In this paper we investigate the polarization properties of a fiber laser with an intracavity c-cut calcite crystal that is capable of producing reconfigurable vectorial output modes. Vectorial modes with radial, azimuthal and generalized cylindrical vector polarizations can be generated by translating one lens within the laser cavity. Detailed studies of the mode polarization evolution show that the modes inside the laser cavity can be spatially homogeneously polarized in one section of the cavity while being spatially inhomogeneously polarized in another section of the cavity, which opens the opportunities for many potential new fiber laser design possibilities and applications. Furthermore, more complicated vectorial vortex output modes are also observed by purposefully introducing angular misalignments.
©2010 Optical Society of America
Laser beams with spatially engineered polarization distribution, particularly those so-called cylindrical vector (CV) beams with cylindrical polarization symmetry , have attracted significant recent interests. These spatially inhomogeneous polarizations have important potential applications in high numerical aperture imaging [2–9], laser machining and nanofabrication [10,11], optical particle trapping [12,13], surface plasmon excitation and nanofocusing [14–20], optical instrumentation [21–23], and many others. Driven by these applications, various passive and active generation methods have been developed . Among those active generation methods, fiber laser sources that can produce vectorial output modes [24–28] are particularly attractive due to its potential of generating high output power and extreme flexibility.
Recently we reported a fiber laser design using the combination of axially symmetric birefringence and a three-lens telescope in the cavity . With the intra-cavity axial birefringence, radially polarized mode and azimuthally polarized mode experience different magnifications through the three-lens telescope inside of the cavity. By translating one of the lenses in the three-lens telescope, one can select one of the vectorial modes to be collimated towards to end mirror and oscillate in the resonator, producing output modes that can be switched between radial and azimuthal polarizations. Record CV beams output power with a fiber laser was reported with this laser cavity design.
In this paper, the polarization modes behavior of this fiber laser cavity is investigated in details. The evolution of the mode polarization inside of the cavity is observed and reported. It is found that the mode polarization inside of the cavity can be spatially homogeneous in one section of the cavity while spatially inhomogeneous in another section of the cavity. This could open opportunities for other specifically nonlinear optical related processes that require polarization and phase matching. In addition, by translating the lens to collimate rays between the ordinary and extraordinary foci, a generalized CV beam output that consists of a linear superposition of the radial and azimuthal polarizations has been obtained. Vectorial modes with more complicated spatial polarization distributions are also observed by creating angular misalignment of the axially symmetric birefringent element.
2. Fiber laser cavity configuration
The fiber laser system used in the experiment is illustrated in Fig. 1 . The details of the setup are described in a recent publication . A 976 nm pump laser is used to pump a 4-meter long erbium doped fiber (LIEKKI Er120-20/125DC with NA = 0.09 ± 0.01 and 20 µm core diameter). The fiber is carefully chosen so that it can support the fundamental mode, and the second higher order modes, i.e., the radially polarized mode (TM01), azimuthally polarized mode (TE01), and the hybrid mode (HE21). For all the experimental studies in this work, a pump power of 1.8 W is used. End mirrors M1 (100% reflectance at 1.6 µm) and M2 (80% reflectance at 1.6 µm) form the cavity where M2 is the output coupler. The c-cut calcite crystal is used to separate the ordinary (corresponding to azimuthal polarization) and extraordinary (corresponding to radial polarization) rays. After the crystal, they are focused at two different foci (as indicated in Fig. 1). A positive lens L6 which can be precisely translated along the optical axis is used to select the polarization to be collimated in the cavity. The polarization state that is collimated in the cavity will experience less loss and achieve lasing when the pump is above the threshold. In Fig. 1, radial polarization output mode measured by an infrared camera (Indigo MerlinTM) is shown in the inset as an example. Azimuthal polarization output was also observed by translating L6 to the right about 0.6 mm to collimate the rays near the ordinary focus .
