Fiber core mode leakage induced by refractive index variation in high-power fiber laser
1. IntroductionHigh-power fiber lasers have undergone unprecedented development over the last decade[1] and have evolved from watt-level laboratory curiosities into multi-kilowatt-scale systems with high market penetration. Their outstanding characteristics, which include high power scalability and stability, high optical efficiency, excellent beam quality, small footprint, and low weight, have become the trademarks of fiber lasers.
To operate under continuous wave multi-kilowatt power or pulsed high peak power conditions, the optical fibers used in these lasers require a large mode field diameter (MFD) to reduce the impact of nonlinear effects such as stimulated Raman scattering[2,3] and self-phase modulation on the laser performance. However, it is extremely challenging to increase the MFD while maintaining the outstanding beam quality characteristics of the fiber laser. To reduce the capacity of the transverse mode and thus ensure that fiber lasers operate under single-mode conditions or with only a few low-order modes, the refractive index difference between the fiber core and the fiber cladding must be strictly controlled.
Under the weakly guiding approximation, the refractive index changes in the fiber core or the fiber inner cladding cause variations in both the transverse mode capacity and the mode stability. As a result, the guided modes in the core region can leak into the cladding region and propagate with a larger numerical aperture. This irreversible process will lead to deterioration of the laser’s efficiency, power density, and beam quality.
Refractive index variations can be induced by various factors, including excessive fiber bending[4] and splice misalignment;[5] however, these factors are not significantly related to the laser power level. This paper discusses the process of core mode coupling into the cladding and the subsequent propagation of these modes as cladding modes as a result of thermal effects in high-power fiber lasers. Based on the analysis presented here, the relationship between the fiber core leakage loss and the fiber laser’s characteristics (e.g., cooling conditions, beam quality) is described.
In an initial effort to understand this process, the analysis focused on thermally-induced refractive index variations in the fiber core region of high-power fiber lasers. Additionally, the coupling coefficients of several typical low-order linear polarization modes coupled to the cladding modes in double-clad fibers were derived. The physical origins of the mode coupling loss in fiber lasers were discussed and a model of these losses was proposed, before the proportion of leaky modes in fiber lasers and amplifiers was numerically simulated using actual laser parameters. Experimental measurements subsequently provided data that verified these simulation results.
2. Thermally-induced refractive index variation in fiber lasersA schematic illustration of the proposed fiber amplifier model is shown in Fig. 1. Heat is generated in the optical fiber by absorption of the pumping power, which results in changes in the refractive index. Before the analysis can be performed, the power distributions of the core modes in a typical fiber laser amplifier must be calculated. The rate equations that consider the distributions of the transverse and longitudinal gains and the mode fields in a cylindrical coordinate system can be expressed in the form of the following equation set:[6]
 | (1) |
Here, the amplified spontaneous emission (ASE) spectrum is divided into Ms channels and each channel is marked using the subscript i (where i = 1, 2, …, Ms). N2(r, θ, z) and N1(r, θ, z) represent the population densities of the upper and lower levels, respectively.
and
are the pump powers in the forward and backward directions, respectively, and
and
are the signal laser powers of the i-th channel in the forward and backward directions, respectively. σap (σep) and σas (σes) are the absorption (emission) cross-sections of the pump light and the laser, respectively. σap and σep are 2.34×10−24 m2 and 2.34×10−24 m2, respectively. σas and σes of the laser are 6.58 × 10−27 m2 and 3.11 × 10−25 m2, respectively. The loss factor of the pump light, αp, and the loss factor of the laser, αsi, are 2.0 × 10−3 m−1 and 4.0 × 10−4 m−1, respectively. Δλs is the spectral width of the signal laser and
describes the influence of ASE on the mode power. When the seed power is strong enough, the ASE can be neglected and
can be set as 0.[7] φp(r, θ) and φsi(r, θ) are the power normalized intensity distributions of the pump laser and the i-th laser mode, respectively. The boundary conditions for the power normalized intensity distribution of the pump laser and the i-th laser mode are set as
 | (2) |
In these equations, the pump light is considered even distributed and contains numerous high order modes. Therefore, the normalized pump light intensity can be expressed by the area of the inner-cladding Aclad as φp(r, θ) = 1/Aclad. The dij denotes the power coupling coefficient of the i-th mode and the j-th mode.
To compare the theoretical figures with the experimental results, the actual parameters of the fiber amplifier are used to construct the simulation models, and these parameters are listed in Table 1. The power coupling coefficient is neglected by setting dij = 0 during the simulation. The simulation results are shown in Fig. 2, and indicate that a total laser output power of 666 W at 1080 nm is obtained, with 347 W in the LP01 mode and 319 W in the LP11 mode under launched pump power conditions of 750 W@975 nm. The beam quality is given as
and
.
