Simulation of the optical properties of Tm:ZBLAN glass. II. Energy transfer between Tm3+ ions under single- and dual-wavelength excitation

Journal of Non-Crystalline Solids 336 (2004) 135–147

www.elsevier.com/locate/jnoncrysol

Simulation of the optical properties of Tm:ZBLAN glass.II. Energy transfer between Tm3þ ions under single-

and dual-wavelength excitation

Hiroyuki Inoue a,*, Kenichi Moriwaki b, Norikazu Tabata c, Kohei Soga d,Akio Makshima e, Youichi Akasaka f

a Institute of Industrial Science, The University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8505, Japanb Fuji Photo Film, Recording Media Products Div, 2-12-1, Ohgi-cho, Odawara, Kanagawa 250-0001, Japan

c Accenture Corporation, Nihon Seimei Akasaka Daini Bldg, 7-1-16, Akasaka Minato-ku, Tokyo 107-8672, Japand Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo,

7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japane Center for New Materials, Japan Advanced Institute of Science and Technology, 1-1, Asahidai, Tatsunokuchi,

Nomi, Ishikawa 923-1292, Japanf Sprint, Advanced Technology Labs, 30 Adrian Court, Burlingame, CA 94010, USA

Received 8 May 2003

Abstract

The energy transfer rate between Tm3þ ions in ZBLAN glass was estimated from the optical spectra and transition rates on the

basis of a method proposed by Kushida. The optical spectra and radiative transition rates were obtained from the structural models

prepared by molecular dynamic simulation and the crystal field theory. The lifetimes of 3H4,1G4 and

1D2 levels were estimated from

the calculation and the results were experimentally confirmed. The numerical model was also used to predict emission intensities

under dual-wavelength excitation and decay curves under CW excitation.

� 2004 Elsevier B.V. All rights reserved.

1. Introduction

Rare earth ions doped fluoride fibers are promisingcandidates for compact and efficient optical amplifiers

and laser sources [1]. The fluoride fiber doped with

Tm3þ ion is an attractive medium for an upconversion

laser of the blue wavelength region [2–4]. The laser

oscillation of the upconversion at 455 and 480 nm was

first demonstrated at 77 K [2]. The CW laser oscillation

at room temperature at 480 nm has been reported [3,4].

Under a dual-wavelength excitation (1100 and 680 nm),the threshold decreased significantly and the efficiency

increased [4]. The Tm3þ-doped fluoride fiber has been

reported to be an amplifier for S band (1480–1530 nm)

and S+ band (1450–1480 nm). The lifetime of the upper,

* Corresponding author. Tel.: +81-3 5452 6315; fax: +81-3 5452

6316.

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

0022-3093/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.jnoncrysol.2004.02.006

3H4, level of the stimulated emission for the amplifica-

tion is shorter than that of the lower, 3F4, level.

Therefore, the formation of a population inversion isdifficult by direct pumping. In order to solve the prob-

lem, several upconversion pumping methods have been

proposed [5–9]. A signal gain of 25 dB was first dem-

onstrated at 1470 nm when pumped at 450 mW by a

1064 nm laser [5]. The range of the gain has been shifted

to the S band by use of the dual-wavelength (1050 and

1560 nm [6] or 1400 and 1560 nm [7]) pumping methods

or by use of the fiber with high-concentration Tm3þ ions[8,9].

The excitation of the upconversion process is due to

excited state absorption (ESA) and/or energy transfer

between rare earth ions. The direct measurement of the

ESA spectra and the quantitative evaluation of the en-

ergy transfer are still difficult. The energy transfer rate

has been estimated from the concentration dependence

of the lifetime of the energy level of rare earth ions. Thechange of the lifetime according to the ion concentration

mail to: [email protected]

136 H. Inoue et al. / Journal of Non-Crystalline Solids 336 (2004) 135–147

is mainly due to the energy transfer related to theground state under such direct pumping. For high

power CW excitation of optical amplifier and laser

oscillation, it is necessary to estimate the energy trans-

fers, which are related to the levels for the stimulated

emission and populated intermediate levels. Unfortu-

nately, it is difficult to evaluate the energy transfers

related to the intermediate levels experimentally.

Therefore, it is desirable to construct a simulation whichtakes into account conditions such as Tm3þ ion con-

centration, combination of excitation wavelengths and

their intensities, and interaction with other rare earth

ions on an atomic level.

The calculation of the splitting of the energy levels of

rare earth ions in glasses have been proposed [10] and

developed [11,12]. The radiative transition rates between

the levels of rare earth ions can be readily calculated[13–15]. Recently, we have reported that fluorescence

spectra of the rare earth ion in the fluoride glasses can

be predicted by calculation [16–21]. According to

Kushida [22], the energy transfer coefficient can be

estimated from the overlap of the absorption and

emission spectra, electric dipole transition rates and the

spatial distribution of rare earth ions. Therefore, it has

become possible to estimate the ground state absorptionspectrum. ESA spectra, emission spectra and energy

transfer rates from theoretical calculation. In order to

develop the calculation, it is necessary to evaluate the

obtained values of absorption and emission cross-sec-

tion, multi-phonon relaxation rate and the energy

transfer rates. In this paper, we estimate the energy

transfer rates from the obtained spectrum shape and

transition rates from the simulation. The lifetime underdirect pumping, the relaxation after CW excitation, and

the emission under dual-wavelength excitation condi-

tions are calculated and compared with experimental

observation.

