## Abstract

We develop a theoretical framework to analyze the mechanism of refractive index changes (RIC) in double-clad Yb^{3+} doped optical fibers under resonant core or clad pumping, and with signal amplification. The model describes and compares thermal and electronic contributions to the phase shifts induced on the amplified signal at 1064 nm and the probe signal at 1550 nm, i.e. located inside and outside of the fiber amplification band, respectively. The ratio between the thermal and electronic phase shifts is evaluated as a function of the pump pulse duration, the gain saturation, the amplified beam power and for a variety of fiber parameters.

© 2013 Optical Society of America

## 1. Introduction

The understanding of the origin of the refractive index changes (RICs) in intensively-pumped rare-earth doped fibers appears to be very important because the pump- and signal-induced index changes could significantly impact the dynamics of the fiber lasers and amplifiers, the fiber mode structures, the fiber Bragg gratings, and so on [1–11]. Besides, the enhanced nonlinear phase shift could be used for optical switching [1, 3], coherent beam combining [12–19] and adaptative interferometry [20].

Two main mechanisms of the RIC have been discussed recently: the well-known thermal index change *δn _{T}* caused by the heating owing to the matrix absorption and quantum defects [1,2,5,21,22], and the athermal index change

*δn*(also called electronic or “Kramers-Kroning” effect) related to the population change of the active-ion levels with different polarizabilities [1, 3,4,16–18]. The latter effect has been previously investigated mainly for crystal lasers [23,24] and now attracts a great interest for fiber lasers and amplifiers [7, 8, 19, 25, 26].

_{N}This paper is devoted to a comparative study of the thermal and electronic RICs in optical fiber doped by Yb^{3+}-ions leading to phase shifts of the probe beams at 1064 nm and 1550 nm, i.e. inside and outside the fiber amplification band, respectively, under different pumping conditions, different amplified beam powers and for a variety of the fiber parameters. Theoretical calculations are in good agreement with our experimental results reported earlier [4, 15, 17, 19, 20].

## 2. Description of the refractive index changes in the optical fibers

For analyzing the RIC and the phase shift of a probe beam in an active fiber, the fiber is described as in Fig. 1 by the infinite composite cylinder consisting of a Yb^{3+}-doped core (green), a silica-glass pumping cladding (blue) and a plastic cladding (red). The following fiber parameters have been used for calculations: the core radius, *r*_{0} is 1.8 μm; the glass radius, *r*_{1} is 62.5 μm; the external plastic radius, *r*_{2} is 125 μm; Yb-ion doping concentration in the core is 8.56 × 10^{19} cm^{−3}; the fiber length, *l* is 2 m. Other physical parameters of the fibers are given in Table 1. Only calculation results related to aluminum-silicate (AS) fibers are discussed here. The results for phosphate-silicate (PS) fibers can be obtained with the same formalism using the corresponding data shown in Table 1.

The phase shift of the probe beam at *λ* in the optical fiber of length *l* associated with the RIC induced by the pumping beam at *λ _{p}* ∼ 976nm in the presence or absence of an amplified beam at

*λ*∼ 1064nm can be described by the following expression [32] as a fixed testing-beam structure is assumed:

_{A}*β*=

*k*

_{0}Δ

*n*is the variation of the propagation constant

*β*(

*z*) associated with the RIC,

*k*

_{0}= 2

*π*/

*λ*, the refractive index change Δ

*n*is the sum of the thermal and electronic contributions, Δ

*n*(

*z*,

*r*,

*t*) =

*δn*(

_{T}*z*,

*r*,

*t*)+

*δn*(

_{N}*z*,

*r*,

*t*),

*G*(

*r*) is the radial distribution of the probe beam field [33]:

*u*and

*w*are defined by:

*u*and

*w*are linked by the characteristic Eq.: In these relations,

*J*is the Bessel function of order

_{i}*i*,

*K*is the modified Bessel function of order

_{i}*i*,

*A*is the normalization factor,

*n*is the refractive index of the core (

_{i}*i*= 0) and the cladding (

*i*= 1).

