Analytical investigation on transient thermal effects in pulse end-pumped short-length fiber laser

T. Liu; Z. M. Yang; S. H. Xu

doi:10.1364/OE.17.012875

Optics Express
Vol. 17,
Issue 15,
pp. 12875-12890
(2009)
•https://doi.org/10.1364/OE.17.012875

Analytical investigation on transient thermal effects in pulse end-pumped short-length fiber laser

T. Liu, Z. M. Yang, and S. H. Xu

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T. Liu, Z. M. Yang, and S. H. Xu, "Analytical investigation on transient thermal effects in pulse end-pumped short-length fiber laser," Opt. Express 17, 12875-12890 (2009)

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- Lasers and Laser Optics
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About this Article
History
- Original Manuscript: May 26, 2009
- Revised Manuscript: June 30, 2009
- Manuscript Accepted: July 2, 2009
- Published: July 13, 2009

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.

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 [1–3]. Owing to the excellent stability, low noise and all-fiber configurations, much attention has been paid to pulse pumped fiber lasers [4–7]. 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 [11–13]. 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]:

\frac{\partial^{2} T (z, r, t)}{\partial r^{2}} + \frac{1}{r} \frac{\partial T (z, r, t)}{\partial r} + \frac{\partial^{2} T (z, r, t)}{\partial z^{2}} + \frac{Q (z, r, t)}{k} = \frac{ρ c}{k} \frac{\partial T (z, r, t)}{\partial t}

\frac{\partial^{2} T (z, r, t)}{\partial r^{2}} + \frac{1}{r} \frac{\partial T (z, r, t)}{\partial r} + \frac{\partial^{2} T (z, r, t)}{\partial z^{2}} = \frac{ρ c}{k} \frac{\partial T (z, r, t)}{\partial t}

T = T_{h}, t = 0

k \partial T / \partial r = h (T_{h} - T), r = r_{2}

\partial T / \partial z = 0, z = 0, z = l

T_{1} = T_{2}, \partial T_{1} / \partial r = \partial T_{2} / \partial r, r = r_{1}

\partial T / \partial r = 0 r = 0

where

Q (z, r, t) = (2 η α P_{i n} / π ω_{p}^{2}) \exp (- 2 r^{2} / ω_{p}^{2} - α z) g (t)

is the transient heat loading function in fiber. T₁ and T₂ are the temperatures in fiber core and cladding region, respectively. T_h is the heat sink temperature. k denotes the fiber thermal conductivity. h is the convective coefficient. ω_p 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, P_in 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:

\begin{array}{l} θ = T - T_{0} = \sum_{p = 1}^{\infty} \frac{k}{ρ c l N_{p}} J_{0} (β_{p} r) \exp (- \frac{k}{ρ c} β_{p}^{2} t) \int_{0}^{t} g_{0 p} (τ) \exp (\frac{k}{ρ c} β_{p}^{2} τ) d τ + \\ \sum_{m = 1}^{\infty} \sum_{p = 1}^{\infty} \frac{2 k}{ρ c l N_{p}} J_{0} (β_{p} r) \cos (η_{m} z) \exp [- \frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) t] \int_{0}^{t} g_{m p} (τ) \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ \end{array}

where

g_{0 p} (τ) = \int_{0}^{l} \int_{0}^{r_{1}} \frac{Q (z, r, t)}{k} J_{0} (β_{p} r) r d r d z

\int_{0}^{t} g_{0 p} (τ) \exp (\frac{k}{ρ c} β_{p}^{2} τ) d τ = \frac{2 η P_{i n} [1 - \exp (- β l)]}{k π ω_{p}^{2}} \int_{0}^{r_{1}} \exp (- 2 r^{2} / ω_{p}^{2}) J_{0} (β_{p} r) r d r \int_{0}^{t} g (τ) \exp (\frac{k}{ρ c} β_{p}^{2} τ) d τ

h J_{0} (β_{p} r_{2}) - k β_{p} J_{1} (β_{p} r_{2}) = 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 $σ_{i j}^{(p)}$ 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]:

σ_{r r}^{0} = \frac{γ E}{(1 - ν)} [\frac{1}{r_{2}^{2}} \int_{0}^{r_{2}} T_{1}^{'} r d r - \frac{1}{r^{2}} \int_{0}^{r} T_{1}^{'} r d r] = \frac{γ E}{(1 - ν)} \sum_{p = 1}^{\infty} \frac{A_{p}}{β_{p}} [\frac{J_{1} (β_{p} r_{2})}{r_{2}} - \frac{J_{1} (β_{p} r)}{r}]

σ_{θ θ}^{0} = \frac{γ E}{(1 - ν)} [\frac{1}{r_{2}^{2}} \int_{0}^{r_{2}} T_{1}^{'} r d r + \frac{1}{r^{2}} \int_{0}^{r} T_{1}^{'} r d r - T (r)] = \frac{γ E}{(1 - ν)} \sum_{p = 1}^{\infty} \frac{A_{p}}{β_{p}} [\frac{J_{1} (β_{p} r_{2})}{r_{2}} + \frac{J_{1} (β_{p} r)}{r} - β_{p} J_{0} (β_{p} r)]

σ_{r z}^{0} = 0

Here

A_{p} = \frac{k}{ρ c l N_{p}} \exp (- \frac{k}{ρ c} β_{p}^{2} t) \int_{0}^{t} g_{0 p} (τ) \exp (\frac{k}{ρ c} β_{p}^{2} τ) d τ

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.

