Ground-state absorption saturation and thermo-lensing effect as main sources of refractive index...

$: Ground-state absorption saturation and thermo-lensing effect as main sources of refractive index non-linear change in Cr4+:YAG at CW 1.06 /mum excitation$
Ground-state absorption saturation and thermo-lensingeffect as main sources of refractive index non-linear change

in Cr4þ:YAG at CW 1.06 lm excitation

A.V. Kir�yanov*, Yu.O. Barmenkov, M. del Rayo, V.N. Filippov

Centro de Investigaciones en Optica A.C., Apartado Postal 948, Leon 37000, Gto, Mexico

Received 3 June 2002; received in revised form 15 September 2002; accepted 16 September 2002

Abstract

Non-linear change of refractive index in Cr4þ:YAG is investigated experimentally by the Z-scan technique (CW

case) at wavelength 1.06 lm. Theoretical analysis of the processes resulting in non-linear refraction in Cr4þ:YAG allows

one to conclude that the main contributions in non-linear change of the crystal refractive index are the effect of Cr4þ

centers� population perturbation due to ground-state (GS) absorption saturation and the thermo-lensing effect.

� 2002 Published by Elsevier Science B.V.

Keywords: Cr4þ:YAG; Z-scans technique; Absorption saturation; Thermo-lensing effect

1. Introduction

During the last decade, Cr4þ:YAG became a

routine passive Q-switch component in neodymium

and ytterbium lasers ðk � 1 lmÞ allowing their re-

liable giant-pulsed operation [1–3]. Cr4þ:YAG isalso used as an active laser medium for near-IR

spectral range, k � 1.4–1.5 lm [4].

Cr4þ:YAG possesses of some interesting non-

linear features; one of them is the latent anisotropy

of Cr4þ centers [5–8], which appears due to

ground-state (GS) absorption saturation under the

action of powerful resonant ðk � 1 lmÞ radiation.The latter stems from the fact that a linearly po-

larized pump wave interacts, during propagation

along the crystal, with three equivalent resonantly

absorbing linear dipoles (Cr4þ centers) oriented

along the main crystallographic axes of YAG lat-tice [8,9] (Fig. 1). Another interesting phenomenon

found recently in Cr4þ:YAG is anomalously slow

extra-bleaching under CW pumping of moderate

power [10].

Meanwhile, there are points of the Cr4þ:YAG

physics, which were not, to the best of our

knowledge, addressed in literature. One of them

concerns the non-linear change of refractive index,Dn, in Cr4þ:YAG at CW pumping. Only [5,8]

touch this problem, but in both cases short-pulse

(ns) excitation of Cr4þ:YAG is treated. However,

*Corresponding author. Tel.: +52-477-717-5823; fax: +52-

477-717- 5000.

E-mail address: [email protected] (A.V. Kir�yanov).

Optics Communications 213 (2002) 151–162

www.elsevier.com/locate/optcom

0030-4018/02/$ - see front matter � 2002 Published by Elsevier Science B.V.

PII: S0030-4018 (02 )02016-3

mail to: [email protected]

in conditions of CW pumping the two main con-

tributions in non-linear refraction should be pres-

ent: (1) the refractive index change caused by Cr4þ

centers� GS population perturbation (PP) (owing

to GS absorption saturation) and (2) the thermal-

lensing (TL) effect caused by the sample inhomo-

geneous heating [11,12]. Therefore, an analysis ofnon-linear refraction in Cr4þ:YAG and possible

discrimination of their parts answering to the PP

and TL contributions is of great importance. Ad-

dressing of this problem may bring more clarity in

circumstances, where, say, a CW-pumped

Cr4þ:YAG crystal is used as a lasing medium for

femtosecond pulses� generation and account for all

sources of the cavity dispersion change is critical[13].

Experimentally, non-linear change in refractive

index Dn in Cr4þ:YAG (k¼ 1.064 lm) may be

performed applying the Z-scan technique [14].

This technique is well-known method applicable

for measuring non-linear change of refractive in-

dex for a wide class of mechanisms (Kerr non-

linearity, two-photon absorption, etc.). Howeverthe Z-scan technique suffers from definite limita-

tions in some particular cases. For instance, as

shown in [15], there are evident obstacles for sim-

ple interpretation of so-called Z-scans in a me-

dium, where non-linear refraction is accompanied

by notable saturating of resonant absorption.

