Dynamics of Tm–Ho energy transfer and deactivation of the 3 F 4 low level of thulium in fluorozirconate glasses L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messaddeq Citation: Journal of Applied Physics 95, 5451 (2004); doi: 10.1063/1.1704845 View online: http://dx.doi.org/10.1063/1.1704845 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/95/10?ver=pdfcov Published by the AIP Publishing [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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D. da Vila, L. Gomes,a) and L. V. G. Tarelho Centro de Lasers e Aplicac¸ões, IPEN-SP, Travessa R 400, Cidade Universita´ria, CEP 05508-900, São Paulo, Brazil S. J. L. Ribeiro and Y. Messaddeq Laboratório de Materiais Fotoˆnicos, Instituto de Quı´mica de Araraquara, UNESP, Sa˜o Paulo, Brazil ~Received 18 August 2003; accepted 24 February 2004! The mechanism involved in the Tm31 (3F4)→Ho31 (5I7) energy transfer and Tm31 (3H4 , 3H6) →Tm31 (3F4 , 3F4) cross relaxation as a function of the donor and acceptor concentrations was investigated in Tm–Ho-codoped fluorozirconate glasses. The experimental transfer rates were determined for the Tm→Ho energy transfer from the best fit of the acceptor luminescence decay using an expression which takes into account the Inokuti–Hirayama model and localized donor-to-acceptor interaction solution. The original acceptor solution derived from the Inokuti– Hirayama model fits well the acceptor luminescence transient only for low-concentrated systems. The results showed that a fast excitation diffusion that occurs in a very short time (t!g22) reduces the mean distance between an excited donor (D* ) and the acceptor~A!. A localized donor-to-acceptor interaction takes place, leading to an exponential decay of donors as an average of the microscopic rate equation solution of eachD* –A pair separated by distanceR that contributes in addition to the Inokuti–Hirayama solution. The observation that the experimental transfer rates were always much bigger than the one predicted by the diffusion model, in which the energy transfer process is assisted by excitation migration among donors state, reinforces the existence of a fast excitation diffusion among donor ions before the energy transfer to acceptor already observed in Yb:Er:ZBLAN. The fast excitation diffusion effect was observed to dominate both the Tm→Tm cross relaxation and Tm→Ho energy transfer ions from3H4 and 3F4 thulium states, respectively. ©2004 American Institute of Physics.@DOI: 10.1063/1.1704845# I. INTRODUCTION Rare-earth-doped fluoride glasses have been reported as a promising glass for optical applications because of their characteristics, such as low phonon energy and good capac- ity to accept lanthanide dopants. The low phonon energies of the fluoride glasses~,600 cm21! allow the observation of rare-earth lasers emissions~about 30! ~Refs. 1–3! in a large optical range. Particularly, laser emissions near 1.5 and 2.0 mm have important technological applications in telecommu- nications and medicine, respectively. In the telecommunica- tions area, Pr31, Tm31 and Er31, with respective laser emis- sions at 1.3, 1.4, and 1.55mm, are candidates for amplification operations.4–6 These wavelengths are in the re- gion where the silica glass, used to carry large volumes of information, has a minimal optical loss. Because of the low phonon energies of the fluoride glasses~,600 cm21!, the multiphonon relaxation of excited states of rare earths is strongly reduced, improving the efficiency of the rare-earth fluorescence and increasing the amplification gain. In the praseodymium-doped fiber amplifier~PDFA! and amplifiers based on thulium-doped fibers, amplification gains of 40 dB (Pr31) and 30 dB (Tm31) have been obtained.1 Fluoride glasses can be melted over a wide range of compositions. In this work we used fluorozirconate glass ~ZBLAN ! (ZrF4– BaF2– LaF3– AlF3– NaF) as matrix, Tm31 as doping, and Ho31 as codoping ions. The Tm31 is a four- level laser which has important advantages for amplification purposes since the lower laser level3F4 remains unpopu- lated. Ho31 (5I7) may play an important role in the ‘‘deactivation’’ of the 3F4 lower laser level of Tm31. Tm:Ho:ZBLAN fluoride glass is also an attractive material for infrared laser fibers because of its efficient energy trans- fer ~ET! Tm31 (3F4)→Ho31 (5I7), which is followed by important5I7→5I8 laser emission of Ho31 around 2mm. The present article reports on our studies of the mecha- nism involved in the Tm31 (3F4)→Ho31 (5I7) ET and Tm31 (3H4 , 3H6)→Tm31 (3F4 , 3F4) cross-relaxation~CR! processes observed in Tm:Ho:ZBLAN glasses, using several combinations of Tm31 and Ho31 concentrations. In this sense, first, a microscopic analysis was performed consider- ing the five possible ET processes, which occur for Tm:Ho:ZBLAN glasses: ~i! Tm31 (3F4 , 3H6)→Tm31 (3H6 , 3F4) migration, ~ii ! Tm31 (3F4)→Ho31 (5I7) energy transfer, ~iii ! Ho31 (5I7)→Tm31 (3F4) backtransfer,~iv! Tm31 (3H4 , 3H6)→Tm31 (3F4 , 3F4) cross relaxation, and ~v! Tm31 (3H4 , 3H6)→Tm31 (3H6 , 3H4) migration. The donor-to-donor, donor-to-acceptor, and acceptor-to-donor en- ergy transfer constants were determined and applied to esti- mate the expected transfer rates for the case of an energy transfer assisted by a diffusion model. The expected transfera!Electronic mail: lgomes@net.ipen.br JOURNAL OF APPLIED PHYSICS VOLUME 95, NUMBER 10 15 MAY 2004 54510021-8979/2004/95(10)/5451/13/$22.00 © 2004 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 186.217.234.225 On: Tue, 14 Jan 2014 13:05:55 http://dx.doi.org/10.1063/1.1704845 rate values were compared to the experimental transfer rates obtained