 crystal wereused. The eigenstates and eigenvalues were ob-tained from the diagonalization of the crystal-fieldHamiltonian. The resulting 364 (14C3) eigenstatesof the 4f12 configuration were a linear combinationof basis states SLJM of the form

jwi ¼XSLJM

aðSLJMÞj4f12aSLJMi: ð4Þ

These states will henceforth be referred to asSLJM.

The electric-dipole transition rate between aninitial level SLJM and a final level S0L0J 0M 0, thatwas calculated according to the full Judd–Ofelttheory [31,32,35] can be written as

AðSLJM ; S0L0J 0M 0Þ ¼ e2

4pe0

64p4�mm3

3hc3nðn2 þ 2Þ2

9

hSLJM jPEDjS0L0J 0M 0i�� 2;

hSLJM jPEDjS0L0J 0M 0i

¼X

aSLJMa0S0L0J 0M 0aðaSLJMÞa0ða0S0L0J 0M 0Þ�

� ð4fnaSLJM jPEDj 4fna0S0L0J 0M 0Þ�;

ð4fnaSLJM jPEDj4fna0S0L0J 0M 0Þ

¼Xkqk

ð�1ÞqþqþJ�MþS0þL0þJþk½J ; J 0�1=2

� ½k� 1 k kq �ðq þ qÞ q

� �

� J k J 0

�M qþ q M 0

� �J J 0 kL0 L S

� �Akq

� Nðk; kÞð4fnaSLjU kj4fna0S0L0Þ;

Nðk; kÞ ¼ 2Xn0l0

½l; l0�ð�1Þlþl0 1 k kl l0 l

� �

� l 1 l0

0 0 0

� �l0 k l0 0 0

� �

� h4f jrjn0l0ihn0l0jrkj4fiDðn0l0Þ ; ð5Þ

where �mm is the frequency of the transition, n is therefractive index, c is the velocity of light and q isthe polarization of the transition. The doubly re-

duced matrix elements of the spherical tensor op-erator U k have been tabulated by Nielson andKoster [39] The values of the refractive index n atwavelength k was calculated using the relationn ¼ Aþ B=k2 taking A ¼ 1:50 and B ¼ 3500 ðnm2Þfor ZBLAN glass [40]. For a determination of theNðkk) coefficients, it is necessary to know the valueof the radial integrals h4f jrjn0l0i and h4f jrkjn0l0i,and the energy for n0l0 configuration, Dðn0l0Þ. Then0l0 configurations were assumed to be 5d and 5gconfigurations. The radial integrals and energy ofthe 5d and 5g states of the free-ion state of the Er3þ

ion were estimated from the DV-Xa calculation. Aphenomenological parameter, bðk; k), was intro-duced, because the correction was also necessaryfor the value of Nðk; k)Nðk; kÞ ¼ bðk; kÞNDV-Xaðk; kÞ: ð6ÞThe magnetic-dipole transition rate was obtainedby use of the method in the literature [41].

The entire transition rate between Stark levels,AðSLJM ; S0L0J 0M 0Þ, could be obtained. The spon-taneous emission rate from the SLJ state to theS0J 0L0 state, AðSLJ ; S0L0J 0Þ, was estimated. J þ 1=2Stark levels exist in the initial SLJ state. Thedensity of the ions of the Stark level, nSLJM , in theSLJ state at a temperature T was assumed onthe basis of the Maxwell–Boltzmann distribution

nSLJM ¼ expð�DESLJM=kT ÞPM expð�DESLJM=kT Þ

; ð7Þ

where DESLJM is the energy difference between theSLJM level and the lowest Stark level in the SLJstate. The total density of the initial SLJ statewas set as unity. The spontaneous emission rateAðSLJ ; S0L0J 0Þ can be written as

AðSLJ ; S0L0J 0Þ ¼XM

nSLJMXM 0

AðSLJM ; S0L0J 0M 0Þ:

ð8ÞThe relation between the obtained value ofAðSLJM ; S0L0J 0M 0Þ ð¼ AðS0L0J 0M 0; SLJMÞÞ and thecross-section, rSLJM�S0J 0L0M 0 , of absorption andemission attributed to the transition is

AðSLJM ; S0L0J 0M 0Þ ¼ 8p�mm2n2

c2

ZrSLJM�S0L0J 0M 0dm;

ð9Þ


where m is the frequency. The Gaussian shapefunction, f ðkÞ, where k is the wave number, with50 cm�1 of FWHM (full width at half maximum)was assumed for the generation of the spectralshapes at room temperature. The cross-section,rSLJM�S0J 0L0M 0 , is given by

rSLJM�S0L0J 0M 0 ðkÞ ¼ c� 106

8p�mm2n2AðSLJM ; S0L0J 0M 0Þf ðkÞ:

ð10ÞThe cross-section of absorption or emission fromthe SLJ state to the S0L0J 0 state, rSLJ�S0J 0L0 , can bewritten as

rSLJ�S0L0J 0 ðkÞ ¼XM

nSLJMXM 0

rSLJM�S0L0J 0M 0 ðkÞ: ð11Þ

The averages of the cross-section of absorptionand emission of 300 spectra obtained from 300structural models were compared with the ob-served spectra.

