4. Simulations and experiments for a fiber combiner with a single pump port
The ray tracing simulations were carried out with the commercially available software Zemax (Radiant Zemax, LLC) in the non-sequential mode. Detailed information about ray tracing in tapered cylindrical fibers can be found in Ref  and . The ray tracing method is applicable due to the large cross sections of the employed fibers compared to the applied wavelength of 976 nm. The 3-dimensional simulation model of the fiber combiner was based on the setup depicted in Fig. 1 with the approximation of a parallel fiber arrangement of the IF and TF. For the PFF a fully filled condition was always assumed, meaning that all possible pump light rays, independent of the NA and the transversal position in the fiber core, carry equal power. For the geometrical shape of the taper in the longitudinal direction, a simplified linear shape was assumed in the simulations, instead of the measured parabolic shape. As already mentioned, the FL was set to 1.99.
4.1 Simulations of the pump coupling efficiency
The pump coupling efficiency in dependence of the converging taper length (TL) and the taper ratio (TR) of the IF for a 1064nm high power isolator with an NA of 0.22 is depicted in Fig. 2(a)
Fig. 2 (a) Pump coupling efficiency (CE) with respect to the taper ratio (TR) and the converging taper length (TL) and (b) a comparison of the pump coupling efficiencies without intermediate fiber (IF) and with IF for different fiber parameters, IF Ø: IF cladding diameter.
. The simulations show that an increasing TL leads to higher coupling efficiencies at a constant TR. For example at a constant TR of 6 a TL of 5 mm leads to a theoretical maximum pump coupler coupling efficiency of 86%, whereas for a TL of 20 mm 96.4% were calculated. Furthermore, Fig. 2(a) shows that the TR can be reduced, if the TL is increased to maintain a certain coupling efficiency level. For instance, for a TL of 20 mm, a coupling efficiency of 85% can already be obtained at a TR of 2 instead of a TR of 5.5 at a TL of 5 mm. The improved coupling behavior at longer TLs can be explained by the increasing number of bounces of the pump light rays at the lateral surface of the converging taper portion. Hence, for shorter TLs it is necessary to taper more than for longer TLs in order to compensate for the shorter interaction length of the converging taper portion with the TF. The maximum theoretically obtainable pump coupling efficiency was limited to 97.3% due to different loss mechanisms, which will be discussed in Section 4.3.
In the following section we discuss the impact of the intermediate fiber on the pump coupling efficiency and the taper parameters. Thus, for comparison the fiber combiner was also simulated without the IF, which means that the tapered PFF was directly connected to the TF, assuming the same FL and also a NA of 0.22. Figure 2(b) illustrates that the coupling efficiency can be increased and the TR reduced, if an IF is inserted between the PFF and the TF. For a TR of 2.5 at a TL of 20 mm the coupling efficiencies with and without IF are 61.2% and 90.1%, respectively. The moderate coupling efficiencies without the employment of an IF at low TR can be explained by the presence of a depressed refractive index of the cladding of the PFF, blocking the power transfer from the IF to the TF, as already discussed in Section 2. Thus, without IF, the pump light rays with a low NA cannot escape from the core of the PFF, and a considerable fraction of power will be transmitted via the diverging taper portion. A further increase of the pump light NA, due to the increase of the TR up to 10 at a TL of 20 mm for the PFF and the IF, results in a successive approximation of the Polarization Maintaining Optical Circulator efficiencies. However, even at a TR of 10 and a TL of 20 mm (with IF) a 2.5% higher pump coupling efficiency can be obtained. That means for a hypothetical available input pump power of 1 kW, a reduction in power loss of 25 W can be essential to prevent thermal damage of the fiber combiner. Additionally, it must be taken into account that a TR of 10 corresponds to a considerable reduction of the mechanical stability due to the fiber diameter tapering from 125 µm to 25 µm. Furthermore, Fig. 2(b) clearly shows that the insertion of an IF with a TL of 10 mm already yields better pump coupling efficiencies than a PFF with a TL of 20 mm, especially for low TR.
A further increase of the pump coupling efficiency up to 97.8% can be realized by inserting an IF with a TL of 20 mm and diameter of 105 µm, which is perfectly adapted to the core diameter of the PFF, and thus, no pump brightness loss occurs. Note that for all of the following simulations and experiments, we only used the fiber component containing an inserted IF with a cladding diameter of 125 µm.
4.2 Simulations for the impact of the pump light input NA on the pump coupling efficiency
In the next simulation step we figure out, how the pump coupling efficiency changes with the pump light input NA depending on TR and TL. For these simulations three types of PFFs with a core NA of 0.15, 0.22 and 0.30 were investigated, assuming for each PFF a fully filled pump light condition. The TR was considered in the range from 1 to 10 at a TL of 5 mm
Fig. 3 Pump coupling efficiency with respect to the taper ratio at a converging taper length of (a) 5 mm and (b) 20 mm for a PFF with a pump light input NA of 0.15, 0.22 and 0.30.
) and 20 mm (Fig. 3(b)). From both figures it can be seen that at lower TRs the coupling efficiency increases with NA, since the pump light rays with a higher NA have more bounces with the lateral surface of the converging taper portion. However, the pump coupling behavior changes with increasing TR, since a TR of much higher than 2 leads to pump light rays with a NA far above 0.46, which cannot couple into the TF, if the TL is too short. The occurring pump power losses will be discussed in Section 4.3. E.g., for a low TL of 5 mm and a TR of 7 the coupling efficiency for an input NA of 0.15 was simulated to be 10% higher than for an input NA of 0.30. In contrast, with a longer TL of 20 mm the coupling efficiency seems to be less sensitive to variations of the pump light input NA. Thus, it appears that for the combiner design, the pump coupling efficiency should not be significantly influenced by the pump light input NA in the range of 0.15 to 0.30, if a sufficient TL is considered.
If the pump light input NA gets closer to the NA of the TF of 0.46, it can be advantageous to use a straight IF portion in addition to the converging taper to obtain a highly efficient pump light transfer into the TF as described in Ref . An alternative approach to the straight IF portion is an increased TL, i.e. for a pump light input NA of 0.46 a theoretical pump coupling efficiency of about 90% can be achieved, if the TL is at least 40 mm.