Moreover, an absorbing medium, even without

taking into account of the saturation effect, issubjected to significant thermal loading in cir-

cumstances of CW irradiation, which leads to

formation of a thermal lens in it. The correspon-

dent change in refractive index, as shown in [16],

can be analyzed with the Z-scan technique at se-

rious limitations only.

Study of non-linear change of refractive index

in Cr4þ:YAG is, therefore, a complicate task.

These crystals are characterized by significant

saturable GS absorption, which obviously leads todeviations of the probe beam shape from Gauss-

ian. The latter inserts uncertainties in the corre-

spondent Z-scans� interpretation [15]. In addition,

GS absorption in Cr4þ:YAG itself (as well as

possible excited-state (ES) absorption and/or re-

sidual losses) results, via the sample heating, to

non-linear TL effect [12,13,17]. Thus, one might

expect mixing of the PP and TL effects inCr4þ:YAG, which complicates the analysis of the

overall Dn. Let us also note that the time-resolved

Z-scans method [16,25] cannot, unfortunately, be

reliably applied in case of Cr4þ:YAG crystal, since

the characteristic response times of the PP and TL

effects are too close (�4 and �13 ls, respectively)in this case; see Section 3.4.

Misunderstanding of the latter features was acause of uncertainties and even errors occurred in

[18,19], where the Z-scan technique is applied for

study of Cr4þ:YAG. Ref. [18] addresses evaluation

of GS and ES absorption cross-sections in

Cr4þ:YAG using the open-aperture Z-scans. How-

ever, authors of [18] tried to fit their experimental

Z-scans traces, a priori supposing presence of ES

absorption in Cr4þ:YAG and ignoring the provenfact [20] that its cross-section (at k¼ 1.06 lm) tends

to be zero; in addition, they ignored the role of

Cr4þ:YAGnon-linear anisotropy [5–8]. Besides, the

experiment [18] was performed with a CW train of

femtosecond pulses as the probe, thus leading to the

sample heating, whichwas not taken into account in

the numerical modeling. The mentioned causes led

to misestimating of Cr4þ:YAG GS absorptioncross-section and overestimating of ES absorption

cross-section and also to the doubtful conclusion

that these parameters depend on pump intensity

and Cr4þ:YAG preparation. One more luck of [18]

is absence of any information about the probe beam

waist and focusing features of the Z-scan experi-

ment, which does not allow any check of the results.

There was also an attempt [19] to estimate non-linear refractive index change in Cr4þ:YAG from

the closed-aperture Z-scans. However, in that

Fig. 1. Schematic of interaction of probe radiation with

Cr4þ:YAG. Ep gives orientation of probe beam polarization

azimuth with respect to linear resonant dipoles (Cr4þ centers).

152 A.V. Kir�yanov et al. / Optics Communications 213 (2002) 151–162

preliminary study we, unfortunately, ignored the

same important features: estimation of Dn in

Cr4þ:YAG was done without taking into account

of the TL effect and slow extra-bleaching. Besides,

a probe beam in that investigation was not purely

Gaussian, which also led to additional overesti-mating of Dn (see, for instance [21]).

An aim of the present paper is to report on a

thorough experimental study and modeling of

non-linear change of refractive index in Cr4þ:YAG

by its probing with CW radiation at wavelength

1.06 lm applying the Z-scan technique and to

correct the results [19]. The study was performed

for a special orientation of the probe beam po-larization azimuth (Ep), which coincides with ori-

entation of one of the Cr4þ centers� group in YAG

(Fig. 1, h ¼ 0Þ. Such geometry allows one to avoid

the role of non-linear absorption anisotropy (and

birefringence) in Cr4þ:YAG. Comparison of the

data obtained experimentally with the PP and TL

effects� modeling allowed us to conclude that

contributions in overall refractive index change ofboth effects are of the same sign (a positive non-

linear lens is formed in Cr4þ:YAG as their ap-

pearance). The modeling analysis showed that the

TL effect in Cr4þ:YAG at CW pumping is com-

patible (or dominant, incidence power exceeding

�1 W) with the PP effect.