Fig. 1. The energy diagram of a Tm3þ ion and the excitation and

relaxation processes for dual-wavelength excitation schemes.

2. Experimental conditions

2.1. Glass preparation and measurement of optical prop-

erties

The base glass has the following molar compositions:

52.7ZrF4 Æ 19.9BaF2 Æ 4LaF3 Æ 3AlF3 Æ 19.9NaF Æ 0.5InF3.

Samples of different Tm3þ concentrations were prepared

from batches containing from 0.2 to 1.0 mol% TmF3.

The batches were melted in a gold crucible at 900 �C for15 min and cast into a preheated aluminum mold. All

of these processes were conducted in a glove box under

an atmosphere of dry nitrogen gas. The glass obtained

was cut into a 10 · 25 · 5 mm shape and polished on all

faces.

The excitation source for the spectroscopic measure-

ments was an optical parametric oscillator pumped by

the THG line of a pulsed Nd:YAG laser. The emissionfrom the sample was focused on the entrance slit of a

1 m spectrometer and detected with a photomultiplier

tube. All of the measurements were carried out at room

temperature. The emission decay at 790, 465 and 355 nm

was measured under the excitation at 820, 500 and 450

nm, respectively. These emissions corresponded to the

transitions of 3H4–3H6,

1G4–3H6 and

1D2–3F4 by direct

pumping. The energy diagram of the Tm3þ ion is shownin Fig. 1.

Further measurements for the sample with 0.6 mol%

Tm3þ ion were performed under the dual-wavelength

excitation condition by using the OPO laser and a dye

laser with DCM dye. The excitation schemes are shown

in Fig. 1. In scheme 1, the Tm3þ ion was excited to the3H5 level by the first pulse laser, k1, at 1210 nm. After

the multi-phonon relaxation to the 3F4 level, the Tm3þ

ion was excited to the 1G4 level by the second pulse

laser, k2 at 630 nm. The emission intensity around

480 nm (1G4–3H6 transition) was measured by changing

the delay time of k2 to k1. In scheme 2, the Tm3þ ion

was excited to the 3F3 level by the first pulse laser, k1 at670 nm. After the relaxation to the 3H4 or

3F4 level, the

Tm3þ ion was excited to the 1D2 or 1G4 level by the

second pulse laser, k2, at 630 nm. The emission inten-sities around 480 nm (1G4–

3H6 transition) and 450 nm

(1D2–3F4 transition) were measured by changing the

delay time of k2 to k1.

H. Inoue et al. / Journal of Non-Crystalline Solids 336 (2004) 135–147 137

2.2. Calculation of the optical properties

The structure of the Tm3þ-doped ZBLAN glass was

simulated using the molecular dynamics (MD) tech-

nique. The structure models with 393 ions (Zr4þ: 53,

Ba2þ: 20, La3þ: 3, Al3þ: 3, Naþ: 20, Tm3þ: 1, F�: 293)have been reported [21]. It is necessary to estimate

absorption and emission spectra, energy transfer rates

and multi-phonon relaxation rates of the transitions,which were related to the optical properties. All of the

absorption and emission spectra can be estimated on the

basis of the crystal field theory [23], the Judd–Ofelt

theory [13,14] and the structure model prepared from

MD simulation [24,25]. The reported spectra of the

Tm3þ ion in ZBLAN glass [21] are shown in Fig. 2.

Each energy level of the rare earth ion is described by

three quantum numbers of S, L and J . These quantumnumbers represent total spin, total orbital and total

angular momentum. According to Kushida [22], the

resonant energy transfer coefficient, WET, between two

rare earth ions A and B can be expressed as

WETððSLJ ; S00L00J 00Þ ! ðS0L0J 0; S000L000J 000ÞÞ

¼ 1

ð2J þ 1Þð2J 00 þ 1Þ2p�h

� �� jhSLJ ; S00L00J 00jH jS0L0J 0; S000L000J 000ij2So; ð1Þ

where these ions are in the state SLJ of A ion and S00L00J 00

of B ion before and in the state S0L0J 0 and S000L000J 000 afterthe energy transfer. So is obtained from the overlap

integral of the normalized line shape function for the

Fig. 2. The observed (a) and calculated (b) absorption cross-section of

Tm3þ-doped ZBLAN glass [21].

individual transitions from SLJ to S0L0J 0 of A ion andfrom S00L00J 00 to S000L000J 000 of B ion. The transfer processes

are classified into three groups, the dipole–dipole, di-

pole–quadrapole and quadrapole–quadrapole processes

and these transfer coefficients are given by

W d–dET ððSLJ ;S00L00J 00Þ ! ðS0L0J 0;S000L000J 000ÞÞ

¼ 1

ð2J þ 1Þð2J 00 þ 1Þ2

3

� �2p�h

� �e2

R3

� �2

�Xk

XAkhSLJ jjU ðkÞjjS0L0J 0i2" #

�Xk

XBkhS00L00J 00jjU ðkÞjjS000L000J 000i2" #

So;

W d–qET ððSLJ ;S00L00J 00Þ ! ðS0L0J 0;S000L000J 000ÞÞ

¼ 1

ð2J þ 1Þð2J 00 þ 1Þ2p�h

� �e2

R4

� �2

�Xk

XAkhSLJ jjU ðkÞjjS0L0J 0i2" #

� h4f jr2Bj4f i2hf jjCð2Þjjf i2hS00L00J 00jjU ð2ÞjjS000L000J 000i2So;