from a macroscopic analysis of the measured lumi- nescence decay of the5I7 state of Ho31 at ;2 mm and the 3H4 state of Tm31 at ;1.5 mm. The experimental transfer rates and efficiencies were determined for Tm→Ho energy transfer from the best fit of the luminescence decay curve using a theoretical expression derived for the acceptor lumi- nescence decay based on the Inokuti–Hirayama7 approach. We have already shown that the Inokuti–Hirayama solution from the acceptor viewpoint is an important tool to investi- gate the mechanism of resonant energy transfer among triple- ionized rare-earth ions in glasses.8 An expression in which the Inokuti–Hirayama approach is used together with the proposed localized donor-to-acceptor interaction@dipole– dipole (s56)] solution has been used in the determination of experimental transfer rates and efficiencies for Tm–Ho, Tm–Tm cross relaxation and Tm–Tm migration. The tem- poral dynamics of the excited levels of Tm31 and Ho31 ions were measured using a pulsed laser excitation~4 ns! from a tunable optical parametric oscillator~OPO! pumped by the second harmonic of aQ-switched Nd–YAG laser which can pump exactly the level to be investigated. It is important to emphasize that the donor state3F4 of Tm31 ions was selec- tively excited at 1.67mm in the case of the Tm→Ho energy transfer analysis~this excitation does not excite Ho ions!. In this case, the Ho31 luminescence transient was observed at 2 mm from where the most important information was derived. The Tm←Ho backtransfer cannot be experimentally ob- served when exciting Ho ions at 2mm and measuring the Tm31 emission at 1.8mm because the net direction of the energy transfer points to the Ho direction (CTm→Ho 50.95CTm←Ho). However, this mechanism exists and its microscopic parameterCAD can be calculated and compared with the direct transfer constantCDA . As a consequence, an effective transfer parameter is given byCDA(net)5CDA 2CAD . In the case of Tm→Tm cross relaxation, the decay of Tm31 luminescence at 1.47mm from the3H4 excited state was measured after selective laser excitation at 0.78mm. This excitation excites only Tm31 ions in Tm:Ho:ZBLAN and Tm:ZBLAN glasses. Excitation migration~or diffusion! through3H4 excited states of Tm31 plays an important role in the Tm→Tm cross relaxation, but cannot be separately observed. The same argument is also applied to the case of excitation migration~or diffusion! through3F4 excited states of thulium ions, which plays an important role in the Tm →Ho energy transfer. II. EXPERIMENT A. Glass preparation Thulium and holmium fluorides were used, respectively, as codopant and dopant starting materials. They were pre- pared by fluorination of the respective ultrapure oxides ~99.999%! from Alfa Aesar. Other chemicals employed to prepare ZBLAN glasses were the grade reagents~.99.9%! fluorides: ZrF4 ~Fluortran!, BaF2 ~Alfa Aesar!, LaF3 ~reac- tron, Alfa Aesar!, and AlF3 and NaF~Fluortran!. Two series of Tm:Ho:ZBLAN glasses were prepared with the following compositions, wherex andy are given in mol %: ~ i! ~992x!~53ZrF4•20BaF2•4LaF3•3AlF3•20NaF!•xTmF3•1HoF3 ~x50, 0.5, 1, 3, 6, and 9!, ~ ii ! ~992y!~53ZrF4•20BaF2•4LaF3•3AlF3•20NaF!•1TmF3•yHoF3 ~y50, 0.5, 2, 3, and 4!. Tm:Ho:ZBLAN glasses of 5 g were produced by melting processing at 750–800 °C for 2 h in a tubular furnace. A platinum crucible in form of tube was used as sample con- tainer. Annealing at 260 °C for 2 h was performed after cast- ing. The rectangular samples were polished using isopropyl alcohol as lubricant agent. B. Spectroscopic measurements A time-resolved luminescence spectroscopy was em- ployed to measure the acceptor~A! luminescence transient induced by resonant laser excitation of the donor~D! state. The luminescence and lifetimes of excited Tm31 and Ho31 were measured using a pulsed laser excitation~4 ns! from a tunable optical parametric oscillator pumped by the second harmonic of aQ-switched Nd-YAG laser from Quantel. La- ser excitation at 0.78 and 1.67mm was used to excite the3H4 and3F4 states of Tm31, respectively. The luminescence sig- nals, as well the time-dependent luminescence of the donor ~or the activator!, were detected by a InSb~77 K! infrared detector~Judson model J10D! with a fast preamplifier~re- sponse time of 0.5ms! and analyzed using a signal- processing boxcar~PAR 4402! ~luminescence measurements! or a digital 200-MHz oscilloscope from Tektronix~TDS 410! ~time-dependent luminescence measurements!. Both fluores- cence spectrum and decay times were measured at 300 K. III. RESULTS A. Determination of the energy transfer microparameters Schematic energy level diagrams showing some energy states of Tm31 and Ho31 ions with the respective 1.47mm (3H4) and 2 mm (5I7) emission transitions, as well as the possible energy transfer mechanisms for Tm:Ho:ZBLAN glasses, are presented in Figs. 1~a! and 1~b!. The following notations were adopted for the energy transfer processes in- vestigated: ~1! (3F4) Tm→Tm migration for Tm31 (3F4 , 3H6)→Tm31 (3H6 , 3F4), ~2! Tm→Ho ET for Tm31 (3F4)→Ho31 (5I7), 5452 J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 da Vila et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 186.217.234.225 On: Tue, 14 Jan 2014 13:05:55 ~3! Ho→Tm backtransfer~BT! for Ho31 (5I7)→Tm31 (3F4), ~4! Tm→Tm CR for Tm31 (3H4 , 3H6)→Tm31 (3F4 , 3F4), ~5! (3H4) Tm→Tm migration for Tm31 (3H4 , 3H6) →Tm31 (3H6 , 3H4). In order to optimize the Tm→Ho ET process, the con- centration effect of each dopant ion must be investigated in detail. For example, the Tm31 ion at high concentration al- lows efficient Tm→Tm CR in which two Tm31 ions are generated in the3F4 excited state for each Tm31 ion in the 3H4 state@see mechanism 4 in Fig. 1~b!