#### 2.1. Thermal contribution to refractive index changes

The thermal contribution to the RIC in the fiber can be defined by the expression:

where*∂n/∂T*is the thermo-optic coefficient of the silica glass, and

*δT*(

*z*,

*r*,

*t*) is the temperature variation of the fiber.

The temperature evolution in the fiber under pumping and in the presence of amplified signal is described by the following Eq.:

where ${a}_{i}^{2}={K}_{i}/\left({\rho}_{i}{c}_{pi}\right)$ is the thermal diffusivity,*K*is the thermal conductivity,

_{i}*ρ*and

_{i}*c*are the density and heat capacity (

_{pi}*i*= 1 for glass and

*i*= 2 for plastic),

*Q*(

*z*,

*r*,

*t*) is the heat source inside the fiber. In the case of pumping at 976 nm (in the lowest sublevel of

^{2}F

_{5/2}-state) and amplification signal at 1064 nm, the heat source can be described by the sum:

*h*is the Planck’s constant,

*ν*

_{B}_{1}is the frequency of the nonradiative transition between sublevels of the ground state,

*N*

_{2}(

*z*,

*r*,

*t*) is the population of the excited level,

*τ*

_{1}is the relaxation time of the excited level (0.83 ms and 1.245 ms for AS and PS fibers, respectively [29]),

*N*

_{0}is the total population of excited and unexcited ions,

*I*is the intensity of the pumping beam,

_{p}*ν*is the pumping frequency,

_{p}*I*is the luminescence intensity,

_{L}*I*is the amplified beam intensity,

_{A}*ν*is the amplified beam frequency,

_{A}*σ*

_{21}(

*ν*) is the emission cross-section at frequency

*ν*,

*σ*

_{12}(

*ν*) is the absorption cross-section at frequency

*ν*,

*ν*is the effective frequency of the luminescence corresponding to

_{L}*λ*∼ 1.01μm,

_{L}*ν*

_{B}_{3}=

*ν*−

_{p}*ν*,

_{A}*ν*=

_{BL}*ν*−

_{p}*ν*, ${\alpha}_{P}^{nr}$ and ${\alpha}_{L}^{nr}$ are the absorption coefficients of the pumping and amplified beams in the silica glass host, respectively. The first summand in (7) corresponds to the gray losses in the matrix, and the other three summands describe to thermalization of the pump, luminescence, and amplified beam respectively.

_{L}In the case of a linear heat transfer from plastic to air, the boundary condition on the outer surface of the plastic cylinder can be described by the Newton’s law:

*η*is the heat transfer coefficient from plastic to air.

The increase of temperature *δT* (*z*, *r*, *t*) of the fiber (inner cladding and core regions) can be found from (6) – (8) and written by the following expression [34]:

*J*

_{0}is the Bessel function of the zero order,

*r*is the transverse (radial) coordinate of the fiber,

*Z*is described by the following expression:

_{n}*ψ*=

_{n,i,j}*μ*/

_{n}r_{i}*a*and

_{j}*μ*is the

_{n}*n*-th positive root of the Eq.:

*J*

_{1},

*Y*

_{0}and

*Y*

_{1}are the Bessel function of the first order, Neumann functions of the zero and first order, respectively.

When the fiber is pumped, generation of heat is created inside the core and the cladding. This leads to a modification of the refractive index through the thermo-optic effect. But as the temperature is increased, the fiber length and section will expand and these secondary effects will also lead to phase shifts induced on the probe beam. A careful analysis is thus necessary to compare the relative contributions of these two secondary thermal effects with the thermo-optic effect acting directly on the refractive index *n*. Let us define *δφ*_{1}, *δφ*_{2} and *δφ*_{3}, the phase shift contributions arising respectively from the local thermo-optic coefficient, the transverse thermal expansion, and the longitudinal thermal expansion.

The thermal contribution *δφ*_{1} associated with the thermo-optic coefficient is described by the formula:

To calculate the fiber deformation, it is necessary to solve the problem of thermoelasticity in a quasi-static formulation with a given temperature field in the approximation of plain strain.

