Optics Communications 295 (2013) 155–160

Contents lists available at SciVerse ScienceDirect

Optics Communications

0030-40

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/optcom

Thermal model of continuous wave end-pumped passively Q-switched laser

Shuaiyi Zhang, Xia Wang n

Institute of Photonic Information Technology, School of Mathematics and Physics, QingDao University of Science and Technology, QingDao 266061, China

a r t i c l e i n f o

Article history:

Received 18 November 2012

Received in revised form

10 December 2012

Accepted 17 December 2012Available online 24 January 2013

Keywords:

Thermal effect

Passively Q-switched

Transient temperature distribution

18/$ - see front matter & 2013 Elsevier B.V. A

x.doi.org/10.1016/j.optcom.2012.12.066

esponding author. Tel.: þ86 15244231928.

ail address: phxwang@yahoo.com.cn (X. Wan

a b s t r a c t

A more accurate thermal model of continuous wave (cw) end-pumped passively Q-switched laser is

established, by treating the absorption coefficient of the laser medium and the diffractive loss as a

function of instantaneous population inversion density. The interactions between the transient

temperature distribution in laser crystal and the output pulse parameters of cw-end-pumped passively

Q-switched laser are investigated by numerically solving the coupling between the lasers rate

equations and the transient heat conduction equation. The numerical results reveal that when the

quasi-steady-state is reached, the instantaneous variations of the temperature in crystal and the

absorption coefficient can result in the instability of the output pulse train in the passively Q-switched

laser. Under the pump power of 5 W, the variation of output pulse amplitude reaches 0.5%.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Cw end-pumped all-solid-state passively Q-switched lasers withsaturable absorbers have attracted much attention for the merits ofhigh efficiency, simplicity, low cost and reliable operation [1–5].Although the passively Q-switched lasers have been applied innonlinear studies, information storage, medicine, etc., its wide appli-cations are partially limited due to the Q-switching instability whichalso occurs in passively mode-locked laser [6–9]. Li et al. attribute theQ-switch instability to thermal effect in the gain medium of thepassively Q-switch microchip laser and qualitatively investigates thetransient temperature profile [10]. Thermal effects in solid-statedlasers are unavoidable, especially when the laser medium is pumpedby the high-power pump source. The thermal distortions in laser rodshave been investigated in details [11–15], nevertheless thoseresearches are mainly interested in the thermal profile and thermaldistribution in laser medium of the laser equipments which operatein the steady or transient states. Li et al.have developed a thermalmodel to calculate the temperature distribution in an activelyQ-switched laser by solving the coupling equations between the rateequation and transient equation [16]. Until now few reports tookinsight into the interaction between the transient temperature andQ-switched laser parameter in the passively Q-switched laser.

In this paper, a novel model is developed to calculate the thermalprofile in laser medium and analyze the instability of output pulseamplitude in passively Q-switched laser by solving the couplingbetween the rate equation and transient heat equation. Treating theabsorption coefficient of the laser medium as a function of

ll rights reserved.

g).

instantaneous population inversion density, which is commonlydeemed to be a constant in heat equation [17,18], the two integratingrate equations can be obtained. For the first time to our knowledge,the transient temperature distribution and output pulse parametersof the passively Q-switched laser can be attained simultaneouslythrough one comprehensive model.

Numerically solving the abovementioned heat and rate equa-tions, we obtain the dynamics of the transient thermal buildup,the thermal profile in the laser medium, the dependences of theabsorption coefficient, the thermal focal length and pulse ampli-tude. In order to illuminate the influence of the thermal effect onthe output pulse characters, we also compare the results obtainedby our new model with that attained by the traditional rateequation solely under the same conditions.

2. Theoretical model

In this section, the finite difference and the Runge–Kuttamethods are adopted to simulate the thermal effect and theoutput characters of the passively Q-switched laser simulta-neously. As can be seen from Fig. 1, the system with which weare concerned is a line cavity laser consisting of a laser medium,the saturable absorber, and the resonator mirrors with the lengthsof two arms L1, L2 of 3 cm and 5 cm, respectively. The Neodymiumdoped yttrium vanadate laser crystal with Nd3þ doping concen-tration of 1 at% is a-cut into a dimension of 3�3�5 mm3. M1 isHR coated at 1064 nm and AR coated at 808 nm with radii ofcurvature of 20 cm. The flat mirror M2 is the output coupler withtransmission of 10% at laser wavelength of 1064 nm.