#. The energy is then transferred to the Ho31 ions (Tm→Ho ET), which exhibit luminescence near 2mm suitable for technological applica- tions. Another nonradiative energy transfer mechanism pos- sible to the Tm:Ho:ZBLAN system is the (3H4) Tm→Tm migration @see mechanism 5 in Fig. 1~b!#. Figures 2~a! and 2~b! show the fluorescence spectra for the Tm- and Tm:Ho-doped ZBLAN. Emissions around 1.5 and 1.8mm are observed for Tm:ZBLAN@Fig. 2~a!# and Tm:Ho:ZBLAN samples@Fig. 2~b!#. These are attributed to 3H4→3F4 and 3F4→3H6 emissions of Tm31, respectively. Since the Tm–Ho energy transfer partially depopulates the 3F4 excited state of Tm31, a weak residual Tm31 (3F4) luminescence at;1.8 mm ~;20%! is observed in Tm:Ho:ZBLAN. This effect is explained based on the Tm ←Ho backtransfer process. In addition, a strong emission at ;2 mm due to the5I7→5I8 fluorescence of Ho31 ions is observed in Fig. 2~b!. The lifetime of3H4 , 3F4 excited states of Tm31 and 5I7 of Ho31 ions have been measured for single-doped ZBLAN glasses. Figures 3~a!–3~c! show the luminescence decays and best fits. A nonexponential decay with a lifetime of 1.5 ms has been measured for3H4 (Tm31) emission at 1.47mm in Tm~0.5 mol %!:ZBLAN. In this case FIG. 1. Simplified energy level diagram of Tm:Ho:ZBLAN which exhibits five possible energy transfer processes as indicated.~a! exhibits the Tm–Ho transfers according to~1! (3F4) Tm–Tm migration involving the3F4 excited ~donor! and 3H6 ground ~acceptor! states;~2! Tm→Ho energy transfer where the3F4(Tm) state is the donor and the5I7(Ho) state is the acceptor; ~3! Tm←Ho backtransfer~BT! ~the intrinsic reverse transfer!. ~b! exhibits the Tm–Tm transfers according to~4! Tm–Tm cross relaxation being the 3H4 the donor state and the3H6 ground state as the acceptor;~5! (3H4) Tm–Tm migration having the3H4 excited state as the donor and the3H6 ground state as the acceptor. FIG. 2. Infrared luminescence spectrum of Tm31 and Ho31 in ZBLAN glass. ~a! shows the spectrum of Tm~1%! single-doped ZBLAN and~b! shows the spectrum of Tm(x%):Ho(1%):ZBLAN after laser excitation at 0.79mm with 12 mJ at 300 K. FIG. 3. ~a! shows the fluorescence decay of Tm31 ions at 1.47mm after laser excitation at 0.78mm ~12 mJ, 4 ns! measured in Tm~0.5 mol %!:ZBLAN at 300 K ~open circles!. The solid line represents the Inokuti–Hirayama best fit (g512.1 s21/2). ~b! shows the fluorescence decay of the 3F4 state at 1.8mm after laser excitation at 1.671mm ~open circles!. The solid line is the exponential fitting (t58.9 ms).~c! shows the fluores- cence decay of the5I7 state at 2mm after laser excitation at 1.95mm ~open circles!. The solid line is the exponential fitting (t516.4 ms). 5453J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 da Vila et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 186.217.234.225 On: Tue, 14 Jan 2014 13:05:55 a good Inokuti–Hirayama fit was obtained@see solid line of Fig. 3~a!#. The luminescence decay of the3F4 state at 1.8mm is exponential with a lifetime of 8.9 ms in Tm~0.5 mol %!:ZBLAN @see Fig. 3~b!#. The decay of the5I7 state at 2 mm is exponential with a time constant of 16.4 ms ob- served in Ho~0.5 mol %!:ZBLAN @see Fig. 3~c!#. Donor–donor (DD), donor–acceptor (DA), and acceptor–donor ~AD! energy transfer constants (CDD ,CDA ,CAD), critical radii (RDD ,RDA), and critical concentrations (c0) were calculated for~i! (3F4) Tm→Tm migration ~mechanism 1!, ~ii ! Tm→Ho ET ~mechanism 2!, ~iii ! Ho→Tm BT ~mechanism 3!, ~iv! Tm→Tm CR ~mecha- nism 4!, and ~v! (3H4) Tm→Tm migration~mechanism 5!. The following expressions were used for energy transfer con- stant determination: CDD5 RDD 6 tD @~ i! and ~v! processes#, ~1! CDA5 RDA 6 tD @~ ii ! and ~ iv! processes#, ~2! CAD5 RAD 6 tA @~ iii ! process#, ~3! wheretD andtA is the total lifetime of the donor~Tm! and acceptor~Ho! states, respectively, measured for single-doped samples. The critical radiiRDD , RDA , andRAD were calcu- lated using the extended overlap integral method9 since the DD, DA, and AD energy transfer processes for the Tm:Ho system are nonresonant~or phonon assisted!. In this method a nonvanishing overlap integral is produced from a transla- tion of the donor emission spectrum towards the acceptor absorption. The expressions for direct transfer~DA! and backtransfer~AD! are RDA 6 5 6ctD ~2p!4n2 gD low gD up ( n50 ` E semis D ~lN 1!sabs A ~l!dl 3S ( j 50 N P~N2 j ! 1 Pj 2Pj 1D ~4! ~for Tm→Ho ET!, RDD~or AD! 6 5 6ctA ~2p!4n2 gA low gA up ( n50 ` E semis D~or A! 3~lN 2!sabs D ~l!dlS ( j 50 N P~N2 j ! 2 Pj 1Pj 2D ~5! @for Tm→Tm Cr, (3H4) Tm→Tm, (3F4) Tm→Tm migra- tion ~or Tm←Ho BT!#, where c is the light speed,n the refractive index of the medium~1.5!, glow D /glow A and gup D /gup A are the degeneracy of the respective lower and upper levels of the donor/acceptor,N is the total number of phonons in- volved in the energy transfer process, (N2 j ) is the number of phonons emitted~or created! by D in the DA and DD transference~A in AD backtransfer!, and j is the number of phonons absorbed~or annihilated! by A ~D in bothDD trans- ference andAD backtransfer!. lN 1 andlN 2 denote the wave- lengths translations of the emission cross-section spectra by E5@N\v#21. lN 1 denotes the translations due to mul- tiphonon emission byD in the DA transference.lN 2 denotes the translations due to multiphonon absorption byD in the DD transference~A andAD backtransfer!. PN2 j 1 is the prob- ability of multiphonon emission byD, while PN2 j 2 is the probability of multiphonon absorption byD in the D –D transference~A in AD backtransfer!. The electron–phonon coupling constantS0 has been estimated to be;0.31, and the mean phonon energy that couples with the phonon sideband is \v;330 cm21 in ZBLAN glasses.10 The critical concen- tration was calculated using c05S 4p 3 RC 3 D 21 , ~6! whereRC is the critical radius of interaction. Figures 4~a!, 4~b! and 5~a!, 5~b! show the spectral super- position between the emission and absorption cross-section spectra for the phonon-dependent energy transfer mecha- nisms in Tm:Ho:ZBLAN glasses. The strong spectral super- position observed@Figs. 4~a! and 4~b!