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The amplification of a 2 μm concentric vortex beam in a Ho:YAG crystal rod amplifier is simulated and experimentally verified. Different concentric vortex beams are studied and the amplification results are compared with simulation. The integrity of the launched vortex beams is well maintained through the amplification process. Further discussions are provided to increase the gain as well as for power scaling. Due to the nature of Ho:YAG material and the geometry of the rod amplifier, our system shows potential of working as a power amplifier for vortex beams.
© 2016 Optical Society of America
1. Introduction
Optical vortex beams are a special type of optical beam that carries helical phase and it is related to the orbital angular momentum (OAM) state of the beam. A phase term exp(ilθ) is added to the phase front of the beam, with l being the topological charge number and θ being the azimuthal angle [1–3]. This additional phase term gives the vortex beams unique properties such as handedness [4] and self-healing [5]; therefore vortex beams are associated with many applications such as optical tweezers, optical trapping, optical imaging and quantum information processing [2, 6]. In addition, vortex beams are also used in free-space information transfer and communication systems combined with spatial division multiplexing (SDM) to expand the bandwidth of the communication system [7, 8]. As a super-position of vortex beams, concentric vortex beams are interesting mainly due to the propagation properties and their application in communication systems. Optical vortices have been used to control optical filaments using diffractive optic phase plates with high efficiency [9] and diffractive phase plates to create rotating Bessel beams based on concentric phase vortices [10]. Additional realizations of concentric phase functions were used to beam shape light exiting VCSELs [11]. In some cases, additional amplification of the beam is required for directed energy and long distance propagation channels for communication links. So far, the amplification of vortex beams has been demonstrated using rare-earth doped single crystal rod amplifier [12], MOPA configuration [13], parametric amplification [14] and light-acoustic interaction [15]. Optical fiber that supports OAM beams amplification has also been theoretically proposed [16]. However, it is still an open question as to how these novel optical beams behave in an amplification scenario and if the concentric optical vortices maintain their properties.
In this paper, to the best of our knowledge, the amplification of concentric vortex beams has been demonstrated for the first time in a rod amplifier. A rod amplifier is a frontier approach in solid state systems combing the advantages of the conventional bulk crystal materials and optical fiber systems. Typically the seed goes through the rod as in free space while the pump is wave-guided by the rod. However, the design can be rather flexible depending on the applications, such as to increase the modal overlap between the seed and the pump and therefore increasing the gain performance. The rod is long enough to provide sufficient gain to the input seed beam while the integrity of the beam mode maintains well through the rod. The small diameter of the rod helps with the confinement of the pump and it is potentially possible for the rod to work as a power amplifier.
2. Simulation
Since the rod is a highly multimode waveguide, how the beam propagates in the rod, guided or as in free space, will depend on the incident beam shape and launching. Thus, a 3-D beam propagation model will be needed to predict the output shape. Meanwhile, the pump and signal beam will have different intensity distributions at the same cross section all along the rod, which means the pump and signal beam overlap will be different along the rod. Models based on consistent pump and signal beam overlap, either for fiber amplifiers or bulk amplifiers, are not suitable for such scenarios. Consequently, a transversally resolved (TR) rate equation will be combined with Finite difference beam propagation method (FD-BPM) to solve the signal gain. In the following discussion, a brief explanation on the FD-BPM method and the TR rate equation set will be presented along with the simulation procedure. This model follows closed to the model used in [17] with necessary modifications [18].
Assume the coupling between the two polarizations is weak and negligible, which is true for rod waveguides. Then the semi-vector wave equations based on E-field formulation are [19, 20]:
If one-way propagation (in + z direction) is assumed and the electric field E(x, y, z) can be separated into two parts: the axially slowly varying envelope term ϕ(x, y, z) and the fast varying phase term exp(-jkn0z), then the electric field can be expressed as:
Where subscript t represents x or y polarization, n0 is a reference refractive index close to the effective index of the beam in the waveguide, and n0 is chosen so that the envelope varies slowly in the propagation direction. Substituting Eq. (3) into Eqs. (1) and (2), one-way paraxial semi-vector wave equations are obtained:
The refractive index n is a space-dependent 2-D term n(x, y) to represent the waveguide structure of interest. Equations (4) and (5) can be solved numerically by FD approximation at a small distance Δz, by iterating the process as the beam propagates in the + z direction until the desired length is reached. However, solving the wave equations involves directly inverse the operator matrix, which requires a lot of time and computational resources. For this reason, alternating-direction implicit finite difference method (ADIFDM) is adopted here to facilitate the computation. In ADIFDM method, either Eqs. (4) and (5) can be decoupled into two equations, each propagates the wave half a propagation step at only x or y direction. Such a propagation problem in 1-D can be solved with a recurrence method, which does not involve matrix inversion and therefore speeds up the computation. The detailed formulation of ADIBPM and algorithms follow closely the details in [20]. The combination of Eqs. (4) and (5) can be used to solve the propagation of a cylindrically polarized beam in a waveguide, using only one of them corresponds the propagation of a linearly polarized beam. In a radially symmetric waveguide, Eqs. (4) and (5) are essentially the same. If weakly guiding condition is applied to Eqs. (4) and (5), they reduce to the scalar wave equation, which applies to the propagation in an optical fiber. In our case, the seed beam is TE polarized, the pump is randomly polarized, therefore only Eq. (4) is solved numerically to propagate either the seed or pump beam.
