. INTRODUCTIONefractive index changes (RIC) and deformations inducedy intensive pumping and amplified beam in solid-statective elements affect the spatio-temporal and energyharacteristics of laser generation significantly. The ther-al lensing effect (due to thermally induced volume index
hange and deformations) has been actively studied in re-ent years [1–9]. However, as the thermalization energyecreases under narrowband diode-pumping of the activelements with small quantum defect, the electronic lens-ng (caused by population change of ground and excitedtates having different polarizabilities [10]) should beaken into account along with the thermal lensing.
The electronic mechanism of RIC in the laser crystalsnd glasses has been recently investigated both theoreti-ally and in experiments [11–16]. The electronic RIC wasound to dominate over the thermal effects in Yb-dopedrystals and glasses under pulse pumping [11–13]. More-ver, the electronic mechanism may be comparable withhe thermal mechanism in Nd-doped materials [14–17].
This paper is devoted to the study of the lenses inducedn laser disks and rods (based on the broadly used Yb:YAGnd Nd:YAG crystals) by pumping and amplified beams.he estimations of electronic lenses are based on experi-ental measurements of polarizability difference of
oped ions in the excited and ground states [11,13–15].he RIC and deformations (bulging of end faces and bend-
ng of disks) are taken into account. Conditions of domi-ation of one or the other lens components are analyzed.
. ORIGIN OF ELECTRONIC AND THERMALENSES IN YB:YAG AND ND:YAGIC in intensively diode-pumped laser crystals can beritten as the sum of electronic ��ne� and thermal ��nT�
The electronic mechanism of RIC is related to the dif-erence in polarizability �p of active ions in the excited2F5/2 in Yb:YAG and 4F3/2 in Nd:YAG) and ground (2F7/2n Yb:YAG and 4I9/2 in Nd:YAG) states [10] (energy leveliagram, see [1] Fig. 1). The electronic component of RICor cubic crystals can be described (to the approximationf one excited level) by the following expression [11]:
�ne =2�FL
2
n0�p�N2, �2�
here n0 is the unperturbed refractive index; FL= �n02
2� /3 is the Lorentz factor; and �N2 is the change ofopulation of the metastable level.The excited-state population N2 can be found from the
ate equation. An amplified signal, amplified spontaneousmission (ASE), and up-conversion (in Nd:YAG) beingaken into account, one can write [1,6]
�N2
�t= +
�03N0
h�03Ip�r,z,t� −
�21N2
h�21Ia�r,z,t� −
N2
�21
−�ASE
�21N2
2 − �up-convN22, �3�
here �03 and �21 are the absorption and emission cross-ections; �21 is the spontaneous lifetime of the upper en-rgy level; N0 is the ground-state population; (N0+N2N=const; N is the active ion concentration); h islanck’s constant; �03 and �21 are the pump and lasing
requencies; Ip�r ,z , t� and Ia�r ,z , t� are the pump andmplified signal intensities (we suppose the pump beamo be Gaussian, wp to be a radius: Ip�r ,0 , t�I0 exp�−2r2 /wp
2�); �ASE is the constant characterizingSE; �up-conv is the upconversion rate (in Yb:YAGup-conv=0). We chose for the simulation the commonlysed active ion concentrations [1,10]. � is proportional
ASE
010 Optical Society of America
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364 J. Opt. Soc. Am. B/Vol. 27, No. 3 /March 2010 E. Anashkina and O. Antipov
o �21 and a geometrical factor: one can estimate �ASE0.5��21L for the disks (if 2wp�L), and �ASE0.5��21wp
2 /L for the rods (where L is the active elementength) [4]. It is important to state that under nonlasingonditions ASE and upconversion change dynamics of thenduced lens and the effective lifetime �eff=�21/ �1�ASEN2+�up-convN2� becomes smaller than �21. Equation
3) is solved numerically by the Runge–Kutta method.The temperature rise �T, mechanical and thermal
