Simulation of the optical properties of Tm:ZBLAN glass

Simulation of the optical properties of Tm: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 12 June 2001; received in revised form 17 January 2002

Abstract

The optical properties of Tm:ZBLAN glass have been simulated. The simulated optical spectra agreed substantially

with the observed spectra. The population of the energy levels of the Tm3þ ion under dual-wavelength excitation for

upconversion emission and amplification was evaluated by solving the rate equation with the simulated transition

probabilities and spectral shapes of the transitions of which the initial levels were from the 3H6 to3P2 levels. The large

population of the 1G4 level, which is the upper level for 480 nm laser oscillation, was estimated for the conditions of

laser oscillation at room temperature. We also proposed new combinations of the excitation wavelengths for the

amplification of the 1470 nm signal. � 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Many research efforts have been made towarddevelopment of rare earth doped fiber amplifiersand lasers [1]. Currently, Er3þ, Tm3þ and Pr3þ-doped fiber amplifiers provide gain bands from1530–1610, 1450–1510 and 1290–1320 nm, wherethe loss levels are low in silica fiber. In research onthe amplifier of Tm3þ [2–6] and Pr3þ [7–9] -dopedfiber, fluoride glasses, such as ZBLAN (ZrF4–BaF2–LaF3–AlF3–NaF) and PIGZYL (PbF2–

InF3–GaF3–ZnF2–LaF3–YF3), are mainly uti-lized.

The Tm3þ-doped fluoride fiber amplifier is op-erated as a four-level system. The lifetime ofthe upper, 3H4, level of the stimulated emission isshorter than that of the lower, 3F4, level. There-fore, the formation of a population inversion isdifficult by direct pumping. In order to solve theproblem, several upconversion pumping methodshave been proposed [2–6]. A signal gain of 25 dBwas first demonstrated at 1470 nm when pumpedat 450 mW by a 1064 nm laser [2]. Recently, gainsexceeding 30 dB have been achieved by a 1470 nmpumping [4], and the range of the gain has beenshifted to 1475–1510 nm by use of a dual-wave-length (1050 and 1560 nm [5] or 1400 and 1560 nm[6]) pumping method. A schematic energy diagram

Journal of Non-Crystalline Solids 306 (2002) 17–29

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 (02 )01059-1

of the Tm3þ ion is shown in Fig. 1. Another so-lution is the addition of other rare earth ions. Theaddition of the Ho3þ or Tb3þ ion with the Tm3þ

ion is effective in depopulating the 3F4 level [10,11].The advantage of this method is that a high-powercommercially available 800 nm band LD can beused as a pump source.

The Tm3þ-doped fluoride fiber is also an at-tractive medium for an upconversion laser of theblue wavelength region [12–14]. The laser oscilla-tion of the upconversion at 455 and 480 nm wasfirst demonstrated at 77 K [12]. The CW laseroscillation at room temperature at 480 nm hasbeen reported [13,14]. Under a dual-excitation(1100 and 680 nm), the threshold decreased sig-nificantly and the efficiency increased [14].

The excitation of the upconversion process isdue to excited state absorption (ESA) and/or en-ergy transfer between rare earth ions. The directmeasurement of ESA spectra and the quantitativeevaluation of the energy transfer are still difficult.Therefore, it is very difficult to optimize manyconditions such as the Tm3þ ion concentration, thecombination of excitation wavelengths and theirintensities and the combination with other rareearth ions. The probabilities of magnetic andelectric dipole transitions can be readily calculated[15–17]. The calculation of the splitting of the en-

ergy levels of rare earth ions in glasses have beenproposed [18] and developed [19,20]. Recently, wehave reported that fluorescence spectra of the rareearth ion in the fluoride glasses can be reproducedby calculation [21–24]. We consider that optimi-zation and design of the characteristics describedabove can be realized on the basis of the calculatedspectra. In this paper, we simulate the opticalspectra of the Tm3þ ion in ZBLAN glass. Next, wedescribe the population of the energy levels of theTm3þ ion under several excitation conditions. Fi-nally, we propose new dual-wavelength excitationconditions for the upconversion emission and 1470nm amplification.

