Abstract

The transient heat conduction and thermal effects in pulse end-pumped fiber laser are modeled and analytically solved. For the arbitrary temporal shape of pump pulse, a three-dimensional (3D) temperature expression is derived via an integral transform method, and the thermal stress field is deduced through solving the Navier displacement equations. The results show that pulse shape has an important influence on the peak thermal stress and transient phase shift induced by heating of the fiber. Reasonable design for pulse duration and period can reduce thermal effects and optimize the performance of high-power fiber laser.

© 2009 OSA

1. Introduction

Several applications require high average power laser systems and specifically high repetition rate ultra-intense and ultra-short pulse laser systems. In recent years, high power, short-pulse fiber lasers have been developed with ever increasing performances [13]. Owing to the excellent stability, low noise and all-fiber configurations, much attention has been paid to pulse pumped fiber lasers [47]. But during the operation of pulse laser, the fiber medium is simultaneously subject to heating by pulse pump radiation and cooling by surroundings, causing nonuniform temperature distribution followed by thermal stress, thermal birefringence and thermal lensing [8] detrimental to laser performance, and even inducing fiber facet damage [7]. In addition, the time-dependent thermal effects caused by repetitive pump pulse more easily give rise to the fatigue fracture of laser medium and influence the light transmission characteristics.

Since thermal effects depend on the thermal profile inside the fiber, knowledge of the transient temperature distribution as a function of pump-pulse repetition rate, coolant flow, pump energy and laser material property permits one to predict thermal distortion and optimize a large variety of operating parameters. Existing analytical thermal models that describe the temperature and stress in fiber medium are restricted to special cases and approximations, such as continuous wave pump, steady–state conditions and radial two-dimensional (2D) heat flux approximation. 2D simulations for long fibers with small pump absorption coefficient constitute reasonable approximation to reality. However, such treatment may introduce a rather large calculation error in short-length fiber laser. The axial temperature changes significantly due to the exponentially varying thermal loading in the longitudinal direction, so the plain-strain approximation [9] calculating thermal stress is not fully satisfied for short-length fiber. A general analytical calculation describing the transient temperature distribution in a pulse-pumped fiber laser could be little found in literature except the Davis’s [10] research. But his heat conduction model is based on 2D heat flux approximation and excludes any heat source function. Until now a comprehensive analytical investigation on thermal stress in short-length fiber has not yet reported.

Up to now, analytical investigation on transient thermal effects is only available in solid-state lasers [1113]. In view that the analytical methods possess many recognized advantages over numerical methods both in theory and in engineering application. In this paper, the three-dimensional (3D) transient heat conduction are modeled and solved analytically. The thermal stress and strain caused by the temperature filed in the absence of external forces are derived by solving the Navier displacement equations with the method of thermoelastic displacement potential. To the best of our knowledge, this is the first time that such a time-dependent analytical model has been reported in fiber lasers. We validate the model through a fully 3D finite element method (FEM), and apply the model to pulse end-pumped Er3+/Yb3+ co-doped phosphate glass fiber laser. The results show that the transient temperature and thermal stress fields in fiber make fluctuating distributions and their amplitudes are related to pulse duration and pulse period, which can be utilized to reduce the peak temperature and stress. Thus the thermal modeling can be further employed to optimize the performance of high-power fiber laser. Finally, the pump-induced transient thermal phase shift is analyzed. The calculated results demonstrate that the transient thermal effects will have an important influence on the light transmission characteristics and the beam quality of the laser medium.

2. Resolving of three-dimensional transient temperature field

Figure 1 is the schematic illustration of a pulse end-pumped short-length fiber butted against two planar mirrors.

 

Fig. 1 Schematic illustration of a pulse end-pumped short-length fiber.

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With reference to Fig. 1, the transient temperature field of fiber laser under air cooling is obtained as [14]:

2T(z,r,t)r2+1rT(z,r,t)r+2T(z,r,t)z2+Q(z,r,t)k=ρckT(z,r,t)t             0<rr1
2T(z,r,t)r2+1rT(z,r,t)r+2T(z,r,t)z2=ρckT(z,r,t)tr1<rr2
T=Th,                                                               t=0
kT/r=h(ThT),                                                 r=r2
T/z=0,                               z=0,                                                 z=l
T1=T2,                                       T1/r=T2/r,                                     r=r1
T/r=0                                                                                                     r=0
where
Q(z,r,t)=(2ηαPin/πωp2)exp(2r2/ωp2αz)g(t)
is the transient heat loading function in fiber. T1 and T2 are the temperatures in fiber core and cladding region, respectively. Th is the heat sink temperature. k denotes the fiber thermal conductivity. h is the convective coefficient. ωp is Gaussian radius of the pump light. η is the fractional thermal loading, α is the optical absorption coefficient, ρ is the density of the fiber material and c is its the specific heat, Pin is the energy in each pulse and g(t) is laser pulse shape.

The equation system (1a)-(1g) can be solved through the integral transform method introduced by Özisik [14]. The 3D temperature rise expression in fiber is:

θ=TT0=p=1kρclNpJ0(βpr)exp(kρcβp2t)0tg0p(τ)exp(kρcβp2τ)dτ+                                 m=1p=12kρclNpJ0(βpr)cos(ηmz)exp[kρc(βp2+ηm2)t]0tgmp(τ)exp[kρc(βp2+ηm2)τ]dτ
where
g0p(τ)=0l0r1Q(z,r,t)kJ0(βpr)rdrdzgmp(τ)=0l0r1Q(z,r,t)kcos(mπLz)J0(βpr)rdrdz
0tg0p(τ)exp(kρcβp2τ)dτ=2ηPin[1exp(βl)]kπωp20r1exp(2r2/ωp2)J0(βpr)rdr0tg(τ)exp(kρcβp2τ)dτ0tgmp(τ)exp[kρc(βp2+ηm2)τ]dτ=2ηβ2l2Pin[1exp(βl)cos(mπ)]kπωp2(β2l2+m2π2)0r1exp(2r2/ωp2)J0(βpr)rdr0tg(τ)exp[kρc(βp2+ηm2)τ]dτηm=mπ/lNp=r22(k2βp2+h2)J02(βpr2)/2k2βp2
   hJ0(βpr2)kβpJ1(βpr2)     =0
Where J0 and J1 are the zero and first rank Bessel function of the first kind, respectively. More detailed deductions are shown in Appendix A.