tresses of pumped crystal lead to the thermally inducedIC [1–3]:
�nT = � �n
�T��T + �nph = �� �n
�T� + 2n03�TC���T, �4�
here ��n /�T��T is the RIC at constant strain of a virtualerfectly rigid crystal; the term �nph=2n0
3�TC��T ac-ounts for the photoelastic effect [1,2]; C� denotesolarization-averaged “photoelastic constants” [1,2]; andT is the thermal expansion coefficient.The temperature T�r ,z , t� in the laser element can be
ound from the heat equation (thermal load takes place inadiationless transitions) [18]:
Cp
�T
�t− K�T = h�43N4w43 + h�32N3w32 + h�10N1w10,
�5�
here K is the thermal conductivity of the crystal (a func-ion of doping concentration [19]); is density; Cp is spe-ific heat; �43 and w43 are the frequency and rate of theadiationless transition from the upper “upconversion”evel “4” to the pump level “3;” N4 is the population of thepper “upconversion” level (2G9/2+ 4G11/2, 2K15/2+ 2D3/2 orG7/2+ 2K13/2+ 2G7/2 in Nd:YAG); �32 and w32 are the fre-uency and rate of the transition from the pump level “3”o the upper laser level “2;” N3 is the population of theevel “3;” �10 and w10 are the frequency and rate of theransition from the low laser level “1” to the ground state;nd N1 is the low laser level population. As the radiation-ess relaxation time is negligible in comparison with thepper laser level lifetime, one can estimate thermal load
rom rate equations as follows:
Fig. 1. Energy levels of Yb:YAG and Nd:YAG.
N4w43 � �up-convN22, �6�
N3w32 ��03N0
h�03I�r,z,t�, �7�
N1w10 � �ASEN22 + �up-convN2
2 +N2
�21. �8�
Heat sink is presumed to occur through the lateral sur-ace (the end faces are thermo-insulating) for the rods,nd through one end-face (the lateral surface and thether end face are thermo-insulating) for the disks. Aewton-type law of cooling is used for both rod and disk
1,2,18]:
K��T + H�T − T0� = 0, �9�
here T0 is the temperature inside the copper radiator;�s cooled surface normal vector; �� is normal derivative;nd H is the heat-transfer coefficient. The nonstationaryeat equation (5) with the boundary condition (9) on theooled surfaces is solved numerically by the finite differ-nce method.
The thermal stresses and deformations of disks andods can be determined by solving the thermo-elasticroblem in the quasi-static statement with the thermaleld calculated from the heat equation. YAG with a cubicymmetry of the lattice is described as an isotropic crystalith two mechanical coefficients: the elasticity modulus End the Poisson ratio � [20]. We studied the thermo-lastic problem at stresses and equations of compatibilityf deformations. The solution of the thermo-elastic prob-em allows evaluating the contribution to the thermalensing related to changes of the active element shape.
The change of the optical path ��r ,z� of a straight rayparallel to the z axis) between the “z” and “z+dz” planess [1–3]
��r,z� = dz��n + �n0 − 1��zz, �10�
here �zz is the component of the stress tensor �ij. Theerm dz�n0−1��zz in the expression (10) accounts for thexial strain of the elementary volume. In the paraxial ap-roximation, the pumped crystal acts as a thin lens whoseioptric power is given by [1,2]
D = 2���/�r2�r=0, �11�
here � is the optical path difference between an on-axisentral ray �r=0� and a parallel ray shifted by distance r:
� =�0
L
���0,z� − ��r,z�. �12�
o the electronic De and the thermal DT lens dioptric pow-rs are defined by
De = �2 · 2�FL2�p
n0
�
�r2�0
L
�N2�0,z,t� − N2�r,z,t�dz�r=0
,
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E. Anashkina and O. Antipov Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. B 365
DT = 2�dn
dT�eff� �
�r2�0
L
�T�0,z,t� − T�r,z,t�dz�r=0
, �14�
here �dn /dT�eff is the effective thermo-optic coefficientncluding �n /�T, bulging of end faces and the photoelasticffect.