\nabla^{2} u_{r} - \frac{1}{r^{2}} u_{r} + \frac{1}{1 - 2 ν} \frac{\partial}{\partial r} (\frac{1}{r} \frac{\partial}{\partial r} (r u_{r}) + \frac{\partial}{\partial z} u_{z}) = \frac{2 (1 + ν)}{1 - 2 ν} γ \frac{\partial T}{\partial r}

\nabla^{2} u_{z} + \frac{1}{1 - 2 ν} \frac{\partial}{\partial r} (\frac{1}{r} \frac{\partial}{\partial r} (r u_{r}) + \frac{\partial}{\partial z} u_{z}) = \frac{2 (1 + ν)}{1 - 2 ν} γ \frac{\partial T}{\partial z}

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

\nabla^{2} Φ = (1 + ν) / (1 - ν) γ θ_{2}

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:

\nabla^{2} \nabla^{2} L = 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 $σ_{i j}^{'}$ are specified on the boundary, they need to be expressed in terms of Φ and L, and can be derived, respectively, from:

σ_{r r}^{1} = \frac{E}{1 + ν} (\frac{\partial^{2} Φ}{\partial r^{2}} - \nabla^{2} Φ)

σ_{θ θ}^{1} = \frac{E}{1 + ν} (\frac{1}{r} \frac{\partial Φ}{\partial r} - \nabla^{2} Φ)

σ_{z z}^{1} = \frac{E}{1 + ν} (\frac{\partial^{2} Φ}{\partial z^{2}} - \nabla^{2} Φ)

σ_{r z}^{1} = \frac{E}{1 + ν} \frac{\partial^{2} Φ}{\partial r \partial z}

σ_{i j}^{'} = σ_{i j}^{1} + σ_{i j}^{2}

Substituting the expression of θ₂ into Eq. (3), we will obtain the particular solution as:

Φ = \sum_{m = 1}^{\infty} \sum_{p = 1}^{\infty} C_{m p} \cos (\frac{m π}{l} z) J_{0} (β_{p} r)

where

C_{m p} = - \frac{1 + ν}{1 - ν} γ \frac{2 k L^{2}}{ρ c l N_{p} (m^{2} π^{2} + L^{2} β_{p}^{2})} \exp [- \frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) t] \int_{0}^{t} g_{m p} (τ) \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ

According to the characteristics of the expression of Φ, the form of the Love function is assumed as:

L = \frac{1 + ν}{1 - ν} γ \sum_{p = 1}^{\infty} [\frac{D_{1 p}}{β_{p}} J_{0} (β_{p} r) + D_{2 p} r J_{1} (β_{p} r)] \frac{\exp (- β_{p} z)}{- β_{p}}

where D_1p and D_2p are undetermined coefficients. So the thermal stresses caused by θ₂ are obtained as:

σ_{r r}^{1} = \frac{E}{1 + ν} \sum_{m = 1}^{\infty} \sum_{p = 1}^{\infty} C_{m p} [\frac{β_{p}}{r} J_{1} (β_{p} r) + {(\frac{m π}{l})}^{2} J_{0} (β_{p} r)] \cos (\frac{m π}{l} z)

σ_{θ θ}^{1} = \frac{E}{1 + ν} \sum_{m = 1}^{\infty} \sum_{p = 1}^{\infty} C_{m p} [β_{p}^{2} J_{0} (β_{p} r) - \frac{β_{p}}{r} J_{1} (β_{p} r) + {(\frac{m π}{l})}^{2} J_{0} (β_{p} r)] \cos (\frac{m π}{l} z)

σ_{θ θ}^{2} = \frac{E}{1 - ν} γ \sum_{p = 1}^{\infty} [- D_{1 p} \frac{J_{1} (β_{p} r)}{r} + D_{2 p} (1 - 2 ν) β_{p} J_{0} (β_{p} r)] \exp (- β_{p} z)

σ_{r r}^{2} = \frac{E}{1 - ν} γ \sum_{p = 1}^{\infty} {D_{1 p} [- β_{p} J_{0} (β_{p} r) + \frac{J_{1} (β_{p} r)}{r}] + D_{2 p} [(1 - 2 ν) β_{p} J_{0} (β_{p} r) - β_{p}^{2} r J_{1} (β_{p} r)]} \exp (- β_{p} z)

σ_{z z}^{2} = \frac{E}{1 - ν} γ \sum_{p = 1}^{\infty} {D_{1 p} β_{p} J_{0} (β_{p} r) + D_{2 p} β_{p} [r β_{p} J_{1} (β_{p} r) - 2 (2 - ν) J_{0} (β_{p} r)]} \exp (- β_{p} z)

According to the following traction free condition,

σ_{r r}^{'} = 0 σ_{r z}^{'} = 0 r = r_{2}

the coefficients D_1p and D_2p are determined as:

D_{1 p} \exp (- β_{p} z) = \sum_{m = 1}^{\infty} C_{m p}^{'} k β_{p} m π \sin (m π z / l) / l h + [k β_{p}^{2} r_{2} / h + 2 (1 - ν)] D_{2 p} \exp (- β_{p} z)

D_{2 p} \exp (- β_{p} z) = \frac{\sum_{m = 1}^{\infty} C_{m p}^{'} [1 / r_{2} + k m^{2} π^{2} / h l^{2}] \cos (m π z / l) - [1 / β_{p} r_{2} - k β_{p} / h] \sum_{m = 1}^{\infty} C_{m p}^{'} k β_{p} m π \sin (m π z / l) / l h}{2 (1 - ν) / β_{p} r_{2} - β_{p} r_{2} - k^{2} β_{p}^{3} r_{2} / h^{2}}

where

C_{m p}^{'} = \frac{2 k L^{2}}{ρ c l N_{p} (m^{2} π^{2} + L^{2} β_{p}^{2})} \exp [- \frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) t] \int_{0}^{t} g_{m p} (τ) \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ

The total thermal stress is obtained as:

σ_{i j} = σ_{i j}^{0} + σ_{i j}^{1} + σ_{i j}^{2}

The stress, strain and temperature are related by the generalized Hooke’s laws:

ε_{r r} = [σ_{r r} - ν (σ_{θ θ} + σ_{z z})] / E + γ θ

ε_{θ θ} = [σ_{θ θ} - ν (σ_{r r} + σ_{z z})] / E + γ θ

ε_{z z} = [σ_{z z} - ν (σ_{r r} + σ_{θ θ})] / E + γ θ

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) = {\begin{cases} 1 (n - 1) T_{0} \leq t \leq (n - 1) T_{0} + t_{0} \\ 0 (n - 1) T_{0} + t_{0} \leq t \leq n T_{0} \end{cases}

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, T_h = 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, P_in = 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) σ_rr; (b) σ_θθ; (c) σ_zz

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

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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) σ_rr; (b) σ_zz, (c) ε_rr

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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]:

Δ n (z, t) = (\partial n / \partial T) \int_{0}^{\infty} θ (z, r, t) f_{s} (r) 2 π r d r

where f_s(r) is the signal mode intensity, and can be approximated by a Gaussian distribution.

f_{s} (r) = \frac{1}{π r_{1}^{2}} \exp (- \frac{r^{2}}{r_{1}^{2}})

The instantaneous thermal phase shift caused by the varying index of refraction is obtained by integrating △n along z.