2. Experimental conditions

2.1. Glass preparation

Glass with a composition of 52ZrF4 � 20BaF2 �3:5LaF3 � 3AlF3 � 20NaF � 0:5InF3 � 1TmF3 was pre-pared. The InF3 was added to prevent the forma-tion of a black species during melting. Thepowders were mixed and melted in a gold crucibleat 900 �C for 15 min. The melt was cast into apreheated aluminum mold. All of these processeswere conducted in a glove box under an atmo-sphere of dry nitrogen gas. The glass obtained wascut into a 10� 25� 5 mm shape and polished.

Absorption spectrum was measured with a self-recording spectrophotometer (U3410, Hitachi) inthe wavelength range of 200–2600 nm. Fluores-cence spectrum was measured by using the THGline of a pulsed Nd:YAG laser and 471 nm exci-tation of an OPO laser (Mirage 500, Hoya Con-tinuum Inc.). The emission from the sample wasfocused on the entrance slit of a spectrometer(1000M, Spex Industries Inc.) and detected with aphotomultiplier tube (R1477, Hamamatsu Corp.).All of the measurements were carried out at roomtemperature.

2.2. Molecular dynamics simulation of the Tm3þ-doped ZBLAN glass

The structure of the Tm3þ-doped ZBLAN glasswas simulated using the molecular dynamics (MD)

Fig. 1. A schematic energy diagram of the Tm3þ ion.

18 H. Inoue et al. / Journal of Non-Crystalline Solids 306 (2002) 17–29

method. Three hundred ninety three ions (Zr4þ:53,Ba2þ:20, La3þ:3, Al3þ:3, Naþ:20, Tm3þ:1, F�:293)were placed randomly in a cubic cell with periodicboundary conditions. The cell parameter of 1.759nm was determined from the experimental densityof the glass. Simulations were carried out at aconstant volume. The potentials of the Born–Mayer type were used with formal ionic charges,and the parameters used are listed in Table 1. TheCoulomb force was evaluated by the Ewald sum-mation. To obtain the variation of the Tm3þ sitesin the glass structure, MD simulation was per-formed for 300 different sets of random initialcoordinates. The temperature of the simulationwas lowered from 3000 to 300 K with a time stepof 1 fs for 10 000 time steps (10 ps). After 5000time steps (5 ps) at 300 K, the coordinates of thelast step were used for further calculation. Com-putations were made with Hitac SR2201 computerin the Information Technology Center, the Uni-versity 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 [25–27]. The Hamiltoniandescribing the electrostatic field at the Tm3þ ioncan be written as follows:

HCF ¼Xkq

A�kq

Xi

rki CkqðriÞ; 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 glass,and Rj is its distance form the Tm3þ ion. Onlyoperators of rank 2, 4 and 6 are needed to deter-mine relative energies within the f 12 manifold.Once the ion positions and charges are known, theonly undetermined variables are the values of thepowers of the electron radii rki . The values wereobtained by use of a DV-Xa calculation [28] for atrivalent Tm ion to which the 5d and 5g orbitswere added. It is known that the modification ofthe values is necessary for the reproduction of theobservation [29].

rk� �

¼ ak rk� �

DV-Xa; ð3Þ

where 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 eachTm3þ ion. The values of the energy-level parame-ters for SLJ terms determined by Carnall et al. forthe Tm3þ ion of TmðC2H5SO4Þ � 9H2O were used[30]. The eigenstates and eigenvalues were ob-tained from the diagonalization of the crystal-fieldHamiltonian. The resulting 91 eigenstates were alinear combination of basis states SLJM of theform

jwi ¼XSLJM

aðSLJMÞ 4f 12aSLJM��

; 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–Ofettheory [15,16,27] can be written as