2.4. Calculation of the fluorescence line narrowingspectra

Two kinds of the emission spectra of the Er3þ

ions in the ZBLAN glass at low temperatures havebeen reported. One is the FLN spectrum of the4I11=2 ! 4I15=2 transition at 4.2 K excited into thelowest Stark level of the 4I9=2 manifold [42,43].The other is the emission spectrum of the 4S3=2 !4I15=2 transition at 13 K excited at 488 nm from theArþ ion laser, which excited the Er3þ ions into the4F7=2 level [15].

The Er3þ ions in the structural models wereclassified into five groups according to the energyseparation from the lowest Stark level of the 4I15=2state to that of the 4I9=2 state to simulate the ex-citation of the FLN measurement. The emissionspectra from the 4I11=2 level to the

4I15=2 level of theEr3þ ions in each group were calculated using theGaussian shape function with 30 cm�1 of FWHM.The density of the ions of the Stark level in the4I11=2 state, which was the initial level of theemission, was estimated from Eq. (7) at 4.2 K.The calculated emission spectra were comparedwith the observed FLN spectra of the 4I11=2 !4I15=2 transition at 4.2 K. The emission spectra ofthe 4S3=2 ! 4I15=2 transition were calculated in the

same way. The FLN spectra at 13 K were esti-mated by classifying the Er3þ ions according to theenergy separation from the lowest Stark level ofthe 4I15=2 state to that of the 4F7=2 state.

3. Results

3.1. Structural models for the Er3þ-doped ZBLANglass

The structural models for ZBLAN glass dopedwith rare earth ions are described elsewhere [30].The peak of the Re–F pair distribution curve ofthe structural models was at 0.229 nm with 0.023nm of FWHM. The valley of the peak was foundaround 0.31 nm. The average first coordinationnumber was 8.04 in the structural models.

3.2. Optical spectra at room temperature

The observed absorption cross-section is shownin Fig. 1(a). The absorption bands can be ascribedto the transitions from the ground state, 4I15=2, tothe upper levels of the Er3þ ion. Fig. 2 presents aschematic of the Er3þ energy levels. The calculatedabsorption cross-section is shown in Fig. 1(b). Theparameters of ak and bðk; kÞ were determined bythe comparison of the observed and calculatedabsorption spectra. The determined values arelisted in Table 2 together with the values obtainedfrom the DV-Xa method. The values of the oscil-lator strength obtained from the observed andcalculated spectra are listed in Table 3. The valueof the rms deviation, drms, between the observedand the calculated oscillator strengths in Table 3was 4:55� 10�7. The value of the drms found byapplying the Judd–Ofelt theory [31,32] to the Er3þ

ion in ZBLA glass was 1:42� 10�7 [44]. The largervalue of our drms indicated that the agreementbetween the observed and calculated oscillatorstrength was worse than the result applying theusual Judd–Ofelt procedure. As can be seen fromTable 3, the disagreements were mainly due to theoscillator strengths of two large absorption bands,the 4I15=2 ! 2H11=2 and

4I15=2 ! 4G11=2 transitions.The features of the positions, widths and heightsof the absorption band were almost reproduced


in the calculated spectrum. The values of hrki andNðk; kÞ are important factors that determine themagnitude of the splitting and transition rate.Morrison et al. have advocated the equation forthe correction of the value of hrki given by

hrki ¼ 1� rk

skhrkiHF; ð12Þ

where rk is a linear screening factor, the parame-ter, s, is a scaling parameter for the expansion ofthe wave function calculated from the Hartree–Fock expectation when the ion is introduced into asolid. The parameter of ak in Eq. (3) correspondsto ð1� rkÞ=sk. We evaluated the value of s fromthe value in Table 2 by the assumption of rk with0. The values of s were 0.897, 0.754 and 0.741 forhr2i; hr4i and hr6i, respectively. Nðk; kÞ is approxi-mately proportional to the k þ 1 powers of r andinversely proportional to D (n0l0) from Eq. (4).Dorenbos [45] has reported that the 5d states ofthe Er3þ ions in fluorides lie about 67000–