3. Thermal stress and strain fields

In the absence of external forces (e. g. extruded and stretched by piezoelectric ceramics as wavelength modulator), the thermal stress components caused by the temperature rise θ can be obtained by summing up the thermal components σij(p) caused by each term θ(p)of the temperature rise [15].

Since the first term θ1 of the temperature rise expression θ is only a function of r, the thermal stress caused by θ1 in the absence of external forces can be obtained by the displacement method as [16]:

σrr0=γE(1ν)[1r220r2T1rdr1r20rT1rdr]=γE(1ν)p=1Apβp[J1(βpr2)r2J1(βpr)r]
σθθ0=γE(1ν)[1r220r2T1rdr+1r20rT1rdrT(r)]=γE(1ν)p=1Apβp[J1(βpr2)r2+J1(βpr)rβpJ0(βpr)]σzz0=γE(1ν)[2r220r2T1rdrT1]=γE(1ν)p=1Apβp[2J1(βpr2)r2βpJ0(βpr)]
σrz0=0
Here
Ap=kρclNpexp(kρcβp2t)0tg0p(τ)exp(kρcβp2τ)dτ
where E, γ and ν are Young’s modulus, thermal expansion coefficient and Poisson’s ratio

Since the second term θ2 of the temperature rise θ is axisymmetric and varies in both r and z directions, the resulting thermal stress under the traction free condition can be determined by the thermoelastic displacement potential method [16].

The displacement field u(r, z) is governed by Navier displacement equations.

2ur1r2ur+112νr(1rr(rur)+zuz)=2(1+ν)12νγTr
2uz+112νr(1rr(rur)+zuz)=2(1+ν)12νγTz

A particular solution of Eq. (2) is represented by the thermoelastic displacement potential Φ satisfying:

2Φ=(1+ν)/(1ν)γθ2
The homogeneous solution of Eq. (2), i.e., the solution of Eq. (2) when the right-hand sides are zero, will be represented by the Love function L satisfying:

22L=0

The functions Φ and L shall be so constructed that they combine to satisfy all the boundary conditions and thus completely define the displacement field. Since the stresses σij are specified on the boundary, they need to be expressed in terms of Φ and L, and can be derived, respectively, from:

σrr1=E1+ν(2Φr22Φ)σrr2=E1+ν[2r2(Lz)ν2(Lz)]
σθθ1=E1+ν(1rΦr2Φ)σθθ2=E1+ν[1rr(Lz)ν2(Lz)]
σzz1=E1+ν(2Φz22Φ)σzz2=E1+ν[2z2(Lz)(2ν)2(Lz)]
σrz1=E1+ν2Φrzσrz2=E1+ν[2rz(Lz)(1ν)r(2L)]
σij=σij1+σij2
Substituting the expression of θ2 into Eq. (3), we will obtain the particular solution as:
Φ=m=1p=1Cmpcos(mπlz)J0(βpr)
where
Cmp=1+ν1νγ2kL2ρclNp(m2π2+L2βp2)exp[kρc(βp2+ηm2)t]0tgmp(τ)exp[kρc(βp2+ηm2)τ]dτ
According to the characteristics of the expression of Φ, the form of the Love function is assumed as:
L=1+ν1νγp=1[D1pβpJ0(βpr)+D2prJ1(βpr)]exp(βpz)βp
where D1p and D2p are undetermined coefficients. So the thermal stresses caused by θ2 are obtained as:
σrr1=E1+νm=1p=1Cmp[βprJ1(βpr)+(mπl)2J0(βpr)]cos(mπlz)
σθθ1=E1+νm=1p=1Cmp[βp2J0(βpr)βprJ1(βpr)+(mπl)2J0(βpr)]cos(mπlz)σzz1=E1+νm=1p=1Cmpβp2J0(βpr)cos(mπlz)
σθθ2=E1νγp=1[D1pJ1(βpr)r+D2p(12ν)βpJ0(βpr)]exp(βpz)σrz1=E1+νm=1p=1Cmpβpmπlsin(mπlz)J0(βpr)
σrr2=E1νγp=1{D1p[βpJ0(βpr)+J1(βpr)r]+D2p[(12ν)βpJ0(βpr)βp2rJ1(βpr)]}exp(βpz)
σzz2=E1νγp=1{D1pβpJ0(βpr)+D2pβp[rβpJ1(βpr)2(2ν)J0(βpr)]}exp(βpz)σrz2=E1νγp=1{D1pβpJ1(βpr)D2pβp[βprJ0(βpr)+2(1ν)J1(βpr)]}exp(βpz)
According to the following traction free condition,
σrr=0                               σrz=0                                                           r=r2
the coefficients D1p and D2p are determined as:
D1pexp(βpz)=m=1Cmpkβpmπsin(mπz/l)/lh+[kβp2r2/h+2(1ν)]D2pexp(βpz)
D2pexp(βpz)=m=1Cmp[1/r2+km2π2/hl2]cos(mπz/l)[1/βpr2kβp/h]m=1Cmpkβpmπsin(mπz/l)/lh2(1ν)/βpr2βpr2k2βp3r2/h2
where
Cmp=2kL2ρclNp(m2π2+L2βp2)exp[kρc(βp2+ηm2)t]0tgmp(τ)exp[kρc(βp2+ηm2)τ]dτ
The total thermal stress is obtained as:
σij=σij0+σij1+σij2
The stress, strain and temperature are related by the generalized Hooke’s laws:

εrr=[σrrν(σθθ+σzz)]/E+γθ
εθθ=[σθθν(σrr+σzz)]/E+γθ
εzz=[σzzν(σrr+σθθ)]/E+γθ

4. Analytical results and discussions

To apply the above analytical solution, the pulse laser temporal shape is assumed as square incident laser irradiance:

g(t)={1(n1)T0t(n1)T0+t00(n1)T0+t0tnT0
where t0, T0 and n are pulse duration, pulse period and the number of pulse, respectively. Integrating the time term of θ, and the expression is shown in Appendix B.