A SLJM ; S0L0J 0M 0� �¼ e2

4pe0

64p4�mm3

3hc3n n2 þ 2ð Þ2

9

� SLJM PEDj jS0L0J 0M 0� �� 2

Table 1

The potential and the parameters used in MD simulation

Na Ba Tm La Al Zr F

q ¼ 0:03 nm

Z +1 +2 +3 +3 +3 +4 -1

Aij (10�16 J)

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

Tm 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

�:

H. Inoue et al. / Journal of Non-Crystalline Solids 306 (2002) 17–29 19

SLJM PEDj jS0L0J 0M 0� �¼

XaSLJMa0S0L0J 0M 0

a aSLJMð Þa0 a0S0L0J 0M 0� �� 4f naSLJM PEDj j4f na0S0L0J 0M 0� �

4f naSLJM PEDj j4f na0S 0L0J 0M 0� �¼Xkqk

ð � 1ÞqþqþJ�MþS0þL0þJþk J ; J 0� �1=2k½

�1 k k

q �ðq þ qÞ q

� �J k J 0

�M qþ q M 0

!

�J J 0 k

L0 L S

( )AkqN k; kð Þ 4f naSL U k

�� 4f na0S0L0� �

N k; kð Þ ¼ 2Xn0l0

l; l0� �

ð � 1Þlþl0 1 k k

l l0 l

( )

�l 1 l0

0 0 0

!l0 k l

0 0 0

!

�4f jrjn0l0� �

n0l0 rkj j4f� �

Dðn0l0Þ ; ð5Þ

where �mm is a frequency of the transition, n is a re-fractive index, c is the velocity of light, q is thepolarization of the transition, and the doubly re-duced matrix elements of the spherical tensor op-erator U k have been tabulated by Nielson andKoster [31]. For a determination of the Nðk; kÞcoefficients, it is necessary to know the value of theradial integrals h4f jrjn0l0i and h4f jrkjn0l0i, and theenergy for n0l0 configuration, Dðn0l0Þ. The n0l0

configurations were assumed to be 5d and 5gconfigurations. The radial integrals and energy ofthe 5d and 5g states of the free-ion state of theTm3þ ion were estimated from the DV-Xa calcu-lation. A phenomenological parameter, bðk; kÞ,was introduced, because the correction was alsonecessary for the value of Nðk; kÞ.N k; kð Þ ¼ b k; kð ÞNDV-Xa k; kð Þ: ð6ÞThe magnetic dipole transition rate was obtainedby use of the method in the literature [17].

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. There

are 2J þ 1 Stark levels in the initial SLJ state. Thedensity of the ions of the Stark level, nSLJM , in theSLJ state was assumed on the basis of the Max-well–Boltzmann’s 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 state wasset as unity. The spontaneous emission rateAðSLJ ; S0L0J 0Þ can be written as

A SLJ ; S0L0J 0� �¼XM

nSLJMXM 0

A SLJM ; S0L0J 0M 0� �:

ð8ÞThe relationship 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

A SLJM ; S0L0J 0M 0� �¼ 8p�mm2n2

c2

ZrSLJM�S0L0J 0M 0dm;

ð9Þwhere m is a frequency. The Gaussian shape func-tion, f ðkÞ, where k is the wave number, with 50cm�1 of HMFW was assumed for the generation ofthe spectral shapes. 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�S0J 0L0M 0 ðkÞ: ð11Þ

The averages of the spontaneous emission ratesand the cross section of absorption and emissionof 300 spectra obtained from 300 structural modelswere used in the following section.