The thermally induced stresses should be comparedith tensile strength. The maximum stresses appearing
n the disks ��maxdisk� and the rods ��max
rod � are evaluated un-er plane stress approximation for the disks and underlane strain approximation for the rods [20]:
�maxdisk � 2
�TE
4�Tmax, �15�
Table 1. Parameters of the La
Description Symbol
ifference in polarizability of activeons in the excited and ground states
eat transfer coefficient Houng modulus Eensile strengthoisson ratio �
ctive ion concentration at 1% doping
Simulated param
oping concentration for rodsoping concentration for disksength of the rods Lhickness of CW-pumped disks Lhickness of pulse-pumped disks Lump power for rods in CW mode Pump power for rods in pulse mode Pump power for disks Pump radius for disks wp
ump radius for rods in CW mode wp
ump radius for rods in pulse mode wp
onstant characterizing ASE in rods �ASE
onstant characterizing ASE in disks �ASE
�maxrod � 2
�TE
4�1 − ���Tmax, �16�
here �Tmax is the maximum temperature difference be-ween the center and the edge of the crystal. Using pa-ameters from Table 1 one can estimate from (15) and (16)hat for �Tmax=170 K, the rod is destroyed, and forTmax=240 K the disk is destroyed. Under the conditionsdopted in the current work, the laser elements are notestroyed.
. LENSES IN RODShe divergence in the crystal of the pump beam withaussian profile and beam quality factor M2�1 is consid-
366 J. Opt. Soc. Am. B/Vol. 27, No. 3 /March 2010 E. Anashkina and O. Antipov
red [1,2]. The pump beam diameter is assumed to be lesshan the rod absorption length, so the heat flux is radial.he axial heat flux (along z) is ignored, and �2T /�z2 is ne-lected in the heat equation (5).
Numerical calculations show that, under continuous-ave (CW) broad pumping in Yb:YAG, Nd:YAG rods, theain mechanism of lens appearing is the thermal one,
ut under sharply focused pumping the electronic lenshould also be taken into account (see Fig. 2). These re-ults at broad CW pumping agree with experimental mea-urements and earlier theoretical estimation of thermalenses [2,3,5–8,21]. For Yb:YAG and Nd:YAG rods underonlasing conditions, given not too high CW pump inten-ity (when ASE, upconversion, and the pump absorptionaturation are negligible), the population of the upper la-er level is proportional to the pump intensity, and theower of the stationary electronic lens is estimated from13) as follows:
De =16�FL
2�p
n0
P�21
�h�03�
0
L exp�− �z�
wp�z�4 dz, �17�
here P is the input power of the pump beam, and ��03N is the nonsaturated absorption coefficient. Under
hese conditions the thermal lens dioptric power is foundo be
DT = �dn
dT�eff
�P�h
�K �0
L exp�− �z�
wp�z�2 dz, �18�
here �h is the fraction of the absorbed pump power con-erted into heat. For an ideal crystal with quantum effi-iency equal to one, we have �h= ��32+�10� /�03. However,n actual experiments, �h is higher than the quantum de-ect due to parasitic effects. The quantum defect is equalo 9% in Ya:YAG but is equal to 24% in Nd:YAG [1,2,22].
ig. 2. Lenses in CW-pumped rods (under non-lasing condi-ions) as a function of pump beam diameter for constant pumpower.
One can see from (17) and (18) that, for constant beamiameter, the thermal lens is proportional to 1/wp
2 and thelectronic lens is proportional to 1/wp
4. Therefore, with de-reasing pump diameter the electronic lens grows fasterhan the thermal one, but for the broad beam the totalens is determined mainly by the thermal component. Un-er sharply focused pump, some effects such as ASE,eam divergence, and pump absorption saturation lead tohe change of the dependence on beam diameter (thelectronic-to-thermal lens ratio is not proportional to/wp
2), but the electronic lens also predominates the ther-al one.Under lasing conditions, decreasing of the population
nversion by the amplified beam leads to decreasing of thelectronic lens. In experiments, the thermal lens inb:YAG and Nd:YAG can decrease as well, due to the non-
deal quantum efficiency [22]. In Nd:YAG, decrease in thehermal lensing under lasing conditions can also takelace due to the upconversion weakening. However,ithin the framework of an ideal Yb:YAG crystal model,
he thermal lens may increase, but the total lensing mayecrease. If the amplified beam diameter is smaller thanhe pump diameter, a minimum may appear in the spatialistribution of the upper level population N2 [see Fig.(a)]. In this case the electronic lens is defocusing. Themaller the ratio of the amplified beam diameter to theump diameter, the higher is the ratio of the electronicontribution to the total lens. A minimum of the electronicens is specified by the emission saturation. For colli-
ated beams the saturation effects can be described asollows:
dIp
dz= − �03IpN0, �19�
dIa
dz= �21IaN2. �20�
he results of solving the system of equations (3), (19),nd (20) allow calculating the temperature and the popu-ation N2 fields and estimating (for the Gaussian ampli-ed beam profile) the electronic and the thermal lensessee Fig. 3(b)].