Δ ϕ_{1} (t) = \frac{2 π}{λ_{s}} \frac{\partial n}{\partial T} \int_{0}^{l} \int_{0}^{r_{1}} θ (z, r, t) f_{s} (r) 2 π r d r d z

The instantaneous thermal phase shift caused by axial strain field is gotten as [19]:

Δ ϕ_{2} (t) = \frac{2 π}{λ_{s}} \int_{0}^{l} \int_{0}^{r_{1}} (n_{0} - 1) ε_{z z} d r d z

where n₀ is the refractive index of fiber, λ_s is the signal wavelength.

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

Δ ϕ (t) = Δ ϕ_{1} (t) + Δ ϕ_{2} (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:

Δ ϕ (t) = \frac{2 π}{λ_{s}} \frac{\partial n}{\partial T} \frac{l}{π r_{1}^{2}} \sum_{p = 1}^{\infty} \frac{k}{ρ c l N_{p}} \exp (- \frac{k}{ρ c} β_{p}^{2} t) \int_{0}^{t} g_{0 p} (τ) \exp (\frac{k}{ρ c} β_{p}^{2} τ) d τ \int_{0}^{r_{1}} J_{0} (β_{p} r) \exp (- \frac{r^{2}}{r_{1}^{2}}) 2 π r d r

Index temperature coefficient is 9.67 × 10⁻⁶ K⁻¹, and λ_s = 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-T_h, Eq. (1a)-(1g) are simplified as:

\frac{\partial^{2} θ_{i} (z, r, t)}{\partial r^{2}} + \frac{1}{r} \frac{\partial θ_{i} (z, r, t)}{\partial r} + \frac{\partial^{2} θ_{i} (z, r, t)}{\partial z^{2}} + g_{i} (z, r, t) = \frac{ρ c}{k} \frac{\partial θ_{i} (z, r, t)}{\partial t} i = 1, 2

θ_{i} = 0, t = 0

k \partial θ_{2} / \partial r + h θ_{2} = 0, r = r_{2}

\partial θ_{i} / \partial z = 0, z = ​
​
​
​
​ 0, z = L

θ_{1} = θ_{2}, \partial θ_{1} / \partial r = \partial θ_{2} / \partial r, r = r_{1}

\partial θ_{1} / \partial r = 0 r = 0

g_{1} = Q (z � r, t) / k

Inversion formula is:

θ_{i} (z, r, t) = \sum_{m = 0}^{\infty} Z (η_{m}, z) θ_{i m} (r, t) / N (η_{m})

followed by integral transform:

θ_{i m} (r, t) = \int_{0}^{L} Z (η_{m}, z) θ_{i} (z, r, t) d z

where

Z (η_{m}, z) = {\begin{cases} \cos (η_{m} z) \\ 1 \end{cases}

The integral transform of the system (7) by the application of the transform (8b) yields:

\frac{\partial^{2} θ_{i m} (r, t)}{\partial r^{2}} + \frac{1}{r} \frac{\partial θ_{i m} (r, t)}{\partial r} - η_{m}^{2} θ_{i m} (r, t) + g_{i m} = \frac{ρ c}{k} \frac{\partial θ_{i m} (r, t)}{\partial t}

θ_{i m} = 0, t = 0

k \partial θ_{2 m} / \partial r + h θ_{2 m} = 0, r = r_{2}

θ_{1 m} = θ_{2 m}, \partial θ_{1 m} / \partial r = \partial θ_{2 m} / \partial r, r = r_{1}

\partial θ_{1 m} / \partial r = 0 r = 0

where

g_{i m} = \int_{0}^{L} Z (η_{m}, z) g_{i} d z

Multiplying Eq. (9a) by

\int_{r_{i - 1}}^{r_{i}} r R_{i m} (r) d r

and using integration of parts twice on the first term on the right hand side, one gets:

\begin{array}{l} \int_{r_{i - 1}}^{r_{i}} [\frac{ρ c}{k} \frac{\partial θ_{i m} (r, t)}{\partial t}] r R_{i m} (r) d r = \int_{r_{i - 1}}^{r_{i}} [\frac{d^{2} R_{i m} (r)}{d r^{2}} + \frac{1}{r} \frac{d R_{i m} (r)}{d r} - η_{m}^{2} R_{i m} (r)] r θ_{i m} (r, t) d r + \\ {[r R_{i m} (r) \frac{\partial θ_{i m} (r, t)}{\partial r} - r θ_{i m} (r, t) \frac{d R_{i m} (r)}{d r}]}_{r_{i - 1}}^{r_{i}} + \int_{r_{i - 1}}^{r_{i}} r R_{i m} (r) g_{i m} d r \end{array}

R_im(r) in the above equation is chosen to satisfy

\frac{d^{2} R_{i m} (r)}{d r^{2}} + \frac{1}{r} \frac{d R_{i m} (r)}{d r} + (- η_{m}^{2} + λ_{}^{2}) R_{i m} (r) = 0

k \partial R_{2 m} / \partial r + h R_{2 m} = 0, r = r_{2}

R_{1 m} = R_{2 m} r = r_{1}

\partial R_{1 m} / \partial r = \partial R_{2 m} / \partial r r = r_{1}

\partial R_{1 m} / \partial r = 0 r = 0

Solutions of the Eq. (11e) are eigenfunctions R_im(r) corresponding to the eigenvalues β_p

R_{i m p} (β_{p} r) = a_{i m p} J_{0} (β_{p} r) + b_{i m p} Y_{0} (β_{p} r)

where

β_{p}^{2} = - η_{m}^{2} + λ_{}^{2}

a_{i m p}

and

b_{i m p}

are the coefficients.