# predicts an efficient ET for Tm:Ho:ZBLAN. A stronger overlap between emis- sion and absorption cross sections for the (3H4) Tm→Tm migration process in relation to the overlap verified for Tm →Tm CR ~cross-relaxation mechanism! has been obtained and can be observed in Figs. 5~a! and 5~b!. These results indicate also that the donor–donor migration among3H4 thu- FIG. 4. ~a! shows the spectral cross-section superposition between Tm (3H6→3F4) absorption and Tm (3F4→3H6) emission involved in the (3F4) Tm–Tm migration. The dotted line of~a! exhibits the Tm31 emission in- volving one-phonon absorption~or annihilation!. ~b! shows the superposi- tion between the Tm (3F4→3H6) emission and the Ho (5I8→5I7) absorption in the Tm→Ho ET. The dotted line of~b! shows the spectral overlap with Tm31 emission involving one-phonon emission~or creation!. ~c! shows the overlap between the Ho (5I7→5I8) emission and the Tm (3H6→3F4) ab- sorption involved in the Tm←Ho backtransfer. The dotted line of~c! shows the overlap contribution with the Ho31 emission involving one-phonon ab- sorption~or annihilation!. 5454 J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 da Vila et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 186.217.234.225 On: Tue, 14 Jan 2014 13:05:55 lium states is more important than the donor–donor migra- tion among3F4 states when higher excited levels of thulium ions are involved in the energy transfer in Tm:Ho:ZBLAN glasses. Table I shows the phonons numbers and contribution term ~%! to the total probability rate involved in the energy transfer. The microscopic parameters values for the nonradi- ative energy transfer for Tm→Ho ET, Tm←Ho BT, (3F4) Tm→Tm migration, Tm→Tm CR and (3H4) Tm→Tm mi- gration are also presented in Table I. Comparing the several N-phonon contribution terms~%! one verifies that the trans- ference (3F4) Tm→Tm and (3H4) Tm→Tm migrations are not phonon dependent in these quasiresonant processes. The Tm→Tm CR transference is a multiphonon mechanism dominated by two-phonon creation~59%!. The Tm→Ho ET and Ho→Tm BT exhibit a competition between zero- and one-phonon processes. FromCDD andCDA results it can be seen that the (3F4) Tm→Tm migration competes with the Tm→Ho ET (CDD;0.8CDA). A similar behavior in which D→D migration competes with the directD→A energy transfer had been observed in other rare-earth-codoped fluo- ride systems, such as Yb:Er:ZBLAN glasses8 and Tm:Ho:YLF9 crystals. A great difference betweenCDD and CDA values for Tm→Tm CR and (3H4) Tm→Tm migra- tions has been observed in the present work (CDD ;11CDA). This is an obvious result since the (3H4) Tm →Tm migration is a quasiresonant process, while Tm →Tm CR is dominated by a two-phonon creation mecha- nism. A CAD /CDA ratio estimated for the Tm→Ho transfer was found to be 0.05, which indicates that only 5% of the excited Tm31 ions remain in the3F4 state due to the exis- tence of Ho→Tm backtransfer~AD process!. However, we observed a thulium residual luminescence from3F4 state of 25% in Tm:Ho-doped ZBLAN glasses exhibiting a lifetime of ;8.9 ms, similar to the one found in Tm single-doped ZBLAN. B. Analysis of the acceptor and donor luminescence transients The time evolution of5I7 (Ho31) luminescence at;2 mm and 3H4 (Tm31) at ;1.5 mm has been measured and analyzed for two sets of Tm:Ho:ZBLAN glasses: ~i! (992x)(ZBLAN) •xTmF3•1HoF3 (x50.5, 1, 3, 6, and 9 mol %!, ~ii ! (992y)(ZBLAN) •1TmF3•y HoF3 (y50.5, 2, 3, and 4 mol %!. Analysis of the5I7 (Ho31) luminescence transient is impor- tant for understanding the energy transfer involved in Tm–Ho ET, while the analysis of the3H4 (Tm31) lumines- cence decay is important to determine the role of the Tm–Tm migration in the Tm–Tm cross-relaxation process. Figures 6–9 exhibit the luminescent decay curves for the samples of sets~i! and ~ii ! used, respectively. Figures 6~a!– 6~c! and 7~a!–7~c! show the Ho31 luminescence decay curves, and Figs. 8~a!–8~c! and 9~a!–9~c! show the Tm31 FIG. 5. ~a! shows the spectral cross-section overlap between the Tm (3H4 →3F4) emission and Tm (3H6→3F4) absorption involved in the Tm–Tm cross relaxation. The dashed lines of~a! exhibit the overlap contribution of the Tm (3H4) emission involving one-phonon~dotted line! and two-phonon emission ~dashed line!. ~b! shows the overlap between the Tm (3H6 →3H4) absorption and the Tm (3H4→3H6) emission near 0.80mm involved in the (3H4) Tm–Tm migration~solid lines!. The dotted line in~b! repre- sents the Tm emission involving one-phonon absorption~or annihilation process!. TABLE I. Calculated microscopic parameters of the energy transfer processes observed in Tm:Ho:ZBLAN glass considering a dipole–dipole electric interaction (s56). Energy transfer N ~No. of phonons! ~% phonons! Transfer constant ~cm6 s21! Rc ~Å! c0 ~mol %! Tm→Tm ~migration! 0, 1 CDD5(3.27)310239 17.5 0.24 (3F4 , 3H6)→(3F4 , 3H6) ~93, 7! Tm→Ho ~energy transfer! 0, 1, 2 CDA5(4.29)310239 18.4 0.21 (3F4→5I7) ~53, 40, 7! Ho→Tm ~backtransfer! 0, 1, 2 CAD5(0.21)310239 11.5 0.85 (5I7→3F4) ~49, 44, 7! Tm→Tm ~cross relaxation! 0, 1, 2, 3, 4 CDA5(7.83)310241 7.0 3.85 (3H4 , 3H6)→(3H4 , 3F4) ~1, 23, 59, 16, 1! Tm→Tm ~migration! 0, 1 CDD5(88.96)310241 10.5 1.14 (3H4 , 3H6)→(3H6 , 3H4) ~92, 8! 