2.4. Populations of 4f energy levels under dual-excitations

The transition between energy levels is mainlydescribed by five factors, stimulated absorption,


stimulated emission, spontaneous emission, mul-tiphonon relaxation and energy transfer betweenrare earth ions. Stimulated absorption and emis-sion can be estimated from the cross section ofabsorption and emission, respectively. The calcu-lated cross section of absorption and emission andspontaneous emission rate were used for the esti-mation of the three former factors. The multiph-onon relaxation rate has been empirically obtainedas a function of the energy gap to the next lowerlevel and the effective phonon frequency of hostmaterials [32]. The multiphonon relaxation rate,WMP was estimated from the empirical expression

WMP ¼ C expð�aDEÞ; ð12Þwhere DE is the energy gap between the lowestStark component of the level and the highestcomponent of the next lower level. The values usedfor C and a were 1:88� 1010 s�1 and 5:77� 10�3

cm, respectively [32]. The multiphonon relaxationwas only effective to the 3H5 level in the levels ofthe Tm3þ ion. The energy gaps to the next lowerlevel of 3F3,

3F2,3P0,

3P1 and 3P2 levels were sosmall that it was thought that thermal equilibriumhad to be considered on the basis of the Maxwell–Boltzmann’s distribution for the degenerated state.The mutiphonon relaxation rates of other levelswere negligibly small in comparison with thespontaneous emission rate of the levels. Here, theenergy transfer was disregarded by the assumptionof the low concentration of the Tm3þ ion.

The density of the Tm3þ ions in each SLJ stateunder dual-excitation of photons of wave numberk1 and k2 was linked to the factors by the equation

dnSLJdt

¼ �nSLJX

SLJ<S0J 0L0rSLJ�S0L0J 0 ðk1Þqðk1Þ

� nSLJX

SLJ<S0J 0L0rSLJ�S0L0J 0 ðk2Þqðk2Þ

þ nS0L0J 0X

SLJ>S0J 0L0rS0L0J 0�SLJ ðk1Þqðk1Þ

þ nS0L0J 0X

SLJ>S0J 0L0rS0L0J 0�SLJ ðk2Þqðk2Þ

� nSLJX

SLJ>S0J 0L0rSLJ�S0L0J 0 ðk1Þqðk1Þ

� nSLJX

SLJ>S0J 0L0rSLJ�S0L0J 0 ðk2Þqðk2Þ

þ nS0L0J 0X

SLJ<S0J 0L0rS0L0J 0�SLJ ðk1Þqðk1Þ

þ nS0L0J 0X

SLJ<S0J 0L0rS0L0J 0�SLJ ðk2Þqðk2Þ

� nSLJX

SLJ>S0J 0L0A SLJ ; S0L0J 0� �

þXS0L0J 0

nS0L0J 0A S0L0J 0; SLJ� �

� nSLJWMP SLJð Þþ nSLJþ1WMPðSLJ þ 1Þ; ð13Þ

where qðkiÞ is the photon flux of wave number ki.The first four and the next four terms of Eq. (13)represent the density changes by stimulated ab-sorption and emission, respectively. The next twoterms represent the density change by spontaneousemission. The last two terms represent the mul-tiphonon relaxation, where the SLJ þ 1 level is thenext upper level of the SLJ one. The populationsof the levels from 3H6 to 3P2 under dual-wave-length excitation in a steady state were estimatedfrom the rate equation.

3. Results

3.1. Molecular dynamics simulation

The pair distribution function for the Tm–Fpair in the simulated ZBLAN glass at 300 K isshown in Fig. 2 together with the accumulatedcoordination number. The peak of the Tm–F pairwas at 0.229 nm with 0.023 nm of HMFW. Thevalley of the peak was found around 0.31 nm.Thus, we determined the F� ions within this dis-tance as the first coordination polyhedron. Thecoordination number of the Tm3þ ions was dis-tributed from 6 to 10. The numbers of 6, 7, 8, 9and 10 coordinated polyhedra in the 300 TmFn

polyhedra were 2, 62, 162, 71 and 3, respectively.The average coordination number was 8.04. Fig. 3shows three typical examples of simulated poly-hedra. Similar geometries in fluoride crystals couldbe found. The site in Fig. 3(a), of which the co-ordination number was 7, was seen as a pen-tagonal bipyramid geometry in the b-KYb2F7

crystal [33]. The site in Fig. 3(b), of which thecoordination number was 8, was seen as a square


anti-prism in the RbEu3F10 crystal [34]. The site inFig. 3(c), of which the coordination number was 9,was seen as a trigonal prism with the rectangularfaces capped in the EuF3 crystal [35].