Table 1.
Table 1.
 Table 1.
Parameters used for fiber amplifier.
.
| Parameters |
Values |
| Pump power |
750 W |
| Pump length |
975 nm |
| Signal length |
1080 nm |
| Gain fiber diameter |
20/400 μm |
| NA |
0.06/0.46 |
| Absorption coefficient at 975 nm |
1.26 dB/m |
| Length of gain fiber |
25 m |
| Seed power |
30 W |
| Table 1.
Parameters used for fiber amplifier.
. |
When the pump power is absorbed, it generates heat in the fiber. The geometry of the fiber structure is shown in Fig. 3. The radial coordinate is r and the tangential angle is φ. The symbols a, b, and c represent the core, inner cladding, and coating radii, respectively. To determine the radially varying refractive index that results from dn/dT, it is necessary to calculate the temperature distribution in the fiber.
To calculate the temperature profile T(r, φ, z) inside the optical fiber, the following equation is applied using azimuthal φ variations while the longitudinal thermal conduction is ignored:
 | (3) |
where
r is the radial coordinate and
z is the longitudinal coordinate along the fiber axis.
Q(
z) is the thermal load power per unit volume in the fiber laser and
k is the thermal conductivity of silica. The equation assumes that the heat power
Q(
z) is uniform throughout the fiber core and can be acquired from the following expression:
[8]
 | (4) |
where
Pabs(
z) is the absorbed pump power in a differential fiber element of volume. Subject to the aforementioned boundary conditions, which are expressed in Eq. (
4), the equations are then transformed into the subsequent expressions for the temperatures in regions I, II, and III:
 | (5) |
 | (6) |
where
k1,
k2, and
k3 are the thermal conductivities of silica, the cladding, and the coating material, respectively,
Ts is the fiber surface temperature, and
Tc is the coolant temperature.
h represents the convective coefficient. Based on the fiber temperature field
T(
r,
z) and the temperature coefficient of the refractive index
β1 = (d
n/d
T), the resulting refractive index variation field Δ
n(
r,
z) can be calculated as
It should be noted here that the temperature difference between the boundary and the center of the core is small when the pump power is launched at sub-kilowatt levels. For example, when a pump power of 1000 W is launched, the temperature difference is calculated to be 7.06 °C. The corresponding refractive index variation is approximately 8.33 × 10−5, and can thus be neglected. Therefore, the refractive index variation can be considered to have the same value over the entire fiber core area. Based on the simulation results for the fiber amplifier rate equation and Eqs. (4)−(6), the calculated thermallyinduced fiber core refractive index variations along the fiber length are as shown in Fig. 4. Here, the refractive index variation is caused by the thermal effects of the fiber amplifier at multi-kilowatt power levels. According to the mode coupling theory, these variations will cause a mode coupling process between the core modes and the cladding modes and the corresponding power transfer and fiber core losses, which will be discussed in subsequent sections.
3. Mode coupling coefficient between core modes and cladding modesAs described in Section 2, a thermally-induced change in the refractive index occurs when the pump power is absorbed in the fiber, and this leads to the mode coupling process. This paper describes the coupling process between a guided core mode and the cladding modes in a typical step-index fiber in detail. The interactions between the fiber mode order υ and the refractive index variation μ in a step-index fiber can be described using the coupling coefficients, which are defined as follows:[9−11]
Here, Ev and EM are the transverse components of the mode electric fields that are labeled v and μ (further details can be found in Ref. [8]). It should be noted here that the core mode field is much weaker in the cladding region than that in the core region, while the refractive index variation in the core region is several times higher than that in the cladding region. Therefore, the discussion in this paper focuses solely on the coupling process in the fiber core region.
In the weakly guiding approximation, the transverse field components of the core modes in the core region (where r ≤ a) can be expressed under linear polarization conditions as[12]
 | (9) |
where
J1(
ulmr) is the Bessel function of the first kind,
β2 = (2
π/λ)
nefflm is the propagation constant, and
is a normalization coefficient. The transverse electric field components of the cladding modes of azimuthal order
l in the core region (where
r ≤
a) can be expressed as
[9,12]
 | (10) |
where
σ = i
lnefflmZ0. Here,
l is the azimuthal mode number,
is the field normalization constant, and
ζ0 is a parameter related to the dispersion of the cladding modes of azimuthal number
l.
[9,12] This definition is discussed in greater detail in Ref. [
8]. It should be noted that if all cladding modes (with any
l) are included in Eqs. (
7)−(
9), then the azimuthal integral becomes
 | (11) |
where
lcl is the number of cladding modes. For the LP
01 mode, the only nonzero coupling constants are those between the LP
01 core mode and the cladding modes where
lcl = 1.