5455J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 da Vila et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 186.217.234.225 On: Tue, 14 Jan 2014 13:05:55 luminescence decay curves. We observed a nonexponential rise time of Ho31 luminescence at 2mm due to the Tm–Ho ET, followed by an exponential decay in Tm:Ho:ZBLAN. Also, a nonexponential decay was observed for Tm31 (3H4) luminescence at 1.47mm in both Tm ~0.5%!:ZBLAN and Tm(x%):Ho(y%):ZBLAN glasses. The Ho31 and Tm31 luminescence transients were ana- lyzed using the donor solution obtained by Inokuti– Hirayama@for a dipole–dipole interaction (s56)] ~Ref. 7! and the acceptor solution based on the Inokuti–Hirayama model:8 Ī D~ t !5expS 2gAt2 t tD D ~ for the donor decay!, ~7! Ī A~ t !5expS 2 t tA D2expS 2gAt2 t tD D ~ for the acceptor transient!, ~8! where g ~s21/2! is the energy transfer constant,tD is the intrinsic total lifetime of an isolated donor~Tm!, andtA is the intrinsic lifetime of the acceptor~Ho!. However, one can see that these solutions do not fit well the 2-mm Ho emission transient~or the 1.47-mm Tm emission! in the Tm:Ho system with an increase of Tm~or Ho concentration!. It is observed also that the values of theg parameter are much larger than the theoretical values expected for each case of Tm(x):Ho(y):ZBLAN. As has been reported before, there is a fast excitation migration among donors that happens in a very short time (t!g2) that modifies the excitation distribu- tion between donors such that an excited donor ion~Tm! feels an apparent increase of the acceptor concentration (cA →cA8 ). Let us now assume that after this fast diffusion there are two types ofD→A interaction that are dependent on the distance~R! involved in theD* –A pair: ~i! If R.RC , the D* →A energy transfer occurs as de- scribed in the Inokuti–Hirayama model: i.e., the excited do- nor interacts with all the acceptors present in the interaction volume in the limit ofV→`. ~ii ! If R,RC , a localizedD* →A energy transfer takes over and the average time solution will be given byf̄D(t). Considering that the decay solution for aD* –A pair is given by fD~ t,R!5exp~2t/tD!exp@2tWDA~R!#, where the transfer rate is given byWDA(R)5CDA /R6 and the normalized distribution functionf (R)dR, which gives the fraction ofD* –A pairs as a function of the separation distanceR, f ~R!dR54pR2xA8 ~12xA8 !~4p/3!R3cA8dR, wherecA8 is the apparent acceptor concentration andxA8 is the correspondent mol fraction, we have FIG. 6. Fluorescence of Ho31 (5I7) at 2mm induced by the Tm→Ho ET by laser excitation of Tm ions at 1.67mm ~10 mJ, 4 ns, 10 Hz! measured for three Tm(x):Ho(1%):ZBLAN samples where x51 mol % ~a!, x 53 mol % ~b!, andx59 mol % ~c!. The dashed lines represents the best fit using the Inokuti–Hirayama solution for the acceptor luminescence transient (Ho31) and the solid line represents the best fit using a solution that also includes the~acceptor! localizedD* –A interaction contribution@Eq. ~12! obtained for the proposed model#. FIG. 7. Fluorescence decay of Ho31 (5I7) at 2 mm excited with laser pulse at 1.67mm ~10 mJ! measured for three Tm(1%):Ho(y):ZBLAN where y 52 mol % ~a!, y53 mol % ~b!, andy54 mol % ~c!. The dashed lines were obtained from the best fit using the Inokuti–Hirayama solution for the ac- ceptor (Ho31) ion and solid lines represent the best fit using a solution that also includes the~acceptor! localized D* –A interaction contribution@ac- cording to Eq.~12!#. 5456 J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 da Vila et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 186.217.234.225 On: Tue, 14 Jan 2014 13:05:55 f̄D~ t !5E fD~ t ! f ~R!dR5exp~2t/tD!E exp~2tCDA /R6! f ~R!dR5exp~2t/tD!E f ~R! ( n50 ` ~ tCDA /R6!n 1 n! dR 5exp~2t/tD!F12tCDA^R26&1~ tCDA!2^R212& 1 2! 2¯G>exp~2t/tD!F12tCDA^R26&1~ tCDA!2^R26&2 1 2! 2¯G 5exp~2t/tD!exp~2tCDA^R26&!, ~9! where ^R26&5*Rmin ` R26 f(R) dR and K5CDA^R26& is the transfer rate expected for the localizedD* –A energy trans- fer model. Using the same procedure for the acceptor tran- sient, we obtain f̄A~ t !5exp~2t/tA!2exp~2t/tD!exp~2tCDA^R26&!. ~10! Considering that the localized interaction competes with the Inokuti–Hirayama interaction approach, one can have the fi- nal solution for the donor decay and the acceptor lumines- cence transient composed by the solutions expressed by Eqs. ~7! and ~9! ~for the donor! and Eqs.~8! and ~10! ~for the acceptor!, respectively given in the following: Ī D~ t !5a expS 2 t tD 2gAt D1b expS 2 t tD 2Kt D ~11! @used in the Tm(3H4) luminescence decay analysis# and Ī A~ t !5aFexpS 2 t tA D2expS 2 t tD 2gAt D G 1bFexpS 2 t tA D2expS 2 t tD 2Kt D G ~12! @used in the Ho(5I7) luminescence transient#. a andb in Eqs. ~11! and ~12! represent the non-normalized contribution of the Inokuti–Hirayama~I! and the localizedD* –A energy transfer~T! solutions, respectively. It is important to note that the final solutions for the donor and acceptor luminescence obtained were built with valid solutions of the localized and nonlocalized interactions ~Inokuti–Hirayama! with no conflicting approach~i.e., both approaches do not include the diffusion among donors, but its apparent effect of acceptor concentration increase induced by the fast excitation diffusion!. The Yokota–Tanimoto11 so- lution was not proper to use to compose the final solution of FIG. 8. Fluorescence decay of Tm (3H4) at 1.47mm emission measured in Tm(x):Ho(1%):ZBLAN with three different Tm concentrations withx 51 mol % ~a!, x53 mol % ~b!, andx59 mol % ~c! ~after laser excitation at 0.78 mm!. The dashed lines were obtained from the best fit using the Inokuti–Hirayama solution for the donor~Tm! ion. The solid lines represent the best fit obtained using a solution that also includes the donor decay due to the localizedD* –A interaction@according to Eq.