W q–qET ððSLJ ;S00L00;J 00Þ ! ðS0L0J 0;S000L000J 000ÞÞ

¼ 1

ð2J þ 1Þð2J 00 þ 1Þ14

5

� �2p�h

� �e2

R5

� �2

� h4f jr2Aj4f i2hf jjCð2Þjjf i2hSLJ jjU ð2ÞjjS0L0J 0i2

� h4f jr2Bj4f i2hf jjCð2Þjjf i2hS00L00J 00jjU ð2ÞjjS000L000J 000i2So;

ð2Þwhere R is a distance between two ions and Xk is the

phenomenological intensity parameter of Judd–Ofelt

theory [13,14]. The electric dipole transition rate,

AðSLJ ; S0L0J 0Þ, from SLJ to S0L0J 0 states can be given by

AðSLJ ; S0; L0; J 0Þ ¼ 64p4�m3

3hc31

2J þ 1vede

2

�Xk

XkhSLJ jjU ðkÞjjS0L0J 0i2: ð3Þ

The values of AðSLJ ; S0L0J 0Þ have already been estimated

[21]. Therefore, without the value of Xk the energy

transfer coefficient can be estimated from the shape ofthe optical spectra, the transition rates and the distance

between Tm3þ ions.

The multiphonon relaxation rate has been empirically

obtained as a function of the energy gap to the next

lower level and the effective phonon frequency of host

materials [26]. The multiphonon relaxation rate, WMPR,

was estimated from the empirical expression

WMPR ¼ C expð�aDEÞ; ð4Þwhere DE is the energy gap between the lowest Starkcomponent of the level and the highest component of

the next lower level. The values used for C and a were

1.88 · 1010 s�1 and 5.77 · 10�3 cm, respectively [26].

Table 1

Parameters of thermal redistribution ratios between several levels, c,and multi-phonon relaxation rates, WMPR

SLJ c WMPR (s�1)

1 3H6

2 3F4

3 3H5 9.36· 1044 3H4 3.8

5 3F3 4.41· 10�3

6 3F2 8.63· 10�2

7 1G4

8 1D2

9 1I610 3P0 2.46· 10�2

11 3P1 1.68· 10�1

12 3P2 3.69· 106

The population densities, nð1ÞSLJ and nð1ÞS 0L0J 0 , of the levels after the thermal

redistribution was calculated from the value of c and the population

densities, nð1ÞSLJ and nð1ÞS0L0J 0 , before the thermal redistribution by

nð1ÞSLJ ¼ cðnð0ÞSLJ þ nð0ÞS0L0J 0 Þ nð1ÞS0L0J 0 ¼ ð1� cÞðnð0ÞSLJ þ nð0ÞS0L0J 0 Þ where S0L0J 0 level

is the next lower level of the SLJ level.


The normalized population density of the Tm3þ ionsin a SLJ level, nSLJ , under an excitation condition was

calculated by the equation

dnSLJdt

¼ �Xi

nSLJX

SLJ<S0J 0L0rSLJ�S0J 0L0 ðkiÞqðkiÞ

þXi

nS0L0J 0X

SLJ>S0L0J 0rS0L0J 0�SLJ ðkiÞqðkiÞ

� nSLJX

SLJ>S0L0J 0AðSLJ ; S0L0J 0Þ

þXS0L0J 0

nS0L0J 0AðS0J 0L0; SLJÞ

�XS0J 0L0

XS00 J 00L00

nSLJnS00L00 J 00

n

XS000L000 J 000

WETððSLJ ; S00L00J 00Þ

! ðS00L00J 00; S000L000J 000ÞÞ

þXS0L0J 0

XS00 J 00L00

nS0L0J 0nS00L00 J 00

n

�X

S000L000 J 000WETððS0L0J 0; S00L00J 00Þ ! ðSLJ ; S000L000J 000ÞÞ

� nSLJWMPRðSLJÞ þ nS0L0J 0WMPRðS0L0J 0Þ; ð5Þ

where qðkiÞ is the photon flux of wave number ki, n is thenormalized population density of total Tm3þ ion. The

value is unity in the case of a single-doped rare earth

ion. The first two terms of Eq. (5) represent the popu-

lation density changes by stimulated absorption and

emission, respectively. The next two terms represent the

population density change by spontaneous emission.

The following two terms represent the energy transfer.

The energy transfer coefficient, WETððSLJ ; S00L00J 00Þ !ðS0L0J 0; S000L000J 000ÞÞ is given by the sum of individual term

of Eq. (2). The last two terms represent the multiphonon

relaxation, where the S0L0J 0 level is the next upper level

of the SLJ one.

We assumed the thermal redistribution between (3H4

and 3F3), (3F3 and

3F2), (1I6 and

3P0) and (3P0 and3P1)

levels. The thermal redistribution ratios, c, were esti-

mated based on the Boltzmann distribution and theenergy gap between the levels. The thermal redistribu-

tion ratios and multi-phonon relaxation rates are listed

in Table 1.

The decay curves of the population of the 3F4,3H5,

3H4,1G4 and 1D2 levels were calculated, respectively,

after a pulse excitation with 1 mJ/mm2 power and 10 ns

duration. The population densities of 12 state from 3H6

to 3P2 levels were calculated. The decay curves of thepopulation densities were calculated after the CW exci-

tation at 647 or 363 nm wavelength. The population

densities of the 1G4 and 1D2 levels under the excitation

schemes 1 and 2 in Fig. 1 were calculated and the

emission intensities were estimated from the population

and radiative transition rates.