The effect of radial thermal expansion on the phase shift can be accounted for as follows: increasing the radius of the fiber *r*_{0} leads to a change of the mode and the wavenumber *β*. Then the phase shift *δφ*_{2} associated with this expansion is expressed as:

*δ*is the coefficient of thermal expansion of glass,

_{T}*b*is the Poisson’s coefficient of glass,

The fiber elongation makes the following contribution to the phase change:

*δ*is the thermal expansion coefficient of glass.

_{T}Estimations show that contributions to the phase shift caused by the elongation *δφ*_{3} and the expansion *δφ*_{2} of the fiber are much smaller than the contribution *δφ*_{1} due to changes in refractive index under the influence of temperature. In fact,

*δφ*

_{3}/

*δφ*

_{1}is smaller by more than 2 orders of magnitude at used parameters, and the contribution associated with the thermo-optic coefficient is the main thermal contribution.

#### 2.2. Electronic (population) contribution to refractive index changes

The athermal (electronic) component of the RIC of doped silica fibers *δn _{N}* at the presence of the population change

*δN*of Yb

^{3+}ions can be described by the following expression [23]:

*n*

_{0}is the undisturbed refractive index and Δ

*p*(

*λ*) is the polarizability difference of medium particles at the probe wavelength expressed as:

*σ*(

_{esa}*λ*) is the excited-state absorption cross-section at

*λ*.

In Yb-doped materials, the well-allowed UV transitions to the 5d-electron shell and the charge-transfer transition are characterized by oscillator forces that are several orders of magnitude higher than the forces of optical transitions inside the 4f-electron shell. As a consequence, in the IR spectrum band, the polarizability difference Δ*p*(*λ*) is expressed from Eq. (22) as a sum of contributions from the near-resonance transitions (between the ground and excited states) and non-resonance UV transitions [17]. The calculated spectrum of the polarizability difference in Yb-doped fiber within the amplification band is presented in Fig. 2 where one can see that within the Yb-fiber amplification band (around 1060 nm) the non-resonant contributions to the polarizability difference are dominating over the resonant parts. This fact has been confirmed in the experiment [20] for aluminum-silicate fibers.

System of levels of ytterbium ions in the glass can be regarded as quasi two-level system provided thermodynamic equilibrium of the population distribution over the sublevels set in each multiplet is assumed. But the description of this system as a two-level one, leads to the fact that absorption and stimulated emission cross-sections are significantly different functions of the wavelength. Under this assumption and assumption of uniform distribution of the activation, the average population *N̄*_{2}(*z*) of the metastable level, given by:

*P*,

_{p}*P*and

_{L}*P*are powers of the pump, the luminescence and the amplified beam, respectively,

_{A}*S*

_{core}is the area of the core, ${\alpha}_{P}^{nr}$, ${\alpha}_{L}^{nr}$ and ${\alpha}_{A}^{nr}$ are coefficients of nonresonant losses for the pump, luminescence and amplified waves, respectively.

*ζ*is the Langevin source of the luminescence:

*ν*is the effective luminescence frequency. In these expressions, the parameter Γ

_{L}*, is defined as follow:*

_{p,L,A;j}*I*is the intensity of the pump, luminescence and amplified beams respectively.

_{p,L,A}Calculations showed that one can neglect the luminescence power *P _{L}* in solving this problem, so it is no longer taken into account in the following theoretical estimations.

For the fiber under simulations, the saturation powers defined as ${P}_{p,A}^{\text{sat}}=\frac{{S}_{\text{mode}}h{\nu}_{p,A}}{\left[{\sigma}_{12}\left({\nu}_{p,A}\right)+{\sigma}_{21}\left({\nu}_{p,A}\right)\right]{\tau}_{1}}$ are 530 mW, 0.41 mW and 8.2 mW for the clad pumping at 976 nm, and core pumping at 976 nm and 1064 nm, respectively.

## 3. Comparison of thermal and electronic contributions to the phase shift of a probing beam during pumping without strong amplified beam

The ratio of the thermal and electronic contributions to RIC was found to depend on the powers of the pump and amplified beams. In this section we consider this ratio in the case of a small amplified beam power, so in the limit *P _{A}* → 0, without saturation of the population inversion.