61000 cm�1 above the 4f configuration. The valueof the Dð5dÞ calculated from the DV-Xa methodwas 125500 cm�1. Considering the difference be-tween the calculated and observed values of Dð5dÞ,the values of s were from 0.81 to 0.75. It has beenfound that the spectra, which correspond to theobservation, can be obtained by use of the 4f wavefunction expanded around s ¼ 0:90–0:74. Our es-timated values of s are similar to those of s � 0:75obtained for several lanthanides on the basis of thepoint charge approximation [46]. From the ap-proximation used in the calculation, such as MDsimulation with 2-body potentials and the pointcharge model, a more detailed discussion is beyondthe scope of the present work.

The observed emission spectrum for an excita-tion wavelength at 255 nm is shown in Fig. 3 to-gether with the calculated emission spectra fromthe 4D5=2,

2P3=2,2H11=2,

4S3=2 and4F9=2 levels. The

observed band could be assigned by the emissionspectra from the five initial levels. The calculatedemission band from the 4S3=2 level to the

4I15=2 levelwas located around 18500 cm�1 (540 nm) with a

Fig. 2. A schematic of the energy levels of the Er3þ ion.

(a)

(b)

Fig. 1. The observed (a) and the calculated (b) absorption

cross-section of the Er3þ ion in ZBLAN glass.


shoulder in the low wave number side, whereas theobserved spectrum had two peaks of almost equalheight at 18 450 and 18200 cm�1 in this range. Thecalculated emission bands from the 2P3=2 level tothe 4I11=2 and 4I13=2 levels were located around21 400 and 25000 cm�1 with a peak or a shoulderin the high wave number side, whereas the shoul-der could not be seen for the peaks in the observedspectrum. These discrepancies on the spectra seemto show the discrepancies of the splitting of the

energy level and the distribution of the Starkcomponents and/or the transition rate. Thoughthere was still some room for the evaluation of theempirical parameters of ak and bðk; kÞ, the spectralshape and the emission intensity ratio of the sameinitial level could be reproduced substantially.

3.3. Emission spectra at low temperatures

The homogeneous and inhomogeneous broad-ening obscures the transitions between individualStark levels of rare earth ions in glasses. Thehomogenous width narrows by cooling the sam-ples to the liquid nitrogen or liquid helium tem-perature. A narrow laser line can excite the subsetof the sites of rare earth ions in the glasses selec-tively at low temperatures. Therefore, the narrowemission spectrum can be observed from the Er3þ

ions excited using a narrow laser line at low tem-peratures. This FLN technique provides us withthe information on the Stark splitting of the energy

Fig. 3. The observed emission spectrum: (a) under 255 nm

excitation at room temperature and the calculated emission

spectra from 4D5=2 level (c), 2P3=2 (d), 2H11=2 (e), 4S3=2 (f) and4F9=2 (g) levels. The spectrum of (b) was summed up the spectra

from (c) to (g).

Table 2

The parameters hrki, Nðt; kÞ and DEðn0l0Þ for 4f, 5d and 5g

electrons calculated by the DV-Xa method and used parameters

of ak and b(t; k)

k hrki ak

2 1:930� 10�21 m2 1.243

4 9:335� 10�42 m4 3.095

6 9:198� 10�62 m6 6.032

t; k Nðt; kÞ b(t; k)

1; 2 �2:962� 10�4 m2 J�1 3.062

3; 2 1:378� 10�24 m4 J�1 6.153

3; 4 1:610� 10�24 m4 J�1 6.153

5; 4 �7:975� 10�45 m6 J�1 7.059

5; 6 �2:261� 10�44 m6 J�1 7.058

7; 6 8:455� 10�65 m8 J�1 13.540

n0l0 DEðn0l0Þ5d 1:25� 105 cm�1

5g 4:31� 105 cm�1

Table 3

The observed and the calculated oscillator strengths

Observed

(10�7)

Calculated (10�7)

MD ED Total

4I13=2 13.57 4.74 9.99 14.734I11=2 4.14 0.02 4.41 4.434I9=2 2.23 0.02 2.39 2.414F9=2 17.47 0.05 15.55 15.604S3=2 49.39 0.00 3.67 42.542H11=2 49.39 0.13 38.74 42.544F7=2 14.62 0.01 14.90 14.914F5=2 6.91 0.00 4.30 6.674F3=2 6.91 0.00 2.37 6.672G9=2 4.99 0.00 5.93 5.934G11=2 81.78 0.11 72.10 72.214G9=2 20.16 0.58 18.98 19.56

MD: Magnetic–dipole transition and ED: Electric–dipole

transition.


levels and the site-to-site energy variations of rareearth ions in glasses. The calculated absorptionand emission spectra at room temperature de-scribed in the previous section could substantiallyreproduce the observation. For the next step, theFLN spectra at 4.2 and 13 K were calculated inorder to examine a more detailed Stark splitting ofthe energy levels and the site-to-site energy varia-tions.