It is necessary to estimate the relation between the

Tm3þ concentration, CTm, and the distance, R, between

the Tm3þ ions so that the calculated decay rate is

compared with the observed one. We evaluated the

relation from the concentration dependence on the up-

conversion intensity of the Er3þ ion in ZBLAN glass

and the relation can be approximated by the following

equation [27]:

R ¼ 1

�� 1

e

�1

CTm

� �1=3

: ð6Þ

3. Results

3.1. Decay rates under direct pumping

The emission decays from the 3H4,1G4 and

1D2 levels

were measured under direct pumping. Fig. 3 shows the

reciprocal of the lifetime i.e. the decay rate as a function

of the Tm3þ concentration. The first e-folding time was

set as a time window. The first e-folding time is the time

required for the emission intensity to decrease to 1=e ofits initial intensity. The decay curve was fitted to a single

exponential for the time-window and the decay rate was

obtained. The error-bar in Fig. 3 represents the varia-

tion of the obtained decay rate by shifting the position

of the time-window in the decay curve.

The decay rates of these levels increased with

increasing Tm3þ ion concentration. The decay rate of

the 3H4 level increased from 6.0 · 102 s�1 for the Tm3þ

concentration of 0.2 mol% to 1.3 · 103 s�1 for 1.0 mol%.

The decay rate increased 7.0 · 102 s�1 by the energy

transfer, when we assumed that the dependence of the

Tm3þ concentration on the decay rate was only due to

the energy transfer. The increases of the decay rates of

the 1G4 and 1D2 levels were 2.3 · 103 s�1 and 6.0 · 103s�1, respectively.

Fig. 3. The observed decay rates,�, from 3H4 (a),1G4 (b) and

1D2 (c) levels under direct pumping. An error-bar represents the variation of the decay

rate in the decay curve. The solid line represents the calculated decay rate. The observed decay rates shown in �, M and � have been reported by

Oomen [28]. The decay rates shown in � and M were observed after CW excitation at 647 nm and those shown in � were observed after CW

excitation at 363 nm.


The relation between the decay rate and distance Rwas estimated using Eq. (5). Using Eq. (6), the distanceR between Tm3þ ions was converted into the Tm3þ

concentration. The calculated decay rates are shown by

solid lines in Fig. 3. As can be seen from the figure, the

calculated decay rates versus the Tm3þ concentration

are in close agreement with experimental data.

The differences between the observed and calculated

decay rates at 0.2 mol% for the 3H4,1G4 and

1D2 levels

were 16 s�1, 4.3 · 102 s�1 and 2.9 · 103 s�1, respectively.The differences at the Tm3þ concentration were mainly

due to the difference of the radiative transition rates.

The amount of increase of the calculated decay rate by

energy transfer (the difference in the decay rates between

0.2 and 1.0 mol%) was 1.0 · 102 s�1 for the 3H4 level and

the increase was much smaller than the observed one.

The amounts of increase for the 1G4 and1D2 levels were

4.8 · 103 s�1 and 4.9 · 103 s�1, respectively. As can beseen from Fig. 3(a), the calculated decay rate of the 3H4

level at low concentration agreed well with the observed

one. However, the effect of the energy transfer at higher

concentration was underestimated. On the contrary, the

concentration dependence of the decay rate for the 1D2

level in Fig. 3(c) was almost in close agreement with the

observed one except at low concentration, where the

calculated values were lower than the observed one. Thedifferent error tendencies are due to the different energy

transfer rates of each level. Since several approximations

were used in the calculation of the position of the energy

levels, the widths of the energy level splitting, and the

transition rates, the uncertainties in the energy transfer

rates were unavoidable. The calculated decay rates for3F4 and

3H5 levels were almost independent of the Tm3þ

concentration. The decay rates for the 3F4 and 3H5

levels were 1.0 · 102 s�1 and 9.4 · 104 s�1 respectively.

3.2. Emission from 1G4 and 1D2 levels under a dual-

wavelength excitation

Fig. 4 shows the emission intensity around 480 nm

(1G4–3H6 transition) under a dual-wavelength excita-

tion. Scheme 1 in Fig. 1 shows the excitation process.Tm3þ ions were excited to the 3H5 level by the first pulse

laser, k1 at 1210 nm. Through the multi-phonon relax-

ation from the 3H5 level to the 3F4 level, the population

of the 3F4 level increased and the intensity of the emis-

sion from the 1G4 level increased with increasing the

amount of Tm3þ ions, which were excited by the second

Fig. 4. The observed (a) and calculated (b) emission intensities around

480 nm under the dual-wavelength excitation of scheme 1 in Fig. 1

(k1 ¼ 1210 nm and k2 ¼ 630 nm).