2. Experimental

2.1. Experimental set-up

The single-beam Z-scan technique scheme [14]

is shown in Fig. 2. A tested sample, Cr4þ:YAG

crystal of the length L¼ 0.3 cm with non-saturated

(T0in) and whole-saturated (T0fin) transmittance of31% and 85%, respectively, was probed by a beam

with purely TEM00 spatial distribution, which is

launched from a CW ytterbium fiber laser (k¼1.06 lm) of 8 W of maximum output power. The

beam was collimated and then focused onto the

sample by a set of lenses 1–3. Non-linear change of

the Cr4þ:YAG sample transmission and the cor-

respondent change of refractive index Dn weredetermined by measuring so-called ‘‘open-’’ and

‘‘closed-aperture’’ Z-scans, respectively [14,22].

The scans were obtained by splitting of the beam

passed the sample by a glass wedge (2�) and

simultaneous recording of the two resultant beamsby photodetectors PD0 and PD1, with an aperture

of transmission S in front of the latter. First of the

signals gives, after normalization on input power,

open-aperture Z-scans, T0ðZÞ, and the second one,

after normalization on T0ðZÞ, gives closed-aper-

ture Z-scans, T1ðZÞ (Z is the longitudinal coordi-

nate of the Cr4þ:YAG sample with respect to the

lens 3 focal point). The input beam power wasvaried by a set of neutral filters set between the

laser and the Z-scan stage. An important chain of

the scheme is a chopper placed between the lenses

1 and 2, which allowed to keep the same pump

intensity on the crystal, but significantly (1/60)

decrease thermo-loading. The latter was important

to avoid additional heating of the sample due to

non-ideal cooling conditions at the edges. The set-up was automated by the use of a movable stage

with the Cr4þ:YAG sample. Therefore, we were

able to simultaneously record the signals from the

photodetectors and a voltage signal from a linear

rheostat connected with the movable stage, which

provided a current coordinate Z. The recorded

signals were collected and analyzed with a per-

sonal computer equipped with an acquisitionboard and the LabVIEW program.

2.2. Experimental results

The Cr4þ:YAG crystal was, as a primary step,

tested for determining the reliable values of its

transmition coefficient in non-saturated, T0in, andwhole-saturated, T0fin, states. Not entering in de-tails, note that we used for this aim a set-up

Fig. 2. Layout of experimental set-up.

A.V. Kir�yanov et al. / Optics Communications 213 (2002) 151–162 153

described in [8,23], where a tested sample is irra-

diated by a pulsed (10 ns) neodymium laser (k ¼1:06 lm) with a practically rectangular beam

shape inside the crystal. From this experiment, we

determined T0in � 31% and T 0fin � 85%. Let us

mention that Z-scan experiments have to be donewith a purely Gaussian probe beam, thus resulting,

at high pump power, in integral saturated trans-

mittance a bit less than 85% (owing to non-whole

GS absorption saturation at the wings of Gaussian

distribution).

The Z-scan measurements were done with a

probe beam from the ytterbium fiber laser, which

profiles, at the place of the lens 3, are shown in

Fig. 3 for two incidence powers, P ¼ 0:5 and 5 W.It is seen that a purely Gaussian distribution of the

beam is kept with high accuracy within the pump

interval (it is quite important to ensure validity of

the Z-scan technique to estimate Dn [21]). Apply-

ing the Gaussian beam algebra we determined the

beam waist at focus to be x0 ¼ 14:5 lm.

Fig. 4 shows the open-aperture Z-scans, T 0ðZÞ,for different incidence powers, P ¼ 0:8, 1.45, 3.2,and 6.4 W. The data of non-normalized trans-

mittance, T0max, corresponding to the open-aper-

ture Z-scans� maxima versus incidence power is

shown in Fig. 5 by curve 1.

Fig. 6 demonstrates the closed-aperture

Z-scans, T 1ðP ; ZÞ, where a pinhole (S¼ 0.1)

diaphragm is set before the photodiode PD1, for

P¼ 0.8, 1.45, 3.2, and 6.4 W. It is seen that anoscillation between the T1ðP ; ZÞ extrema is of the

‘‘valley-peak’’ type, ensuring the correspondent

non-linear lens being positive. One can also note

more asymmetry of the latter scans at input powerFig. 3. Experimental probe beam profiles at P ¼ 0:5 W (full

circles) and 5 W (empty circles) W. Solid line – Gaussian fit.

Fig. 4. Experimental open-aperture Z-scans T0ðZÞ for different incidence powers P.


increase. The dependence of a half-magnitude ofthe oscillation T 1maxðP Þ is plotted in Fig. 5 by

curve 2. The latter data were used to calculate Dnin Cr4þ:YAG.