The calculated FLN spectra for the 4I11=2 !4I15=2 transition of the Er3þ ions, which were clas-sified by the excitation energy between the lowestStark level of the 4I15=2 state and the

4I9=2 state, areshown in Fig. 4. The FLN spectra are all nor-malized to the same amplitude. Moreover, eachspectrum was divided into eight components (A–H) by the terminal Stark levels of the 4I15=2 state.The remarkable features of the calculated FLNspectra were as follows: (1) The bandwidths ofthe Stark components increased in order from Ato H. The peaks of the A–D components were

higher than those of the E–H components. (2)The energy differences between the Stark compo-nents increased with an increase in the excitationwave number. (3) The position of the A compo-nent shifted to the higher wave number side withan increase in the excitation wave number and thepositions of the other components (B–H) shifted tothe lower wave number side with an increase in theexcitation wave number. The A component wasclearly separated from the other components un-der higher excitation wave number. The distribu-tion of the positions of the A–H components withthe excitation wave number is plotted in Fig. 5.The zero for the abscissa was taken at Em ¼12354 cm�1 where the absorption cross-sectionwas a maximum. The zero for the ordinate waschosen to be at E0 ¼ 10194 cm�1, the average ofthe A components.

The observed FLN spectra at 4.2 K reported byZemon et al. [42,43] are shown in Fig. 6. Theyfound that the peaks of the A–D components weresharp and intense. The position of the B–D com-ponents shifted to the lower wave number sidewith an increase in the excitation wave number,whereas the position of the F–H components

Fig. 4. The calculated FLN spectra for the 4I11=2 ! 4I15=2transition. The excitation energy of the spectrum (a), (b), (c), (d)

and (e) were 12 309.7, 12 339.1, 12 368.5, 12 397.9 and

12 427:2� 14:7 cm�1, respectively.

Fig. 5. The distribution of the position of the A–H components

of the 4I15=2 level with the excitation energy. The zero for the

ordinate E0 ¼ 10194 cm�1 and the zero for the abscissa

Em ¼ 12354 cm�1.


shifted to the higher wave number side with anincrease in the excitation wave number. Severaldiscrepancies were found between the calculatedand observed spectra. The peak position of thecalculated absorption band of the 4I15=2 ! 4I9=2transition was 56 cm�1 lower than the observedone. The calculated FWHM for the lowest Starkcomponent of the 4I9=2 level was about 70 cm�1

and the width was about twice as broad as theobserved one. The magnitude of the energy dif-ference of the A–C components was larger thanthose of the observed spectra. The position of theF–H components in the calculated spectra shiftedto the lower wave number side with an increase inthe excitation wave number, whereas the positionof the F–H components in the observed spectrashifted to the higher wave number side with anincrease in the excitation wave number. However,some of the features of the calculated spectra couldbe found in the observed spectra.

The calculated FLN spectra at 13 K for the4S3=2 ! 4I15=2 transitions of the Er3þ ions, whichwere classified by the excitation energy betweenthe lowest Stark level of the 4I15=2 and the 4F7=2

manifold, are shown in Fig. 7. The FLN spectraare all normalized to the same amplitude. Eachspectrum was divided into eight components (A–H) by the terminal Stark components of the 4I15=2level. The distribution of the positions of the A–Hcomponents with the excitation wave number isplotted in Fig. 8. The zero for the abscissa wastaken at Em ¼ 20463 cm�1 where the absorptioncross-section was a maximum. The zero for theordinate was chosen to be at E0 ¼ 18558 cm�1, theaverage of the A components. The shapes ofthe FLN spectra of the 4S3=2 ! 4I15=2 transitionwere clearly different from those of the 4I11=2 !4I15=2 transition. The shape of the FLN spectrum isdetermined by the selectivity of the Er3þ ions in theexcitation, the transition rate and the Stark split-ting of the terminal level. The terminal 4I15=2 levelis common to the 4S3=2 ! 4I15=2 and the 4I11=2 !4I15=2 transitions. From the position of the Starkcomponent in Figs. 5 and 8, the difference of the

Fig. 7. The calculated FLN spectra for the 4S3=2 ! 4I15=2transition. The excitation energy of the spectrum (a), (b), (c), (d)

and (e) were 20 403.8, 20 433.4, 20 463.0, 20 492.6 and

20522:2� 14:8 cm�1, respectively.