An exemplary laser medium is Er3+/Yb3+ co-doped phosphate glass fiber. The thermal properties [17] are k = 0.55 W·m−1·K−1, η = 0.36, Th = 300 K, h = 10 W·m−2·K−1 and the other parameters are taken as ρ = 3.2 g·cm−3, c = 960 J·kg−1·K−1, γ = 9.6 × 10−6 K−1, ν = 0.27, E = 56.4 GPa, r1 = 2.7 μm, r2 = 62.5 μm, l = 1 cm, Pin = 1W.

4.1. Transient temperature distribution

Figure 2 shows the temperature distribution from the analytical solution at the time of 0.1 s with the pulse duration more than 0.1 s. As a verification of the analytical thermal model, a time-dependent 3D temperature finite element analysis was implemented in a commercial software package Ansys [18], as shown in Fig. 3 . From the comparison of Fig. 2 (b) with Fig. 3 (b), it is noted that the results from our analytical thermal model are consistent with those from numerical solutions of a 3D finite element analysis within the calculation error.

 

Fig. 2 The temperature distribution from the analytical solution at the time of 0.1 s. (a) The 3D temperature distribution; (b) The temperature distribution along the active fiber axial coordinate.

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Fig. 3 A time-dependent temperature finite element analysis at the time of 0.1 s. (a) The temperature distribution of fiber end surface at the pump side; (b) The temperature distribution along the active fiber axial coordinate.

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Figure 4 shows the temperature distribution of fiber end surface at the pump side along the active fiber radial coordinate at different times with the interval of 1ms when T0 = 0.1 s, t0 = 0.01 s. From the graph 4(a), a transient thermal diffusive process with time is clearly displayed by the evolution of the ten curves. During the first pulse pump space time, as shown in Fig. 4(b) and Fig. 4(c), the temperature distribution in fiber tends to be flat and the temperature difference between the center and edge gradually decreases because of the air convection and the heat conduction of fiber medium. After the first pulse pump ends, the temperature of each position in fiber medium nearly reaches the same, but there is still about 2 °C temperature rise compared with the initial value due to the insufficiency of heat dissipation, which is deposited in laser medium as residual heat. The fiber core temperature will come back to 300 K if the first pulse pump space time is long enough, about 50 s, as shown in Fig. 4(d).

 

Fig. 4 The temperature distribution of fiber end surface at the pump side along the active fiber radial coordinate. (a) During the first pulse pump time; (b) The first 5 ms during the first pulse pump space time; (c) The last 10 ms during the first pulse pump space time; (d) The long enough pump space time and t0 = 0.01, interval: 1 s.

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The transient temperature variation in the centre of fiber end surface as a function of time is plotted in Fig. 5 when T0 = 0.1 s, t0 = 0.01 s. Graph 5(b) and 5(c) are the enlarged drawings of the part of graph 5(a). As shown in figures, the temperature fluctuates with the time and the peak value gradually increases. After a long enough time (about 50 s), the peak temperature becomes stable and the temperature field makes a periodical change, as shown in Fig. 5(c). Figure 6 shows the same the transient temperature distribution under different pulse duration with the same pulse period 0.1 s. Compared with the short pulse duration, the long pulse duration induce a higher temperature rise because of the longer time of pump injection. Furthermore, if fiber medium is under steady-state pump, the maximum temperature will exceed 1000 K in the same conditions according to Liu et al [17]. So, compared with the stationary state, the pulse pumping configuration displays more advantages to reduce heat deposition, and selecting appropriate pulse duration can efficiently avoid fiber fracture.

 

Fig. 5 The transient temperature variation in the centre of fiber end surface as a function of time. Graph b and c are the enlarged drawings of the part of graph a.

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Fig. 6 The transient temperature under different pulse duration. Red: t0 = 0.06, Blue: t0 = 0.01.

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4.2. Transient thermal stress and strain fields

Figure 7 and 8 show the 3D thermal stress and strain fields from the analytical solution at the time of 0.1 s with the pulse duration more than 0.1 s. Comparing our analytical model with the plain-strain approximation (see Appendix C), as shown in Fig. 9 , we found that the radial thermal stress distribution does not induce any changes due to the small thermal gradient between the center and the edge of the cladding region. But the axial thermal stress distribution makes a great change due to considering of axial strain in our model from the comparison of two models, which correspondingly induces differences of strain distributions. The axial stress is about 10 times of the radial one in our analytical model, which owe to the significantly varying temperature field caused by the high gain coefficient of active fiver in the longitudinal direction. So our analytical model shows better accuracy of evaluating thermal stress and strain in fiber medium, and can be used to investigate end effects caused by heat disposition because of consideration of axial strain.

 

Fig. 7 The 3D radial, tangential and axial thermal stress distributions at the time of 0.1 s. (a) σrr; (b) σθθ; (c) σzz

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Fig. 8 The 3D radial, tangential and axial thermal strains distributions at the time of 0.1 s. (a) εrr; (b) εθθ; (c) εzz

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Fig. 9 The 3D the radial, axial thermal stress and radial strain distribution at the time of 0.1 s under the plain-strain assumption. (a) σrr; (b) σzz, (c) εrr

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The radial thermal stress in the centre of fiber end surface as a function of time is plotted in Fig. 10 under different pulse laser temporal shapes. From three figures, we found that the pulse duration and period have an important influence on the peak thermal stress. Reasonable design for the pulse duration and period can effectively reduce thermal stress and optimize the performance of high-power fiber laser.

 

Fig. 10 The radial thermal stress in the centre of fiber end surface as a function of time. (a)T0 = 0.1 s, t0 = 0.01 s; (b) T0 = 0.001 s, t0 = 0.0001 s; (c) T0 = 0.0001 s, t0 = 0.00001 s.

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5. Thermal phase shift

Heating of the fiber induces a phase change in the signal via two effects, namely, by changing the index of refraction (coefficient δn / δT) and by axial strain field of the fiber.

The change in the effective index of the signal mode is given by [10]:

Δn(z,t)=(n/T)0θ(z,r,t)fs(r)2πrdr
where fs(r) is the signal mode intensity, and can be approximated by a Gaussian distribution.
fs(r)=1πr12exp(r2r12)
The instantaneous thermal phase shift caused by the varying index of refraction is obtained by integrating △n along z.
Δϕ1(t)=2πλsnT0l0r1θ(z,r,t)fs(r)2πrdrdz
The instantaneous thermal phase shift caused by axial strain field is gotten as [19]:
Δϕ2(t)=2πλs0l0r1(n01)εzzdrdz
where n0 is the refractive index of fiber, λs is the signal wavelength.