3. Modeling and discussion

As shown in [15,25–27], a simple measuring ofan oscillation between the T 1ðP ; ZÞ extrema (Fig.

6) is not enough to calculate properly Dn in the

conditions of strong GS absorption saturation and

thermal loading of the sample. In this section, amodel of these effects is given for a CW-pumped

Cr4þ:YAG crystal.

3.1. The PP effect

The system of equations addressing the

problem of propagation of a CW resonant laser

beam in a Cr4þ:YAG crystal in case of coinci-dence of the beam polarization azimuth with

orientation of one of the Cr4þ groups (Fig. 1) is

as follows:

(a) (b)

Fig. 5. Dependencies of maxima of open-aperture Z-scans T0 (a) and half-magnitude of the ‘‘valley-peak’’ oscillation of closed-

aperture Z-scans T1max (b) on incidence power P. Curves 1,2 – experiment; curves 3–6 – modeling.

Fig. 6. Experimental closed-aperture Z-scans T1ðZÞ for different incidence powers P.


oIoz0

¼ � rpN0I1þ ðI=IsÞ

�þ c

�I ; ð1Þ

owoz0

¼ � drpN0I2 1þ ðI=IsÞð Þ : ð2Þ

Set of Eqs. (1) and (2) is the reducing to the CW

case and the particular geometry accepted of the

system derived in [8]. In Eqs. (1) and (2), I and ware, respectively, the intensity and phase of CW

radiation propagating along Cr4þ:YAG (z0 is the

current coordinate inside the sample and it should

not be confused with the sample coordinate Z);

Is ¼ hv=rps is the characteristic saturating intensity(hv is the energy quanta at wavelength kexc ¼1:06 lm, rp is the Cr

4þ:YAG GS absorption cross-

section (transition 3B1–3A2), and s is the lifetime of

Cr4þ centers in the ES); N0 is the concentration of

Cr4þ centers of one of the three allowed orienta-

tions; c is the coefficient of residual (non-satu-

rated) losses, and d is the atomic resonance

detuning parameter connected with the ratio

between real and imaginary parts of Cr4þ:YAG

polarizability (i.e., with presence of Cr4þ centers in

YAG lattice).

The steady-state population of Cr4þ centers in

the GS, N, is connected with the ratio of the

incidence and saturating intensities (I ; Is) as

N ¼ N0

1þ ðI=IsÞ: ð3Þ

The values of the model basic parameters are

listed in Table 1. It should be noted that data forrp available in literature vary by more than an

order of magnitude (from 3:5 10�19 to 6 10�18

cm2 [28–31]). We use in Table 1 the value, which

allows the best fitting of the experimental depen-

dence T0maxðP Þ by theory (see curve 1, Fig. 5). The

parameter d is taken to be 0.2 [8,24,32,33]. Con-

centration N0 of Cr4þ centers in YAG is calculated

as

N0 ¼ � ln T0in=T 0finð ÞrpL

; ð4Þ

where L is sample length.

Table 1

Cr4þ:YAG (##1–20) and characteristic setup (##21–25) parameters

1 Refractive index n0 ¼ 1:8

2 Thermal index gradient dn=dT ¼ 9:8 10�6 K�1

3 Thermal conductivity j ¼ 0:13 W=cm K

4 Specific heat C ¼ 600 J=kg K

5 Thermal expansion coefficient, [1 0 0] a ¼ 8:2 10�6 K�1

6 Poisson�s ratio m ¼ 0:3

7 Photoelastic coefficients Cr ¼ 0.017; Ct ¼ �0:0025

8 Density q ¼ 4:56 g=cm3

9 Concentration of Cr4þ centers N0 ¼ 4:5 1018 cm�3

10 Atomic resonance detuning parameter d ¼ 0:2

11 GS absorption cross-section rp ¼ 6:8 10�19 cm2

12 ES absorption cross-section rESA ¼ 0

13 Residual losses c ¼ 0:661

14 Length L¼ 0.3 cm

15 Diameter £ ¼ 1 cm

16 Initial and final transmittance T0in ¼ 0:31, T0fin ¼ 0:85

17 Cr4þ lifetime (30 �C) s ¼ 4:0 10�6 s

18 Saturation intensity Is ¼ 67 kW=cm2

19 Fluorescence quantum efficiency e ¼ 0:9

20 Stokes losses n ¼ 0:35

21 Incidence polarization azimuth orientation h ¼ 0�22 Incidence pump power P¼ 0.1–8.0 W

23 Energy of pump quanta ht ¼ 1:86 10�19 J

24 Pump wavelength kexc ¼ 1:06 10�4 cm

25 Beam waist x0 ¼ 14:5 10�4 cm


Note that modeling of the PP effect is not

thorough, since it does not accounts for the ES

absorption in Cr4þ:YAG (which is negligible at

k ¼ 1:06 lm [20]) and some small part contribut-

ing the GS absorption due to the weak r-transi-tion 3B1–

3E (rr is shown [20] to be about 1/30 ofrp). The modeling does not also account for the