3.2. Optical spectra

The observed absorption cross section is shownin Fig. 4(a). The absorption bands can be ascribed

to the transitions from the ground state, 3H6, tothe upper levels of the Tm3þ ion. The oscillatorstrengths and Judd–Ofelt intensity parameters arelisted in Table 2. The values were comparable to

Fig. 4. The observed (a) and calculated (b) absorption cross

section of Tm3þ-doped ZBLAN glass.

Fig. 3. The first coordination polyhedra in the structural

models. The F coordination numbers were: (a) 7, (b) 8 and (c) 9.

Fig. 2. The pair distribution curve of the Tm–F pair and ac-

cumulated coordination number in the structural models.

Table 2

The Judd–Ofelt intensity parameters and the observed and the

calculated oscillator strength of the Tm3þ ion in ZBLAN glass

Judd–Ofelt parameters (10�24 m2)

X2 X4 X6

2.12 1.50 1.02

Oscillator strength (10�7)

Measured Calculated

3F4 14.8 15.53H5 12.0 11.2

4.2a

3H4 19.6 18.13F2 þ 3F2 26.3 25.91G4 6.9 5.511D2 18.4 18.0

aMagnetic dipole transition.


those reported by McDougall et al. [36]. The val-ues of refractive index n at wavelength k is calcu-lated using the relation n ¼ Aþ B=k2 takingA ¼ 1:50 and B ¼ 3500 nm2 for ZBLAN glass [37].The value of the rms deviation, rrms, between theobserved and the calculated oscillator strengthsfrom the Judd–Ofelt intensity parameters as de-fined in Ref. [38] was 2:376� 10�7.

The parameters ak and bðk; kÞ were determinedby the comparison of the observed and calculatedabsorption spectra. The determined values are lis-ted in Table 3 together with the values obtainedform the DV-Xa method. The calculated absorp-tion cross section is shown in Fig. 4(b), and thecalculated values of oscillator strength are listed inTable 2. The value of the rrms between the observed

Table 3

The parameters of hrki and Nðk; kÞ for the Tm3þ ion calculated

by use of a DV-Xa method and the determined values of the

modification parameters of ak and bðk; kÞk hrki ak

2 1:846� 10�21 m2 0.933

4 8:637� 10�42 m4 3.487

6 8:318� 10�62 m6 7.057

k, k Nðk; kÞ bðk; kÞ1, 2 �2:759� 10�4 m2 J�1 2.891

3, 2 1:253� 10�24 m4 J�1 7.410

3, 4 1:458� 10�24 m4 J�1 7.059

5, 4 �7:122� 10�45 m6 J�1 13.867

5, 6 �2:015� 10�44 m6 J�1 6.611

7, 6 7:444� 10�65 m8 J�1 12.411

n0l0 Dðn0l0Þ (cm�1)

5d 1:29� 105

5g 4:36� 105

Fig. 5. The observed (thin line) and calculated (thick line) ab-

sorption cross section of Tm3þ-doped ZBLAN glass: (a) 3H6–3F4, (b)

3H6–3H5, (c)

3H6–3H4, (d)

3H6–3F3 &

3F2, (e)3H6–

1G4

and (f) 3H6–1D2 transitions.