Fig. 5. The observed (a) and calculated (b) relative emission intensities

around 480 nm: M and 450 nm: � under the dual-wavelength exci-

tation of scheme 2 in Fig. 1 (k1 ¼ 670 nm and k2 ¼ 630 nm). The

intensity around 480 nm shown inO was obtained under k2 ¼ 635 nm,

� was under k2 ¼ 632 nm respectively.


pulse laser at 630 nm. The lifetime of the 3H5 level was

estimated from the time required for the emission

intensity to increase to 1� 1=e of its maximum intensity.The lifetime of the 3H5 level was about 1.3 · 10�5 s,

hence, the decay rate was 7.7 · 104 s�1. The multi-pho-

non relaxation rate of the 3H5 level used in the calcu-

lation was 9.4 · 104 s�1. It was found that the estimated

multi-phonon relaxation rate of the 3H5 level was a close

approximate of the observed one. As the lifetime of the3H5 level is very short and the emission from the 3H5

level is in the infrared region, it is difficult to measure thedecay curve via direct excitation of the 3H5 level. The

dual-wavelength excitation made it possible to measure

the lifetime of the 3H5 level. Moreover, since the de-

crease of the population of the 3F4 level could be esti-

mated from the calculation, it would be possible to

measure simultaneously the lifetime of the 3F4 level, if

the delay time for the second pulse laser could be further

controlled.Fig. 5 shows the emission intensity around 480 nm

(1G4–3H6 transition) and 450 nm (1D2–

3F4 transition)

under another dual-wavelength excitation shown in

scheme 2 of Fig. 1. Tm3þ ions were excited to the 3F3 level

by the first pulse laser, k1, at 670 nm. The emission from

the 1D2 level to the 3F4 level by the second pulse laser

increased with increasing the population of the 3H4 level.

Through the relaxation from the 3H4 level to the 3F4

level, the emission from 1G4 level increased. The observed

emission intensity from the 1D2 level increased and

reached a maximum at 10�5 s of delay between the firstand second laser pulse. Then the intensity decreased. The

increase seems to be due to the relaxation from 3F3 level

to 3H4 level and the decrease represented the lifetime of

the 3H4 level. Since we assumed that the equilibrium

between 3F3 and 3H4 levels was achieved immediately,

the increase could not be reflected in the calculation.

Also, the decrease of the emission intensity from the 1D2

level in the simulation shifted to the longer delay time,and this coincides with the fact that calculated decay rate

of the 3H4 level was smaller than that of the observed

ones. The emission intensity ratio between the 1D2 and1G4 levels was also dependent on the ESA spectra shapes.

The calculated ESA spectra related to the second exci-

tation are shown in Fig. 6. The ESA cross-section of the

transition from the 3H4 level to 1D2 level is greatly de-

creased with decreasing wavelength beyond 650 nm. Onthe contrary, the ESA cross-section of the transition from3F4 level to 1G4 level was almost flat in the wavelength

from 645 to 630 nm. Therefore, it was expected from the

calculation that the proportion excited to the 1G4 level by

the second excitation increases with shorter second

excitation wavelength, leading to an increased ratio of the

Fig. 6. The calculated ESA cross-section for 3H4–1D2 (––) and 3F4–

1G4 (– – –) transitions.


emission intensity from the 1G4 level. The observed

intensity ratio of the second excitation at 630 nm was

closely predicted by the calculation with the second

excitation at 632 nm. By using the dual-wavelengthexcitation process, we can obtain the optical parameters

difficult to measure with conventional methods. In

addition, the theoretical model is capable of providing

good estimates of the optical parameters.

4. Discussion

4.1. The coefficient of the energy transfer between Tm3þ

ions

As the energy transfer processes between the energy

levels of rare earth ions are very complicated, it is dif-

ficult to evaluate each energy transfer rate experimen-

tally even if we can utilize a dual-wavelength excitation

process. The energy transfer coefficients were estimatedfrom the absorption and emission spectra, the transition

rate and the distance between Tm3þ ions, according to

the method proposed by Kushida [22]. The energy

transfer process and the transfer coefficients, WET, for

each level at 1.0 mol% of the Tm3þ ion are shown in Fig.

7. For simplicity, energy transfer process, whose coeffi-

cient is over 0.1 s�1, is shown. We cannot find the energy

transfer process of the 3H4 level related to the groundstate, such as the (3H4,

3H6)fi (3F4,3F4) process. The

concentration dependence of the 3H4 level decay rate

cannot be explained by the energy transfer process.

However, we can find very high cross-relaxation coeffi-

cient of the (3F3,3H6)fi (3H5,

3F4) process in Fig. 7(d).

Due to the thermal equilibrium relationship between the

populations of the 3F3 and 3H4 levels, the decrease in

population of the 3F3 level by the cross-relaxation re-sults in the decrease in population of the 3H4 level. It is

found that the concentration dependence on the decay

rate of the 3H4 level is dependent on not only the cross-relaxation coefficient, but also the thermal redistribution

ratio. The evaluation of the thermal redistribution will

be a future task.

We can classify the cross-relaxation according to the

combination of two initial levels of the energy transfer

process. For example, we consider the combination of3F4 and 3H5 levels. We can find the energy transfer

processes of (3F4,3H5)fi (3H6,

3F3) and (3F4,3H5)fi

(3H6,3H4) in Fig. 7(a) and (3F4,

3H5)fi (3H5,3F4),

(3F4,3H5)fi (3F3,

3H6), (3F4,3H5)fi (3H4,

3H5) and

(3F4,3H5)fi (3F2,

3H6) in Fig. 7(b). The rate equation of

the normalized population density for these processes

can be written as follows:

dn3F2

dt¼ n3F4

n3H5

nðWETðð3F4;

3H5Þ ! ð3F2;3H6ÞÞÞ;

dn3F3

dt¼ n3F4

n3H5

nðWETðð3F4;

3H5Þ ! ð3H6;3F3ÞÞ

þ WETðð3F4;3H5Þ ! ð3F3;

3H6ÞÞÞ;dn3H4

dt¼ n3F4

n3H5

nðWETðð3F4;

3H5Þ ! ð3H6;3H4ÞÞ

þ WETðð3F4;3H5Þ ! ð3H5;