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

\sum_{i = 1}^{2} \int_{r_{i - 1}}^{r_{i}} r R_{i m p} (β_{p} r) R_{i m q} (β_{q} r) d r = {\begin{cases} 0 \\ N_{m p} \end{cases}

Without loss of generality, we set one of the nonvanishing cofficients, say a_1mp equal to unity, according to Eq. (11b)-Eq. (11d), a_2mp, b_2mp, and β_p can be obtained:

a_{2 m p} = 1

h J_{0} (β_{p} r_{2}) - k β_{p} J_{1} (β_{p} r_{2}) = 0

N_{m p} = r_{2}^{2} (k^{2} β_{p}^{2} + h^{2}) J_{0}^{2} (β_{p} r_{2}) / 2 k^{2} β_{p}^{2}

In view of Eq. (11a), Eq. (10) can be written as

\int_{r_{i - 1}}^{r_{i}} [\frac{ρ c}{k} \frac{\partial θ_{i m} (r, t)}{\partial t} + λ_{}^{2} θ_{i m} (r, t)] r R_{i m p} (r) d r = {[r R_{i m p} (r) \frac{\partial θ_{i m} (r, t)}{\partial r} - r θ_{i m} (r, t) \frac{d R_{i m p} (r)}{d r}]}_{r_{i - 1}}^{r_{i}} + \int_{r_{i - 1}}^{r_{i}} r R_{i m p} (r) g_{i m} d r

According to Eq. (9b)-Eq. (9d) and summing up the resulting expression from fiber core to cladding, one gets:

\sum_{i = 1}^{2} \int_{r_{i - 1}}^{r_{i}} [\frac{ρ c}{k} \frac{\partial θ_{i m} (r, t)}{\partial t} + λ_{}^{2} θ_{i m} (r, t)] r R_{i m p} (r) d r = \sum_{i = 1}^{2} \int_{r_{i - 1}}^{r_{i}} r R_{i m p} (r) g_{i m} d r

Defining

θ_{m p} (t) = \sum_{i = 1}^{2} \int_{r_{i - 1}}^{r_{i}} r R_{i m p} (r) θ_{i m} (r, t) d r

and

g_{m p} (t) = \sum_{i = 1}^{2} \int_{r_{i - 1}}^{r_{i}} r R_{i m p} (r) g_{i m} d r = \int_{0}^{r_{1}} r R_{1 m p} (r) g_{1 m} d r

we have

\frac{ρ c}{k} \frac{d θ_{m p}}{d t} + λ_{}^{2} θ_{m p} = g_{m p}

Solution of Eq. (12) is

θ_{m p} (t) = θ_{m p} (0) \exp (- \frac{k}{ρ c} λ^{2} t) + \exp (- \frac{k}{ρ c} λ^{2} t) \int_{0}^{t} \frac{k}{ρ c} g_{m p} \exp (- \frac{k}{ρ c} λ^{2} t) d τ = \frac{k}{ρ c} \exp (- \frac{k}{ρ c} λ^{2} t) \int_{0}^{t} g_{m p} (τ) \exp (\frac{k}{ρ c} λ^{2} t) d τ

θ_{i m} (r, t)

can be expanded in the Fourier series as follows:

θ_{i m} (r, t) = \sum_{p = 1}^{\infty} c_{m p} (t) R_{i m p} (r)

According to the orthogonality,

c_{m p} (t)

can be obtained as:

c_{m p} (t) = \frac{\sum_{i = 1}^{2} \int_{r_{i - 1}}^{r_{i}} r R_{i m p} (r) θ_{i m} (r, t) d r}{N_{m p}} = \frac{θ_{m p} (t)}{N_{m p}}

So the

θ_{i} (z, r, t)

can be gotten as:

θ_{i} (z, r, t) = \sum_{m = 0}^{\infty} \sum_{p = 1}^{\infty} \frac{Z (η_{m}, z)}{N (η_{m})} \frac{θ_{m p} (t)}{N_{m p}} R_{i m p} (r)

Appendix B

\int_{0}^{t} g (τ) \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ = \sum_{n = 2}^{N} \int_{(n - 2) T_{0}}^{(n - 2) T_{0} + t_{0}} \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ + \int_{(n - 1) T_{0}}^{t} \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ

(n - 1) T_{0} \leq t \leq (n - 1) T_{0} + t_{0}

\int_{0}^{t} g (τ) \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ = \sum_{n = 1}^{N} \int_{(n - 1) T_{0}}^{(n - 1) T_{0} + t_{0}} \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ

(n - 1) T_{0} + t_{0} \leq t \leq n T_{0}

Appendix C

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

σ_{r r}^{″} = \frac{γ E}{(1 - ν)} [\frac{1}{r_{2}^{2}} \int_{0}^{r_{2}} θ r d r - \frac{1}{r^{2}} \int_{0}^{r} θ r d r]

σ_{θ θ}^{″} = \frac{γ E}{(1 - ν)} [\frac{1}{r_{2}^{2}} \int_{0}^{r_{2}} θ r d r + \frac{1}{r^{2}} \int_{0}^{r} θ r d r - θ]

σ_{z z}^{″} = \frac{γ E}{(1 - ν)} [\frac{2}{r_{2}^{2}} \int_{0}^{r_{2}} T_{1}^{'} r d r - T_{1}^{'}]

ε_{r r}^{″} = \frac{(1 - ν^{2})}{E} (σ_{r r}^{″} - \frac{ν}{1 - ν} σ_{θ θ}^{″}) + (1 + ν) γ θ

ε_{θ θ}^{″} = \frac{(1 - ν^{2})}{E} (σ_{θ θ}^{″} - \frac{ν}{1 - ν} σ_{r r}^{″}) + (1 + ν) γ θ

ε_{z z}^{″} = 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

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

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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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Fig. 7 The 3D radial, tangential and axial thermal stress distributions at the time of 0.1 s. (a) σ _rr ; (b) σ_θθ; (c) σ _zz

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

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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) σ _rr ; (b) σ _zz , (c) ε_rr