Investigation of the thermal and electronic contributions to the phase shift of the fiber was carried out using analytical and numerical method. Coefficients *μ _{n}* of Eq. (9) were numerically calculated by solving Eq. (11) and then Eq. (9) was numerically integrated by the predictor-corrector finite-difference method scheme. Calculations showed that, depending on the conditions, thermal or electronic contributions can be the dominant effect.

Graphics of the distribution of RIC on the distance from the fiber axis are shown on Fig. 3. The orange trace gives the steady-state electronic RIC contribution and one can see a modification of the core refractive index of 9 × 10^{−7}. The thermal RIC contribution is given after 0.5 s in the green trace and in thermal steady-state in the red trace. One can see that the core refractive index change due to the thermal effect is 2 × 10^{−7} after 0.5 s, so much lower than the electronic contribution and then reaches 8 × 10^{−6} in steady-state, which is here much higher than the electronic contribution. Fig. 3 clearly illustrates that there should exist a time for which the two contributions are equal: this is the alignment time estimated to 0.757s as explained below.

In dynamics, the ratio of the thermal *δφ _{T}* and the electronic

*δφ*contributions could be characterized through the alignment time

_{N}*t*that is defined as the time needed for these two contribution to become equal:

_{m}The analytical estimate of the alignment time of electronic and thermal contributions *t _{m}* is obtained from (30) under assumptions of rectangularly switching in time heating

*Q*(

*t*),

*K*

_{2}/

*η*

_{2}≫ 1 and ${\alpha}_{P}^{nr}={\alpha}_{L}^{nr}={\alpha}_{A}^{nr}=0$. After some computations,

*t*is found to be:

_{m}If *Q*(*r*) is uniform over the cross-section of the fiber core (pumping to the clad), i.e. it does not depend on *r*, then relation (33) implies *ξ* = 0.38 at *λ* = 1550nm and *ξ* = 0.69 at *λ* = 1060nm. It is worth to notice that (31) is the upper limit of the equalizing time, i.e. the time for which the thermal contribution accounted through the term *n* = 1 in the expression (9) becomes equal to the electronic contribution to the phase shift.

To observe the alignment of thermal and electronic contributions to the phase shift in a finite time, it is necessary that the ratio *ε* of the electronic steady-state contribution and thermal steady-state contribution to the phase shift is less than one:

From (34), one can get the condition for the critical value of the heat transfer factor *η _{cr}*, corresponding to

*ε*= 1:

*η*>

*η*) the RIC electronic contribution always predominates over the thermal contribution even in steady-state.

_{cr}When one substitutes the parameters mentioned above (included in Eq. (35)), *η _{cr}* takes the value 0.003 cal/(cm

^{2}K s) and this value clearly obeys

*K*

_{2}/(

*η*

_{cr}r_{2}) ≈ 12.3 ≫ 1. This result suggests that, when the heat transfer coefficient

*η*<

*η*, one can use Eq. (31) to find the alignment time of the contributions. For the calculations presented here, the heat transfer coefficient from plastic to air is assumed to be

_{cr}*η*= 0.000118cal/(cm

^{2}K s) [35], if not otherwise stated. Discussing kinetics of the thermal and electronic contributions to the RIC, we have noted that the heat diffusion time of the core ${t}_{\text{core}}\approx 4{r}_{0}^{2}/{a}_{1}^{2}\approx 16\mu \text{s}$ is much less than the Yb-ion life time of the level

^{2}F

_{5/2},

*τ*

_{1}= 830μs and the time for which the general temperature balance between the fiber and its surrounding is achieved ${t}_{\text{bal}}\approx \frac{{r}_{2}{K}_{2}\left(1-p\right)}{2{a}_{2}^{2}\eta}\approx 18\hspace{0.17em}\text{s}$ (this relationship is determined by the decrement of the first dominating term in Eq. (9)).

Figure 4 shows the temporal behavior of the RIC at 1064 nm on the fiber axis simulated for two cladding pump powers: 145 mW in Fig. 4(a), and 100 W in Fig. 4(b). The comparison of these Figs. highlights a strong increase of the total RIC with increase of the pump power due to income of both the thermal and electronic contributions. However, income of electronic RIC at growing pump power is less than the increment of the thermal component. At high pump power, the population becomes saturated but heat continues to rise. The electronic contribution dominates at the beginning, while the thermal RIC dominates in the steady-state.