~11!#. FIG. 9. Fluorescence decay of Tm (3H4) at 1.47mm emission measured in Tm(1%):Ho(y):ZBLAN with three different Ho concentrations withy 52 mol % ~a!, y53 mol % ~b!, andy54 mol % ~c! ~after laser excitation at 0.78 mm!. The dashed lines were obtained from the best fit using the Inokuti–Hirayama solution for the donor~Tm! ion and the solid lines were obtained using an expression that also includes the donor decay due to the localizedD* –A interaction@Eq. ~11!#. 5457J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 da Vila et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 186.217.234.225 On: Tue, 14 Jan 2014 13:05:55 the donor and acceptor luminescence transients because it is valid only for the cases where the diffusion process is con- sidered a perturbation in the directD* –A energy transfer, which is not the case in this work because the fast excitation diffusion cannot be assumed as a perturbation, but it causes the main physical effect. The normalized fractions of each contributionI and T, respectively, for the Inokuti–Hirayama and localizedD* –A energy transfer solutions were obtained using that I 5 a a1b , T5 b a1b . Considering the proposed model, the mean decay time of donor (Tm31) fluorescence is obtained using the expression tTm5 *0 ` Ī D~ t !dt Ī D~0! , where Ī D(0)5a1b, given Kt~exp!5tTm 215F S a a1bD S 1 g2D1S b a1bD S 1 K D G21 . ~13! Figures 6 and 7 show the best fit of 2-mm Ho lumines- cence by laser excitation of the Tm31 (3F4) state at 1.67 mm. Figures 8 and 9 show the best fit of 1.47-mm Tm lumi- nescence by laser excitation of the Tm31 (3H4) state at 0.78 mm. The 1.67-mm ~Tm! and 2-mm ~Ho! luminescence tran- sients are better described using the solution given by Eqs. ~11! and ~12!, respectively, in which the localizedD* –A energy transfer solution takes part. The best fit parameters of Tm→Ho ET obtained from the Ho31 (5I7) luminescence curve fitting using Eq.~12! and the predicted values of g~theor! and Kd ~from the diffusion model! are given in Table II. There it is seen thatg~expt! andK(expt) are always bigger thang~theor! predicted by the Inokuti–Hirayama ap- proach@given by Eq.~14!#. Consequently, the experimental values of the transfer rateg2(expt) are always much bigger than the transfer rate (Kd) predicted from the diffusion model @given by Eq.~15!#. The obtained fitting parameters for the case of Tm(3H4) luminescence decay analysis using Eq. ~11! are given in Table III. Theoretical values of the transfer rate predicted by the Inokuti–Hirayama model~g! can be calculated using the expression g~ theor!5 4p3/2 3 cA~CDA!1/2. ~14! Any diffusing excitation by an electric dipole–dipole in- teraction between donors on a cubic lattice has a trapping radius defined as the distance at which theD→A transfer rate is equal to the rate ofD→D transfer. This givesRT 50.676(CDA /D)1/4. The diffusion coefficient~cm2/s! is given byD5 1 2(4pcD/3)4/3cDD and the energy transfer rate, which is associated with excitation diffusion and trapping ~here is assumed an unit trapping efficiency! derived from the random walk treatment,11 is given byKD54pDRTcA . This can be related to the microscopic parameters of the interaction by Kd521cAcD~CDD 3 CDA!1/4, ~15! where cA is the acceptor concentration~Ho! for the Tm →Ho ET and~Tm! for the Tm→Tm CR, andcD is the donor concentration~Tm! for the Tm→Ho ET and Tm→Tm CR. The theoretical values ofg andKd transfer parameters were calculated for the Tm–Ho ET and Tm–Tm Cr and are given in Tables II and III for comparison with the experimental values. A moderate decrease of the measured lifetime of the 5I7(Ho31) state in Tm(x):Ho(1%) samples with Tm31 con- centration increase (x51 – 9 mol %) was observed in Table II. This lifetime decrease~from 16 to 2 ms! was attributed to the upconversion of the5I7 state by an energy transfer mechanism~ETU! involving the3F4 state of Tm31 ~see sche- matic diagram of Fig. 10!. Excited-state absorption~ESA! TABLE II. Experimental values of ET parameters obtained for the best fit of the 2-mm luminescence of Ho31 (5I7) using Eq.~8! for the Inokuti–Hirayama ~acceptor! and Eq.~12! for a solution that includes the localizedD* –A interaction contribution for two sets of Tm(x):Ho(y):ZBLAN. Theoretic values of g parameters andKd were obtained from the microscopic theory of energy transfer based on the random walk problem involving excitation migration by diffusion through donors states~diffusion model!. Tm:Ho: ZBLAN ~mol %! Transfer parameter ~s21/2! K (103 s21) ~expt!b a ~expt!b b ~expt!b Kt (103 s21) ~expt!c Kd (103 s21) ~theor!d tHo ~ms! ~expt!b tTm ~ms! ~expt!c~x! ~y! g ~theor!a g ~expt!b 0.5 1 101.2 111.661.3 412.062.7 0.84 0.22 15.7 1.2 16.460.1 63.8 1 1 100.9 151.461.2 308.861.4 0.83 0.18 27.5 2.4 16.460.6 36.4 3 1 99.7 281.262.1 162.862.7 0.77 0.27 91.0 6.9 12.460.2 11.0 6 1 94.2 275.369.9 221.962.1 0.34 0.70 137.0 12.4 4.860.2 7.3 9 1 96.4 223.062.1 316.362.5 0.21 0.85 153.8 19.5 2.060.2 6.5 1 0.5 50.6 140.961.8 127.064.4 0.82 0.24 24.5 1.2 13.860.5 40.8 1 2 200.7 215.764.2 184.363.0 0.53 0.49 72.5 4.7 15.460.6 13.8 1 3 299.3 280.668.1 289.763.3 0.38 0.64 145.0 6.9 15.860.7 6.9 1 4 396.9 281.0615 389.963.3 0.27 0.76 188.7 9.2 12.561.7 5.3 aCalculated values using the Inokuti–Hirayama theory~Ref. 7!. Deviation estimation of these calculated parameters was 10% considering the propagation error in the absorption measurements. bExperimental data obtained from the best fit of the acceptor luminescence transient described by Eq.~8!. cObtained from Eq.