Fig. 1. Schematic diagram of Nd:YVO4 laser.

Fig. 2. Model of Nd:YVO4 crystal.

S. Zhang, X. Wang / Optics Communications 295 (2013) 155–160156

2.1. Transient heat equation and focal length of the thermal lens

As can be seen from Fig. 2, the pump power is focused into thegain medium with a spot size of a few hundred microns along theaxis (z direction). The large thermal gradient arises from the heatdeposition within a very small volume near the pumping facet ofthe laser crystal in longitudinally pumped system, resulting inthermal lens and strong aberrations at the pumping facet. Theheat diffusion equation for an anisotropic cubic laser crystal isgiven by [17]

rc@2Tðx,y,z,tÞ

t2þKx

@2Tðx,y,z,tÞ

@x2þKy

@2Tðx,y,z,tÞ

@y2

þKz@2Tðx,y,z,tÞ

@z2¼�S x,y,z,tð Þ ð1Þ

where T(x,y,z,t) is the temperature in laser medium, r is the massdensity, c is the specific heat, Kx, Ky and Kz are three differentthermal conductivities along the x, y and z orientations, respec-tively, and S(x,y,z,t) devotes the heat generation rate per cubicunit volume in watts per meter. For simplification, we assumethat the thermal property of the laser medium is independent oftemperature distribution. The heat generation rate S(x,y,z,t) isassumed to be the same as the shape of the pumping lightabsorption, which could be expressed as the Gaussian functionalong the resonator axis. In the case of end pumping, the heatsource for a laser medium with the dimension of a� b� l can bethen written as [17]

S x,y,z,tð Þ ¼2ZPina½NðtÞ�

po2p

ð1�e�a½NðtÞ�lÞ�1e�2½ðx�a=2Þ2þðy�b=2Þ2 �=o2p e�a½NðtÞ�z

ð2Þ

where, Pin is the incident pump power, Z is the fractional thermalload, op is the Gaussian beam waist of the pumped light, N is theinstantaneous population inversion density in laser medium, anda, which is commonly deemed to be a constant, represents theabsorption coefficient as a function of N(t), and can be given by[10]

a½NðtÞ� ¼ sabs½Nt�NðtÞ� ð3Þ

where Nt is the total doping concentration of active ions in lasercrystal, N(t) is the instantaneous population inversion density ofthe gain medium that can be obtained by the rate equation, andsabs is the absorption cross section at the pump wavelength.

In our theoretical analysis, Eq. (1) can be solved numerically bythe finite difference method. Part of the absorbed pumping energywas assumed to transfer into heat dissipated locally in the lasercrystal due to the quantum defect mechanism. The copper heat

sink around the periphery of the composite crystal was cooled bywater at a constant temperature. Considering that the thermalconductivity of the heat sink is much greater than that of thecrystal, the temperature at the surrounding surfaces of the crystalwas supposed to be the same as water temperature of 289 K, thusthe boundary conditions are

T �a

2,y,z,t

� �¼ T

a

2,y,z,t

� �¼ 289 ð4Þ

T x,�b

2,z,t

� �¼ T x,

b

2,z,t

� �¼ 289 ð5Þ

@Tðx,y,z,tÞ

@z

����z ¼ 0

¼ h T0�Tð Þ

@Tðx,y,z,tÞ

@z

����z ¼ l

¼�h T0�Tð Þ ð6Þ

here h is the heat transfer coefficient between the laser mediumsurface and air and T0 is the initial condition with a value of 289 K.