3. Experimental and numerical results
The polarization evolution in the cavity is studied when the laser produces radial or azimuthal polarization outputs. Besides measuring the output polarization, the polarization states in front and after the fiber are also measured. The measurements are done by putting a pellicle in the cavity at different locations (indicated in Fig. 1). The pellicle is an uncoated very thin polymer membrane with high transmittance and low reflectance that does not affect the fiber laser system output polarization state. With an IR camera and a linear polarizer, the polarization states at those locations are analyzed. In addition, we translate lens L6 to collimate the rays between the foci of radial and azimuthal polarization. Vectorial modes with linear combination of radial and azimuthal polarization with and without topological charges are observed.
3.1 Polarization evolution in the fiber laser cavity
We first examine the polarization in the cavity when the output is radially polarized. A pellicle is inserted between L3 and L4 with the surface of the pellicle rotated by about 45° with respect to the optical axis. Ideally this angle needs to be very small to reduce the difference in reflectance for s- and p-polarizations. However, this was not possible for us due to the relatively large aperture size of the pellicle (2 inches) and a very tight space between L3 and L4. The incident angle we use is very close to the Brewster’s angle (for polymer with index of refraction of 1.47, the Brewster’s angle is 55.8°) where the p-polarization has reflectance nearly zero. This needs to be taken into considerations for the polarization analysis in this part.
The beam profile is recorded with the infrared camera and the polarization is analyzed by rotating a linear polarizer in front of the camera (shown in Fig. 2 ). Figures 2(a)-(c) are the measurements for the beam coming from the fiber end. Figure 2(a) is the beam profile without passing through the linear polarizer, and Figs. 2 (b) and (c) are the beams after a linear polarizer. The polarizer transmission axis is indicated by the white arrow. For radial polarization the right and left sides of the beam that are p-polarized will be extinguished by the pellicle at an incident angle near Brewster’s angle, leaving the top and bottom parts that are s-polarized. Figures 2(b) and (c) further confirm that the polarization of the beam reflected by the pellicle is mostly vertical. Similarly, the polarization state of the beam back to the fiber is measured, and the results are shown in Figs. 2(d)-(f). From these measurements we conclude that, between L3 and L4, the polarization states in both directions between the fiber and the output mirror are radial polarizations, which is expected from the self-consistent requirement of laser theory.
Similarly, the polarization states between L3 and L4 are examined for azimuthal polarization output (shown in Fig. 3 ). Compared with Fig. 2, the beam profile without passing through linear polarizer becomes two horizontal spots. This is because for azimuthal polarization the top and bottom parts are p-polarized with respect to the pellicle. Thus, at incident angle close to the Brewster’s angle, the left and right parts are reflected. The polarization direction measurements with a linear polarizer confirmed this conclusion. Please note that the polarization mode from the crystal back to the fiber exhibits stronger asymmetry. The images are saturated in order to bring up the two-spot patterns.
For radial and azimuthal polarization output, the polarization states between M1 and the dichroic mirror (see Fig. 1) are also measured by using the same pellicle. In this case the incident angle on the pellicle is adjusted to be very small (~10°) to minimize the difference in reflectance for s-polarization and p-polarization since there is sufficient space between M1 and the dichroic mirror. The experimental results show the beam profile for both cases are nearly a Gaussian and the polarization states are linear (see Fig. 4 ). Figures 4 (a)-(c) show the measurements corresponding to radial polarization output and Figs. 4 (d)-(f) for azimuthal polarization output. Notice that the orientations of the linear polarization corresponding to the radial and azimuthal polarization outputs have approximately 45° angle with respect to each other. The linear polarization state near M1 is also verified by directly inserting a linear polarizer between M1 and the dichroci mirror. When the polarizer axis is rotated to be the same as the measured axis of linear polarization, the CV beams output profile and polarization at M2 is maintained to be either radial or azimuthal, which again confirms the linear polarization states at this end of the cavity.
