Excited-state absorption and fluorescence dynamics in Er:CaF2

Author’s Accepted Manuscript Excited-state absorption and fluorescence dynamics in Er:CaF2 C. Labbé, J.L. Doualan, R. Moncorgé, A. Braud, P. Camy www.elsevier.com/locate/jlumin

PII: DOI: Reference:

S0022-2313(18)30068-1 https://doi.org/10.1016/j.jlumin.2018.04.007 LUMIN15522

To appear in: Journal of Luminescence Received date: 12 January 2018 Revised date: 20 March 2018 Accepted date: 4 April 2018 Cite this article as: C. Labbé, J.L. Doualan, R. Moncorgé, A. Braud and P. Camy, Excited-state absorption and fluorescence dynamics in Er:CaF 2, Journal of Luminescence, https://doi.org/10.1016/j.jlumin.2018.04.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Excited-state absorption and fluorescence dynamics in Er:CaF2 C. Labbé, J.L. Doualan, R. Moncorgé*, A. Braud, P. Camy Centre de recherche sur les ions, les Matériaux et la Photonique (CIMAP), UMR 6252 CEACNRS-ENSICAEN, Université de Caen Normandie, 6 Blvd Maréchal Juin, 14050 Caen, France *

Corresponding author. [email protected]

Abstract: Emission, ground- and excited-state absorption spectra of Er:CaF2 single crystals were registered from the visible to the mid-infrared. These spectra were calibrated in unit of cross section and analyzed to derive purely radiative lifetimes and branching ratios. Fluorescence decays were then recorded to determine effective emission lifetimes and branching ratios including non-radiative multiphonon relaxations. These fluorescence data along with population ratios and emission intensity measurements performed versus pumping rates were finally confronted with a rate equation model to derive energy transfer rates. Keywords: Erbium, CaF2, fluorescence, excited-state absorption, laser 1. Introduction Depending on the host material and the energy of the involved lattice phonons, most of the Er3+ energy levels lying in the mid-infrared up to the green already gave rise to more or less efficient laser emissions, the most important ones (see in Fig. 1) being the well-known emission around the telecom wavelength of 1.55µm (4I13/24I15/2 transition), the emission around 2.8µm (4I11/24I13/2), the emission around 4.5µm (4I9/24I11/2), the emission around 3.5µm (4F9/24I9/2) and the emission around 550nm (4S3/2 4I15/2). Other rare earth ions like Ho3+ or Pr3+ also exhibit such extraordinary multi-wavelength emission properties. However, Er3+ ions present the advantage of having emitting levels which can be all excited (see in Fig. 1) either directly or via multi-step absorption processes with the aid of standard and commercially available semiconductor laser diodes and diode-pumped solid-state lasers operating around 1500nm, 980nm, 800nm, 650nm and 530nm. The aim of this article is to report on the majority of these excitation and emission processes in the case of Er:CaF2. Indeed, CaF2 is a crystal which has been known since a long time but which is arising more and more interest nowadays because of a number of attractive properties. It can be grown rather easily in very large size with an excellent optical quality. Moreover, when it is doped with a rare-earth ion like Nd3+ [1], Er3+ [2], Tm3+ [3], or Yb3+ [4], it gives rise to quite broad near- and mid-infrared optical bands suitable both for diode pumping and the production of ultra-short pulses, thus characteristics close to that found with standard laser glasses, but with much better thermo-mechanical properties, which is a crucial issue in the development of high peak power laser chains operating at high repetition rates [5, 6].

Fig. 1: Energy levels and main excitation and energy transfer processes Several spectroscopic and laser emission properties of Er:CaF2 single crystals and transparent ceramics were already published in the past [2, 7-14], but only very few results were reported concerning the emission lifetimes and the branching ratios of the various possible emission transitions, almost nothing on the excited-state absorption and emission cross sections in the absorption domain of the various emitting levels, and nothing of the energy transfer mechanisms and associated transfer rates occurring at high dopant concentration. The article is divided as follows. It starts (section 2) with an analysis of the ground-state absorption spectra within the framework of the Judd-Ofelt formalism and the derivation of radiative lifetimes and branching ratios. Then it is reported (section 3) the result of a full investigation and calibration of the excited-state absorption and emission transitions which occur within the spectral range of the ground-state absorption of each possible emitting level. Section 4 is devoted to fluorescence decay measurements and the derivation of effective emission lifetimes and branching ratios including non-radiative multiphonon relaxations without energy transfers. Section 5 concentrates on the determination of energy transfer rates through a confrontation of registered population ratio and fluorescence saturation curves versus pumping rates for different pump wavelengths and different dopant concentrations with a specific rate equation model. 2. Radiative lifetimes and branching ratios As it was demonstrated in the case of Yb3+ ions [4, 15], and in accordance with what was implicitly admitted for Er3+ [2, 10] above an ion concentration exceeding about 0.5at%, it is assumed that Er3+ ions in CaF2 enter the lattice in the same kind of slightly different local sites giving rise, at least at room temperature, to rather broad optical transitions having similar oscillator strengths. Consequently, absorption spectra can be calibrated either in terms of oscillator strengths or absorption cross sections, as if only one kind of luminescent centers