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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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Equations (115)

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\frac{\partial^{2} T (z, r, t)}{\partial r^{2}} + \frac{1}{r} \frac{\partial T (z, r, t)}{\partial r} + \frac{\partial^{2} T (z, r, t)}{\partial z^{2}} + \frac{Q (z, r, t)}{k} = \frac{ρ c}{k} \frac{\partial T (z, r, t)}{\partial t}

0 < r \leq r_{1}

\frac{\partial^{2} T (z, r, t)}{\partial r^{2}} + \frac{1}{r} \frac{\partial T (z, r, t)}{\partial r} + \frac{\partial^{2} T (z, r, t)}{\partial z^{2}} = \frac{ρ c}{k} \frac{\partial T (z, r, t)}{\partial t}

r_{1} < r \leq r_{2}

T = T_{h}, t = 0

k \partial T / \partial r = h (T_{h} - T), r = r_{2}

\partial T / \partial z = 0, z = 0, z = l

T_{1} = T_{2}, \partial T_{1} / \partial r = \partial T_{2} / \partial r, r = r_{1}

\partial T / \partial r = 0 r = 0

Q (z, r, t) = (2 η α P_{i n} / π ω_{p}^{2}) \exp (- 2 r^{2} / ω_{p}^{2} - α z) g (t)

\begin{array}{l} θ = T - T_{0} = \sum_{p = 1}^{\infty} \frac{k}{ρ c l N_{p}} J_{0} (β_{p} r) \exp (- \frac{k}{ρ c} β_{p}^{2} t) \int_{0}^{t} g_{0 p} (τ) \exp (\frac{k}{ρ c} β_{p}^{2} τ) d τ + \\ \sum_{m = 1}^{\infty} \sum_{p = 1}^{\infty} \frac{2 k}{ρ c l N_{p}} J_{0} (β_{p} r) \cos (η_{m} z) \exp [- \frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) t] \int_{0}^{t} g_{m p} (τ) \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ \end{array}

g_{0 p} (τ) = \int_{0}^{l} \int_{0}^{r_{1}} \frac{Q (z, r, t)}{k} J_{0} (β_{p} r) r d r d z

g_{m p} (τ) = \int_{0}^{l} \int_{0}^{r_{1}} \frac{Q (z, r, t)}{k} \cos (\frac{m π}{L} z) J_{0} (β_{p} r) r d r d z

\int_{0}^{t} g_{0 p} (τ) \exp (\frac{k}{ρ c} β_{p}^{2} τ) d τ = \frac{2 η P_{i n} [1 - \exp (- β l)]}{k π ω_{p}^{2}} \int_{0}^{r_{1}} \exp (- 2 r^{2} / ω_{p}^{2}) J_{0} (β_{p} r) r d r \int_{0}^{t} g (τ) \exp (\frac{k}{ρ c} β_{p}^{2} τ) d τ

\int_{0}^{t} g_{m p} (τ) \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ = \frac{2 η β^{2} l^{2} P_{i n} [1 - \exp (- β l) \cos (m π)]}{k π ω_{p}^{2} (β^{2} l^{2} + m^{2} π^{2})} \int_{0}^{r_{1}} \exp (- 2 r^{2} / ω_{p}^{2}) J_{0} (β_{p} r) r d r \int_{0}^{t} g (τ) \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ

η_{m} = m π / l

N_{p} = r_{2}^{2} (k^{2} β_{p}^{2} + h^{2}) J_{0}^{2} (β_{p} r_{2}) / 2 k^{2} β_{p}^{2}

h J_{0} (β_{p} r_{2}) - k β_{p} J_{1} (β_{p} r_{2}) = 0

σ_{r r}^{0} = \frac{γ E}{(1 - ν)} [\frac{1}{r_{2}^{2}} \int_{0}^{r_{2}} T_{1}^{'} r d r - \frac{1}{r^{2}} \int_{0}^{r} T_{1}^{'} r d r] = \frac{γ E}{(1 - ν)} \sum_{p = 1}^{\infty} \frac{A_{p}}{β_{p}} [\frac{J_{1} (β_{p} r_{2})}{r_{2}} - \frac{J_{1} (β_{p} r)}{r}]

σ_{θ θ}^{0} = \frac{γ E}{(1 - ν)} [\frac{1}{r_{2}^{2}} \int_{0}^{r_{2}} T_{1}^{'} r d r + \frac{1}{r^{2}} \int_{0}^{r} T_{1}^{'} r d r - T (r)] = \frac{γ E}{(1 - ν)} \sum_{p = 1}^{\infty} \frac{A_{p}}{β_{p}} [\frac{J_{1} (β_{p} r_{2})}{r_{2}} + \frac{J_{1} (β_{p} r)}{r} - β_{p} J_{0} (β_{p} r)]

σ_{z z}^{0} = \frac{γ E}{(1 - ν)} [\frac{2}{r_{2}^{2}} \int_{0}^{r_{2}} T_{1}^{'} r d r - T_{1}^{'}] = \frac{γ E}{(1 - ν)} \sum_{p = 1}^{\infty} \frac{A_{p}}{β_{p}} [\frac{2 J_{1} (β_{p} r_{2})}{r_{2}} - β_{p} J_{0} (β_{p} r)]

σ_{r z}^{0} = 0

A_{p} = \frac{k}{ρ c l N_{p}} \exp (- \frac{k}{ρ c} β_{p}^{2} t) \int_{0}^{t} g_{0 p} (τ) \exp (\frac{k}{ρ c} β_{p}^{2} τ) d τ

\nabla^{2} u_{r} - \frac{1}{r^{2}} u_{r} + \frac{1}{1 - 2 ν} \frac{\partial}{\partial r} (\frac{1}{r} \frac{\partial}{\partial r} (r u_{r}) + \frac{\partial}{\partial z} u_{z}) = \frac{2 (1 + ν)}{1 - 2 ν} γ \frac{\partial T}{\partial r}

\nabla^{2} u_{z} + \frac{1}{1 - 2 ν} \frac{\partial}{\partial r} (\frac{1}{r} \frac{\partial}{\partial r} (r u_{r}) + \frac{\partial}{\partial z} u_{z}) = \frac{2 (1 + ν)}{1 - 2 ν} γ \frac{\partial T}{\partial z}