~12!. dCalculated values using the diffusion model~Ref. 11!. Deviation estimation of these calculated parameters was 10% considering the propagation error in the absorption measurements. 5458 J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 da Vila et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 186.217.234.225 On: Tue, 14 Jan 2014 13:05:55 can not account for this observed upconversion effect be- cause it occurs in a much longer time~;2.3 ms! than the laser pulse ~4 ns!. For samples of set ~ii !, Tm(1%):Ho(y):ZBLAN where y50,0.5,2,3,4 mol %, the lifetime of 5I7 (Ho31) is within 15 and 16 ms~see data of Table II!. However, a small reduction of this lifetime is ob- served for the cases when the Ho31 concentration is higher than 3 mol %. In addition, we observed for the samples of set ~ii ! a small reduction of the luminescence lifetime of the 3H4 (Tm31) state with an increase of Ho31 concentration ~see data of Table III!. This suggests that a small fraction of Tm31 ions in the3H4 state can transfer energy to the5I5 state of Ho31. The results presented in Tables II and III showed that the best fit values of theg parameter@or g~expt!# is always higher than the predict value ofg by the Inokuti–Hirayama model @or g~theor!# for the Tm→Ho ET. The results ob- served for the Tm→Ho ET and presented in Table II confirm the existence of fast excitation diffusion among donor (Tm31) ions before energy transfer to the acceptor (Ho31). A similar conclusion was also obtained when observing the results of Tm→Tm cross relaxation exhibited in Table III. These two ET processes—Tm–Ho ET and Tm–Tm CR— have an experimental transfer rateKt(expt) about 40 times bigger than the transfer rate (Kd) predicted by the diffusion model.11 Based on this observation, we concluded that the Tm–Ho ET and Tm–Tm CR are a fast-excitation diffusion- dependent energy transfer mechanism similar to the one ob- served for the Yb→Er energy transfer in ZBLAN glass.8 As a consequence, the hypothesis on the occurrence of fast ex- citation diffusion occurrence before the directD* –A reso- nant energy transfer is reinforced. IV. DISCUSSION AND CONCLUSIONS In this work the transfer parameterKd was obtained in- dependently of solving the diffusion-assisted energy transfer equation~which has no analytical solution! from the random walk problem description, Eq.~15!. This transfer rate value gives the highest limit of the transfer rate for a random- excitation migration-assisted energy transfer and is valuable for comparison with the experimental~and theoretical! trans- fer ratesg2 ~or K! to evidence the existence of a fast excita- tion diffusion among donors as the main diffusion mecha- nism responsible for the highest values observed for the Tm→Tm and Tm→Ho transfer rates. The analysis of the luminescence decay of Tm(3H4) at 1.47mm and Tm(3F4) at 1.8mm in low single-doped Tm~0.5 mol %!:ZBLAN showed that the Inokuti–Hirayama model~donor solution for a dipole–dipole interaction! gives good fitting of the lumines- cence transient with transfer parameterg~expt! equal to 12.2 s21/2, which is higher than the expected valueg(theor) 56.8 s21/2 @see data of Table III~first row!#. This result shows that the fast excitation diffusion is already present in this case of low Tm31 concentration besides the Inokuti– Hirayama model, indicating that the migration is not occur- ring ~for dipole–dipole interactions,s56). In this case, the TABLE III. Experimental values of ET parameters obtained for the best fit of the 1.47-mm luminescence of Tm31 (3H4) using Eq.~7! for the Inokuti– Hirayama~donor! and Eq.~11! that includes the solution of the localizedD* –A interaction contribution for two sets of Tm(x):Ho(y):ZBLAN. Theoretic values ofg parameters andKd were obtained from the microscopic theory of energy transfer based on the random walk problem involving excitation migration by diffusion through donors states~diffusion model!. Tm:Ho ~mol %! Transfer parameter ~s21/2! K (103 s21) ~expt!b a ~expt!b b ~expt!b Kt (103 s21) ~expt!c Kd (103 s21) ~theor!d tTm ~ms! ~expt!c~x! ~y! g ~theor!a g ~expt!b 0.5 0 6.8 12.160.4 1.860.1 0.95 0.05 0.15 0.08 6660.8 0.5 1 6.8 11.960.4 1.960.1 0.92 0.09 0.16 0.08 6420.3 1 1 13.6 40.260.4 3.360.9 0.82 0.20 1.79 0.33 559.7 3 1 40.4 18264.0 25.060.7 0.64 0.40 29.5 2.9 33.9 6 1 76.4 357611 153.261.6 0.31 0.73 145 10.3 6.9 9 1 117.2 231.567.6 395.662.5 0.09 0.97 256.4 24.3 3.9 1 0.5 13.7 42.360.6 2.760.2 0.88 0.16 1.89 0.33 529.9 1 2 13.6 53.360.6 2.460.2 0.96 0.11 2.79 0.33 358.2 1 3 13.5 66.660.4 3.460.2 1.11 0.07 4.36 0.32 229.2 1 4 13.4 73.360.7 10.360.7 0.98 0.09 33.6 0.32 178.5 aCalculated values using the Inokuti–Hirayama theory~Ref. 7!. Deviation estimation of these calculated parameters was 10% considering the propagation error in the absorption measurements. bExperimental data obtained from the best fit of the donor luminescence and rate equation given by Eq.~7!. cObtained from Eq.~11!. dCalculated values using the diffusion model~Ref. 11!. Deviation estimation of these calculated parameters was 10% considering the propagation error in the absorption measurements. FIG. 10. Schematic energy level diagram showing the energy transfer by upconversion~ETU mechanism! involving Tm and Ho ions after Tm exci- tation at 0.78mm in Tm:Ho:ZBLAN. 