The inhomogeneous temperature distribution in crystal leadsto the stress, the strains and the displacement of the crystalresulting in the changes of refractive index inside the crystal. Forthe light propagating along the resonator axis, the beam distortedand the variation of wave front appeared. The optical pathdifference (OPD) for one pass through the crystal can be writtenas [17]

OPD x,y,tð Þ ¼ n0az

Z c

0½Tðx,y,z,tÞ�T0�dzþ

Z l

0

@n

@TT x,y,z,tð Þdz ð7Þ

where n and n0 are the refractive indices of the laser medium atT and at the room temperature, respectively, and az is the thermalexpansion coefficient. In the pumped region, it is common toconsider the laser crystal to be a thermally induced sphericalconvex lens. The focal length of the lens could be derived fromEq. (7)

f ðx,y,tÞ ¼ ðx2þy2Þ=2ðOPDo�OPDðx,y,tÞ ð8Þ

where OPDo denotes the OPD in the center of the laser crystal.The focal length of the thermal lens is estimated by using thefollowing parameters qn/qT¼3.0�10�6, n0¼1.9573 and az¼4.43�10�6 K�1[17].

2.2. The diffractive loss

According to the aberration diffraction theory, the diffractiveloss caused by the thermal effect of the gain medium can bewritten as [19]

dT ¼ 1�

R rb

0 exp ikDf rð Þ� �

exp � 2r2

o2g

� �rdr

R rb

0 exp � 2r2

o2g

� �rdr

��������

��������

2

ð9Þ

where Dj(r)is the wave aberration referred to as a referencesphere

Df rð Þ ¼pZPin

bl

1þ lnðr2b=o

2pÞ ðr

2ro2pÞ

r2

o2pþ ln r2

b=r2

ðr2Zo2

pÞ

8<:

9=; ð10Þ

b¼ 2pKco2p=ðdn=dTþaznÞ ð11Þ

where rb is the efficient radius of the laser crystal, op is the1/e2 radius of the Gaussian pump beam, og is the radius of theTEM00 mode at the position of active medium which can beobtained by the ABCD transmission matrix theory

Table 1The parameters for theoretical calculation [14,17,21].

Parameters Values Parameters Values

Kx 5.23 W/m/K Ky 5.10 W/m/K

Kz 5.10 W/m/K r 4.22 g/cm3

c 0.59 Jg�1 K�1 se 8.2�10–19 cm2

Lc 8 cm hgp 2.46�10–19 J

ls 0.26 cm Ns0 2�1017 cm3

l 0.5 cm ts 3.2�10�6 s

ns 1.81 t 9.8�10�5 s

R 0.9 L 0.02

op 320 mm sg 4.3�10–18 cm2

3 4 5 6 7 8 9 10 11x 105

340

340.5

341

341.5

342

342.5

Time (ns)

Tem

pera

ture

(K)

Fig. 3. Temperature evolution in the center of the pumped surface (x¼0, y¼0,

z¼0) underPin¼5 W.

S. Zhang, X. Wang / Optics Communications 295 (2013) 155–160 157

og ¼

ffiffiffiffiffiffilB

p

r1�

AþD

2

� �2" #�1=4

ð12Þ

here l is the oscillation wavelength. It is obvious that thediffractive loss dT is the function of the radius of the TEM00 modeog which depends on the transmission matrix. The transmissionmatrix under the influence of the thermal focal length f(x,y,z,t)relates to a and N. Therefore, both the instantaneous populationinversion density N and the diffractive loss dT which depend onthe thermal focal length are mutually influenced. The heat andthe rate equations are coupled together by the absorber coeffi-cient which is the function of the population inversion density N.

2.3. The coupled rate equations

From the above analysis, the coupled rate equations of thepassively Q-switched laser with Cr4þ:YAG saturable absorber canbe modified as follows [20]:

dfdt¼

ftr

2sNlþ2seNels�2sgNgls�2seNs0ls�ln1

R

� ��L�dT

� ð13Þ

dN

dt¼�gscfN�

N

t þWp Nð Þ ð14Þ

dNg

dt¼�sgcfNgþ

Ns0�Ng

tsð15Þ

here f is the photon density in the laser cavity, tr is the cavityround-trip time, tr¼2n1lþ2nslsþ2(Lc� l� ls)/vc, in which Lc, ls andl are the lengths of the laser cavity, Cr4þ:YAG saturable absorberand Nd:YVO4 gain medium, respectively, vc is the velocity of lightin vacuum, ns is the refractive index of Cr4þ:YAG, se is theabsorption cross-section of the excited state, sg is the absorptioncross-section of ground state of the saturable absorber, Ng and Ne

are the population densities of the ground state and the excitedstate respectively, Ns0 is the total population density of thesaturable absorber; R is the reflectivity of the output couple, L isthe intrinsic loss, ts is the excited-state lifetime of the saturableabsorber, and t is the stimulated-radiation lifetime of the gainmedium; the pump rate Wp is the function of N and can be givenby