\nabla^{2} Φ = (1 + ν) / (1 - ν) γ θ_{2}

\nabla^{2} \nabla^{2} L = 0

σ_{r r}^{1} = \frac{E}{1 + ν} (\frac{\partial^{2} Φ}{\partial r^{2}} - \nabla^{2} Φ)

σ_{r r}^{2} = \frac{E}{1 + ν} [\frac{\partial^{2}}{\partial r^{2}} (\frac{\partial L}{\partial z}) - ν \nabla^{2} (\frac{\partial L}{\partial z})]

σ_{θ θ}^{1} = \frac{E}{1 + ν} (\frac{1}{r} \frac{\partial Φ}{\partial r} - \nabla^{2} Φ)

σ_{θ θ}^{2} = \frac{E}{1 + ν} [\frac{1}{r} \frac{\partial}{\partial r} (\frac{\partial L}{\partial z}) - ν \nabla^{2} (\frac{\partial L}{\partial z})]

σ_{z z}^{1} = \frac{E}{1 + ν} (\frac{\partial^{2} Φ}{\partial z^{2}} - \nabla^{2} Φ)

σ_{z z}^{2} = \frac{E}{1 + ν} [\frac{\partial^{2}}{\partial z^{2}} (\frac{\partial L}{\partial z}) - (2 - ν) \nabla^{2} (\frac{\partial L}{\partial z})]

σ_{r z}^{1} = \frac{E}{1 + ν} \frac{\partial^{2} Φ}{\partial r \partial z}

σ_{r z}^{2} = \frac{E}{1 + ν} [\frac{\partial^{2}}{\partial r \partial z} (\frac{\partial L}{\partial z}) - (1 - ν) \frac{\partial}{\partial r} (\nabla^{2} L)]

σ_{i j}^{'} = σ_{i j}^{1} + σ_{i j}^{2}

Φ = \sum_{m = 1}^{\infty} \sum_{p = 1}^{\infty} C_{m p} \cos (\frac{m π}{l} z) J_{0} (β_{p} r)

C_{m p} = - \frac{1 + ν}{1 - ν} γ \frac{2 k L^{2}}{ρ c l N_{p} (m^{2} π^{2} + L^{2} β_{p}^{2})} \exp [- \frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) t] \int_{0}^{t} g_{m p} (τ) \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ

L = \frac{1 + ν}{1 - ν} γ \sum_{p = 1}^{\infty} [\frac{D_{1 p}}{β_{p}} J_{0} (β_{p} r) + D_{2 p} r J_{1} (β_{p} r)] \frac{\exp (- β_{p} z)}{- β_{p}}

σ_{r r}^{1} = \frac{E}{1 + ν} \sum_{m = 1}^{\infty} \sum_{p = 1}^{\infty} C_{m p} [\frac{β_{p}}{r} J_{1} (β_{p} r) + {(\frac{m π}{l})}^{2} J_{0} (β_{p} r)] \cos (\frac{m π}{l} z)

σ_{θ θ}^{1} = \frac{E}{1 + ν} \sum_{m = 1}^{\infty} \sum_{p = 1}^{\infty} C_{m p} [β_{p}^{2} J_{0} (β_{p} r) - \frac{β_{p}}{r} J_{1} (β_{p} r) + {(\frac{m π}{l})}^{2} J_{0} (β_{p} r)] \cos (\frac{m π}{l} z)

σ_{z z}^{1} = \frac{E}{1 + ν} \sum_{m = 1}^{\infty} \sum_{p = 1}^{\infty} C_{m p} β_{p}^{2} J_{0} (β_{p} r) \cos (\frac{m π}{l} z)

σ_{θ θ}^{2} = \frac{E}{1 - ν} γ \sum_{p = 1}^{\infty} [- D_{1 p} \frac{J_{1} (β_{p} r)}{r} + D_{2 p} (1 - 2 ν) β_{p} J_{0} (β_{p} r)] \exp (- β_{p} z)

σ_{r z}^{1} = \frac{E}{1 + ν} \sum_{m = 1}^{\infty} \sum_{p = 1}^{\infty} C_{m p} β_{p} \frac{m π}{l} \sin (\frac{m π}{l} z) J_{0} (β_{p} r)

σ_{r r}^{2} = \frac{E}{1 - ν} γ \sum_{p = 1}^{\infty} {D_{1 p} [- β_{p} J_{0} (β_{p} r) + \frac{J_{1} (β_{p} r)}{r}] + D_{2 p} [(1 - 2 ν) β_{p} J_{0} (β_{p} r) - β_{p}^{2} r J_{1} (β_{p} r)]} \exp (- β_{p} z)

σ_{z z}^{2} = \frac{E}{1 - ν} γ \sum_{p = 1}^{\infty} {D_{1 p} β_{p} J_{0} (β_{p} r) + D_{2 p} β_{p} [r β_{p} J_{1} (β_{p} r) - 2 (2 - ν) J_{0} (β_{p} r)]} \exp (- β_{p} z)

σ_{r z}^{2} = \frac{E}{1 - ν} γ \sum_{p = 1}^{\infty} {D_{1 p} β_{p} J_{1} (β_{p} r) - D_{2 p} β_{p} [β_{p} r J_{0} (β_{p} r) + 2 (1 - ν) J_{1} (β_{p} r)]} \exp (- β_{p} z)

σ_{r r}^{'} = 0 σ_{r z}^{'} = 0 r = r_{2}

D_{1 p} \exp (- β_{p} z) = \sum_{m = 1}^{\infty} C_{m p}^{'} k β_{p} m π \sin (m π z / l) / l h + [k β_{p}^{2} r_{2} / h + 2 (1 - ν)] D_{2 p} \exp (- β_{p} z)