3H4ÞÞþ WETðð3F4;

3H5Þ ! ð3H4;3H6ÞÞÞ;

dn3H5

dt¼ � n3F4

n3H5

nðWETðð3F4;









3H6ÞÞ

þ n3F4n3H5

nWETðð3F4;


dn3F4

dt¼ � n3F4

n3H5

nðWETðð3F4;









3H6ÞÞ

þ n3F4n3H5

nWETðð3F4;


dn3H6

dt¼ n3F4

n3H5

nðWETðð3F4;





þ WETð3F3;3H5Þ ! ð3F3;




3H6ÞÞÞ:ð7Þ

Fig. 7. The calculated energy transfer coefficients, WET, and their processes at the Tm3þ concentration of 1.0 mol%: (a) 3F4, (b)3H5, (c)

3H4, (d)3F3,

(e) 3F2, (f)1G4 and (g) 1D2 levels.


Fig. 7 (continued)



These equations can be summarized as follows:

dn3F2

dt¼ n3F4

n3H5

n

XS;L;J

½WETðð3F4;3H5Þ ! ð3F2; SJLÞÞ

þ WETðð3F4;3H5Þ ! ðSJL; 3F2ÞÞ�;

dn3F3

dt¼ n3F4

n3H5

n

XS;L;J

½WETðð3F4;3H5Þ ! ð3F3; SJLÞÞ

þ WETðð3F4;3H5Þ ! ðSLJ ; 3F3ÞÞ�;

dn3H4

dt¼ n3F4

n3H5

n

XS;L;J

½WETðð3F4;3H5Þ ! ð3H4; SLJÞÞ

þ WETðð3F4;3H5Þ ! ðSLJ ; 3H4ÞÞ�;

dn3H5

dt¼ � n3F4

n3H5

n

XS;L;J ;S0 ;L0 ;J 0

WETðð3F4;3H5Þ ! ðSLJ ; S0L0J 0ÞÞ;

dn3F4

dt¼ � n3F4

n3H5

n

XS;L;J ;S0 ;L0 ;J 0

WETðð3F4;3H5Þ ! ðSLJ ; S0L0J 0ÞÞ;

dn3H6

dt¼ n3F4

n3H5

n

XS;L;J

½WETðð3F4;3H5Þ ! ð3H6; SJLÞÞ

þ WETðð3F4;3H5Þ ! ðSJL; 3H6ÞÞ�;

ð8ÞWe can see that the change of the population by the

energy transfer is dependent on the population of two

initial levels and the sum of the energy transfer coeffi-

Fig. 8. The sum of the energy transfer coefficients for the combination of tw

(1) 3H6, (2)3F4, (3)

3H5, (4)3H4, (7)

1G4 and (8) 1D2 levels.

cients. The sum of the coefficients for the combination oftwo initial levels at 1.0 mol% is shown in Fig. 8. The 3F3

and 3F2 levels were included into the 3H4 level by con-

sidering the redistribution coefficients, since the popu-

lations of the 3F3 and3F2 levels are connected with that

of the 3H4 level. Under direct excitation to the 3H4 level,

the populations of other levels are much smaller than

those of the 3H4 and 3H6 levels. Therefore, we may

consider only the energy transfer coefficients of(3H6,

3H4) in Fig. 8. The populations of the 3F4 and3H5

levels increase by the energy transfer from the 3H4 and3H6 levels. When the population of the 3H4 level in-

creases under high power excitation condition, we have

to consider the energy transfer coefficients of (3H4,3H4)

in Fig. 8 together with (3H6,3H4). In this case, the pop-

ulations of the 1D2 and 1G4 levels also increase. As

under a dual-wavelength excitation the populations ofthe excited levels increase, the effect of the energy

transfer, whose initial levels were excited, is enhanced. It

will be possible to measure the change of population

according to the combination of some energy transfers

by controlling population of each level. The combina-

tion of the dual-wavelength excitation using pulse laser

and/or CW laser is useful for controlling population of

each level.

o initial levels. The numerals of the on the abscissa indicate the levels:


4.2. Decay process under CW excitation

It can be expected that the decay curve after CW

excitation differs from that under the direct excitation by

pulse laser. Oomen [28] has reported two-exponential

decay curve for higher Tm3þ concentrations in ZBLAN

glass after CW excitation. In the relaxation of 3H4 and1G4 levels after the CW excitation of 647 nm, the decay

curve becomes two-exponential for the Tm3þ concen-trations of 0.5 mol% and higher. The decay curve of 1D2

level is two-exponential for all Tm3þ concentrations and

a third decay process can be discerned for 1 mol%. After

the 363 nm excitation, the decay curve of the 1D2 level

is two-exponential for the Tm3þ concentrations of

0.5 mol% and higher. These decay rates are plotted in

Fig. 3.