Fig. 6. (a) The observed emission spectra under 355 nm exci-

tation. (b) The calculated emission spectra of 1D2–3F4,

1D2–3H5

and 1G4–3H6 transitions. The intensity of the 1D2–

3H5 transi-

tion is depicted at 20 times than that of the 1D2–3F4 transition.


oscillator strengths and the calculated strengthsusing parameters in Table 3 was 2:574� 10�7. Eachabsorption band is shown in Fig. 5. As can be seenfrom the value of the rrms and the figures, the po-sition, width and height of the absorption bandwere reproduced substantially. The observed emis-sion spectrum for an excitation wavelength at355 nm is shown in Fig. 6 together with the cal-culated emission spectra from the 1D2 and 1G4

levels. The intensity ratio of emissions from the 1D2

and 1G4 levels could not be evaluated in this cal-culation. However, the observed shape of thespectra could be decomposed by the calculatedspectra. The position and shape of the calculatedemission spectrumof the 1G4–

3F4 transition around15 500 cm�1 (650 nm) was also correspondent tothe observed emission spectrum for an excitationwavelength at 471 nm. The ESA spectra from the3F4 to

3P1 levels and emission spectra from otherlevels were obtained and used in the followingdiscussion.

4. Discussion

4.1. Values of hrki and Nðk; kÞ

The values of hrki and Nðk; kÞ are importantfactors that determine the magnitude of the split-ting and transition rate. Morrison and Leavitthave advocated the equation for the correction ofthe value of hrki given by [29]

rk� �

¼ 1� rk

skrk� �

HF; ð14Þ

where rk is a shielding factor of the crystalline fieldby the outer shells, s is a scaling parameter for theexpansion of the wave function calculated fromthe Hartree–Fock expectation when the ion is in-troduced into a solid. The parameter ak in Eq. (3)corresponds to ð1� rkÞ=sk. We evaluate the valueof s from the value in Table 3 by the assumption ofrk with 0. The values of s were 1.035, 0.732 and0.722 for hr2i, hr4i and hr6i, respectively. Nðk; kÞ isapproximately proportional to the k þ 1 powers ofr and inversely proportion to Dðn0l0Þ from Eq. (5).Dorenbos [38] has reported that the 5d states ofthe Tm3þ ions in fluorides lie about 60 000 cm�1

above the 4f configuration, whereas the value ofthe Dð5dÞ calculated from the DV-Xa method was129 000 cm�1. Considering the difference betweenthe values of Dð5dÞ, the values of s were from0.862 to 0.733. Our estimated values are similar tothose of s � 0:75 obtained for several lanthanides[39]. In their calculation, the values of r2 � 0:8,r4 � r6 � 0:1 have been assumed. A more detaileddiscussion is beyond the scope of the present workbecause of the approximation used in our calcu-lation, such as the MD simulation with two-bodypotentials and the point charge model. It is no-ticeable that not only the transition rate but alsothat of wavelength dependence of all transitionscan be obtained by this calculation.

4.2. Populations of energy levels

The populations of the 3F4,3H4,

1G4,1D2 and

1I6 levels under dual-wavelength excitation areestimated. The populations of the 3F4,

3H4 and1G4 levels are shown by the contour line in Fig. 7.The total population of the Tm3þ ion in this cal-culation was set at unity. Each power of the exci-tation of wave number k1 and k2 was assumed tobe 10 mW for the area of a 4 lm diameter. Thereare two regions with the high population of the1G4 state in Fig. 7(c). One is the region around thecombination of 8850 and 8650 cm�1 (1130 and1156 nm) which is marked with an a in Fig. 7(c).The other is the region around the combination of14 710 and 8870 cm�1 (680 and 1127 nm) which ismarked with a b in Fig. 7(c). Laser oscillationbetween the 1G4 and

3H6 levels has been reportedunder both excitation conditions at room tem-perature. On the contrary, laser oscillation only at77 K has been reported on the region marked witha c in the figure. The population of the region wasclearly lower than those of the above regions. Thisindicates that it is difficult to form or keep thepopulation inversion between the 1G4 and 3H6

states at room temperature by the condition c inthe figure. The maximum population of the 1D2

state was about 6.6% of that of the 1G4 state. Thisvalue was lower than that marked with the c inFig. 7(c). Transitional laser oscillation between the1D2 and