slow extra-bleaching of Cr4þ:YAG under CW ex-

citation (see [10] and discussion below, Section

3.4), since neither physical origin of the latter

process, nor its features are still clear.

Eqs. (1) and (2) may be numerically calculated

for a Gaussian beam as incidence with the initial

intensity and phase given as

I0 r; Zð Þ ¼ 2Ppx2 Zð Þ exp

�� 2r2

x2 Zð Þ

�; ð5Þ

w0 r; Zð Þ ¼ kr2

2R Zð Þ ; ð6Þ

where P is the input power (P ¼ 0–8 W), r is the

current beam radius along the Gaussian intensityenvelope, xðZÞ and RðZÞ are, respectively, the

spot-size and radius of curvature of the focused

beam; Z is the position of Cr4þ:YAG with respect

to the focal point (Z ¼ 0, R0 RðZ ¼ 0Þ ¼ 1 and

x0 xðZ ¼ 0Þ) and k ¼ 2p=kexc (kexc ¼ 1:06 lm is

the pump wavelength).

The closed-aperture Z-scans T 1ðP ; ZÞ are nu-

merically generated by using a formalism based onKirchhoff integral calculation over a pinhole S of

the radial size r1 [14,22,34]

EaðP ; ZÞ ¼ ikZ r1

0

EoutðP ; Z; rÞ2prdrðd � ZÞ ; ð7Þ

where

EoutðP ; Z; rÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIoutðP ; Z; rÞ

pexp

�� r2

w2ðZÞ

�

þ ikr2

2RðzÞ þikr2

2ðd � ZÞ

��

exp f � iwoutðP ; Z; rÞg ð8Þ

is the laser beam electric field at the sample output.

The values IoutðP ; Z; rÞ and woutðP ; Z; rÞ dwPP

ðP ; Z; rÞ are the solutions of Eqs. (1) and (2). The

parameters xðZÞ and RðZÞ at the entrance are

governed by the well-known Gaussian beam

algebra; d is the distance between the diaphragm

and focal point.

One can easy calculate now the open- and

closed-aperture Z-scans, T0ðP ; ZÞ and T1ðP ; ZÞ.The dependence T0ðP ; ZÞ is found as an integrated(on r) transmittance of the Cr4þ:YAG crystal,

which is dependent on the incidence power and

sample coordinate only

T0ðP ;ZÞ ¼R10

IoutðP ; Z; rÞrdrR10

I0ðP ; Z; rÞrdr: ð9Þ

T1ðP ;ZÞ is written as

T1ðP ;ZÞ ¼ EaðP ; ZÞE�aðP ; ZÞ

EaðP ; 0ÞE�aðP ; 0ÞT0ðP ; ZÞ

: ð10Þ

The results of numerical calculations for

Cr4þ:YAG with parameters listed in Table 1 are

shown in Fig. 7.

It is seen that a positive non-linear lens (an

oscillation between the T1ðP ; ZÞ extrema is of the

‘‘valley-peak’’ type [14]) is formed in the sample atCW pumping, which is the appearance of the PP

effect. It is seen that an oscillation� amplitude has a

characteristic dependence versus incidence power

P: at low input intensities, I < Is, it grows, but atexceeding Is the growth is saturated. The latter

feature is the obvious consequence of the GS

absorption saturation.

Fig. 7. Theoretical closed-aperture Z-scans T1ðZÞ at PP effect

modeling. Curves 1–6 correspond to incidence powers P ¼0:2 ð1Þ, 0.8 (2), 1.6 (3), and 3.2 (4) W.


The dependences of T 0max and T 1max versus

pump power P are given in Fig. 5 by curves 3 and

4, correspondingly.

3.2. The TL effect

One can deduce from comparison of the ex-

perimental and theoretical dependencies T 1maxðP Þ(compare curves 2 and 4 in Fig. 5) that there is no

quantitative agreement between them.