Wp½NðtÞ� ¼ Pinð1�e�sabs ½Nt�NðtÞ�lÞ=hgppw2pl ð16Þ

in which hgp is the single-photon energy of the pump light.Summarizing our new model, we solve the coupling of the

heat and rate equations in the passively Q-switched laser bychanging the absorption coefficient a into variable as the functionof population inversion density in the laser medium. Correspond-ingly, the heat generation rate S(x,y,z,t), the thermal focal length,the diffractive loss and the pump rate Wp in heat and rateequations turn into the variable, respectively. All of those vari-eties make the temperature distribution in laser medium andoutput pulse parameter of the laser influence to each other.

3. Numerical results

The coupling between the heat and rate equations can besolved by treating the two equations under the same time circle.During our calculation, the time step is 1 ns. The correspondingparameters used for numerical simulation are listed in Table 1.

3.1. The thermal distribution

The transient temperature evolution process in the center ofthe pumped surface (x¼0, y¼0, z¼0) of the Nd:YVO4 lasermedium is numerically calculated at the pumping power of 5 W

as shown in Fig. 3. It can be seen that, it takes approximately10 ms for the temperature to reach a quasi-steady-state. Thetemperature in the center of Nd:YVO4 laser medium is 342.25 K,thus yielding a temperature rise TR¼53.25 K.

When the thermal field reaches the quasi-steady-state, the radialheat flow will be equal to the heat input in the crystal, and thetemperature will maintain the oscillatory behavior between themaximum temperature Tmax¼342.25 K and the minimum tempera-ture Tmin¼342.12 K in every period. The temperature instability canbe calculated from DT=Trise ¼ ðTmax�TminÞ=Trise¼0.26%, as shown inFig. 4(a). The quasi-steady-state temperature and the Q-switchedpulse have the same repetitive period with the value of 16 ms asshown in Fig. 4(a) and (b).

Fig. 5 shows the thermal field distribution on the pump facet (x–y

section, z¼0) at t¼10 ms when the temperature distribution reachesthe quasi-steady-state. It shows that the large temperature rise occursin the central region of the pump facet, which is the incidence area ofthe pump source (about 320 mm). Also in the central region thetemperature decrease very rapidly with the distance from theincidence area causing a large temperature gradient.

Fig. 6 shows the three dimensional temperature distribution inthe x�z section and o1104 along the z axial direction at t¼10 ms.The highest temperature in the center of the pumped facet was342.25 K, resulting in a temperature rise of 53.25 K. The temperaturediminishes along the z axial direction. And the temperature rise ofthe point of (x¼0, y¼0, z¼5 mm) is 3.49 K, which indicates thatthermal effect mainly exists in the foreside of the crystal.

3.2. The diffraction loss

The inversion population density in the laser medium willmaintain an oscillatory behavior during the Q-switched laser

-1.5-0.75

00.75

1.5

-1.5

-0.75

0

0.75

1.5290

300

310

320

330

340

350

x (mm)y (mm)

Tem

pera

ture

(K)

Fig. 5. Three dimensional temperature distribution in the x–y plane at z¼0 with

pump power of 5 W.

01

23

45

-1.5-0.75

00.75

1.5290

300

310

320

330

340

350

Tem

pera

ture

(K)

Fig. 6. Three dimensional temperature distribution in the x–z plane at y¼b/2.

1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1x 106

342.15

342.2

342.25

Time (ns)

Tem

pera

ture

(K)

1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1

x 106

0

1

2

3

4x 1014

Time (ns)

Tem

pera

ture

(K)

Fig. 4. (a) Temperature oscillation in the crystal within the time interval of 0.1 ms

and (b) the time evolution of the intra-cavity photon number density.