Fig. 6. The observed FLN spectra at 4.2 K reported by Zemon

et al. [42]. The excitation wavelengths of spectra (a), (b), (c) and

(d) were 808.3, 806.8, 805.8 and 804.3 nm (12 371.6, 12 394.6,

12 410.0 and 12 433:1 cm�1), respectively.


shapes of the spectra was mainly due to a differ-ence between the selectivity of the Er3þ ions in theexcitation. The remarkable features of the FLNspectra for the 4S3=2 ! 4I15=2 transitions are asfollows: (1) The bandwidths of the Stark compo-nents increased in order from A to H. (2) Theemission bandwidths became narrower and theintensity in the lower wave number side increasedwith an increase in the excitation wave number. (3)The positions of all components kept or increasedwith an increase in the excitation wave number.The emission spectrum for the 4S3=2 ! 4I15=2 tran-sition, which is shown in Fig. 9 was only observed

under 488 nm excitation at 13 K [15]. The shape ofthe spectrum was quite similar to the calculatedspectrum under 20522 cm�1 (487.3 nm) excitation.The positions of the peaks in the calculated spec-trum were about 100 cm�1 higher.

4. Discussion

4.1. The relation between the splitting of the energylevels and the crystal field parameters

In the previous section we showed that thecalculation could reproduce the observed opticalspectra at room temperature and at low tempera-tures to some extent. In this calculation, the FLNspectra for the 4I11=2 ! 4I15=2 and 4S3=2 ! 4I15=2transitions showed different shapes. Unfortu-nately, the FLN spectra for the 4S3=2 ! 4I15=2transition have not been reported. The reportedshape of the spectrum for the 4S3=2 ! 4I15=2 tran-sition [15] is clearly different from the FLN spectrafor the 4I11=2 ! 4I15=2 transition. This indicates thatthe excitation using a narrow laser line at a lowtemperature is composed the subset of the Er3þ

ions, and the selected Er3þ ions to the subset aredifferent in excitation processes, i.e., 4I15=2 ! 4I9=2and 4I15=2 ! 4F7=2 transitions. In other words, theposition of the lowest Stark level of the 4I9=2 statefrom that of the 4I15=2 state is not correlated muchwith that of the 4F7=2 state for each Er3þ ion. Theenergy splitting is determined by the crystal fieldHamiltonian. Therefore, the splitting of these twolevels was caused by the different part of the crystalfield parameters. Here, the characteristics of thesplitting of each level is examined from the calcu-lated energy splitting of 18 levels between 4I15=2and 4D7=2 levels. On the absorption spectrum, the4G9=2;

2K15=2 and4G7=2 levels and

2K13=2;2P1=2 and

4G5=2 levels were located around 360 and 300 nm,respectively. These two sets of three levels wereexcluded from the examination since they stronglyoverlapped each other.

The width of the splitting of each level is definedas an energy difference between the highest andlowest Stark components of the calculated level.The correlation coefficients between the width ofthe splitting of the 4I15=2 level and that of the other

Fig. 9. The observed spectrum at 13K reported byMortier et al.

[15]. The excitation wavelengths was 488 nm (20 491:8 cm�1).

Fig. 8. The distribution of the position of the A–H components

of the 4I15=2 level with the excitation energy. The zero for the

ordinate E0 ¼ 18558 cm�1 and the zero for the abscissa

Em ¼ 20463 cm�1.


levels for each Er3þ ion are listed in Table 4. Ascan be seen from the table, there are 5 levels ofwhich the correlation coefficient was larger than0.9. In particular, the highest value of the corre-lation coefficient of the 4I13=2 level was 0.98. Thehigh correlation coefficient showed that the split-ting of the level was similar to that of the 4I15=2level. More noticeable was that there were 2 levels,4I9=2 and 4G11=2, of which correlation coefficientwas less than 0.5. The value for the 4I9=2 level wasonly 0.38, while the value for the 4F7=2 level was0.88. It was found that there were several ways ofsplitting the energy levels. The difference betweenthe correlation coefficient of the 4I9=2 and 4F7=2

levels, which were terminal levels of the excitationof the FLN spectra in Figs. 4 and 7, correspondsto the difference of the selectivity of the Er3þ ionsin the excitation.