was actually active. Then, these spectra can be analyzed within the framework of the JuddOfelt (J.O.) formalism to derive parameters which can be used, in turn, to calculate radiative lifetimes and branching ratios. This is what was performed here by recording absorption spectra between 450nm and 1700nm, in order to cover most of the important optical transitions, and for CaF2 crystals with Er3+ dopant concentrations of 0.49at%, 2.57at%, 4.49at% and 10.34at%, as measured by the induction-coupled plasma technique. Averaging over the spectra led to the J.O. parameters:  1.22,  0.98, 1.72 (in unit of 10-20 cm2) and to the radiative lifetimes and branching ratios reported in Table 1. R i Table 1 : Radiative lifetimes  R and branching ratios  ij for ij transitions with i,j =

1

1,2,3,4,5,6 for levels 4I15/2, 4I13/2, 4I11/2, 4I9/2, 4F9/2, 4S3/2, respectively; energies Ei of the lower lying Stark sublevels and partition functions Z i associated with the multiplets.

 RI 4

 7.7ms ,  R I11/2  6.8ms ,  R I9/2  9.1ms 4

13/2

4

 R F  1.0ms ,  R S  0.59ms 4

4

9/2

3/2

 21R  1 31R  0.84 , 32R  0.16

41R  0.54 , 42R  0.44 , 43R  0.018

51R  0.89 , 52R  0.04 , 53R  0.06 , 54R  0.003 61R  0.67 , 62R  0.27 , 63R  0.022 , 64R  0.03 , 65R  0.0003 E14 I

15/2

 0 , E14 I

13/2

 6573cm1 , E14 I

11/2

 10291cm1 , E14 I

 12416cm1 9/2

E14 F  15396cm1 , E14 S  18544cm1 9/2

Z 4I

15/2

3/2

 7.94 , Z 4 I

13/2

 8.62 , Z 4 I

11/2

 8.76 , Z 4 I

 4.28 9/2

Z 4 F  7.32 , Z 4 S  3.34 9/2

3/2

1 We also report in Table 1 the energy Ei of the lower lying Stark sublevel of each multiplet and the partition functions Z i associated with these multiplets according to the Stark sublevel positions which can be found in [16] and by using the expression:

Zi 

 Ein  Ei1  exp    n 1  kT 

i 1/2

(1)

where gn stands for the degeneracy of each Stark sublevel n with gn = 2 for Er3+ as Kramers n ion, and Ei for the energy of the considered Stark sublevel in number i+1/2 for a low enough - less than cubic - local site symmetry, which is the case here with CaF2 [16].

3. ESA and Emission cross section spectra As already noticed, depending on the host material and the energy of the involved lattice phonons, most of the Er3+ energy levels can lead to more or less efficient laser emissions. In most cases, optical pumping can be efficiently achieved by exciting into the 4I9/2 and 4I11/2 multiplets with Ti:Sapphire or semiconductor diode lasers around 800nm and 980nm, or into the 4I13/2 multiplet with color-center or semiconductor diode lasers around 1.5µm. Added to the fact that each of the considered emitting levels can absorb light at the pump and/or the emission wavelengths, it is worth to register the emission and excited-state absorption (ESA) spectra in the ground-state absorption domain associated with each of these multiplets. This is what was performed here in the case of Er:CaF2 around 550 nm, for the 4S3/2 energy level, around 650nm for 4F9/2, around 810nm for 4I9/2, around 980nm and 2800nm for 4I11/2 and around 1550nm for 4I13/2. 3.1. Techniques and methods The method used to register these spectra is the same as used and described in the past in the case of Er:LiYF4 [17]. Briefly, the method consists in recording what is called excitedstate absorption difference spectra by using a pump-probe technique [18] based on two CW light sources, a probe lamp (a tungsten-halogen lamp source) and a pump laser, where each is modulated at different frequencies. Moreover, in the course of the calibration of these spectra in cross section unit and of the extraction of the final excited-state absorption data from of the emission and ground-state absorption spectra, a population ratio is defined and measured assuming that only the 4I11/2 and 4I13/2 metastable levels (with lifetimes of several milliseconds) can be significantly populated, whatever is the excitation wavelength. This population ratio is given by pwhere, according to the notations used in Fig. 1, N2 stands for the population density of level 4I13/2 and N3 for the population density of level 4 I11/2. Its variation with the pump rate and the dopant concentration and its confrontation with rate equations will be used in the following section to derive important energy transfer parameters. The population ratio which will be used for the calibration of all the excited-state absorption spectra was derived in fact from the data obtained around 2800nm. Indeed, in this wavelength domain, there is no ground-state absorption but only 4I11/24I13/2 stimulated emission and 4I13/24I11/2 excited-state absorption. Consequently, the expression for the excited-state absorption difference signal I() – the variation of the transmitted probe light intensity caused by the pump beam - reduces to the typical expression for a small signal gain and is written as [17] :