Despite both dependencies T 1maxðP Þ demon-

strate a positive non-linear lens induced in

Cr4þ:YAG, neither amplitude of an oscillation

T1max, nor its characteristic dynamics versus P iscompatible. We supposed that such a discrepancy

is connected with prevailing, especially at high-

power CW excitation, of a thermal non-linear lens

(the TL effect) over the lens due to the GS ab-

sorption saturation (the PP effect).

The TL problem in a thermally loaded

Cr4þ:YAG crystal is treated usually [11,12,17] by

numerical solving the heat-transport equation,which allows one to get temperature distributions

across the Cr4þ:YAG sample and to evaluate a

thermal lens in the area, where the distributions

demonstrate a quadratic (versus radius r) depen-

dence. It is not our aim to give here a thorough

numerical modeling of the TL effect in Cr4þ:YAG.

Fortunately, it can be addressed by a simpler way.

Basing on the approach developed in [26,35,36],the focal length Fth of a thermal lens in a crystalline

absorbing sample at CW pumping is calculated as

FthðP ; ZÞ ¼4jpx2ðZÞ

nPabsðds=dT Þ; ð11Þ

where the optical path length change, ds=dT , is

determined as

dsdT

¼ dndT

þ ðn0 � 1Þð1þ tÞa þ n30aCr;t

: ð12Þ

Here, Pabs is the absorbed power, n is the fractional

heat load parameter (which gives the part of inci-

dent power transformed, due to non-radiative re-

laxation processes, to heat), n0 is the refractive

index of a ‘‘cold’’ sample, dn=dT , j, m, a, and Cr;t

are, correspondingly, the temperature gradient ofrefractive index, thermal conductivity, Poisson�s

ratio, thermal expansion coefficient, and photo-

elastic coefficients of the material (see Table 1).

It is seen from Eq. (12) that ds=dT is a sum of

three contributing terms. Equally with the evident

source of refractive index change given by thermaldispersion (dn=dT ), there are thermal expansion

(second term) and photoelastic (third term) con-

tributions. All the terms are present in the case of

CW pumped Cr4þ:YAG crystal, since YAG lattice

is subjected to thermally induced deformations.

Let us evaluate values of the parameters enter-

ing formulas (11) and (12).

Unfortunately, there are not reliable data inliterature for Cr4þ:YAG crystal; however, it may

be a good approximation the use of data for

Nd3þ:YAG, since in sense of the thermal loading

problem, these materials are virtually identical. As

it seen from Table 1, dn=dT � 9:8 10�6 K�1;

whereas the term characterizing the crystal thermal

expansion, ðn0 � 1Þð1þ tÞa, is estimated to be

�8:5 10�6 K�1 (YAG is very isotropic material,and coefficient a only slightly depends on its cut

orientation [37]; we took in Table 1 its value for

the [1 0 0] cut, which is the case of our study). One

can see that the first two terms in (12) are very

close in magnitude. Regarding the third term, let

us note that it is different for r- and t-possible

polarization orientation and also for different

propagations of light along the crystal, resulting inthermally induced birefringence in YAG. For the

[1 1 1] cut YAG, the correspondent values are

n30aCr � 0:8 10�6 K�1 and n30aCt � �0:1 10�6

K�1; and it is known that for the [1 0 0] cut YAG

these parameters are of the same order, differing

from the above values by 15–20% [38,39]. We

underlined in Section 1 that it is not the aim of this

study to investigate non-linear anisotropy and bi-refringence of Cr4þ:YAG; thus, we use further as

an estimate for the [1 0 0] cut Cr4þ:YAG the above

value n30aCr. In any case, it is clear that partial

weight of the last term in (12) is within �5% of a

sum of the first two terms and cannot lead to se-

rious errors in final results.

Hence, we can conclude that ds=dT � 19:310�6 K�1; and only thermal dispersion (dn=dT )and thermal expansion ðn0 � 1Þð1þ tÞa are the

important sources of thermally induced change of

refractive index in Cr4þ:YAG.