So the total instantaneous thermal phase shift rise experienced by the signal is given as:

Δϕ(t)=Δϕ1(t)+Δϕ2(t)
Comparing formula (5) with (6), we found that thermal phase shift caused by axial strain field amounts to less than 1% of the one caused by the varying index of refraction. We only include the thermal index change, which induces the thermal phase shift as follows:
Δϕ(t)=2πλsnTlπr12p=1kρclNpexp(kρcβp2t)0tg0p(τ)exp(kρcβp2τ)dτ0r1J0(βpr)exp(r2r12)2πrdr
Index temperature coefficient is 9.67 × 10−6 K−1, and λs = 1.53 μm. The phase shift rise induced by heating as a function of time is shown in Fig. 11 when T0 = 0.1 s, t0 = 0.01 s. Graph 11(b) is the enlarged drawing of graph 11(a). The large temperature rise results in a non-negligible phase shift. Subsequently, the varying thermal phase shift will have an influence on the light transmission characteristics and the beam quality of the laser medium. It is pointed out that the maximum value of transient phase shift is about one order of magnitude smaller than the stationary-state one owing to the higher temperature rise under steady state.

 

Fig. 11 The phase shift rise induced by thermal effects as a function of time. Graph b is the enlarged drawing of graph a

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6. Conclusions

In conclusion, a time-dependent analytical solution is derived to investigate the transient temperature and thermally induced stress and strain in pulse end-pumped fiber laser. Assuming the pulse laser temporal shape as square incident laser irradiance and using Er3+/Yb3+ co-doped phosphate glass fiber as an exemplary laser medium, we calculate the transient temperature, thermal stress and strain distributions. The results show that pulse shape has an important influence on the peak thermal stress, and reasonable design for the pulse duration and period can be utilized to reduce thermal stress and optimize the performance of high-power fiber laser. At last, the pump-induced transient thermal phase shift is analyzed. The calculated results demonstrate that the varying thermal phase shift will affect the light transmission characteristics and the beam quality of the laser medium.Appendix A

Let θ=T-Th, Eq. (1a)-(1g) are simplified as:

2θi(z,r,t)r2+1rθi(z,r,t)r+2θi(z,r,t)z2+gi(z,r,t)=ρckθi(z,r,t)t            i=1,2
θi=0,                 t=0
kθ2/r+hθ2=0,                             r=r2
θi/z=0,             z=​ ​ ​ ​ ​0,                     z=L
θ1=θ2,                 θ1/r=θ2/r,                 r=r1
θ1/r=0                               r=0
g1=Q(zr,t)/kg2=0
Inversion formula is:
θi(z,r,t)=m=0Z(ηm,z)θim(r,t)/N(ηm)
followed by integral transform:
θim(r,t)=0LZ(ηm,z)θi(z,r,t)dz
where
Z(ηm,z)={cos(ηmz)1ηm={mπ/l01N(ηm)={2/l1/l(m0)(m=0)
The integral transform of the system (7) by the application of the transform (8b) yields:
2θim(r,t)r2+1rθim(r,t)rηm2θim(r,t)+gim=ρckθim(r,t)t
θim=0,                                                               t=0
kθ2m/r+hθ2m=0,                                                 r=r2
θ1m=θ2m,                                       θ1m/r=θ2m/r,                                     r=r1
θ1m/r=0                                                                                                     r=0
where
gim=0LZ(ηm,z)gidz
Multiplying Eq. (9a) by ri1rirRim(r)drand using integration of parts twice on the first term on the right hand side, one gets:
ri1ri[ρckθim(r,t)t]rRim(r)dr=ri1ri[d2Rim(r)dr2+1rdRim(r)drηm2Rim(r)]rθim(r,t)dr+[rRim(r)θim(r,t)rrθim(r,t)dRim(r)dr]ri1ri+ri1rirRim(r)gimdr
Rim(r) in the above equation is chosen to satisfy
d2Rim(r)dr2+1rdRim(r)dr+(ηm2+λ2)Rim(r)=0
kR2m/r+hR2m=0,                                                 r=r2
R1m=R2m                                                       r=r1
R1m/r=R2m/r                                                 r=r1
R1m/r=0                                                                                                     r=0
Solutions of the Eq. (11e) are eigenfunctions Rim(r) corresponding to the eigenvalues βp
Rimp(βpr)=aimpJ0(βpr)+bimpY0(βpr)
whereβp2=ηm2+λ2, aimp and bimp are the coefficients.

The eigenfunctions Rimp(r) satisfy the following orthogonality condition [20]

i=12ri1rirRimp(βpr)Rimq(βqr)dr={0Nmpififpqp=q
Without loss of generality, we set one of the nonvanishing cofficients, say a1mp equal to unity, according to Eq. (11b)-Eq. (11d), a2mp, b2mp, and βp can be obtained:
a2mp=1b2mp=0
   hJ0(βpr2)kβpJ1(βpr2)     =0
Nmp=r22(k2βp2+h2)J02(βpr2)/2k2βp2
In view of Eq. (11a), Eq. (10) can be written as
ri1ri[ρckθim(r,t)t+λ2θim(r,t)]rRimp(r)dr=[rRimp(r)θim(r,t)rrθim(r,t)dRimp(r)dr]ri1ri+ri1rirRimp(r)gimdr
According to Eq. (9b)-Eq. (9d) and summing up the resulting expression from fiber core to cladding, one gets:
i=12ri1ri[ρckθim(r,t)t+λ2θim(r,t)]rRimp(r)dr=i=12ri1rirRimp(r)gimdr
Defining
θmp(t)=i=12ri1rirRimp(r)θim(r,t)dr
and
gmp(t)=i=12ri1rirRimp(r)gimdr=0r1rR1mp(r)g1mdr
we have
ρckdθmpdt+λ2θmp=gmp
Solution of Eq. (12) is
θmp(t)=θmp(0)exp(kρcλ2t)+exp(kρcλ2t)0tkρcgmpexp(kρcλ2t)dτ=kρcexp(kρcλ2t)0tgmp(τ)exp(kρcλ2t)dτ
θim(r,t) can be expanded in the Fourier series as follows:
θim(r,t)=p=1cmp(t)Rimp(r)
According to the orthogonality, cmp(t) can be obtained as:
cmp(t)=i=12ri1rirRimp(r)θim(r,t)drNmp=θmp(t)Nmp
So the θi(z,r,t) can be gotten as:
θi(z,r,t)=m=0p=1Z(ηm,z)N(ηm)θmp(t)NmpRimp(r)
Appendix B