I ( )  Al ( N 2  N 3 ) g ( ) I with

 g ( )   p em ( )  (1   p ) esa ( )

(2) (3)

thus to a linear combination of the associated stimulated emission and excited-state absorption cross section spectra  em ( ) and  esa ( ) . l stands for the sample thickness and A for a setup parameter.

The emission cross section  em ( ) is independently obtained by recording an emission spectrum and by using the Fuchtbauer-Ladenburg expression which applies in the case of an isotropic or a cubic system like CaF2, i.e. i j  em ( ) 

 5 ijR I ij ( ) 2 i 8 n c R   I ij ( )d 

(4)

where Iij stands for the measured unpolarized emission intensity (in W/cm2) corresponding to R R the 4I11/24I13/2 emission transition, ij  32  0.16 for the associated radiative branching I i ratio and  R   R 11/2  6.8ms for the radiative lifetime of the emitting level i = 4I11/2, as reported above in Table 1. 4

The excited-state absorption cross section spectrum  esa ( ) corresponding to the reverse transition is obtained in turn from the stimulated emission cross-section spectrum by using the reciprocity expression, i.e. j i i j  esa ( )   em ( )

Zi  Z 4 I

where

 8.76

11/2

1 ZL  E 1I 4

11/2

 E14 I

13/2

and

 hc Zi exp   Zj  kT

Z j  Z 4I

13/2

 1 1      ZL   

(5)

 8.62 , as reported in Table 1, and

 3718 cm1 for the inverse of the so-called « zero-line » wavelength.

The population ratio p is then obtained by looking for the best linear combination of  em ( ) and  esa ( ) which fits to the bandshape of the I() excited-state absorption difference R i spectrum. With such a procedure, it is even not necessary to know the ratio ij /  R entering in the Fuchtbauer expression to derive the p value.

In the other spectral domains investigated in this work, i.e. around 550 nm, 650nm, 810nm, 980nm and 1550nm, the procedure is slightly different since in the excited-state absorption difference spectra there might be contributions from ESA and stimulated emission but also from ground-state absorption (GSA). For these cases, still assuming that only the ground- and excited-levels 4I15/2, 4I13/2 and 4I11/2 are populated enough to give rise to significant ESA signals, the expression for the excited-state absorption difference signal must be adapted (see Exps (5) and (6) in [17]) and use must be made of the GSA data directly expressed in cross section unit. 3.2. Experimental results The results obtained around 2800nm are reported in the Figs. 2a and 2b. Fig. 2a displays the  em ( ) and  esa ( ) spectra derived from the above two expressions (4) and (5). Fig. 2b shows the I() spectral data and the fit of a linear combination of the two cross section spectra to these data by using p  0.5. From this result, the best adjustment is found in turn I R I R for 32 /  R 11/2  25 , in very good agreement with the value 32 /  R 11/2  23.5 (see in Table 1) which is obtained from the J.O. analysis of the ground-state absorption spectra. 4

4

1.0

(a)

4

I11/2

4

I13/2

Cross section (10

-20

2

cm )

0.8 0.6

em

0.4 0.2 0.0

esa

-0.2 -0.4

4

4

I13/2

I11/2

-0.6

2500

2600

2700

2800

2900

3000

Wavelength (nm) 0.15

Gain cross section g (10

-20

2

cm )

(b)

0.10

gmeasured

p=0.5

g=SE-(1- )ESA

0.05

0.00

-0.05 2500

2600

2700

2800

2900

3000

Wavelength (nm)

Fig. 2: (a) Stimulated emission and excited-state absorption cross section spectra, (b) Measured gain (excited-state absorption difference) spectrum and adjustment with p = 0.5, around 2800nm.