In order to examine the relation between thesplitting of each energy level and the crystal fieldparameters, the crystalline field parameters wereclassified into three groups by their order; A2q, A4q

and A6q terms. The energy splitting of each levelcalculated only from the parameter of each groupis compared with the original splitting, which wascalculated from full sets of the crystal filed pa-rameters. If the contribution of a certain group of

the crystal field parameter is large for the splittingof an energy level, the energy calculated only fromthe group must be correlated with the originalvalue of the energy splitting. The correlation co-efficients of the calculated value from each pa-rameter group to the original calculated value arelisted in Table 5. It is found that the relation be-tween the splitting of the energy level and thecrystal field is not identical at each level. The en-ergy levels could be roughly divided into fourgroups according to the combination of the valuesof the three correlation coefficients of CA2q, CA4q

and CA6q. The first group was represented by the4I15=2 level. The values of CA2q, CA4q and CA6q of the4I15=2 level were 0.69, 0.54 and 0.40, respectively.The values of the coefficients decreased in order ofCA2q, CA4q and CA6q. The values of the coefficientsfor 4I13=2,

4I11=2 and 4F9=2 levels were similar tothose of the 4I15=2 level. The second group wasrepresented by the 4S3=2 level. The value of CA2q

was more than 0.9 and the values of CA4q and CA6q

were less than 0.2. We classified 4S3=2;4F7=2;

4F3=2;2P3=2, and

4D7=2 levels into this group. Thethird group was composed of 4F5=2;

4G7=2;2H9=2

Table 5

The correlation coefficients between the energy positions of

Stark components of each level calculated from individual A2q,

A4q and A6q terms and the energy positions calculated from the

full set of the crystal field parameters

Energy level CA2q CA4q CA6q

4I15=2 0.69 0.53 0.404I13=2 0.80 0.49 0.194I11=2 0.88 0.36 0.114I9=2 0.18 0.63 0.684F9=2 0.87 0.40 0.114S3=2 1.00 0.08 0.002H11=2 0.33 0.86 0.374F7=2 0.98 0.14 0.164F5=2 0.92 0.27 0.004F3=2 0.98 0.03 0.002G9=2 0.46 0.57 0.574G11=2 0.21 0.97 0.164G9=2;

2K15=2;2G7=2 – – –

2P3=2 0.99 0.01 0.012K13=2;

4G5=2;2P1=2 – – –

4G7=2 0.96 0.36 0.132D5=2 0.92 0.34 0.002H9=2 0.89 0.19 0.364D5=2 0.54 0.65 0.004D7=2 1.00 0.15 0.02

Table 4

The correlation coefficient between the width of the splitting of

the 4I15=2 level and those of other level

Energy level Correlation coefficient

4I13=2 0.984I11=2 0.964I9=2 0.384F9=2 0.924S3=2 0.842H11=2 0.644F7=2 0.884F5=2 0.884F3=2 0.772G9=2 0.744G11=2 0.414G9=2;

2K15=2;2G7=2 –

2P3=2 0.892K13=2;

4G5=2;2P1=2 –

4G7=2 0.892D5=2 0.932H9=2 0.924D5=2 0.794D7=2 0.87


and 2D5=2 levels. The values of the correlationcoefficients were the middle range of the values forthe first and second groups. The fourth group wasrepresented by the 4I9=2 level. The value of CA4q wasthe highest in the three coefficients. The values ofthe coefficients for 2H11=2;

2G9=2;4G11=2 and 4D5=2

levels were similar to that of the 4I9=2 level. Thevalues of the correlation coefficients of the widthof the splitting in Table 4 were in the order of first,third, second and fourth groups, because the 4I15=2level belonged to the first group.

4.2. The effect of the first-coordination number ofthe Er3þ ions on the splitting of the 4I9=2 level

The value of the crystal field parameter Akq iscalculated from Eq. (2) on the basis of the crystalfield theory and point charge approximation. Thevalue of the Akq term with a higher k converges atthe shorter distance from the central rare earthion. Therefore, the value of the Akq term with ahigher k is dominated by the closer structurearound the rare earth ion. It is expected that theshort-range structure like the first-coordinationnumber of the rare earth ion appears in the split-ting of the level in the fourth group. The structuralmodels contained the Er3þ ions of which the first-coordination numbers were 7, 8 and 9. The split-ting of the 4I9=2 level, which was one of the energylevels in the fourth group, must represent the in-formation of the short-range structure around theEr3þ ions. The average widths of the splitting ofthe 4I9=2 level for the 7-, 8- and 9-fold coordinationwere 204, 183 and 170 cm�1, respectively. Thewidth tended to decrease with an increase in thecoordination number. For the examination of thissplitting, the emission spectrum, of which the ter-minal level was the 4I9=2 level, was estimated. Inaddition, it is necessary that the emission bandoverlaps with others as little as possible. Figs. 10and 11 show the calculated emission spectra of the2H11=2 ! 4I9=2 and