0tg(τ)exp[kρc(βp2+ηm2)τ]dτ=n=2N(n2)T0(n2)T0+t0exp[kρc(βp2+ηm2)τ]dτ+(n1)T0texp[kρc(βp2+ηm2)τ]dτ
(n1)T0t(n1)T0+t0
0tg(τ)exp[kρc(βp2+ηm2)τ]dτ=n=1N(n1)T0(n1)T0+t0exp[kρc(βp2+ηm2)τ]dτ
(n1)T0+t0tnT0

Appendix C

According to the plain-strain approximation, the thermal stress and strain can be expressed as:

σrr=γE(1ν)[1r220r2θrdr1r20rθrdr]
σθθ=γE(1ν)[1r220r2θrdr+1r20rθrdrθ]
σzz=γE(1ν)[2r220r2T1rdrT1]
εrr=(1ν2)E(σrrν1νσθθ)+(1+ν)γθ
εθθ=(1ν2)E(σθθν1νσrr)+(1+ν)γθ
εzz=0

Acknowledgement

This research was supported by the Guangdong Science and Technology Program (2005A10602001), the Guangzhou Science and Technology Program (2006Z2-D0161).

References and links

1. C. Lecaplain, C. Chedot, A. Hideur, B. Ortac, and J. Limpert, “High-average power femtosecond pulse generation from a Yb-doped large-mode-area microstructure fiber laser,” Proc. of SPIE 6873, 68730S1–68730S5 (2008)

2. M. Eichhorn and S. D. Jackson, “High-pulse-energy actively Q-switched Tm3+-doped silica 2 microm fiber laser pumped at 792 nm,” Opt. Lett. 32(19), 2780–2782 (2007). [CrossRef]   [PubMed]  

3. Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

4. B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009). [CrossRef]  

5. Z. Y. Dai, Z. S. Peng, Y. Z. Liu, and Z. H. Ou, “Research on SBS and pulse pumped hybrid Q-switched Er3+/Yb3+ co-doped fiber laser,” Proc. of SPIE 6823, 68231C1–68231C4.

6. S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

7. C. G. Ye, P. Yan, M. Gong, and M. Lei, “Pulsed pumped Yb-doped fiber amplifier at low repetition rate,” Chin. Opt. Lett. 3, 249–250 (2005).

8. V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008). [CrossRef]  

9. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001). [CrossRef]  

10. M. K. Davis, M. J. F. Digonnet, and R. H. Pantell, “Thermal effects in doped fobers,” J. Lightwave Technol. 16(6), 1013–1023 (1998). [CrossRef]  

11. E. H. Bernhardi, A. Forbes, C. Bollig, and M. J. D. Esser, “Estimation of thermal fracture limits in quasi-continuous-wave end-pumped lasers through a time-dependent analytical model,” Opt. Express 16(15), 11115–11123 (2008). [CrossRef]   [PubMed]  

12. W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44(7), 3162–3170 (1973). [CrossRef]  

13. F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).

14. M. N. Özisik, Heat Conduction (Wiley, New York, 1980).

15. Z. G. Li, X. L. Huai, L. Wang, and Y. J. Tao, “Influence of longitudinal rise of coolant temperature on the thermal strain in a cylindrical laser rod,” Opt. Lett. 34(2), 187–189 (2009). [CrossRef]   [PubMed]  

16. Y. Takeuchi, Thermal Stress (Science, 1977).

17. T. Liu, Z. M. Yang, and S. H. Xu, “3-Dimensional heat analysis in short-length Er3+/Yb3+ co-doped phosphate fiber laser with upconversion,” Opt. Express 17(1), 235–247 (2009). [CrossRef]   [PubMed]  

18. Ansys Finite Element Software Package, http://www.ansys.com/

19. C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994). [CrossRef]  

20. P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer 131(1), 011304–0113047 (2009). [CrossRef]  

References

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  1. C. Lecaplain, C. Chedot, A. Hideur, B. Ortac, and J. Limpert, “High-average power femtosecond pulse generation from a Yb-doped large-mode-area microstructure fiber laser,” Proc. of SPIE 6873, 68730S1–68730S5 (2008)
  2. M. Eichhorn and S. D. Jackson, “High-pulse-energy actively Q-switched Tm3+-doped silica 2 microm fiber laser pumped at 792 nm,” Opt. Lett. 32(19), 2780–2782 (2007).
    [CrossRef] [PubMed]
  3. Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).
  4. B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
    [CrossRef]
  5. Z. Y. Dai, Z. S. Peng, Y. Z. Liu, and Z. H. Ou, “Research on SBS and pulse pumped hybrid Q-switched Er3+/Yb3+ co-doped fiber laser,” Proc. of SPIE 6823, 68231C1–68231C4.
  6. S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).
  7. C. G. Ye, P. Yan, M. Gong, and M. Lei, “Pulsed pumped Yb-doped fiber amplifier at low repetition rate,” Chin. Opt. Lett. 3, 249–250 (2005).
  8. V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
    [CrossRef]
  9. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
    [CrossRef]
  10. M. K. Davis, M. J. F. Digonnet, and R. H. Pantell, “Thermal effects in doped fobers,” J. Lightwave Technol. 16(6), 1013–1023 (1998).
    [CrossRef]
  11. E. H. Bernhardi, A. Forbes, C. Bollig, and M. J. D. Esser, “Estimation of thermal fracture limits in quasi-continuous-wave end-pumped lasers through a time-dependent analytical model,” Opt. Express 16(15), 11115–11123 (2008).
    [CrossRef] [PubMed]
  12. W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44(7), 3162–3170 (1973).
    [CrossRef]
  13. F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).
  14. M. N. Özisik, Heat Conduction (Wiley, New York, 1980).
  15. Z. G. Li, X. L. Huai, L. Wang, and Y. J. Tao, “Influence of longitudinal rise of coolant temperature on the thermal strain in a cylindrical laser rod,” Opt. Lett. 34(2), 187–189 (2009).
    [CrossRef] [PubMed]
  16. Y. Takeuchi, Thermal Stress (Science, 1977).
  17. T. Liu, Z. M. Yang, and S. H. Xu, “3-Dimensional heat analysis in short-length Er3+/Yb3+ co-doped phosphate fiber laser with upconversion,” Opt. Express 17(1), 235–247 (2009).
    [CrossRef] [PubMed]
  18. Ansys Finite Element Software Package, http://www.ansys.com/
  19. C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
    [CrossRef]
  20. P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer 131(1), 011304–0113047 (2009).
    [CrossRef]