5459J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 da Vila et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 186.217.234.225 On: Tue, 14 Jan 2014 13:05:55 luminescence behavior is described by 95%~a! of the Inokuti–Hirayama contribution and 5%~b! by the exponen- tial term. The luminescence decays of3F4 and 5I7 excited states of Tm and Ho ions in Tm~0.5 mol %! and Ho~0.5 mol %!:ZBLAN were observed be pure exponentials with lifetimes of 8.9 and 16.4 ms, respectively. The use of a dipole–dipole (s56) interaction for the energy transfer cases and systems used in this work is justified because the concentration level of dopants is not high enough to evidence the higher order of dipole interactions~i.e., @Tm# and @Ho# <6 mol %). We have observed that a dipole–dipole (s56) interaction dominates the Er–Er cross-relaxation process in- volving 4S3/2 and 4I15/2 in YLF crystals ~and Er:ZBLAN glasses! for Er concentration up to 20 mol %. Deviation from the Inokuti–Hirayama donor solution for the dipole–dipole (s56) interaction starts when the Er concentration reaches 40 mol % YLF crystals. In this case, a fast component decay of the 4S3/2 excited state appears and the fitting has to con- sider the Inokuti–Hirayama donor solutions for dipole– dipole and dipole–quadrupole (s58) interactions. However, the main contribution still comes from the dipole–dipole in- teraction. The results on the luminescence decay of5I7(Ho31) at ;2 mm and3H4 (Tm31) at ;1.5 mm showed that the Tm →Ho ET is dominated by the acceptor luminescence solu- tion based on the Inokuti–Hirayama model, while both Tm →Tm CR and (3H4) Tm→Tm migrations are dominated by the donor solution of the Inokuti–Hirayama model. The ex- perimental transfer parameterg~expt! is larger than predicted by diffusion theory for the two-sample set of codoped Tm- :Ho:ZBLAN samples. This fact corroborates the assumption that a fast excitation diffusion modifies the excitation distri- bution among excited donors and acceptor ions produced initially by laser excitation att50. After the excitation dif- fusion, two types ofD* –A interaction are expected. First, one expects aD* –A interaction whereD* interacts with all the acceptors inside of the interaction volumeV leading to the Inokuti–Hirayama-type solution~contribution fraction A!. Second, theD* ion can transfer its excitation to the nearest-neighborA ion, giving rise to the exponential decay ~localizedD* –A interaction!. The a and b parameters ob- tained from the best fit using Eq.~11! for 3H4 (Tm31) lumi- nescence at;1.5 mm @or Eq. ~12! for 5I7 (Ho31) lumines- cence at;2 mm# can be also calculated assuming the existence of a critical radius (RC) and using the distribution function f (R)dR betweenD andA ions and assuming that a localizedD* –A energy transfer occurs only ifD* –A pairs are separated by a distanceR,RC . On the other hand, the Inokuti–Hirayama transfer predominates in the case ofR .RC . Obviously, one must assume the opposite for the case of a localizedD* –A interaction. According to this, I 5E RC ` f ~R!dR5expS 24pcA8 3 ~RC 3 2Rm 3 ! D , ~16! T512expS 24pcA8 3 ~RC 3 2Rm 3 ! D , ~17! whereRm is the minimum distance between rare-earth ions in ZBLAN estimated to be equal to 3.78 Å. HerecA8 is the apparent acceptor concentration felt by the donor after the fast excitation diffusion occurrence that modified the initial D* –A distribution. This apparent acceptor concentration (cA8 ) was obtained from the relation cA8 /cA 5g(expt)/g(theor). Figure 11~a! exhibits the Inokuti– Hirayama contribution~I! due to the fast excitation diffusion for the (3H4) Tm–Tm CR ~open circles! and Tm→Ho ET ~open squares! as a function of the apparent acceptor concen- tration (cA8 ) for set ~i! @Tm(x):Ho(1%):ZBLAN #. Figure 11~b! exhibits the data for theD* –A localized interaction contribution ~T!. The solid and dotted lines were used for representing the best fit of the data of (3H4) Tm–Tm and (3F4) Tm–Ho interactions, respectively. The results show that the critical radius that triggers the localized (3H4) Tm–Tm interaction is equal to 4.76 Å@andRc55.09 Å for the (3F4) Tm–Ho interaction#. Table IV showsI andT values obtained from5I7 (Ho31) luminescence at;2 mm and I and T values obtained from 3H4 (Tm31) luminescence at;1.5 mm. The values ofI and T were plotted as a function of Tm31 concentration (x 50.5, 1, 3, 6, and 9 mol %!, as well as a function of the total @Tm(x)#1@Ho(y)# concentration@or (x1y)51.5, 2, 3, 5, 7, and 10 mol %#. These plots are shown in Figs. 12~a! and 12~b! usingI andT obtained from a best fit of Ho emission at 2 mm. Figures 13~a! and 13~b! were obtained usingI andT FIG. 11. Inokuti–Hirayama~I! and localizedD* –A interaction~T! contri- butions observed in Tm(x%):Ho(1%):ZBLAN after laser excitation.~a! shows the contribution~I! for the (3H4) Tm–Tm cross-relaxation process ~open circles! and for Tm→Ho ET ~open squares!. ~b! shows the contribu- tion ~T! of the localizedD* –A interaction that occurs after the fast excita- tion diffusion, for the above interaction systems. The solid and dotted lines represent the best fit obtained using Eqs.~16! @~a!# and~17! @~b!#. cA8 is the apparent acceptor concentration calculated using the ratio between the ex- perimental and theoretical values of the energy transfer constant~g!. 5460 J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 da Vila et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 186.217.234.225 On: Tue, 14 Jan 2014 13:05:55 obtained from a best fit of Tm emission at 1.47mm. Both figures show that when the@Tm31# or (@Tm31#1@Ho31#) concentration increases, the contribution of the rate equation solution~T! also increases and the Inokuti–Hirayama contri- bution ~I! decreases~I equalsT for Tm3154.5 mol %). This result is consistent with the assumption that for low- concentrated systems fast excitation diffusion can produce a less localizedD* –A interaction: i.e., in this case, we have I .T. The increase of the Tm31 or Ho31 concentration fa- vors the production ofD* –A pairs having a separation dis- tanceR