S. Zhang, X. Wang / Optics Communications 295 (2013) 155–160158

operation. At the same time, the absorption coefficient whichdepends on the inversion density will also have the same variationswith it. Fig. 7 gives the correspondence between the time evolutionsof N and the transient temperature distribution at the center of thepump surface (x¼0, y¼0, z¼0). With the increase in the inversionpopulation density, the absorption coefficient decreases accordinglyand keeps an oscillatory behavior. The maximum and minimumvalues of a are amax¼4.94 cm�1 and amin¼1.74 cm�1, respectively,and the differences of the absorption coefficient can be calculatedfrom Da¼ amax�amin¼3.2 cm�1. The reason for that the absorptioncoefficient cannot reach the initial value of 5.3 cm�1 is that inversiondensity N does not decrease to zero after the release of the laser pulse.

It can be seen from Eqs. (1) and (2) that the heat source dependson the absorption coefficient, and the thermal focal length, thus theheat source will also oscillate in a relative period. Fig. 8 depicts thefocal length of the thermally induced lens as the function of time atPin¼5 W. The thermal focal length oscillates between 53.2 cm and58.9 cm, leading to a focal length difference of 5.7 cm.

1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1x 106

1

2

3

4

5

Abs

orb

coef

ficie

nt (c

m-1

)

1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1

x 106

0

5

10

15x 1017

Time (ns)

Inve

rsio

n de

nsity

(cm

-3)

Fig. 7. (a) Absorb coefficient evolution of the laser medium underPin¼5 W and

(b) inversion population of laser medium as a function of t.

4 5 6 7 8 9 10x 105

0

50

100

150

Foca

l len

gth

(cm

)

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10x 105

50

55

60

Time (ns)

Focl

a le

ngth

(cm

)

Fig. 8. Focal length variation of laser crystal when the incident pump power is

5 W.

4.15x 1014

S. Zhang, X. Wang / Optics Communications 295 (2013) 155–160 159

The curve in Fig. 9 shows the diffractive loss variations as afunction of t under the pump power of 5 W, which is also regarded asa constant in the traditional model [21]. Because it is induced bythermal effect, the diffractive loss has similar character with focallength and also keep an oscillatory behavior between dTmax¼0.0135and dTmin¼0.0023.

1 2 3 4 5 6 7 8 9 10

x 105

4

4.05

4.1

Time (ns)

Pho

to d

ensi

ty (c

m-3

)

Fig. 11. Simulation pulse train of the passively Q-switched laser under the pump

3.3. Instability of Q-switched pulse train

Fig. 10 shows the simulation pulse train of the passivelyQ-switched Nd:YVO4/Cr4þ:YAG laser under the pump power of5 W. It can be obviously seen that when the interaction betweenthe transient heat equation and the rate equation is considered, thereappears to be an envelope of pulse amplitudes. The instability of thepulse amplitudes which leads to the variation of output power asdemonstrated in [8] can be given by Df=f¼ ðfmax�fminÞ=f¼0.5%.In order to compare with Fig. 10, we also calculate the rate equationaccording to the traditional methods under the same condition [22],without taking the influence of thermal effect into considering.

9 9.2 9.4 9.6 9.8 10 10.2 10.4 10.6x 105

2

4

6

8

10

12

14x 10-3

Time (ns)

Diff

ract

ive

loss

0.0112

Fig. 9. Diffractive loss variations as a function of t under the pump power of 5 W.

1 2 3 4 5 6 7 8 9 10 11x 105

3.2

3.21

3.22

3.23

3.24

3.25

3.26

3.27

3.28

3.29

3.3x 1014

Time (ns)

Pho

to d

ensi

ty (c

m-3

)

Fig. 10. Output pulse trains obtained by our new model under the pump power of

5 W.

power of 5 W without considering thermal effect.

As shown in Fig. 11, the pulse train is obviously very trim, which isnot accurate.

Meanwhile, we can also find that, as for passively Q-switchedoperation, the impact to the output pulse train owing to thetransient temperature profile is not severe. That is because thepulse repetition rate of the passively Q-switched is generally veryhigh, which leads that the variation of the instantaneous popula-tion density N(t) contains less proportion of the total dopingconcentration Nt. And then it results in negligible change of theabsorption coefficient a. So when designing the thermal model ofthe passively Q-switched operation, it usually can be consideredto have similar characteristics as cw state operation.