The increase of the input signal power increases heating and decreases of the population inversion. So, the thermal contribution increases whereas the electronic contribution decreases

Figure 5 shows the time behaviors of all the contributions to the phase shift on the probe beam at 1550 nm for the case of cladding pumping [Fig. 5(a)] and core pumping [Fig. 5(b)], and without signal amplification. The orange curve is the electronic contribution to the phase shift, and in each case, after a rapid transient, this electronic phase shift saturates. Green, blue and red curves show respectively the thermal contributions to phase shift arising from the thermo-optic coefficient, the elongation of the fiber, and the lateral expansion of the fiber. As expected, the thermo-optic effect on the phase shift is much larger than the two other thermal effects.

The comparative analysis of all the contribution to the phase shift on the probe beam at 1550 nm for the cladding or core pumping (in the absence of an amplified beam) highlights several stages in the RIC dynamics: domination of the electronic component at the beginning, alignment of the electronic and thermal contribution, and domination of the thermal RIC in steady-state.

It should be noticed that the relative electronic contribution to the phase shift is less than the relative electronic contribution to RIC. This is explained by the fact that (1) the electronic phase shift is proportional to the integral of the product of the RIC and the mode profile of the probe beam, and (2) the thermal RIC is located on whole fiber structure whereas the electronic RIC is located only inside the core of the fiber. As the mode of the probe beam is present outside the core too, this leads to the decrease of the electronic contribution to the phase shift.

The expression of the phase shift associated with the electronic part of RIC for the cases of core and clad pumping (without an amplified signal) can be found from Eq. (1) and with the steady-state solutions of (24) to (27):

Theoretical estimate of the electronic part of the phase shift given by expression (36) coincides with the value obtained by numerical modeling. It is 0.75*π* for 145 mW cladding pumping (the pump intensity does not depend on the transverse coordinate) and is 2.87*π* for similar core pumping. The experimental value of the phase shift in the core pumped Yb-doped fiber was about 3.8*π*[4,17]. This difference can be explained [16] by taking into account the doping profile which is uniform in the simulation case and was gaussian-like in the experiment but with mean value given in table 1. The alignment time of the thermal and electronic phase shift for clad pumping with power of 145 mW is estimated to be *t _{m}* = 839 ms from expression (31) and is close to the calculated value of 757 ms (the intersection point of the green and orange curves on Fig. 5(a)).

## 4. Comparing of thermal and electronic contributions to the phase shift of a probing beam during pumping in the presence of the amplified beam

The presence of the amplified beam which depopulates the Yb^{3+} ions excited state can strongly change the ratio of RIC contributions. One can see from the Figs. 4(a)–4(b) that, with increase of the amplified beam power, the thermal contribution to RIC increases, while the electronic contribution decreases. The thermal loading increases due to growing rate of the cycle: the excited states populated by the pump are depopulated by the amplified beam. This process is more efficient for higher pump power levels causing stronger effect of the amplified beam on the thermal and electronic RICs.

The total phase shift shown in Fig. 6 also depends on power of the pumping and amplified beams. The electronic component dominates over the thermal one during the transient stage. However, in steady-state, the thermally induced phase shift is higher than the electronic contribution. An increase of the amplified beam power leads to depopulation of the excited level of the Yb^{3+} ions and a decrease of the electronic contribution. The thermal component continues to increase due to higher rate of the excited state population - depopulation cycle. For a given set of parameters, the alignment time of the electronic and thermal components of the phase shift is less for cladding pumping than for the core pumping.

The alignment time of the phase-shift components shown in Fig. 7 depends on the pump and amplified beam powers, the fiber length and the heat transfer coefficient *η*.

The increase of the amplified beam power reduces the population inversion and increases the heat evolution, so the alignment time is reduced. Reduction of the pump power causes a decrease of both excited state population and amplified beam power, leading to a strong reduction of the thermal contribution and a less drastic decrease of the electronic contribution. Therefore, the alignment time of both contributions increases with the reduction of the pump power. Lower values of heat-transfer coefficient *η* provide higher thermal effect, and so the electronic contribution could no longer dominate over the thermal one, the alignment time increases. The estimation results are in good qualitative agreement with direct measurements of the phase shift induced in Yb-doped fiber by 1064 nm radiation [36].