2009

B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
[CrossRef]

Z. G. Li, X. L. Huai, L. Wang, and Y. J. Tao, “Influence of longitudinal rise of coolant temperature on the thermal strain in a cylindrical laser rod,” Opt. Lett. 34(2), 187–189 (2009).
[CrossRef] [PubMed]

T. Liu, Z. M. Yang, and S. H. Xu, “3-Dimensional heat analysis in short-length Er3+/Yb3+ co-doped phosphate fiber laser with upconversion,” Opt. Express 17(1), 235–247 (2009).
[CrossRef] [PubMed]

P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer 131(1), 011304–0113047 (2009).
[CrossRef]

2008

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

E. H. Bernhardi, A. Forbes, C. Bollig, and M. J. D. Esser, “Estimation of thermal fracture limits in quasi-continuous-wave end-pumped lasers through a time-dependent analytical model,” Opt. Express 16(15), 11115–11123 (2008).
[CrossRef] [PubMed]

2007

M. Eichhorn and S. D. Jackson, “High-pulse-energy actively Q-switched Tm3+-doped silica 2 microm fiber laser pumped at 792 nm,” Opt. Lett. 32(19), 2780–2782 (2007).
[CrossRef] [PubMed]

F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).

2006

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

2005

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

C. G. Ye, P. Yan, M. Gong, and M. Lei, “Pulsed pumped Yb-doped fiber amplifier at low repetition rate,” Chin. Opt. Lett. 3, 249–250 (2005).

2001

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

1998

1994

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

1973

W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44(7), 3162–3170 (1973).
[CrossRef]

Ballato, J.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Bass, M.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Bernhardi, E. H.

Bollig, C.

Brown, D. C.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

Chen, Y.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Davis, M. K.

Digonnet, M. J. F.

Dong, F. J.

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Dong, W.

F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).

Dong, X. Y.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Eichhorn, M.

Esser, M. J. D.

Forbes, A.

Gong, M.

Gong, M. L.

B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
[CrossRef]

Gruber, R.

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

Hoffman, H. J.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

Hu, S. L.

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Huai, X. L.

Huang, F.

F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).

Jackson, S. D.

Jain, P. K.

P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer 131(1), 011304–0113047 (2009).
[CrossRef]

Jia, W. W.

F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).

Kai, K. G.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Koechner, W.

W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44(7), 3162–3170 (1973).
[CrossRef]

Lei, M.

Li, Z. G.

Liu, Q.

B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
[CrossRef]

Liu, T.

Liu, Y. G.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Lu, F. Y.

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Lu, Y. F.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Mccomb, T.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Merazzi, S.

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

Pantell, R. H.

Peng, B.

B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
[CrossRef]

Pfistner, C.

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

Richardson, M.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Rizwan-uddin,

P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer 131(1), 011304–0113047 (2009).
[CrossRef]

Siegman, A. E.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Singh, S.

P. K. Jain, S. Singh, and Rizwan-uddin, “Analytical solution to transient asymmetric heat conduction in a multilayer annulus,” J. Heat Transfer 131(1), 011304–0113047 (2009).
[CrossRef]

Sudesh, V.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90(3-4), 369–372 (2008).
[CrossRef]

Sun, T. T.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Tao, Y. J.

Wang, H. J.

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Wang, L.

Wang, Y. F.

F. Huang, Y. F. Wang, W. W. Jia, and W. Dong, “Modeling and resolving calculation of thermal effect in face-pumped high power heat capacity disk laser,” Proc. SPIE 6823, 6823111–6823118 (2007).

Wang, Z.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Weber, R.

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

Wever, H. P.

C. Pfistner, R. Weber, H. P. Wever, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd: YAG, Nd: GSGG, and Nd: YLF rods,” IEEE J. Quantum Electron. 30(7), 1605–1615 (1994).
[CrossRef]

Xie, C. X.

S. L. Hu, C. X. Xie, F. Y. Lu, F. J. Dong, H. J. Wang, S. M. Zhang, and X. Y. Dong, “Analysis the dynamics of pulse pumped Yb-doped double-clad fiber laser,” Acta Photon. Sin. 34, 333–335 (2005).

Xu, S. H.

Yan, P.

Yang, P.

B. Peng, M. L. Gong, P. Yang, and Q. Liu, “Q-switched fiber laser by all-fiber piezoelectric modulation and pulsed pump,” Opt. Commun. 282(10), 2066–2069 (2009).
[CrossRef]

Yang, Z. M.

Ye, C. G.

Yuan, S. Z.

Y. G. Liu, C. S. Zhang, T. T. Sun, Y. F. Lu, Z. Wang, S. Z. Yuan, K. G. Kai, and X. Y. Dong, “Clad-pumped Er3+/Yb3+-doped short pulse fiber laser with high average power output exceeding 2 W,” Acta Phys. Sin. 55, 4679–4685 (2006).

Zhang, C. S.

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Figures (11)

Fig. 1
Fig. 1

Schematic illustration of a pulse end-pumped short-length fiber.

Fig. 2
Fig. 2

The temperature distribution from the analytical solution at the time of 0.1 s. (a) The 3D temperature distribution; (b) The temperature distribution along the active fiber axial coordinate.

Fig. 3
Fig. 3

A time-dependent temperature finite element analysis at the time of 0.1 s. (a) The temperature distribution of fiber end surface at the pump side; (b) The temperature distribution along the active fiber axial coordinate.