The fractional heat load parameter n is deter-

mined by the Stokes losses in the sample, i.e., by the

ratio of exciting (pumping) and fluorescence wave-

length, kexc=kfluor, and fluorescence quantum effi-

ciency, e

n ¼ 1� ekexc

kfluor

: ð13Þ

In case of Cr4þ:YAG, e � 0:9 and kfluor ¼ 1.45–

1.5 lm [4], giving n � 0:35:In circumstances of GS absorption saturation,

formula (11) should be generalized for taking into

account of the dependence of absorption coeffi-

cient on input intensity

FthðP ; ZÞ ¼4jpx2ðZÞ

n 1� T0ðP ;ZÞð ÞP ðds=dT Þ ; ð14Þ

where T0ðP ; ZÞ is the transmittance of the sample,

which is automatically generated at numerical

simulation of the PP effect (see above).Using formulas (11)–(14), we can now calculate

a thermal lens, or its dioptric power DP ¼ 1=Fth inCr4þ:YAG (see Fig. 8). It should be noted that,

despite a thermal lens of rather high dioptric

power is formed in Cr4þ:YAG under CW pump-

ing (Fig. 8), this does not essentially influence on

the beam spot inside the crystal (see, for instance

[12]), but strongly changes the beam wave frontcurvature. Assuming the sample to be a thin lens,

one can calculate the wave front behind it as

follows:

R�ðP ; ZÞ ¼ FthðP ; ZÞRoutðP ; ZÞFthðP ; ZÞ � RoutðP ; ZÞ

; ð15Þ

where Fth is determined by formula (14) and

RoutðP ; ZÞ is the wave front curvature in the place

of Cr4þ:YAG (Rout is generated numerically at

simulation of the PP effect). The correspondentchange in non-linear phase shift, dwTL, should be

introduced at calculation of the Kirchhoff integral

for obtaining thermally modified T1ðP ; ZÞ Z-scans.The results of the latter modeling are given in

Figs. 5 and 9.

Numerically generated dependence T1maxðPÞowing to the TL effect is shown in Fig. 5 by curve

5, whereas overall dependence T 1maxðP Þ caused byinterfere influence of the non-linearities (PP+TL)

is plotted by curve 6.

Fig. 9 presents the resultant closed-aperture Z-

scans, which are obtained for different powers of

incidence (P ¼ 0:8, 3.2, and 6.4 W) with taking

into account of: (i) the PP effect only (curves 1); (ii)

the TL effect only (curves 2), and (iii) the overall

TL+PP effect (curves 3). Comparing theoreticalZ-scans in Fig. 9 (curves 3) and the correspondent

experimental scans (Fig. 6), one may conclude

on full agreement between the experiment and

theory.

Hence, we can conclude that (i) a positive overall

non-linear lens is formed in Cr4þ:YAG at CW

excitation, which stems from the PP and TL effects

and (ii) its thermal part determined by the TL effect)strongly prevails the electronic part (the PP effect),

especially at high (>1 W) power of incidence.

Our estimates show that the latter conclusions

are valid not only for the treated Cr4þ:YAG

sample, but for similar crystals having initial

transmittance T0in of less than �93–95%.

3.3. Evaluation of DnðIÞ in Cr4þ:YAG

Let us finally discuss the details of determining

the non-linear phase shift, DwNLover ¼ dwPP þ dwTL,

and the correspondent change of refractive index

Dn in Cr4þ:YAG which are the results of influence

of the PP and TL effects.

Fig. 8. Theoretical values of thermal lens dioptric power DP at

TLmodeling for incidence powers P ¼ 0:2 (1), 0.8 (2), 1.6 (3), and

3.2 (4) W.


Averaged (on input intensity) non-linear phase

shift DwNLover (when the sample is set at Z ¼ 0) is

calculated as

DwNLoverðP Þ ¼

Z 1

0

dwPPðP ; 0; rÞ��

þ dwTLðP ; 0; rÞ�I0ðP ; 0; rÞrdr

� Z 1

0

I0ðP ; 0; rÞrdr: ð16Þ

Therefore, overall change of refractive index Dnin Cr4þ:YAG at the beam waist is given as follows:

DnðPÞ ¼ DwNLoverðP ÞLeff

kexc

2p; ð17Þ

where Leff is the effective sample length

Leff ¼1� e�ðrN0þcÞL

rN0 þ c: ð18Þ

Since P ¼ I0px20=2 (I0 is the input intensity in

maximum of Gaussian distribution), formula (17)gives the wished dependence Dn ¼ DnðI0Þ.

The results of numerical calculations are given

in Fig. 10.