4. Conclusion

In conclusion, by considering the instantaneous variations ofthe population inversion density, the absorption coefficient andthe diffraction loss, we integrate the transient heat equation andthe rate equation of the passively Q-switched operation for thefirst time. A new more accurate model is developed to calculatethe thermal profile in laser medium and theoretically analyze theinstability of the output pulse amplitude in passively Q-switchedlaser. The dynamics of thermal buildup and thermal effectinfluence on the output power stability of the Q-switched laserare also obtained. The numerical results reveal that the instanta-neous population inversion density has a certain impact on theabsorption coefficient and makes the diffraction loss keep anoscillatory behavior, resulting in the instable output character-istics of the Q-switched laser.

Acknowledgment

This work is supported by NSFC (No. 11274189), the DoctoralFoundation of Shandong Province (No. BS2012DX010), theScience and Technology Project of Qingdao (No. 11-2-4-3-(1)-jch), and the Doctoral Found of QUST (Grant No. 0022520).

References

[1] Yung-Fu Chen, S.W. Tsai, S.C. Wang, Optics Letters 25 (2000) 1442.[2] S. Zhang, E. Wu, H. Zeng, Optics Communications 231 (2004) 365.

S. Zhang, X. Wang / Optics Communications 295 (2013) 155–160160

[3] S.P. Ng, D.Y. Tang, A.Q. Liu, L.J. Qin, X.L. Meng, Optics Communications259 (2006) 256.

[4] Yung-Fu Chen, Jian-Lung Lee, Hung-Dau Hsieh, Sheng-Wei Tsai, IEEE Journalof Quantum Electronics 38 (2002) 312.

[5] J. Dong, Optics Communications 266 (2003) 337.[6] Guiqiu Li, Shengzhi Zhao, Kejian Yang, Dechun Li, Jing Zou, Optics Express

13 (2005) 1178.[7] T.T. Kajava, A.L. Gaeta, Optics Letters 21 (1996) 1244.[8] K. Yang, S. Zhao, G. Li, H. Zhao, IEEE Journal of Quantum Electronics 40 (2004)

1252.[9] J.J. Zayhowski, C. Dill III., Journal of Optics Letters 19 (1994) 1427.

[10] Jianlang Li, Jun Dong, Musha Mitsurua, Akira Shirakawa, Ken-ichi Ueda,Optics Communications 270 (2007) 63.

[11] C. Pfistner, R. Weber, H.P. Weber, S. Merazzi, R. Gruber, IEEE Journal ofQuantum Electronics 30 (1994) 1605.

[12] H. Nadgaran, M. Sabaian, Pramana Journal of Physics 67 (2006) 1119.

[13] P. Shi, W. Chen, L. Li, A. Gan, Applied Optics 46 (2007) 4046.[14] Z. Ma, D. Li, J. Gao, N. Wu, K. Du, Optics Communications 275 (2007) 179.[15] M. Sabaeian, H. Nadgaran, L. Mousave, Applied Optics 47 (2007) 2317.[16] T. Li, S.Y. Zhang, S.Z. Zhao, K.J. Yang, Z. Zhuo, Optics Communications 283

(2010) 3070.[17] Z. Xiong, G. Li, Nicholas Moore, W.L. Huang, G.C. Lim, IEEE Journal of

Quantum Electronics 39 (2003) 979.[18] W. Koechner, Journal of Applied Physics 7 (1973) 3162.[19] W. Xie, S.C. Tam, Y.L. Lam, J. Liu, H. Yang, J. Gu, W. Tan, Applied Physics

30 (2000) 5382.[20] X. Zhang, S. Zhao, Q. Wang, Q. Zhang, L. Sun, S. Zhang, IEEE Journal of

Quantum Electronics 33 (1997) 2286.[21] Shengzhi Zhao, Lei Chen, Hongming Zhao, Guiqiu Li, Lu Zhang,

Zhenxiang Chen, Huanchu Chen, Kejian Yang, Optical Materials 27 (2004) 481.[22] J.J. Degnan, IEEE Journal of Quantum Electronics 31 (1995) 1890.