ig. 3. (a) The spatial distribution of the population inversion inhe middle-rod plane z=L/2 for various input amplified beamowers given in W. (b) Lenses in the Yb-doped rod under CWumping as a function of the input amplified beam power. Theump beam diameter is twice as large as the amplified beamiameter.
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E. Anashkina and O. Antipov Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. B 367
In a pulsed mode, the pulse duration �pulse is supposedo be comparable with the relaxation time of the elec-ronic lens �eff. Numerical calculations demonstrate thathe electronic lens predominates over the thermal lens foroth Yb:YAG and Nd:YAG crystals under nonlasing con-itions (see Fig. 4). This fact was also noted in [17]. Thelectronic lens predomination can be explained within theramework of a simple model independent of pump diam-ter, and taking into account nonradiative transitionsrom level 3 to level 2, and from level 1 to level 0. For ahort pulse �pulse��T (where �T�wp
2 /4� is the character-stic thermal relaxation time on beam radius, �=K /Cp ishe thermal diffusivity) the spatial derivatives are ne-lected in the heat equation (5) and without ASE, upcon-ersion, and saturation effects the thermal lens becomes
DT = � �32 + �10
�03t −
�10
�03�21�1 − exp�−
t
�21���
8�dn
dT�eff
�P
�c�
0
L exp�− �z�
wp�z�4 dz. �21�
The rate equation (3) (for �ASE=0, �up-conv=0, N0�N)s integrated; after that the electronic lens is estimatedrom (13) by the following expression:
De =16�FL
2�p
n0
P�21
�h�03�1 − exp�−
t
�21���
0
L exp�− �z�
wp�z�4 dz.
�22�
n this approach, the lenses have the same dependence onhe beam diameter �1/a0
4�, but the rise-time behavior isifferent [compare expressions (21) and (22)].
. LENSES IN DISKSfixed disk is deformed under the action of mechanical
tresses, so the phase front of a test beam can be distortedven in the absence of pumping [4,9]. Powerful pumpingntroduces additional deformations and lenses, which arenalyzed here.It is assumed that an ideal mirror is deposited on the
nd face z=L of the disk. The disk is pumped by a power-ul collimated beam with the radius wp significantlyreater than the disk thickness wp�L. For the axial heatux the heat equation (5) is one-dimensional. The thermo-
ig. 4. Lenses in pulse-pumped rod (under non-lasing condi-ions) as a function of a pump pulse duration.
lastic problem is solved under the plane stress approxi-ation with two types of boundary conditions: fixed andonfixed end face z=L. It is assumed that heat sink at theoundary is not disturbed under disk deformations. Thenfluence of the disk bulging on bending is neglected. Theroblems of bulging and bending are considered indepen-ently [20]. In this model, the thermal lens component re-ated to the effective thermo-optical coefficient can beritten as [2]
DT = 4� �n
�T+ 2n0
3�TC + �T�n0 − 1��1 + ��� � �
�r2�0
L
�T�0,z,t� − T�r,z,t�dz�r=0
, �23�
here the term �T�n0−1��1+�� is related to thermaltrength. The lenses in the disk calculated for two pas-ages of the test beam; the thermal lens component DTEq. (23)] and the electronic lens power De are twice thosen the analogous expressions (13) and (14) in the rod.