It is seen that non-linearities stemming from the

PP and TL effects are of a different nature (com-

pare curves 1 and 2). The part of Dn caused by the

PP effect (curve 1) obviously demonstrates satu-

ration, which is not surprising. Meanwhile, un-saturated character of overall Dn (curve 3) at high

pumps is because of the TL effect contribution

(curve 2), which partial weight increases due to

(a)

(b)

(c)

Fig. 9. Theoretical closed-aperture Z-scans T1ðZÞ for differentincidence powers P. Curves 1–3 correspond to modeling of:

PP effect only (1), TL effect only (2), and overall PP+TL ef-

fect (3).

Fig. 10. Dependencies of Dn versus normalized intensity (I=Is).Curves 1–3 correspond to modeling of: PP effect only (1), TL

effect only (2), and overall PP+TL effect (3).


residual (non-saturated) losses (c) present in the

Cr4þ:YAG sample.

Therefore, in conditions of strong GS absorp-

tion saturation and high thermo-loading (as in the

case of CW-pumped Cr4þ:YAG), one should be

careful at evaluating DnðIÞ by a simple measuringof the T1p�v oscillation magnitude; otherwise,

definite mistakes might arise. Even though GS

absorption saturation [15] is accounted for, the TL

effect may cause additional (and big) phase shift in

the beam wave front, resulting in extra-deforma-

tions of the T1ðZÞ Z-scans (see, for instance

[40,41]).

3.4. Remaining questions

Comparing shapes of the experimental open-

aperture Z-scans T0ðP ; ZÞ (Fig. 4) and the theo-

retical ones (Fig. 7(a)), one may conclude that

widths of the first peaks are bigger than those

numerically calculated. Inspite of this feature, as

we cleared up, does not essentially influence on theshape and amplitude of the correspondent closed-

aperture Z-scans T1ðZÞ (compare Figs. 6 and 9)

and, therefore, does not result in notable uncer-

tainties in DnðIÞ measurement (the correspondent

error is within 15%), this fact should be addressed

in more detail.

We found out experimentally that widening of

the experimental T 0ðZÞ traces is due to some spe-cific slow extra-bleaching (�15–20%) during CW

irradiation of Cr4þ:YAG. Fig. 11 clearly demon-

strates that this extra-bleaching process is fitted, at

least, by two characteristic times lying in millisec-

onds and seconds ranges.

Remind that PP effect in Cr4þ:YAG has char-

acteristic response time s ¼ 4 ls, which is consid-

erably less than the times seen in Fig. 11. From theother side, inhomogeneous heating (TL effect)

across the Cr4þ:YAG disc cannot be a physical

mechanism responsible for the slow extra-bleach-

ing as well. Solution of the heat-transport equation

for Cr4þ:YAG gives the temperature rise inside the

crystal DT ¼ 4� and 21�, respectively, for the

crystal positions Z¼ 0 and �0.5 cm and pump

P ¼ 6:4 W. These changes could make the open-aperture Z-scans T0ðZÞ only narrower (tempera-

ture dependent lifetime of Cr4þ centers, sðT Þ, leads

to an increase of saturating intensity IsðT Þ ¼hv=rsðT Þ). The characteristic response time of the

TL effect in Cr4þ:YAG at CW pumping is deter-

mined by thermal properties of the YAG lattice

and is estimated to be [35]

sth ¼x2qC4j

; ð19Þ

where q and C are the density and specific heat of

YAG (see Table 1), giving sth � 13 ls, which is

also much less than characteristic times of the slow

extra-bleaching (see Fig. 11).

Thus, physical origin of the slow extra-bleach-

ing in Cr4þ:YAG at CW pumping is not still clear;

its investigating exhibits a separate problem, which

should be addressed elsewhere.

4. Conclusions

We report on experimental study of non-linear

change of refractive index Dn in Cr4þ:YAG at CW

excitation at wavelength 1.06 lm applying the Z-

scan technique. Theoretical analysis of experi-mental data allows to conclude that the main

contributions in overall refractive index change in

Cr4þ:YAG are the TL and PP effects. Both effects

result in the same (positive) sign of a self-induced

non-linear lens formed in the crystal, but the

TL effect is shown to prevail the PP one at higher

(>1–2 W) power of excitation.

Fig. 11. Slow extra-bleaching of Cr4þ:YAG at CW pumping.


Acknowledgements

A.V.K. and Yu.O.B. wish to thank CONACyT

(Mexico) for partial support of this work via

Projects #32269-E and #32195-E, respectively.

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