The results obtained in the other spectral domains are reported in the figures 3 to 7. Examining these figures, several remarks can be made. According to Fig. 3, in the green spectral range, ESA mainly occurs around 560nm due to 4I11/2 4G7/2 and 4I13/2 2H9/2 transitions, and around 580nm due to a 4I11/2 4G9/2, 2K15/2 transition. No significant ESA occurs around the main emission peak at about 538.3nm.

0.8

gsa 4I 15/2

2 cm )

0.2

- esa , gsa , em (10

0.4

-20

0.6

4

S3/2

em 4

4

S3/2

I15/2

0.0 -0.2

- esa

-0.4 4

4

I11/2

-0.6

G7/2

4

530

4

I11/2

2

G9/2 + K15/2

2

I13/2

-0.8

4

H9/2

540

550

560

570

580

590

600

Wavelength (nm)

Fig. 3: Emission, ground- and excited-state absorption cross section spectra around 550nm In the red range (Fig. 4), ESA occurs between 630nm and 670nm due to 4I13/2 4F5/2+4F3/2 transitions, and between 680nm and 740nm due to 4I11/2 2H9/2 and 4I13/2 4F7/2 transitions. No significant ESA occurs around the long wavelength emission peak located at 670nm. On the contrary, ESA does occur around the other emission peaks located at 651nm and 656nm.

- esa , gsa , em (10

em

gsa

0.4

-20

2 cm )

0.6

4

4

I15/2

4

F9/2

4

F9/2

I15/2

0.2

0.0 esa -0.2

4

I11/2

4

G11/2 4

-0.4 600

4

620

4

I13/2

640

4

I11/2

2

I13/2

4

F7/2

H9/2

4

F5/2 + F3/2

660

680

700

720

740

760

Wavelength (nm)

Fig. 4: Emission, ground and excited-state absorption cross section spectra around 650nm Around 820nm (Fig. 5), strong ESA occurs in the whole spectral range corresponding to the 4I9/2 4I15/2 emission transition, which is not favorable for this emission. However, it can be noticed that a strong overlap exists around 800-810nm between 4I15/24I9/2, 4I13/22H11/2 and 4I11/24F5/2+4F3/2 GSA and ESA transitions. Knowing that in most of the Er-doped materials, including CaF2, level 4I9/2 is a very short-lived emitting level (see in the following section), the above mentioned spectral overlap will be favorable for up-conversion excitation, after two-step absorption of 800nm pump photons and subsequent multiphonon relaxations

from the above lying multiplets 2H11/2, 4F7/2 and 4F5/2, of the 4S3/2 emitting level. Such upconversion excitation process was indeed implemented in the past in the case of Er:LiYF4 [19] for which room‐temperature continuous wave laser emission was achieved at 551 nm (4S3/24I15/2 emission transition) after excitation by a Ti:sapphire laser at 810 nm. 0.2 4

em

I9/2

gsa

2

cm )

4

I15/2

0.1

4

4

I9/2

I15/2

- esa , gsa , em (10

-20

0.0 -0.1 4

2

I13/2

-0.2

4

H11/2

4

I 13/2

S 3/2

esa

-0.3 -0.4 -0.5

4

4

I11/2

4

F5/2+ F3/2

-0.6 760

780

800

820

840

860

880

900

Wavelength (nm)

Fig. 5: Emission, ground- and excited-state absorption cross section spectra around 800nm Around 980nm (Fig. 6), strong ESA also occurs in the whole spectral range corresponding now to the 4I11/2 4I15/2 emission transition, which is again not favorable for this emission. Similarly, a strong overlap also exists between 4I15/24I11/2 and 4I11/24F7/2 GSA and ESA transitions. But here, the GSA cross-section around 980nm is more than four times larger than that around 810nn. Consequently, the up-conversion process associated with the consecutive absorption of two 980nm pump photons will be much more efficient than with 810nm photons. This was nicely demonstrated in the case of Er:LiLuF4 to achieve green laser emission around 552nm [20] by using pump photons at 974nm.