D_{2 p} \exp (- β_{p} z) = \frac{\sum_{m = 1}^{\infty} C_{m p}^{'} [1 / r_{2} + k m^{2} π^{2} / h l^{2}] \cos (m π z / l) - [1 / β_{p} r_{2} - k β_{p} / h] \sum_{m = 1}^{\infty} C_{m p}^{'} k β_{p} m π \sin (m π z / l) / l h}{2 (1 - ν) / β_{p} r_{2} - β_{p} r_{2} - k^{2} β_{p}^{3} r_{2} / h^{2}}

C_{m p}^{'} = \frac{2 k L^{2}}{ρ c l N_{p} (m^{2} π^{2} + L^{2} β_{p}^{2})} \exp [- \frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) t] \int_{0}^{t} g_{m p} (τ) \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ

σ_{i j} = σ_{i j}^{0} + σ_{i j}^{1} + σ_{i j}^{2}

ε_{r r} = [σ_{r r} - ν (σ_{θ θ} + σ_{z z})] / E + γ θ

ε_{θ θ} = [σ_{θ θ} - ν (σ_{r r} + σ_{z z})] / E + γ θ

ε_{z z} = [σ_{z z} - ν (σ_{r r} + σ_{θ θ})] / E + γ θ

g (t) = {\begin{cases} 1 (n - 1) T_{0} \leq t \leq (n - 1) T_{0} + t_{0} \\ 0 (n - 1) T_{0} + t_{0} \leq t \leq n T_{0} \end{cases}

Δ n (z, t) = (\partial n / \partial T) \int_{0}^{\infty} θ (z, r, t) f_{s} (r) 2 π r d r

f_{s} (r) = \frac{1}{π r_{1}^{2}} \exp (- \frac{r^{2}}{r_{1}^{2}})

Δ ϕ_{1} (t) = \frac{2 π}{λ_{s}} \frac{\partial n}{\partial T} \int_{0}^{l} \int_{0}^{r_{1}} θ (z, r, t) f_{s} (r) 2 π r d r d z

Δ ϕ_{2} (t) = \frac{2 π}{λ_{s}} \int_{0}^{l} \int_{0}^{r_{1}} (n_{0} - 1) ε_{z z} d r d z

Δ ϕ (t) = Δ ϕ_{1} (t) + Δ ϕ_{2} (t)

Δ ϕ (t) = \frac{2 π}{λ_{s}} \frac{\partial n}{\partial T} \frac{l}{π r_{1}^{2}} \sum_{p = 1}^{\infty} \frac{k}{ρ c l N_{p}} \exp (- \frac{k}{ρ c} β_{p}^{2} t) \int_{0}^{t} g_{0 p} (τ) \exp (\frac{k}{ρ c} β_{p}^{2} τ) d τ \int_{0}^{r_{1}} J_{0} (β_{p} r) \exp (- \frac{r^{2}}{r_{1}^{2}}) 2 π r d r

\frac{\partial^{2} θ_{i} (z, r, t)}{\partial r^{2}} + \frac{1}{r} \frac{\partial θ_{i} (z, r, t)}{\partial r} + \frac{\partial^{2} θ_{i} (z, r, t)}{\partial z^{2}} + g_{i} (z, r, t) = \frac{ρ c}{k} \frac{\partial θ_{i} (z, r, t)}{\partial t} i = 1, 2

θ_{i} = 0, t = 0

k \partial θ_{2} / \partial r + h θ_{2} = 0, r = r_{2}

\partial θ_{i} / \partial z = 0, z = ​
​
​
​
​ 0, z = L

θ_{1} = θ_{2}, \partial θ_{1} / \partial r = \partial θ_{2} / \partial r, r = r_{1}

\partial θ_{1} / \partial r = 0 r = 0

g_{1} = Q (z � r, t) / k

g_{2} = 0

θ_{i} (z, r, t) = \sum_{m = 0}^{\infty} Z (η_{m}, z) θ_{i m} (r, t) / N (η_{m})

θ_{i m} (r, t) = \int_{0}^{L} Z (η_{m}, z) θ_{i} (z, r, t) d z

Z (η_{m}, z) = {\begin{cases} \cos (η_{m} z) \\ 1 \end{cases}

η_{m} = {\begin{cases} m π / l \\ 0 \end{cases}

\frac{1}{N (η_{m})} = {\begin{cases} 2 / l \\ 1 / l \end{cases}

\begin{array}{l} (m \neq 0) \\ (m = 0) \end{array}

\frac{\partial^{2} θ_{i m} (r, t)}{\partial r^{2}} + \frac{1}{r} \frac{\partial θ_{i m} (r, t)}{\partial r} - η_{m}^{2} θ_{i m} (r, t) + g_{i m} = \frac{ρ c}{k} \frac{\partial θ_{i m} (r, t)}{\partial t}

θ_{i m} = 0, t = 0

k \partial θ_{2 m} / \partial r + h θ_{2 m} = 0, r = r_{2}

θ_{1 m} = θ_{2 m}, \partial θ_{1 m} / \partial r = \partial θ_{2 m} / \partial r, r = r_{1}

\partial θ_{1 m} / \partial r = 0 r = 0

g_{i m} = \int_{0}^{L} Z (η_{m}, z) g_{i} d z

\begin{array}{l} \int_{r_{i - 1}}^{r_{i}} [\frac{ρ c}{k} \frac{\partial θ_{i m} (r, t)}{\partial t}] r R_{i m} (r) d r = \int_{r_{i - 1}}^{r_{i}} [\frac{d^{2} R_{i m} (r)}{d r^{2}} + \frac{1}{r} \frac{d R_{i m} (r)}{d r} - η_{m}^{2} R_{i m} (r)] r θ_{i m} (r, t) d r + \\ {[r R_{i m} (r) \frac{\partial θ_{i m} (r, t)}{\partial r} - r θ_{i m} (r, t) \frac{d R_{i m} (r)}{d r}]}_{r_{i - 1}}^{r_{i}} + \int_{r_{i - 1}}^{r_{i}} r R_{i m} (r) g_{i m} d r \end{array}