4S3=2 ! 4I9=2 transitions. As canbe seen from the figures, it was estimated that theEr3þ ions of the 7-fold coordination show thecharacteristic shape in both calculated spectra. Inthe spectra of the 4S3=2 ! 4I9=2 transition, the peakposition for the Er3þ ions of the 7-fold coordina-tion was located in the low wave number side on

the spectrum, but in the high wave number side forthe 9-fold coordination. It was estimated that theshape of the spectrum for the 4S3=2 ! 4I9=2 transi-tion differed by the coordination number of theEr3þ ions. The shape of this spectrum seems to beinfluenced by not only the coordination numberof the Er3þ ions but also by the symmetry ofthe coordination. Based on the simulation, amore detailed examination would be possible by

Fig. 10. The calculated spectra of 2H11=2 ! 4I9=2 transition of

the Er3þ ions, of which the first-coordination numbers were:

(a) 9, (b) 8 and (c) 7.

Fig. 11. The calculated spectra of 4S3=2 ! 4I9=2 transition of the

Er3þ ions, of which the first-coordination numbers were: (a) 9,

(b) 8 and (c) 7.


comparison of these spectra with the observationand selective excitation of the Er3þ ions.

4.3. Requirements for the measurement of a well-defined FLN spectrum

For improvement and verification of thismethod, it is necessary to measure the spectrum ofother transitions at a low temperature. In order tomeasure a well-defined FLN spectrum at a lowtemperature, there are three important require-ments.

An example of the measurement of the splittingof the 4I15=2 level is explained. Fig. 12(a) shows aschematic of the absorption and emission process.The sample cooled at a low temperature is irradi-ated with light of a finite line width, and a part ofthe Er3þ ions in the sample is excited to a Starkcomponent of a SLJ level. The width of thesplitting of the 4I15=2 level of the excited Er3þ ionshas to be more narrowly distributed than that ofthe entire Er3þ ions in the sample. This is the firstrequirement. For this requirement, the energyseparation from the lowest Stark component of the4I15=2 level to the Stark component of the SLJ levelhas to be correlated with the width of the splittingof the 4I15=2 level. In the SLJ level, one Starkcomponent (usually the lowest one) has to be

separated from other Stark components for theselective excitation. This requirement is the secondone. After the transition from the SLJ level to aS0L0J 0 level, which is an initial level for the emis-sion, the transition from the lowest Stark compo-nent of the S0L0J 0 level to the 4I15=2 level is observedas a FLN spectrum. In this transition the energyseparation from the lowest Stark component of theS0L0J 0 level to that of the 4I15=2 level also has to becorrelated with the width of the splitting of the4I15=2 level. This requirement is the third one. Table6 shows the correlation coefficients between thewidth of the splitting of the 4I15=2 level and theenergy separation from the lowest component ofthe 4I15=2 level to that of the upper levels. Rela-tively large values of the coefficients were obtainedfor the 2H11=2;

4D7=2;4F3=2;

4I11=2;4I9=2 and 4G11=2

levels. One can see that two levels may be chosenfrom these levels as a terminal level of the excita-tion and an initial level of the emission. Fig. 13shows the simulated absorption spectra of the2H11=2;

4F3=2;4I11=2;

4I9=2 and4G11=2 levels at 4.2 K.

Since Stark components in the absorption spectraof the 2H11=2;

4I11=2 and 4G11=2 levels stronglyoverlapped each other, these three levels are ex-cluded from the candidates as a terminal level ofthe excitation. The 4F3=2 and

4I9=2 levels remain asthe terminal levels of the excitation when tunable

Fig. 12. Schematics of the excitation and the emission process for the measurements of the splitting of the levels: (a) 4I15=2, (b)4I9=2 at a

low temperature.


laser excitation was considered. Zemon et al.[42,43] have measured the FLN spectra of the4I11=2 ! 4I15=2 emission by the 4I15=2 ! 4I9=2 exci-tation. The combination of the levels is one of theevaluated combinations for the well-defined FLNspectrum.