## 5. The electronic and thermal phase shifts for the amplified pulse train

The repetitively-pulsing amplified signal combined with CW pumping is an operational regime commonly used with real amplification systems. The numerical simulation of the phase shift at the signal wavelength of 1060 nm highlights interesting dynamical behavior with antiphase behavior of the electronic and thermal phase shift contributions during the signal pulse propagation [Fig. 8]. The reduction of the excited state population during the amplified pulses causes negative RIC, while the thermal load induced by the stimulated transitions to the ground-state leads to the increase of the refractive index. For amplified pulses with duration in nanoseconds - microseconds time scale, i.e. much shorter than the excited-state life time, the decrease of the electronic index change is stronger than the income of the thermal RIC. As a result the total index decreases during the amplified pulses, while the time-averaged refractive index increases due to increase of the thermal contribution.

## 6. Conclusion

We have developed a theoretical model for the description of thermal and electronic mechanisms of refractive index changes in Yb-doped optical fiber. The first or second effects can dominate depending on parameters of the system. At the initial stage of the pumping or pulse amplification, the electronic contribution dominates for any types of Yb-doped fibers. The thermal contribution becomes more significant at CW operation, specially when the amplitudes of the amplified and pumping beams (in saturation conditions) are increased, when fiber core diameter is increased, and when the heat transfer coefficient to air is decreased. The electronic index change predominates over the thermal contribution during amplified signal pulses with duration less than the excited-state life time even in the case of CW pumping.

## Appendix. Particular solutions of the heating problem

1. To give insight on the thermal behavior, one can simplify the system by considering a fiber without a plastic coating and assume simpler boundary conditions such as constant temperature at the cladding-air interface or no heat transfer from cladding to air (the fiber is perfectly insulated). In that way, we end up with a simple cylinder whose analytical solutions can be found in the literature [37].

We assume that the heat source *Q*(*r*, *t*) is uniformly distributed in the core (*r* ≤ *r*_{0}) and has a rectangular time profile of duration *τ _{p}*. When the temperature at the external boundary of the glass is fixed (ideal heat sink at the external boundary), the expression of the temperature on the axis (

*r*= 0) of the activated fiber could be evaluated from Eqs. (9) – (13) as:

*μ*(parameter without dimension) is the n-th positive root of the Eq.

_{n}*J*

_{0}(

*μ*) = 0 and

*S*(

_{n}*t*,

*τ*,

_{p}*τ*

_{1}) is given by:

*δT*goes to zero.

Another limiting case occurs when the heat flow at the external boundary of the glass is equal to zero (heat-insulated external glass boundary). The temperature on the axis of the activated fiber (*r* = 0) is expressed as:

*μ*(parameter without dimension) is n-th positive root of the Eq.

_{n}*J*

_{1}(

*μ*) = 0 and

*U*(

*t*,

*τ*) is given by:

_{p}*δT*has a permanent term after the switch-off of the heat source, and this term increases linearly with the heat duration

*τ*.

_{p}2. In general, the population and the heat source depend on the transverse coordinate *r*, the boundary condition at the external plastic-clad is arbitrary and the steady-state solution on the axis (*r* = 0, *t* → **∞**) can be expressed from (9) as:

*Q*does not depend on the transverse coordinate (cladding pumping for instance), this solution can be simplified to:

For a rectangular heat source, the temperature on the core axis can be found from (9) as:

*μ*(parameter in s

_{n}^{−1/2}) is n-th positive root of the Eq. (11).

In the case of the cladding pumping, the heat source *Q* can be assumed to independent of *r*, and the solution (43) simplified to:

This work was supported by European “ERA-NET RUS INTENT”, FEDER-Wallonia “Mediatic” and FP7 IRSES projects, the Interuniversity Attraction Pole program IAP PVII-35 of the Belgian Science Policy and by the Ministry of Education and Science of the Russian Federation and Russian Academy of Sciences.

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