[9,12] For the LP
11 mode, the phase expressions are expanded, and the azimuthal integral can then be expressed as
In the above equations, δcl−l is equal to 1 when lcl = 2, and is equal to 0 when lcl ≠ 2. Therefore, the only nonzero coupling constants are those between the LP11 mode and the second-order cladding modes. The mode coupling coefficients of the LP11 even mode and the LP11 odd mode are a real number and an imaginary number, respectively. This means that the cladding mode sub-wave that is produced by the LP11 even mode has a 90° phase difference with that produced by the LP11 odd mode at the same position in the fiber laser. Simulations were performed to calculate the coupling coefficients of the LP01 mode to the first-order cladding modes and the LP11 mode to the second-order cladding modes, the results are shown in Fig. 5.
Figure 5 shows that the coupling coefficients of the LP01 mode and the LP11 mode when coupling into the even cladding mode are larger than the corresponding coefficients when coupling into the neighboring odd mode. The coupling coefficient of the LP11 mode into the cladding modes is much higher than that of the LP01 mode. In Fig. 5(b), the coupling process of the LP11 mode into the low numbers of the second-order cladding modes is shown to be very strong; the main part of the core leakage losses is taken up into the cladding modes, and this will be discussed in Section 4. To avoid the interference phenomenon of different cladding modes coupling to the same wave, the simulation process must also consider the spectral width of the fiber laser used in our experiments. Figure 6 shows the relationship between the variations in the coupling coefficients of the LP01 mode and the LP11 mode to a specific cladding mode and the signal laser wavelength over the range from 1077 nm to 1083 nm when our experiment was performed at 1080 nm with a 3 dB bandwidth of approximately 1 nm.
4. Mode coupling and core mode loss process in fiber amplifierBecause the mode coupling coefficients have been obtained, the field amplitude can then be calculated using Eq. (13). The increment in the cladding mode’s lv amplitude (where l is the cladding mode order and v is the cladding mode number) is expressed on the basis of the general coupledmode equations that describe the variations in the forward- and backward-propagating amplitudes of a mode that result from the presence of other modes near a dielectric variation in the fiber[9,11]
 | (13) |
where
Alv is the amplitude of the forward cladding mode
lv,
Aμ is the amplitude of the forward core mode
u,
Bμ is the amplitude of the backward core mode
u,
β = (2
π/λ)
neff is the propagation constant, and
Ku−1v is the coupling coefficient between
u and
lv. Based on Eq. (
13), we combine mode coupling, including fiber core mode coupling and clad mode coupling, with the rate equations. With further refinement, the new rate equations contain amplitude factors of core modes and clad modes as follows:
 | (14) |
where
and
are the power of the
i-th fiber mode. Their corresponding complex amplitudes are
and
, respectively, where
.
and
are the complex amplitudes of the k-th clad modes propagating in the forward and backward directions, respectively.
Kij(
z)
t is the transverse coupling coefficient for modes from
i-th to
j-th at position
z. The factor
stands for the excess fiber core power induced by the gain. The factors
and
represent the increased or reduced amplitude of the fiber core induced by the coupling among the fiber modes and clad modes. And
is the gain induced by mode coupling.
The amplitude attenuation of the core modes and the gain of the cladding modes caused by mode coupling along the fiber axis induced by the refractive index variations is calculated. Then, the intensity of the cladding modes at the fiber’s output end is acquired. Figure 7 shows the gains of the first-order and the second-order cladding modes at the output end of the fiber laser used in this study. Importantly, this cladding mode power indicates the leakage losses of the LP01 and LP11 modes.
Figure 8 shows the power distributions of the LP01 and the LP11 modes along the length of the fiber. The simulation results with and without mode coupling are indicated by the dashed line and the solid line, respectively. From Fig. 8, the mode-coupling power loss of the LP01 mode is close to 1 W (0.288%), and the power loss of the LP11 mode is 21.75 W (6.81%) with 750 W pump power launched into the gain fiber. The total signal (LP01 +LP11) power loss is 22.73 W, with a proportion arriving at 3.4%, and the mode coupling loss occurs mainly in the coupling process of the LP11 mode into the second-order cladding modes.
Combined Fig. 5 and Fig. 8, some conclusions can be obtained. The mode coupling coefficient in the double clad fiber between core modes and clad modes decreases with rising ordinal number of clad modes. In another word, the bigger the ordinal number of the clad modes, the larger the difference between the field distributions of the clad mode and the core mode, resulting in a smaller mode coupling coefficient. The coupling coefficient of the LP11 mode with the clad modes is much bigger than that of the LP01 mode. This is mainly due to the much more similar effective refractive index of the LP11 mode with the clad mode, which in essence indicates a great similarity in the field distribution. Consequently, in the double clad fiber laser, the fundamental mode is the most stable mode because high order modes are easily submitted to all kinds of disturbance and transform into other modes, which is shown in the simulation result in Fig. 8. When the fiber laser is influenced by external thermal disturbance, the propagation of the LP01 mode is relatively stable while the propagation of the LP11 mode suffers from large fluctuation caused by strong coupling effect.