Fig. 9 shows the calculated decay curves of the 1G4

and 1D2 levels after the CW excitation of 647 nm to-

gether with the calculated decay curve under the direct

excitation by the pulse laser. Within the calculation, the

decay curve of the 3H4 level was single-exponential. The

calculated decay curve of the 1G4 level became two-

exponential of 0.5 mol% and higher. The calculated

Fig. 9. The calculated decay curve for 1G4 (a) and 1D2 (b) levels after

the CW excitation at 647 nm. The thin lines represent the decay line

under the direct excitation by pulse laser. Arrows indicate the bending

point between the first and second parts in the decay curve: (– � – � – �)0.1 mol%, (– – –) 0.5 mol% and (––) 1.0 mol%.

decay curve of the 1D2 level was more than two-expo-nential curve for all Tm3þ concentrations and a third

component could be distinguished at more than 0.5

mol%. As can be seen from the figure, the slope of the

first part of the decay curve was similar to that of

the decay curve under the pulse excitation. The slopes of

the second and third parts were smaller than that of the

first part. Fig. 10 shows the population change of the

energy level after the CW excitation of 647 nm forthe Tm3þ concentration of 1 mol%. As can be seen from

the figure, the decay curve of 3H5 level was also two-

exponential. The slope of the decay curve of the 3H5

level was similar to that of 3H4 level even though the

decay rate of 3H5 level under direct excitation was more

than 10 times faster than that of 3H4 level. As the

population of the 3H4 level was high, the 3H5 level was

populated by the energy transfer of (3H4,3H6) process in

Fig. 8. Therefore, the population change of the 3H5 level

can be described approximately as follows:

dn3H5

dt¼ �k3H5

n3H5þ n3H4

n3H6

nWETðð3H4;

3H6Þ

! ð3H5; SLJÞÞ; ð9Þ

where k3H5is the decay rate under the pulse excitation.

When the population of the 3H5 level decreases, the ef-

fect of the second term becomes dominant. If the change

of the population of the 3H6 level is negligible, the decay

rate is expected to be similar to the decay rate of the 3H4

level. In this condition, Eq. (9) can be written as

dn3H5

dt¼ �k3H5

n3H5þ n3H4

n3H6

nWETðð3H4;

3H6Þ

! ð3H5; SLJÞÞ ¼ �k3H4n3H5

: ð10Þ

Therefore, the ratio between the populations of the 3H4

and 3H5 levels is given as follows:

n3H5

n3H4

¼ n3H6

nWETðð3H4;

3H6Þ ! ð3H5; SLJÞÞk3H5

� k3H4

; ð11Þ

Fig. 10. The normalized population densities after the CW excitation

at 647 nm at the Tm3þ concentration of 1.0 mol%.

Fig. 11. (a) The calculated decay curve for 1D2, (b) level after the CW

excitation at 363 nm. The thin lines represent the decay line under the

direct excitation by pulse laser. Arrows indicate the bending point

between the first and second parts in the decay curve: (– � – � – �) 0.1

mol%, (– – –) 0.5 mol% and (––) 1.0 mol%.


where the value of WET is 104 s�1, k3H5� k3H4

is 8.7 · 104s�1. Therefore, the value of the ratio, N3H5

=N3H4, is ex-

pected to be about 1 · 10�3. This ratio is confirmed in

Fig. 10. It is found that the 1G4 level is populated by the

energy transfer of the (3H4,3H4) and (3H5,

3H4) processes

in Fig. 8. The effect of the (3H5,3H4) process is about

seven times smaller than that of (3H4,3H4) process, be-

cause the population of the 3H5 level is three order

smaller than that of the 3H4 level. Therefore, the pop-ulation change of the 1G4 level can be described

approximately as follows:

dn1G4

dt¼ �k1G4

n1G4þ n3H4

n3H4

nWETðð3H4;

3H4Þ

! ð1G4; SLJÞÞ; ð12Þ

where k1G4is the decay rate under the pulse excitation.

When the population of the 1G4 level is high, the pop-

ulation of the 1G4 level decays according to the first term

of Eq. (12). When population of the 1G4 level decreases,

the effect of the second term becomes dominant. In this

case the decay rate is expected to be twice of the decay

rate of the 3H4 level. The slope of the second part of the

decay curve of the 1G4 level was 1.0 · 103 s�1 and the

decay rates of the 3H4 level was 6.9 · 102 s�1. The valueof the slope of the decay curve of the 1G4 level is a little

smaller than the expected value. It is found that the 1D2

level is populated by the energy transfer of the (1G4,1G4)

and (3H4,1G4) processes in Fig. 8. The transfer coeffi-

cient of the (3H4,1G4) process is about eight times larger

than that of (1G4,1G4) process and the population of the

3H4 level is higher than that of 1G4 level. Therefore, the

decay curve of the 1D2 level can be described as follows:

dn1D2

dt¼ �k1D2

n1D2þ n3H4

n1G4

nWETðð3H4;

1G4Þ

! ð1D4; SLJÞÞ; ð13Þwhere k1D2

is the decay rate of the 1D2 level under the

pulse excitation. When the population of the 1D2 level is

high, the population of the 1D2 level decays according tothe first term of Eq. (13). When the population of the1D2 level decreases, the effect of the second term be-

comes dominant. In this case, the slope is the sum of the

decay rate of the 1G4 and3H4 levels, that is, 6.6 · 103 s�1.

When the slope of the decay curve of the 1G4 level

changes to the second part, the slope of the 1D2 level

also changes. In this case, the slope is expected to be

1.7 · 103 s�1. The slopes of the second and third parts ofthe decay curve of the 1D2 level were 4.7 · 103 and

1.4 · 103 s�1, respectively.

Fig. 11 shows the calculated decay curves of the 1D2

levels after the CW excitation of 363 nm together with

the calculated decay curve under the direct excitation by

the pulse laser. The calculated decay curves of the 1D2

level were two-exponential for all Tm3þ concentrations.

The change to the second part shifted to lower popula-tion than that of the 647 nm excitation. The second

slope was similar to that of the 647 nm excitation, except

for 1.0 mol%. As the population of the 1D2 level is

higher under the direct excitation at 363 nm, the effect of

the energy transfer of the (3H4,1G4) process becomes less

obvious.