3.2 Generation of generalized cylindrical vector output
In Fig. 1, we used the rim rays to draw two distinctively separated foci corresponding to the radial and azimuthal polarization to illustrate the working principle of this fiber laser design. In reality, the two foci are connected as a continuous caustic zone with the near axis rays being included. If we translate lens L6 such that its front focal point is located in the middle of the two foci for the rim rays, then both radial and azimuthal polarization could oscillate simultaneously in the cavity, producing a generalized cylindrical vector beam  with the local polarization pointing with certain angle between the radial and azimuthal directions. This is confirmed experimentally (see Fig. 5 ). The overall beam intensity profile is shown in Fig. 5 (a). The beam profiles after a linear polarizer are shown in Figs. 5(b)-(d). From these measurements we calculated the local polarization direction which is illustrated with those black arrows superimposed on Fig. 5 (a). In this case, it is found that the local polarization points at approximated 30° away from the radial direction and the polarization distribution maintains cylindrical symmetry. The radial and azimuthal components superimpose to each other and result in a generalized polarization state as we expected.
3.3 Vectorial vortex modes generation with angular misalignment
For the experiments described above, the c-cut calcite crystal is carefully aligned with the three-lens telescope and the cavity. However, if the calcite crystal is intentionally misaligned angularly while lens L6 focuses in the middle of the two foci, complicated polarization vortex structures are observed. One of such example is shown in Fig. 6 . Figure 6(a) is the output beam profile which still exhibits a donut shape with larger dark core. However, the beam after a linear polarizer has a 4-lobe pattern and the pattern does not follow the rotation of the linear polarizer (Fig. 6(b)-(e)). Apparently, the polarization of this output mode is no longer cylindrically symmetric.
The observed patterns can be phenomenologically explained by the linear superposition of radially polarized and azimuthally polarized modes with opposite topological charges. Assuming the radial and the azimuthal polarizations have the same Laguerre Gaussian profile with opposite topological charges + 1 and −1 respectively; these two components can be written as:30]. In our simulation we choose and. In order to perform coherent addition of both fields, the radial and azimuthal polarizations are decomposed into x- and y-components as:Fig. 7 . Figure 7(a) is the simulated output beam profile, while Figs. 7(b)-(e) show the beam profiles after passing through a linear polarizer with transmission axis indicated by the white arrow. The simulation is able to repeat the general pattern of the experimental results shown in Fig. 6. Experiment results of donut output modes with 6-lobe pattern after the linear polarizer have also been observed. Numerical simulation of a linear superposition of radial and azimuthal polarizations with positive and negative spiral phase with topological charge of 2 are consistent with experimental observations. However, the physical explanation of these observations is still under investigation.
4. Discussions and conclusions
In conclusions, we reported the polarization behavior of a fiber laser with c-cut calcite crystal that is capable of producing reconfigurable vectorial output modes. The evolution of the mode polarization in the laser cavity has been investigated. The vectorial self-consistency condition within this cavity is confirmed by observing the polarization in both directions between the output mirror and the erbium doped fiber. It is also found that the erbium doped fiber performs a polarization mode conversion function in the cavity that allows the conversion between a spatially homogeneous polarization state at one end of the fiber and a spatially inhomogeneous polarization states at the other end of the fiber . As long as the fiber is kept mechanically stable, such as mode conversion is observed to be very stable once the cavity is aligned to produce the desired vectorial output mode.
In addition, by translating one of the intracavity lenses, both radial and azimuthal polarizations can oscillate simultaneously, producing generalized CV beam output directly. If angular misalignment is introduced to the birefringent crystal at this point, more complicated vectorial output modes have also been observed. We phenomenologically explained the observed complex polarization patterns of these donut shape vectorial modes as linear combinations of orthogonally polarized vectorial vortex beams with different topological charges. However, the exact underlying physical process that gives rise to such phenomena needs further investigation.
The findings reported here contribute to the understanding of the polarization dynamics and design issues of vectorial fiber lasers and demonstrate the rich polarization phenomena within a fiber laser cavity that could be further exploited. For example, the spatial separation of the spatially homogeneous and inhomogeneous polarization states in the same laser cavity is very intriguing. Such a spatial separation may enable new fiber laser cavity designs. Vectorial laser modes with shorter wavelengths could be achieved by inserting nonlinear optical crystal in the part of cavity with linear polarization states to perform frequency conversion such as frequency doubling, a nonlinear optical process cannot be directly performed to the spatially inhomogeneous states due to the requirements on polarization and phase matching.
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