3F4 levels has been reported at 77 K [12].When total excitation power was 100 mW, the


maximum population of the 1G4,1D2 and 1I6

states became 3.1, 3.6 and 12.8 times higher thanthose of a 20 mW excitation, respectively. Thepopulation of the 1I6 level under a 100 mW exci-tation was 2.3 times higher than that of the 1D2

level. Therefore, the 1I6 level has a larger possi-bility as an initial level for emission at the shortwavelength, such as 344 nm (1I6–

3F4) and 451 nm

(1I6–3H4), in comparison to the 1D2 level under

high-pump power.

4.3. Evaluation of the characteristics of 1470 nmamplification

To evaluate the 1470 nm amplification, wepaid special attention to the formation of the

Fig. 7. (a) The distribution of the population of 3F4 (a),3H4 (b) and

1G4 (c) state under excitation of wave number k1 and k2. Thepositions marked with a, b and c are the combination of 8850 and 8650 cm�1, 8870 and 14710 cm�1, and 14 784 and 15 454 cm�1,

respectively.


population inversion between the 3H4 and 3F4

levels. The population of the 3F4 state (Fig. 7(a))was subtracted from that of 3H4 state (Fig. 7(b)).The residual population is shown in Fig. 8. Somepositive regions exist. These regions indicate theexcitation condition for the formation of popula-tion inversion under a 20 mW excitation. Thesignal gain of amplification was then estimated fornine conditions of the excitation wavelength whichare listed in Table 4 and depicted in Fig. 8. The

combinations from (1) to (4) are extracted fromFig. 8. Conditions (6) and (7) are the single-wavelength excitation which were previously re-ported [2–4]. Combinations (5), (8) and (9) aresimilar combinations of dual-wavelength excita-tion which were previously reported [5,6,40]. Fig. 9shows the ground state absorption cross sectionand the cross section of the ESA from the 3F4,

3H5

and 3H4 levels. The absorption at 6000, 8170,12 790 and 14 610 cm�1 (1667, 1224, 782 and 684nm) corresponds to the transitions from theground state (3H6 level) to the 3F4,

3H5,3H4 and

3F3 levels, respectively. The excitation at 9470 and7170 cm�1 (1056 and 1395 nm) corresponds to theESA from the 3F4 to

3F2 levels and of the 3F4 to3H4 levels. The Tm

3þ ion excited to the 3F3 and3F2

levels quickly relaxes to the 3H4 level. The excita-tion by 9470 and 7170 cm�1 decreases the popu-lation of the 3F4 state and increases that of the

3H4

state. In addition, all pump sources hardly overlapwith ESA spectra to the 1G4 and

1D2 levels.The signal gain was estimated under the fol-

lowing conditions: the Tm3þ ion concentrationwas 1000 ppm. The input signal power and totalpump power were �30 dBm and 100 mW. Thecore diameter of the fiber was 4 lm. We assumedthat the signal and pump were propagated to thesame direction in the core homogeneously. Thefiber length was 10 m and there was no scatteringloss. The terms of stimulated absorption andemission for the signal photon were added to

Fig. 8. The population difference between 3H4 and3F4 states,

nð3H4Þ � nð3F4Þ.

Table 4

The excitation conditions and the maximum values of the signal gain for the 1470 nm amplification

k1 k2 Maximum gain

(dB)Wave number (cm�1)

(wavelength (nm))

Power (mW) Wave number (cm�1)

(wavelength (nm))

Power (mW)