4

0.3

I15/2

4

I11/2

4

I11/2

4

I15/2

- esa ,gsa , em (10

-20

2

cm )

em

gsa

0.2 0.1 0.0 -0.1 -0.2

esa

-0.3

4

-0.4

I11/2

880

900

920

940

960

980

4

F7/2

1000 1020 1040

Wavelength (nm)

Fig. 6: Emission, ground- and excited-state absorption cross section spectra around 980nm

In the end, as shown in Fig. 7, in the mid-infrared spectral range around the wavelength of 1550nm, no ESA occurs, except at the very long-wavelength end of the 4I13/24I15/2 emission spectrum above about 1625nm where the 4I13/24I9/2 ESA transition takes place. 0.8

em , gsa, em-esa (10

4

I15/2

I13/2 4

I13/2

4

I15/2

gsa

-20

2

cm )

4

0.6

0.4

0.2

em

0.0 em -esa -0.2 1400

4

I13/2

1450

1500

1550

1600

1650

4

I9/2

1700

1750

1800

Wavelength (nm)

Fig. 7: Emission, ground- and excited-state absorption cross section spectra around 1550nm 4. Effective emission lifetimes and branching ratios including non-radiative multiphonon relaxation Branching ratios and emission lifetimes reported in Table 1 do not account for the nonradiative multiphonon relaxations between the adjacent energy levels. Such non-radiative relaxations added to energy transfers occurring between ions at high enough dopant concentrations usually increase depopulation of energy levels, what is resulting in shortened emission decays and drastic changes of branching ratios. Such effective emission decays, with i the time constants noted  f , and branching ratios noted  ij are those which must be used in the usual rate equations needed to be solved to simulate or model fluorescence dynamics and/or laser operation. The effective fluorescence time constants  f were obtained from the experimental fluorescence decays which were registered after direct excitation of each of the considered emission levels by using a standard wavelength tunable OPO (Optical Parametric Oscillator) pumped by a Q-switched Nd:YAG laser. When the fluorescence decays were slightly nonexponential, which occurred for the higher doped samples, the time constants were obtained by calculating the average values given by the expression: i

f 



1 I (t )dt I (0) 0

(6)

where I(t) stands for the fluorescence intensity at time t. The results are reported in Table 2.

Table 2: Effective fluorescence lifetime  if of each emitting level i versus dopant concentration knowing that 1%Er  2.45x1020 Er ions/cm3 Emitting level 0.49at%Er 2.57at%Er 4.49at%Er 10.34at%Er 4 S3/2 22µs 15µs 11µs 5µs 4 F9/2 160µs 159µs 149µs 146µs 4 I9/2 6.6µs 5.9µs 5.4µs 3.1µs 4 I11/2 5.6ms 7.5ms 7.3ms 4.7ms 4 I13/2 9ms 8.2ms 5.7ms 2.4ms

Comparing these values between each other and with the purely radiative ones gathered in Table 1, several remarks can be made. Emitting levels 4I13/2 and 4I11/2 are not subject to nonradiative multiphonon relaxations and do not vary significantly, except for the highest one, with the dopant concentration. Emitting level 4I9/2 is drastically subject to non-radiative multiphonon relaxation, which is expected due to its proximity with the level 4I11/2 lying just below it, but is rather insensitive to dopant concentration, thus to energy transfers, at least up to about 4.49%Er. Emitting level 4F9/2 is also subject to non-radiative multiphonon relaxation but not so much as the latter, and it is quite insensitive to energy transfers. Emitting level 4S3/2 is both strongly sensitive to multiphonon-relaxation and dopant concentration, thus to energy transfers. Using the lifetime values obtained with the lower doped sample (0.49at%Er), i.e. the sample for which up-conversion and cross-relaxation energy transfers are supposed to be sufficiently reduced to be neglected, branching ratios  ij including multiphonon non-radiative relaxations, but without energy transfers, can be obtained by using the following expressions:

and thus and

ij   AijR  AiNR  if if j = i-1

(7a)

ij  A 

(7b)

R i ij f

if j < i-1

ij ijR  1 1       if  Ri   if  Ri  if j = i-1  ij ijR  if j < i-1  if  Ri

(8a) (8b)

Such branching ratios are reported in Table 3. It worth noting here that the branching ratios for the optical transitions occurring between adjacent multiplets, like the already operated 4I11/2  4I13/2 emission transition around 2.8µm [2, 7, 8, 10] or the 4F9/2  4I9/2 currently studied one around 3.5 µm [21], can have much higher branching ratios including multiphonon relaxation than the purely radiative ones, with values of 0.308 against 0.16 for the first emission transition and of 0.84 against 0.003 for the second one.