Fig. 4
Fig. 4

The temperature distribution of fiber end surface at the pump side along the active fiber radial coordinate. (a) During the first pulse pump time; (b) The first 5 ms during the first pulse pump space time; (c) The last 10 ms during the first pulse pump space time; (d) The long enough pump space time and t0 = 0.01, interval: 1 s.

Fig. 5
Fig. 5

The transient temperature variation in the centre of fiber end surface as a function of time. Graph b and c are the enlarged drawings of the part of graph a.

Fig. 6
Fig. 6

The transient temperature under different pulse duration. Red: t0 = 0.06, Blue: t0 = 0.01.

Fig. 7
Fig. 7

The 3D radial, tangential and axial thermal stress distributions at the time of 0.1 s. (a) σ rr ; (b) σθθ; (c) σ zz

Fig. 8
Fig. 8

The 3D radial, tangential and axial thermal strains distributions at the time of 0.1 s. (a) εrr ; (b) εθθ ; (c) εzz

Fig. 9
Fig. 9

The 3D the radial, axial thermal stress and radial strain distribution at the time of 0.1 s under the plain-strain assumption. (a) σ rr ; (b) σ zz , (c) εrr

Fig. 10
Fig. 10

The radial thermal stress in the centre of fiber end surface as a function of time. (a)T0 = 0.1 s, t0 = 0.01 s; (b) T0 = 0.001 s, t0 = 0.0001 s; (c) T0 = 0.0001 s, t0 = 0.00001 s.

Fig. 11
Fig. 11

The phase shift rise induced by thermal effects as a function of time. Graph b is the enlarged drawing of graph a

Equations (115)