Finally, the splitting of the 4I9=2 level, whichrepresents the short-range structure around theEr3þ ions, is examined again. A schematic of theabsorption and emission process is shown in Fig.12(b). In this case, the excitation and emissionprocesses are from the 4I15=2 level to the SLJ leveland from the S0L0J 0 level to the 4I9=2 level, respec-tively. Therefore, the energy separations of bothtransitions have to be correlated with the width ofthe splitting of the 4I9=2 level. The correlation co-efficients of the energies and the width of thesplitting of the 4I9=2 level are listed in Table 7.Most coefficients for the excitation process weresmall. The highest value was 0.50 for the4I15=2 ! 4G7=2 transition. Unfortunately, this exci-tation is difficult, since a tunable laser of about a294 nm wavelength is necessary. On the otherhand, there were several transitions for the emis-sion, of which the coefficients were high enough.

The value of the coefficient for the 2H11=2 ! 4I9=2transition was 0.87. This high coefficient value in-dicated that the width of each Stark component ofthe emission spectra for the 2H11=2 ! 4I9=2 transi-tion in Fig. 11 was narrower than that for the4S3=2 ! 4I9=2 transition in Fig. 10, whose coefficientvalue was 0.49. The shape of the FLN spectrum isalso dependent on the transition rate between theStark levels. The relation between the transitionrate and the crystal field parameter must be dif-ferent from the relation between the splitting andthe crystal field parameter. The discussion for therelation between the transition rate and the crystalfield parameter as well as the measurement of theFLN spectra for the transition which have notbeen observed yet will be a problem for futureconsideration.

The characteristics of the splitting of each leveldiscussed here is dependent on the values of thecrystal field parameters obtained from the struc-tural models and the phenomenological parame-ters ak and bðk; kÞ. Therefore, better agreementbetween the observed and the calculated spectra isdesired. Moreover, a more sufficient examinationof the different chemical compositions of glass anddifferent glass systems seems to be necessary.However, the importance of considering thecharacteristics of the splitting of the level in theFLN measurement will not change.

5. Conclusion

It is useful to estimate various optical spectrafrom a structural model not only for the analysisof the structure around rare earth ions in glassesbut also for the prediction of various opticalproperties which cannot be measured. In this studyaimed at the establishment of this simulationmethod, the optical properties of Er3þ-dopedZBLAN glass were investigated. Absorption andemission spectra of Er3þ-doped ZBLAN glass atroom temperature were estimated from structuralmodels prepared by using molecular dynamicssimulation on the basis of crystal field theory andpoint-charge approximation. The calculated ab-sorption and emission spectra substantially agreedwith the observed ones. Furthermore, it was found

Table 6


the 4I15=2 level and the transition energy, which was obtained

from the energy from the lowest Stark component of that of the4I15=2 level and each level

Energy level Correlation coefficient

4I13=2 0.214I11=2 0.584I9=2 0.574F9=2 0.034S3=2 0.222H11=2 0.724F7=2 0.324F5=2 0.294F3=2 0.622G9=2 0.394G11=2 0.574G9=2;

2K15=2;2G7=2 –

2P3=2 0.432K13=2;

4G5=2;2P1=2 –

4G7=2 0.222D5=2 0.222H9=2 0.184D5=2 0.334D7=2 0.63


Fig. 13. The calculated absorption cross-section of levels: (a) 2H11=2, (b)4F3=2, (c)

4I11=2 , (d)4I9=2, (e)

4G11=2 at 4.2 K.


that the emission spectra at low temperatures alsocould be reproduced to some extent. Based on theJudd–Ofelt theory, the transition rate between theenergy levels could be estimated. In addition, itwas possible to estimate the shape of the spectrumby using this method. Therefore, the optical spec-tra obtained will be utilized for the detailed designof optical properties.

From the relation between the splitting of theenergy level and the crystal filed parameters cal-culated from the Er3þ ion in each structural model,the following items were examined. The relationwas different at individual levels. On the basis ofthis relation, the energy levels could be classifiedinto several groups. The splitting of the grouprepresented by the 4I9=2 level was dominated bythe short-range structure around the Er3þ ions. Thedifference of the first-coordination number of theEr3þ ions appeared in the calculated emissionspectra of the transitions to the 4I9=2 level. Thetransition process for the measurement of the well-defined FLN spectrum was estimated from thecharacteristics of the splitting of the energy levels.

The measurement of emission spectra for varioustransitions and prediction of optical properties willbe the next steps for the establishment of thismethod.

Acknowledgements

This study was financially supported by aGrant-in-Aid from the Ministry of Education withthe contract number #09450239. The authorswould like to thank Morita Chemical IndustriesCo., and Central Glass Co., for the supply of flu-orides.

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Documents

Simulation of the optical properties of Er:ZBLAN glass