(1) 14 610 (684) 85 9470 (1056) 15 30.4

(2) 12 790 (782) 85 9470 (1056) 15 33.6

(3) 14 610 (684) 90 7170 (1395) 10 25.2

(4) 12 790 (782) 90 7170 (1395) 10 28.3

(5) 9470 (1056) 60 8170 (1224) 40 19.7

(6) 9470 (1056) 100 – – 0.0

(7) 7170 (1395) 100 – – 0.005

(8) 9470 (1056) 60 6000 (1667) 40 22.4

(9) 7170 (1395) 60 6000 (1667) 40 21.1


Eq. (13). The population of each state was thenestimated. The transmission and amplificationfactors of the signal and pumped sources wereestimated from the populations of the states ateach position of the fiber. The estimated wave-length dependence of the gain from 1420 to 1520nm is shown in Fig. 10. The maximum gains andthe combination of input power are listed in Ta-ble 4. In the condition of single-wavelength exci-tation, the gain was not obtained in spite of thesufficient gain reported. Since there is hardly anyground state absorption at 9470 and 7170 cm�1,an additional mechanism is required for theevaluation of the gain. In the conditions of (8)and (9), the gain of about 20 dB was estimated.When the pump source of wave number 7170cm�1 was used, the population of the 3F4 statewas high for the stimulated emission from the 3H4

level to the 3F4 level. The insufficient populationinversion caused a small gain at the shorterwavelength side, and the gain curve became flat inthe region of 1470–1510 nm. The equivalent orexceeding gain of the conditions of (8) and (9)were estimated for those of (1)–(5). The maximumvalues of the gains for (1) and (2) were over 30dB. When the pump sources of the wave number

14 610 or 12 790 cm�1 were used, the gains becamemaximums at 85 or 90 mW of the sources, re-spectively. This indicates that the excitation fromthe ground state to the 3H4 level is most impor-tant for the formation of the population inver-sion. It seems to be advantageous for theexcitation under low pump power to use awavelength with a large absorption cross section,such as the combination of (1) and (2). Thus, thehigh gain and the flat gain curve are worthy offurther study, though many problems remain forthe comparison with the real amplifier. The con-centration dependence in consideration of energytransfer seems to be the next problem to consider.

Fig. 10. The signal gain as a function of the signal wavelength

for the conditions: (a) from (1) to (5) and (b) (8) and (9).

Fig. 9. The calculated cross section of the ground state ab-

sorption, 3H6, and the ESA of 3F4,3H5 and

3H4 states.


5. Conclusion

The optical properties of Tm:ZBLAN glasshave been investigated. The optical spectra of theTm3þ ion in ZBLAN glass were simulated fromthe calculated energy levels of 4f electrons and thetransition rate between the levels. The energy levelsand their splitting were calculated from a crystalfield Hamiltonian on the basis of a point chargeapproximation, which was applied to the struc-tural model simulated by use of an MD technique.The transition rates were estimated based on thefull Judd–Ofelt theory. The estimated absorptionand emission spectra agreed substantially with theobserved spectra.

The population of the SLJ states under dual-wavelength excitation was evaluated by solving therate equation with the simulated optical proper-ties. The large population of the 1G4 state, which isthe initial level for 480 nm emission, was esti-mated for the conditions of laser oscillation atroom temperature. It is concluded that the opti-cal properties can be characterized on the basisof the estimated population of the SLJ states.The optical amplification of 1470 nm signal wasevaluated from the population difference betweenthe 3H4 and the 3F4 states. Finally, we proposednew conditions of dual-wavelength excitation forthe amplification of 1470 nm. The signal gainover 30 dB was expected under the 100 mWexcitation of wave number 14 610 and 9470 cm�1

(684 and 1056 nm), and 12 790 and 9470 cm�1 (782and 1056 nm). It is advantageous to examinecharacteristics in various conditions using thistechnique without experimental restrictions. Itwill be possible to examine the concentrationdependence of rare earth ions and the combina-tion of the rare earth ions systematically, if theenergy transfer between them can also be evalu-ated.

Acknowledgements

This study is financially supported by a Grant-in-Aid from the Ministry of Education with thecontract number #09450239. The authors wouldlike to thank for the supply of fluorides by Morita

Chemical Industries Co., Ltd and Central GlassCo., Ltd.

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Documents

Simulation of the optical properties of Tm:ZBLAN glass