i 0 Table 3 : Emission lifetimes  f   i and branching ratios  ij derived for the 0.49%Er

sample and for the ij transitions with i,j = 1,2,3,4,5,6 for levels 4I15/2, 4I13/2, 4I11/2, 4I9/2, 4F9/2, 4 S3/2, respectively

 20  

4

I13./2

 9ms ,  30   4 I  5.6ms ,  40   4 I  6.6µs 11/2 9/2

 50  

 160µs ,  60   R S3/2  22µs 4

4

F9/2

 21  1

31  0.692 , 32  0.308 41  0.00038 , 42  0.0003 , 43  0.9993

51  0.145 , 52  0.006 , 53  0.009 , 54  0.84

61  0.026 , 62  0.01 , 63  0.001 , 64  0.001 , 65  0.962 5. Up-conversion and cross-relaxation energy transfer rates According to the past literature, at high dopant concentrations, the fluorescence dynamics in Er-doped materials can be strongly influenced, for the most important ones, by two types of energy transfer mechanisms: - Up-conversion energy transfers transferring part of the excitation pump energies of levels 4I13/2 and 4I11/2 at 1500nm and 980nm into excitation energies of levels 4I9/2 and 4S3/2 (after rapid non-radiative multiphonon relaxation from levels 4F7/2 and 2H11/2) and following the energy transfer mechanisms: 4

I13/2, 4I13/2  4I15/2, 4I9/2, noted ETU1 (see in Fig. 1) with the transfer rate W2214 4

and

I11/2 , 4I11/2  4I15/2 , 4S3/2, noted ETU2 with the transfer rate W3317

- Cross-relaxation energy transfers depopulating levels 4S3/2 and 4I15/2 at the benefit of levels 4I9/2 and 4I13/2, according to the two non-discernable energy transfer mechanisms : 4

S3/2, 4I15/2  4I9/2, 4I13/2

and

4

S3/2, 4I15/2  4I13/2, 4I9/2 (see in Fig. 1)

energy transfer mechanisms noted CR1 and CR2 (see in Fig. 1) with the respective transfer rates W6142 and W6124 or the combined one noted W61 since transfers are equivalent and cannot be distinguished the one from the other. Among all these transfer rates, only the latter in fact can be directly estimated from the 4S3/2 fluorescence decay. Indeed, if  6 is the effective emission lifetime of this level at a given dopant concentration NT (in ions/cm3), and  6 its effective emission lifetime at a weak dopant concentration (0.49%Er in the present case) for which cross-relaxation energy transfers can be neglected, the energy transfer rate W61 can be determined by using the expression: 0

W61 

1 1 1    N T   6  60 

(9)

This expression can be considered as an approximation [22] but it gives a rather good estimate for W61 . Using the 4S3/2 lifetime values and the dopant concentrations reported in Table 3, it is then found the W61 values reported in Table 4. The estimation of the up-conversion transfer rates is more delicate. It necessarily passes through a confrontation of some experimental data with a rate equation model. Two approaches were thus adopted. The former consisted in recording the fluorescence intensity of level 4I11/2 at 980nm and of level 4I13/2 at 1550nm versus the pumping rate noted Rp (in ions.cm-3.s-1) and given by the expression

Rp 

Pabs V .h p

(10)

where Pabs is the absorbed pump power, V  d .0 is the pumped volume and 0 the beam waist radius in the sample of thickness d . These measurements were performed with the two highly doped samples (4.5at%Er and 10.34at%Er) and for two pump wavelengths (799nm and 968 nm). The results obtained for the 980nm emission of the 4I11/2 level for 10.34at%Er:CaF2 are reported in the figure 8. 2

10.34 at%Er:CaF2

1.2 1.0 0.8 0.6 0.4 0.2

4

I11/2 population density (10

19

-3 ions. cm )

1.4

0.0 0.0

0.2

0.4

0.6

0.8 19

1.0 -3

1.2

1.4

-1

Pumping rate Rp (10 ions.cm .s )

Fig. 8: Calibrated 4I11/2 population density versus pumping rate Rp after 968 nm excitation; dots are for experimental data, solid line for theoretical fit. Saturation of this fluorescence intensity occurs because of the bleaching of the ground-state and because of the up-conversion processes which increase with the dopant concentration and the pumping rate. This calibration results from the adjustment of the curve with the variation which is obtained from the resolution of the following rate equations for different upconversion transfer rates W2214 and W3316 while keeping the cross-relaxation one W61 equal to the above estimated value. The second approach consisted in the determination of the above mentioned population ratio p for different pump wavelengths and different dopant concentrations, versus the pumping rate Rp (Exp. 10). The results obtained for two pump wavelengths at 968nm and 799nm and for 10.34at%Er:CaF2 are reported in the figure 9.