\frac{d^{2} R_{i m} (r)}{d r^{2}} + \frac{1}{r} \frac{d R_{i m} (r)}{d r} + (- η_{m}^{2} + λ_{}^{2}) R_{i m} (r) = 0

k \partial R_{2 m} / \partial r + h R_{2 m} = 0, r = r_{2}

R_{1 m} = R_{2 m} r = r_{1}

\partial R_{1 m} / \partial r = \partial R_{2 m} / \partial r r = r_{1}

\partial R_{1 m} / \partial r = 0 r = 0

R_{i m p} (β_{p} r) = a_{i m p} J_{0} (β_{p} r) + b_{i m p} Y_{0} (β_{p} r)

\sum_{i = 1}^{2} \int_{r_{i - 1}}^{r_{i}} r R_{i m p} (β_{p} r) R_{i m q} (β_{q} r) d r = {\begin{cases} 0 \\ N_{m p} \end{cases}

\begin{array}{l} i f \\ i f \end{array}

\begin{array}{l} p \neq q \\ p = q \end{array}

a_{2 m p} = 1

b_{2 m p} = 0

h J_{0} (β_{p} r_{2}) - k β_{p} J_{1} (β_{p} r_{2}) = 0

N_{m p} = r_{2}^{2} (k^{2} β_{p}^{2} + h^{2}) J_{0}^{2} (β_{p} r_{2}) / 2 k^{2} β_{p}^{2}

\int_{r_{i - 1}}^{r_{i}} [\frac{ρ c}{k} \frac{\partial θ_{i m} (r, t)}{\partial t} + λ_{}^{2} θ_{i m} (r, t)] r R_{i m p} (r) d r = {[r R_{i m p} (r) \frac{\partial θ_{i m} (r, t)}{\partial r} - r θ_{i m} (r, t) \frac{d R_{i m p} (r)}{d r}]}_{r_{i - 1}}^{r_{i}} + \int_{r_{i - 1}}^{r_{i}} r R_{i m p} (r) g_{i m} d r

\sum_{i = 1}^{2} \int_{r_{i - 1}}^{r_{i}} [\frac{ρ c}{k} \frac{\partial θ_{i m} (r, t)}{\partial t} + λ_{}^{2} θ_{i m} (r, t)] r R_{i m p} (r) d r = \sum_{i = 1}^{2} \int_{r_{i - 1}}^{r_{i}} r R_{i m p} (r) g_{i m} d r

θ_{m p} (t) = \sum_{i = 1}^{2} \int_{r_{i - 1}}^{r_{i}} r R_{i m p} (r) θ_{i m} (r, t) d r

g_{m p} (t) = \sum_{i = 1}^{2} \int_{r_{i - 1}}^{r_{i}} r R_{i m p} (r) g_{i m} d r = \int_{0}^{r_{1}} r R_{1 m p} (r) g_{1 m} d r

\frac{ρ c}{k} \frac{d θ_{m p}}{d t} + λ_{}^{2} θ_{m p} = g_{m p}

θ_{m p} (t) = θ_{m p} (0) \exp (- \frac{k}{ρ c} λ^{2} t) + \exp (- \frac{k}{ρ c} λ^{2} t) \int_{0}^{t} \frac{k}{ρ c} g_{m p} \exp (- \frac{k}{ρ c} λ^{2} t) d τ = \frac{k}{ρ c} \exp (- \frac{k}{ρ c} λ^{2} t) \int_{0}^{t} g_{m p} (τ) \exp (\frac{k}{ρ c} λ^{2} t) d τ

θ_{i m} (r, t) = \sum_{p = 1}^{\infty} c_{m p} (t) R_{i m p} (r)

c_{m p} (t) = \frac{\sum_{i = 1}^{2} \int_{r_{i - 1}}^{r_{i}} r R_{i m p} (r) θ_{i m} (r, t) d r}{N_{m p}} = \frac{θ_{m p} (t)}{N_{m p}}

θ_{i} (z, r, t) = \sum_{m = 0}^{\infty} \sum_{p = 1}^{\infty} \frac{Z (η_{m}, z)}{N (η_{m})} \frac{θ_{m p} (t)}{N_{m p}} R_{i m p} (r)

\int_{0}^{t} g (τ) \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ = \sum_{n = 2}^{N} \int_{(n - 2) T_{0}}^{(n - 2) T_{0} + t_{0}} \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ + \int_{(n - 1) T_{0}}^{t} \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ

(n - 1) T_{0} \leq t \leq (n - 1) T_{0} + t_{0}

\int_{0}^{t} g (τ) \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ = \sum_{n = 1}^{N} \int_{(n - 1) T_{0}}^{(n - 1) T_{0} + t_{0}} \exp [\frac{k}{ρ c} (β_{p}^{2} + η_{m}^{2}) τ] d τ

(n - 1) T_{0} + t_{0} \leq t \leq n T_{0}

σ_{r r}^{″} = \frac{γ E}{(1 - ν)} [\frac{1}{r_{2}^{2}} \int_{0}^{r_{2}} θ r d r - \frac{1}{r^{2}} \int_{0}^{r} θ r d r]

σ_{θ θ}^{″} = \frac{γ E}{(1 - ν)} [\frac{1}{r_{2}^{2}} \int_{0}^{r_{2}} θ r d r + \frac{1}{r^{2}} \int_{0}^{r} θ r d r - θ]

σ_{z z}^{″} = \frac{γ E}{(1 - ν)} [\frac{2}{r_{2}^{2}} \int_{0}^{r_{2}} T_{1}^{'} r d r - T_{1}^{'}]

ε_{r r}^{″} = \frac{(1 - ν^{2})}{E} (σ_{r r}^{″} - \frac{ν}{1 - ν} σ_{θ θ}^{″}) + (1 + ν) γ θ

ε_{θ θ}^{″} = \frac{(1 - ν^{2})}{E} (σ_{θ θ}^{″} - \frac{ν}{1 - ν} σ_{r r}^{″}) + (1 + ν) γ θ

ε_{z z}^{″} = 0

Abstract

1. Introduction

2. Resolving of three-dimensional transient temperature field

3. Thermal stress and strain fields

4. Analytical results and discussions

4.1. Transient temperature distribution

4.2. Transient thermal stress and strain fields

5. Thermal phase shift

6. Conclusions

Appendix C

Acknowledgement

References and links

Cited By

Figures (11)

Equations (115)

Optics Express