The slope of the second part in the decay rate of the1G4 level reported by Oomen [28] was about twice that

of the 3H4 level. It agrees with our expectation. The

slope of the second part in the decay rate of the 1D2 level

reported by Oomen was more than twice that of 1G4

level. If the population of the 1G4 level is higher than

that of our calculation, the slope increases by the

(1G4,1G4) process. It is still difficult to confirm that the

calculated phenomena are identical with those observedby Oomen [28]. However, we could show that the effect

of energy transfer (except for the ground state) in the

decay curve after the CW excitation. Hence, our model

could serve as a starting point although it is still difficult

to theoretically predict all phenomena.

5. Conclusion

The optical spectra and radiative transition rates of

Tm3þ ions in ZBLAN glass were obtained from the

structural models prepared by molecular dynamic sim-

ulation and the crystal field theory. The energy transfer

rate between Tm3þ ions was estimated from the optical

spectra, transition rates and a method proposed by

Kushida. It was found that the energy transfer coeffi-cients could be summarized according to the popula-

tions of two initial levels for the energy transfer

processes. The numerical model for the optical transi-

tions under various excitation conditions was con-

structed using the optical parameters obtained from the

calculation and assumed relation between Tm–Tm dis-


tances and Tm3þ concentration. The lifetimes of 3H4,1G4 and

1D2 levels were evaluated from calculation and

experiment under direct pumping. The lifetime of the3H5 level could be experimentally evaluated by using the

dual-wavelength excitation process and it could be

substantially reproduced by our numerical model. It was

found that the relative intensities and wavelength

dependence between the ESA spectra for 1G4–3H6 and

1D2–3F4 transitions could be experimentally evaluated

by using the dual-wavelength excitation process. It was

also found that the decay processes after CW excitations

could be evaluated by our numerical model. The decay

processes were different from that under pulse excita-

tion. By using various excitation processes, it will be

possible to obtain various optical parameters difficult to

measure with conventional methods. The proposed

numerical model serves as a good starting point toanalyze such complicated excitation and decay pro-

cesses.

References

[1] S. Sudo, Optical Fiber Amplifiers: Materials, Devices, and

Applications, Artech House, Boston, MA, 1997, and references

cited therein.

[2] J.Y. Allain, M. Monerie, H. Poignant, Electron. Lett. 26 (1990)

166.

[3] S.G. Grubb, K.W. Bennett, R.S. Cannon, W.F. Humer, Electron.

Lett. 28 (1992) 1243.

[4] G. Tohmon, J. Ohya, H. Sato, T. Uno, IEEE Photonic. Tech.

Lett. 7 (1995) 742.

[5] T. Komukai, T. Yamamoto, T. Sugawa, Y. Miyajima, Electron.

Lett. 29 (1993) 110.

[6] T. Kasamatsu, Y. Yano, H. Sekita, Opt. Lett. 24 (1999) 1684.

[7] T. Kasamatsu, Y. Yano, T. Ono, Electron Lett. 36 (2000) 1607.

[8] S. Aozasa, T. Sakamoto, T. Kanamori, H. Hoshino, M. Shimizu,

Electron. Lett. 36 (2000) 418.

[9] S. Aozasa, H. Masuda, H. Ono, T. Sakamoto, T. Kanamori, Y.

Ohishi, M. Shimizu, Electron. Lett. 37 (2001) 1157.

[10] S.A. Brawer, M.J. Weber, Phys. Rev. Lett. 45 (1980) 460.

[11] G. Cormier, J.A. Capobianco, Europhys. Lett. 24 (1993) 743.

[12] M.T. Harrison, R.G. Deaning, J. Lumin. 69 (1996) 265.

[13] B.R. Judd, Phys. Rev. 127 (1962) 750.

[14] G.S. Ofelt, J. Chem. Phys. 37 (1967) 511.

[15] M.J. Weber, Phys. Rev. 157 (1967) 262.

[16] K. Soga, H. Inoue, A. Makishima, S. Inoue, Phys. Chem. Glasses

36 (1995) 253.

[17] H. Inoue, K. Soga, A. Makishima, J. Non-Cryst. Solids 222 (1997)

212.

[18] K. Soga, H. Inoue, A. Makishima, J. Non-Cryst. Solids 274 (2000)

69.

[19] K. Soga, H. Inoue, A. Makishima, J. Appl. Phys. 89 (2001) 3730.


270.


17.

[22] T.J. Kushida, Phys. Soc. Jpn. 34 (1973) 1318.

[23] B.R. Judd, Operator Techniques in Atomic Spectroscopy, Prince-

ton University, NJ, 1998.

[24] H. Inoue, A. Makishima, J. Non-Cryst. Solids 161 (1993) 118.

[25] H. Inoue, K. Soga, A. Makishima, I. Yasui, Phys. Chem. Glasses

36 (1995) 1.

[26] M.D. Shinn, W.A. Sibley, M.G. Drexhage, R.N. Brown, Phys.

Rev. B 27 (1983) 6635.

[27] M. Tsuda, K. Soga, K.H. Inoue, S. Inoue, A. Makishima, J. Appl.

Phys. 85 (1999) 29.

[28] E.W.J.L. Oomen, J. Lumin 50 (1992) 317.

Documents

Simulation of the optical properties of Tm:ZBLAN glass. II. Energy transfer between Tm3+ ions under single- and dual-wavelength excitation