Using the coupled mode theory, we have described the interaction between the fiber core modes and the cladding modes, established a power attenuation model under thermally-induced refractive index variation conditions, and then obtained the fiber core loss under conventional cooling conditions[13] at multi-kW power levels.
5. Experimental setup and resultsA double-clad ytterbium-doped fiber (YDF) laser amplifier was designed for thermally-induced core mode leakage measurement, and is depicted in Fig. 9. The fiber amplifier system parameters are listed in Table 1. A total of 42 fiber output laser diodes (LD), which each emit 25 W of pump power to provide a combined pumping power of 1 kW at 975 nm, were used as the pump source for launching into the active fiber. At the end of the amplifier, the laser output end cap was cleaved to an angle of 8°.
To verify the high-power fiber laser core mode leakage caused by the refractive index variations, this study separated the core mode power from the total fiber amplifier output power. A spectroscope that is highly reflective (HR) at 1080 nm and highly transmissive (HT) at 975 nm was placed at the output end of the amplifier to measure the pump components in the laser output and thus obtain both Ppump and Pcore + Pcladding (as shown in Fig. 9(a)). A cladding mode stripper (CMS) was then splice fused at the output end to remove both the cladding modes and the pump power from the output to obtain Pcore (as shown in Fig. 9(b)).
Figure 10(a) shows the output power (Pcore + Pcladding, black curve), the remnant pump power (red curve), and the optical efficiency (P1080/P975, blue curve) with increasing pump power. When 750 W of pump power was launched into this fiber amplifier, 645 W (636 W in our simulation without the seed) of laser output power was obtained at 1080 nm, with a remnant pump power of 8.49 W@975 nm, an optical efficiency of 84.6%, and a beam quality of
and
.
In the system shown in Fig. 9(b), following the removal of all pump power and cladding mode power by the CMS, the remaining pure fiber core mode output reached 562 W (without seed) for the core mode @1080 nm (black curve in Fig. 10(b)) when 750 W pump power was launched into the fiber amplifier. The corresponding core mode optical efficiency (blue curve in Fig. 10(b)) given by Pcore/Ppump was 70.2%.
As shown in Fig. 10(c), the proportion of the cladding mode was 13.1% with low gain, rising to 17.0% (blue curve) at a pump power of 750 W, and this suggests that the cladding mode increases by 3.9% with the increment in the pump power. The leakage of the core mode into the cladding that was caused by bending or fusion splices did not vary dramatically with the increasing gain. Additionally, the fiber laser gainguided phenomenon that occurred at multi-kilowatt levels appeared to be very weak and could thus be ignored. Therefore, the significant temperature variation that occurs with increasing pump power in the fiber amplifier system is the most likely reason for the leakage of the fiber core modes that is discussed in this paper.
If a comparison is made between the simulation results (3.4%) and the experimental results (3.9%), the modecoupling loss constitutes 87.1% of the total loss induced by the increasing pump power, which means that the leakage of the fiber core modes is the main part of the losses in multikilowatt fiber lasers.
6. ConclusionThis study develops a straightforward theoretical model to calculate the core mode losses in a fiber amplifier that demonstrates a specific level of cladding-mode coupling. The theory first provides a method to calculate the thermallyinduced refractive index variation in the fiber laser. Then, a specific method is used to calculate the coupling coefficients and the coupling processes between the core and cladding modes. It is found that in a fiber laser, the thermally-induced refractive index variation is the main cause of the leakage of the fiber core modes. In a 750 W fiber laser, this leakage constitutes 3.4% of the total power.
This leakage process is irreversible and will lead to deterioration in both the laser efficiency and the power density. The analysis presented here indicates that the mode coupling loss of the fiber laser has a close relationship with the strength and the inhomogeneity of the thermal disturbance. Good heat dissipation and fiber temperature distribution compensation technology can significantly inhibit the mode coupling losses. Additionally, because a significant difference exists between the coupling strength of the fundamental mode and that of the higher order modes, improvement of the fiber laser beam quality through approaches such as mode selection techniques and mode filters can also prevent this kind of loss.
An experiment was also designed to measure the thermally-induced fiber amplifier mode leakage corresponding to the quantity in the calculations. There was 562 W core mode power in the total laser output power of 637 W at 1080 nm. The core mode leakage increased with increasing pump power. The experimentally measured results agreed well with the simulated results.