0.7

10.34 at%Er:CaF2 0.6

Population ratio p

0.5 0.4

 po

0.3

exp. data, exc. 799 nm simulation exc. 799 nm

0.2

exp. data, exc. 968 nm simulation exc. 968 nm

0.1 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

22

1.4

1.6

1.8

-3 -1

Pumping rate Rp (10 ions.cm s )

Fig. 9: Population ratio p versus pumping rate Rp for two excitation wavelengths; dots and squares are for experimental data, solid lines for theoretical fits. According to the model and the rate equations which will be presented and worked out a little farther, the population ratio p0 at very low pumping rate is given by:

 p0 

3

(11)  2  32   3 thus, according to the data reported in Table 3, by  p 0  0.67 . On the other hand, the limit which is reached at the high pumping rates is of the order of 0.52. This result is very close to the value of 0.5 obtained with the ESA measurements at 2800nm (see in Fig 2) which were indeed performed at high pumping rates. The numerical model and the rate equations which were used for the calibration of the above experimental data and for the derivation of the above defined up-conversion transfer rates are very close to that used in the past for other Er-doped systems [23]. According to the level and energy transfer notations used above and in Fig. 1 and the lifetimes and branching ratios reported in Table 3, the rate equations which apply in case of pumping around 799nm or 968nm into the 4I9/2 or 4I11/2 level were written as follows: dN 6 N   6  W3317 N 32  W61 N 6 N1   esa N 2 p    esa N 3 p  799 nm 968 nm dt 6

(12)

dN 5 N 6  65 N 5   dt 6 5

(13)

N dN 4 N   i i 4  4  W2214 N 22  W61 N 6 N1   gsa N1 p   em N 4p  799 nm dt i 4 i 5,6

(14)

dN 3 N N   i i 3  3  2W3317 N 32   gsa N1p  ( em   esa ) N 3 p  968 nm dt i 3 i  4,5,6

(15)

N i i 2 N 2 dN 2     2W2214 N 22  W61 N 6 N1 dt i 2 i  3,4,5,6

(16)

N1  N2  N3  N 4  N5  N6  NT

(17)

where the expressions given between square brackets represent the pumping terms when Pp excitation is produced at the pump wavelength p = 799nm or 968nm, where  p  is h p . p2 the pumping flux (in photons.cm-2.s-1) for the incident pump power Pp .  gsa ,  esa ,  em are the ground-state, excited-state absorption and emission cross sections (in cm2) at the considered pump wavelength, as given in the figures 5 and 6. As in [23-25], the energy transfer rates are assumed to be constant with the pumping rate. Solving these equations in the continuous wave regime, i.e. for

dN i  0 with i  1 to 6 , and dt

adjusting the up-conversion rates W2214 and W3317 to reach the best fits to the data (solid lines in Figs 8 and 9), it is obtained for the two highly doped (4.5%Er and 10.34%Er) samples the values which are reported in Table 4. Table 4: Estimated cross relaxation and up-conversion transfer rates (extrapolated values indicated with an asterisk *) Transfer rates W61 W2214 W3317 (10-17cm3s-1) 2.57% 3.1 0.5* 1* 4.49%Er 4.12 0.8 1.76 10.34%Er 6.08 1.00 2.29 The up-conversion transfer rates reported for the 2.57%Er doped sample were extrapolated values assuming that the transfer rates fall down to zero at zero dopant concentration. All these transfer rates are given with an uncertainty of about 20%. According to these results, it is clear that all the transfer rates increase with the dopant concentration with values around 10-17 cm3s-1, as it was reported for several other Er-doped materials such as Er:LiYF4 [26] and Er:ZBLAN [27], for instance. The relative magnitudes of the transfer rates, however, differ from one system to the other. For Er:LiYF4 and Er:CaF2, the cross-relaxation transfer rates W61 appear with larger values than the up-conversion ones, whereas it is the reverse in the case of Er:ZBLAN. On the other hand, the up-conversion transfer rates W3317 determined in the case of Er:CaF2 have larger values than the upconversion transfer rates W2214 , while it is the reverse in the case of Er :LiYF4 and Er:ZBLAN. The relative magnitudes of the two up-conversion transfer rates in the case of Er:CaF2 can be qualitatively explained and supported with the aid of the spectra reported in the figures 6 and 7. Indeed, each type of energy transfers should be roughly proportional to the overlap of the associated emission and ESA spectra (see in Fig. 1), and the overlap between these spectra around 980nm (see in Fig. 6) which is at the origin of the transfer rate