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2T(z,r,t)r2+1rT(z,r,t)r+2T(z,r,t)z2+Q(z,r,t)k=ρckT(z,r,t)t
             0<rr1
2T(z,r,t)r2+1rT(z,r,t)r+2T(z,r,t)z2=ρckT(z,r,t)t
r1<rr2
T=Th,                                                               t=0
kT/r=h(ThT),                                                 r=r2
T/z=0,                               z=0,                                                 z=l
T1=T2,                                       T1/r=T2/r,                                     r=r1
T/r=0                                                                                                     r=0
Q(z,r,t)=(2ηαPin/πωp2)exp(2r2/ωp2αz)g(t)
θ=TT0=p=1kρclNpJ0(βpr)exp(kρcβp2t)0tg0p(τ)exp(kρcβp2τ)dτ+                                 m=1p=12kρclNpJ0(βpr)cos(ηmz)exp[kρc(βp2+ηm2)t]0tgmp(τ)exp[kρc(βp2+ηm2)τ]dτ
g0p(τ)=0l0r1Q(z,r,t)kJ0(βpr)rdrdz
gmp(τ)=0l0r1Q(z,r,t)kcos(mπLz)J0(βpr)rdrdz
0tg0p(τ)exp(kρcβp2τ)dτ=2ηPin[1exp(βl)]kπωp20r1exp(2r2/ωp2)J0(βpr)rdr0tg(τ)exp(kρcβp2τ)dτ
0tgmp(τ)exp[kρc(βp2+ηm2)τ]dτ=2ηβ2l2Pin[1exp(βl)cos(mπ)]kπωp2(β2l2+m2π2)0r1exp(2r2/ωp2)J0(βpr)rdr0tg(τ)exp[kρc(βp2+ηm2)τ]dτ
ηm=mπ/l
Np=r22(k2βp2+h2)J02(βpr2)/2k2βp2
   hJ0(βpr2)kβpJ1(βpr2)     =0
σrr0=γE(1ν)[1r220r2T1rdr1r20rT1rdr]=γE(1ν)p=1Apβp[J1(βpr2)r2J1(βpr)r]
σθθ0=γE(1ν)[1r220r2T1rdr+1r20rT1rdrT(r)]=γE(1ν)p=1Apβp[J1(βpr2)r2+J1(βpr)rβpJ0(βpr)]
σzz0=γE(1ν)[2r220r2T1rdrT1]=γE(1ν)p=1Apβp[2J1(βpr2)r2βpJ0(βpr)]
σrz0=0
Ap=kρclNpexp(kρcβp2t)0tg0p(τ)exp(kρcβp2τ)dτ
2ur1r2ur+112νr(1rr(rur)+zuz)=2(1+ν)12νγTr
2uz+112νr(1rr(rur)+zuz)=2(1+ν)12νγTz
2Φ=(1+ν)/(1ν)γθ2
22L=0
σrr1=E1+ν(2Φr22Φ)
σrr2=E1+ν[2r2(Lz)ν2(Lz)]
σθθ1=E1+ν(1rΦr2Φ)
σθθ2=E1+ν[1rr(Lz)ν2(Lz)]
σzz1=E1+ν(2Φz22Φ)
σzz2=E1+ν[2z2(Lz)(2ν)2(Lz)]
σrz1=E1+ν2Φrz
σrz2=E1+ν[2rz(Lz)(1ν)r(2L)]
σij=σij1+σij2
Φ=m=1p=1Cmpcos(mπlz)J0(βpr)
Cmp=1+ν1νγ2kL2ρclNp(m2π2+L2βp2)exp[kρc(βp2+ηm2)t]0tgmp(τ)exp[kρc(βp2+ηm2)τ]dτ
L=1+ν1νγp=1[D1pβpJ0(βpr)+D2prJ1(βpr)]exp(βpz)βp
σrr1=E1+νm=1p=1Cmp[βprJ1(βpr)+(mπl)2J0(βpr)]cos(mπlz)
σθθ1=E1+νm=1p=1Cmp[βp2J0(βpr)βprJ1(βpr)+(mπl)2J0(βpr)]cos(mπlz)
σzz1=E1+νm=1p=1Cmpβp2J0(βpr)cos(mπlz)
σθθ2=E1νγp=1[D1pJ1(βpr)r+D2p(12ν)βpJ0(βpr)]exp(βpz)
σrz1=E1+νm=1p=1Cmpβpmπlsin(mπlz)J0(βpr)
σrr2=E1νγp=1{D1p[βpJ0(βpr)+J1(βpr)r]+D2p[(12ν)βpJ0(βpr)βp2rJ1(βpr)]}exp(βpz)
σzz2=E1νγp=1{D1pβpJ0(βpr)+D2pβp[rβpJ1(βpr)2(2ν)J0(βpr)]}exp(βpz)
σrz2=E1νγp=1{D1pβpJ1(βpr)D2pβp[βprJ0(βpr)+2(1ν)J1(βpr)]}exp(βpz)
σrr=0                               σrz=0                                                           r=r2
D1pexp(βpz)=m=1Cmpkβpmπsin(mπz/l)/lh+[kβp2r2/h+2(1ν)]D2pexp(βpz)
D2pexp(βpz)=m=1Cmp[1/r2+km2π2/hl2]cos(mπz/l)[1/βpr2kβp/h]m=1Cmpkβpmπsin(mπz/l)/lh2(1ν)/βpr2βpr2k2βp3r2/h2
Cmp=2kL2ρclNp(m2π2+L2βp2)exp[kρc(βp2+ηm2)t]0tgmp(τ)exp[kρc(βp2+ηm2)τ]dτ
σij=σij0+σij1+σij2
εrr=[σrrν(σθθ+σzz)]/E+γθ
εθθ=[σθθν(σrr+σzz)]/E+γθ
εzz=[σzzν(σrr+σθθ)]/E+γθ
g(t)={1(n1)T0t(n1)T0+t00(n1)T0+t0tnT0
Δn(z,t)=(n/T)0θ(z,r,t)fs(r)2πrdr
fs(r)=1πr12exp(r2r12)
Δϕ1(t)=2πλsnT0l0r1θ(z,r,t)fs(r)2πrdrdz
Δϕ2(t)=2πλs0l0r1(n01)εzzdrdz
Δϕ(t)=Δϕ1(t)+Δϕ2(t)
Δϕ(t)=2πλsnTlπr12p=1kρclNpexp(kρcβp2t)0tg0p(τ)exp(kρcβp2τ)dτ0r1J0(βpr)exp(r2r12)2πrdr
2θi(z,r,t)r2+1rθi(z,r,t)r+2θi(z,r,t)z2+gi(z,r,t)=ρckθi(z,r,t)t            i=1,2
θi=0,                 t=0
kθ2/r+hθ2=0,                             r=r2
θi/z=0,             z=​ ​ ​ ​ ​0,                     z=L
θ1=θ2,                 θ1/r=θ2/r,                 r=r1
θ1/r=0                               r=0
g1=Q(zr,t)/k
g2=0
θi(z,r,t)=m=0Z(ηm,z)θim(r,t)/N(ηm)
θim(r,t)=0LZ(ηm,z)θi(z,r,t)dz
Z(ηm,z)={cos(ηmz)1
ηm={mπ/l0
1N(ηm)={2/l1/l
(m0)(m=0)
2θim(r,t)r2+1rθim(r,t)rηm2θim(r,t)+gim=ρckθim(r,t)t
θim=0,                                                               t=0
kθ2m/r+hθ2m=0,                                                 r=r2
θ1m=θ2m,                                       θ1m/r=θ2m/r,                                     r=r1
θ1m/r=0                                                                                                     r=0
gim=0LZ(ηm,z)gidz
ri1ri[ρckθim(r,t)t]rRim(r)dr=ri1ri[d2Rim(r)dr2+1rdRim(r)drηm2Rim(r)]rθim(r,t)dr+[rRim(r)θim(r,t)rrθim(r,t)dRim(r)dr]ri1ri+ri1rirRim(r)gimdr
d2Rim(r)dr2+1rdRim(r)dr+(ηm2+λ2)Rim(r)=0
kR2m/r+hR2m=0,                                                 r=r2
R1m=R2m                                                       r=r1
R1m/r=R2m/r                                                 r=r1
R1m/r=0                                                                                                     r=0
Rimp(βpr)=aimpJ0(βpr)+bimpY0(βpr)
i=12ri1rirRimp(βpr)Rimq(βqr)dr={0Nmp
ifif
pqp=q
a2mp=1
b2mp=0
   hJ0(βpr2)kβpJ1(βpr2)     =0
Nmp=r22(k2βp2+h2)J02(βpr2)/2k2βp2
ri1ri[ρckθim(r,t)t+λ2θim(r,t)]rRimp(r)dr=[rRimp(r)θim(r,t)rrθim(r,t)dRimp(r)dr]ri1ri+ri1rirRimp(r)gimdr
i=12ri1ri[ρckθim(r,t)t+λ2θim(r,t)]rRimp(r)dr=i=12ri1rirRimp(r)gimdr
θmp(t)=i=12ri1rirRimp(r)θim(r,t)dr
gmp(t)=i=12ri1rirRimp(r)gimdr=0r1rR1mp(r)g1mdr
ρckdθmpdt+λ2θmp=gmp
θmp(t)=θmp(0)exp(kρcλ2t)+exp(kρcλ2t)0tkρcgmpexp(kρcλ2t)dτ=kρcexp(kρcλ2t)0tgmp(τ)exp(kρcλ2t)dτ
θim(r,t)=p=1cmp(t)Rimp(r)
cmp(t)=i=12ri1rirRimp(r)θim(r,t)drNmp=θmp(t)Nmp
θi(z,r,t)=m=0p=1Z(ηm,z)N(ηm)θmp(t)NmpRimp(r)
0tg(τ)exp[kρc(βp2+ηm2)τ]dτ=n=2N(n2)T0(n2)T0+t0exp[kρc(βp2+ηm2)τ]dτ+(n1)T0texp[kρc(βp2+ηm2)τ]dτ
(n1)T0t(n1)T0+t0
0tg(τ)exp[kρc(βp2+ηm2)τ]dτ=n=1N(n1)T0(n1)T0+t0exp[kρc(βp2+ηm2)τ]dτ
(n1)T0+t0tnT0
σrr=γE(1ν)[1r220r2θrdr1r20rθrdr]
σθθ=γE(1ν)[1r220r2θrdr+1r20rθrdrθ]
σzz=γE(1ν)[2r220r2T1rdrT1]
εrr=(1ν2)E(σrrν1νσθθ)+(1+ν)γθ
εθθ=(1ν2)E(σθθν1νσrr)+(1+ν)γθ
εzz=0

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