Due to the temperature difference between the endaces �T�z=0��T�z=L�� the disk bends. The test beam isefocused by a convex mirror, and an additional lens termm arises. The problem of bending is considered within
he framework of the theory of bending of thin plates withmall deflection [20]. The solution of this problem allowsne to determine the curvature of the mirror �r (when r0 in the direction of the z axis):
�r =�1 + ���T
2, �24�
here �T depends on time as on parameter:
�T =12�T
L3 �0
L
T�0,z,t� · �z − L/2�dz. �25�
he expressions (24) and (25) allow estimating the diop-ric power of the thermal lens component related to theirror bending �Dm=2�r�. The induced lens is repre-
ented as a sum of three terms. The first of these de-cribes the electronic lens, the second and the third termsescribe the thermal lens:
D = De + DT + Dm. �26�
he dependence of Dm component on pump diameter dif-ers from that for DT and De (without ASE and saturationffects Dm�1/wp
2, whereas De and DT�1/wp4). For the
xed disk end-face, the bending component power Dm isqual to zero.
Analytical and numerical consideration of the lenserms (26) shows that under CW pumping the thermalens predominates the electronic one for both the Yb:YAGnd the Nd:YAG crystal (see Fig. 5), which is in goodgreement with the measurements and with the previousheoretical estimates for thermal lenses [4,9]. This results true for both lasing and nonlasing conditions. If the endace is fixed, the induced lens is focusing. If the end facesre nonfixed, the thermal lens component DT (related tohe effective thermo-optic coefficient ��dn /dT�eff� is focus-ng, but the bending thermal lens is defocusing, so the to-
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368 J. Opt. Soc. Am. B/Vol. 27, No. 3 /March 2010 E. Anashkina and O. Antipov
al lens can be either focusing or defocusing. The thermalens DT depends on the heat transfer coefficient H (theetter the heat conductor, the smaller the DT components), but the bending lens does not depend on it, and itsower Dm is determined by the temperature gradient onhe z axis for r=0.
Under pulse pumping (under the nonlasing condition)he electronic lens predominates the thermal lens compo-ent DT (see Fig. 6). For short pump pulses, whenpulse��T (in disks �T�L2 /�), the thermal and the elec-ronic lensing time dependences are analogous to the ex-ressions for lenses in rods (21) and (22), respectively. Un-er intense pumping, the lenses grow faster [see Fig.(d)]. This occurs due to the decrease of the effective laserevel lifetime as a result of ASE and upconversion [4–6].).he significant contribution of the electronic component
ig. 5. Lenses in CW-pumped disks (under non-lasing condi-ions) as a function of disk thickness for constant pump powernd pump radius.
ig. 6. Lenses in pulse-pumped disks (under non-lasing condi-ions) as a function of a pump pulse duration for fixed end-facend non-fixed end face.
o the total lens is confirmed by experimental measure-ents of RIC under pulse pumping in the Yb:YAG andd:YAG laser crystals [11–15], as well as in other crystals
23]. For the pulse duration �pulse��21, the electronic lensower is stronger than the thermal lens power DT. The to-al lens can be both focusing and defocusing (see Fig. 6),epending on disk thickness and beam diameter.
. CONCLUSIONSummarizing the results, the magnitude of the transientlectronic lens effect is several times stronger than thehermal lens effect in pulse-pumped rods and disks (basedn both Yb:YAG and Nd:YAG crystals) under nonlasingonditions. The electronic lens can also predominate thehermal lens under sharply focused CW end-pumping.owever, the thermal lens predominates the electronic
ens in CW broadly pumped rods and disks. The deforma-ions (bending and bulging of end faces) make a substan-ial contribution to lensing in the disks—the volume RICeing the main mechanisms in rods. The electronic lensffect is weakened under lasing conditions (for compa-able amplified and pumping beam diameters) due to de-reasing of the excited level population. Besides, if the di-meters of the pumping and amplified beams differtrongly, the defocusing contribution of the electronic lensay be significant.
CKNOWLEDGMENTShis research was supported in part by the Russian Foun-ation for Basic Research through grant RFBR-CNRS 07-2-92184 and the program of the Russian Academy of Sci-nce, Nonlinear-Optical Methods and Materials for Nexteneration of the Laser Systems. E. Anashkina alsocknowledges support from the Dynasty Foundation.
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