W3317 is clearly much larger than that found around 1550nm (see in Fig. 7) for the transfer rate W2214 . 6. Conclusion A comprehensive study of the main spectroscopic properties of Er:CaF2 has been realized and presented. The Judd-Ofelt analysis of the ground-state absorption spectra have allowed to derive essential parameters, i.e. radiative lifetimes and branching ratios, in the evaluation of any emission cross sections. Based on energy level positions collected in the existing literature, the partition functions for all the emitting levels have been calculated. These partition functions associated with any emission spectra calibrated in cross section unit can be used to derive absorption cross section spectra corresponding to the reverse transitions by using the reciprocity method. Excited-state absorption and emission cross section spectra have been registered within the absorption range of all the emitting levels providing important information on the possible multi-step excitations and energy transfer processes taking place in Er:CaF2 at a given pump wavelength and dopant concentration. Energy transfer rates associated with the most important cross-relaxation and up-conversion energy transfer processes have been estimated by analyzing fluorescence decays and by fitting experimental data - population ratios and fluorescence intensities versus pumping rates - with a specific rate equation model. A subsequent publication is now being prepared to present a simulation of the laser operating conditions of a newly discovered and promising broadband 3.7 µm emission (transition 4F9/2 4I9/2) of Er:CaF2 [21], based on the data presented here in addition to some complementary emission measurements and a more complete model, as the one recently developed in the case of Er:ZBLAN [28]. References 1. S. Normani, A. Braud, R. Soulard, J. L. Doualan, A. Benayad, V. Ménard, G. Brasse, R. Moncorgé, J. P. Goossens and P. Camy, “Site selective analysis of Nd3+–Lu3+ codoped CaF2 laser crystals” Cryst. Eng. Comm. 18, pp 9016-9025 (2016) and refs therein. 2. C. Labbé, J.L. Doualan, P. Camy, R. Moncorgé, M. Thuau “The 2.8 µm laser properties of Er3+ doped CaF2 crystals”, Opt. Comm. 209, pp 193–199 (2002) 3. P. Camy, J.L. Doualan, S. Renard, A. Braud, V. Ménard, R. Moncorgé, “Tm3+:CaF2 for 1.9 µm laser operation” Opt. Comm. 236, pp 395–402 (2004) 4. V. Petit, J.L. Doualan, P. Camy, V. Ménard, R. Moncorgé, “CW and tunable laser operation of Yb3+ doped CaF2” Appl. Phys. B 78, pp 681–684 (2004) 5. M. Siebold, S. Bock, U. Schramm, B. Xu, J.L. Doualan, P. Camy, R. Moncorgé, “Yb:CaF2 - a new old laser crystal”, Appl Phys B 97, pp 327–338 (2009) 6. I. Tamer, S. Keppler, M. Hornung, J. Körner, J. Hein and M. C. Kaluza, “Spatio-Temporal Characterization of Pump-Induced Wavefront Aberrations in Yb3 + -Doped Materials” Laser & Photonics Reviews DOI: 10.1002/lpor.20170021 7. T. Basiev, Y. V. Orlovskii, M. V. Polyachenkova, P. P. Fedorov, S. V. Kuznetsov, V. A. V. A. Konyushkin, V. V. Osiko, O. K. Alimov, and A. Y. Dergachev, “Continuously tunable cw lasing near 2.75 μmin diode-pumped Er3+:SrF2 and Er3+:CaF2 crystals,” Quant. Electr. 36, 591–594 (2006)

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Fig. 1: Energy levels and main excitation and energy transfer processes Fig. 2: (a) Stimulated emission and excited-state absorption cross-section spectra, (b) Measured gain (excited-state absorption difference) spectrum and adjustment with p = 0.5, around 2800nm. Fig. 3: Emission and excited-state absorption cross section spectra around 550nm Fig. 4: Emission and excited-state absorption cross section spectra around 650nm Fig. 5: Emission, ground- and excited-state absorption cross section spectra around 800nm Fig. 6: Emission, ground- and excited-state absorption cross section spectra around 980nm Fig. 7: Emission, ground- and excited-state absorption cross section spectra around 1550nm Fig. 8: Calibrated 4I11/2 population density versus pumping rate Rp after 968 nm excitation; dots are for experimental data, solid line for theoretical fit. Fig. 9: Population ration p versus pumping rate Rp for two excitation wavelengths; dots and squares are for experimental data, solid lines for theoretical fits.