The TR steady-state rate equations are briefly presented, which follows closed to the form in [21]. The equations are assuming a most general two-level energy system including saturation absorption and saturation emission, which works for either a four-level or a three-level transition that may or may not include excited state absorption. It is also assumed that non-radiative decays are very fast and the transition time can be ignored compared to the laser upper level lifetime τ. In order to be compatible with BPM, the TR rate equations are discretized [17],
Where N(m, k) is the ion concentration at a transversal point (m, k), 1 and 2 stands for the lasing lower and upper levels, respectively. Ps is the signal power and Pp is the pump power. The + sign and – sign represent the beam propagation direction, either + z or –z. In our case a co-pumping scheme is adopted, so both the pump and signal propagate at + z direction. σap and σep are the absorption and emission cross sections at the pump wavelength, respectively. Similarly, σas and σes are the absorption and emission cross sections at the signal wavelength, respectively. Besides, h is the Planck constant, νp is the pump frequency, νs is the signal frequency, αp and αs are the loss/attenuation coefficients of the pump and signal through the SCF, respectively. Γ(m, k) is the power filling factor at a transversal point (m, k) and the discretized area of a point is A(m, k).
As a summary, the model operates based on the following steps until the reaching the end of the rod, shown in Fig. 1.
The parameters for the simulation are as follows: N = 1.3 × 1026 m−3 for 1% Ho3+ doping concentration, σap = 0.90 × 10−20 cm2, σep = 0.50 × 10−20 cm2 at 1908 nm, σas = 0.23 × 10−20 cm2, σes = 1.20 × 10−20 cm2 at 2091 nm, τ = 8.5 ms [22–24], αp = 0.001 cm−1, and αs = 0.001 cm−1. What is worth mentioning is that our model does not take into consideration thermal effects, such as thermal lensing and temperature dependent absorption. For quasi 3-level materials such as Ho and Tm doped crystals and fibers, temperature induced laser wavelength re-absorption has been demonstrated to reduce the laser slope efficiency and laser output power [23, 25].
3. Experiment
Figure 2(a) shows the single-pass, co-pumping amplifier setup utilizing the 1% Ho:YAG rod. The ends of the 1mm diameter × 60 mm long Ho:YAG rod were polished to flat/flat finish and AR coated for both the pump and the seed wavelengths (1.9 – 2.1 μm). The barrel of the Ho:YAG rod was also polished so the rod can be used as a waveguide for the pump light if needed. A Tm:fiber laser (IPG TLR-50) was used as the pump source, which provided single mode 1908 nm laser output. The pump laser was operated in a modulated mode with a pulse repetition frequency of 10 Hz and a duty cycle of 50% to reduce the thermal load in the crystal rod. The output of a CW tunable laser (IPG HPTLM Cr:ZnS/Se) was coupled into a 10/125 polarization-maintaining (PM) fiber and used as the seed delivery fiber to ensure high mode quality. Both the seed laser and pump laser provide more than 40 dB signal to noise ratio (SNR) and very good power stability within +/−1% power fluctuations. The output wavelength of the seed source was tuned to 2091 nm to match the peak emission wavelength of Ho:YAG. L1-L5 are AR-coated lenses for pump and seed collimation and focusing. The focused pump diameter was approximately 400 μm and the pump focal point was located approximately 80 mm before the entrance facet of the rod. The pump beam entered the rod and continued to diverge and filled the entire volume of the rod. The pump absorption was more than 98% with up to 20 W of incident pump power. The prescriptions of lenses L3 and L4 were selected to beam shape the output from the seed source to achieve spatial symmetry of the seed beam across the Ho:YAG rod. Without the concentric phase plate, the focused single-mode seed beam diameter was approximately 200 μm at the focal point located at the center of the rod. D1 and D2 are identical dichroic filters oriented at a 45 degree angle of incidence (AOI) with high transmission at the pump wavelength and high reflection at the seed wavelength. The images of the amplified vortex beam are captured using an IR camera (Ophir Pyrocam IV).
Fig. 2 (a) Experimental setup of the Ho:YAG rod amplifier. (b) The simulated phase wrap; (c) the microscope image of the fabricated phase plate for the 3-lobe concentric vortex. (d) The simulated phase wrap; (e) the microscope image of the fabricated phase plate for the 5-lobe concentric vortex.
The phase plate converts the Gaussian seed beam into the desired concentric vortex beam shape. The phase plate can be designed for any wavelength. In the current effort, the phase plate was designed to operate at 2091 nm. The concentric vortex phase plate consists of an inner vortex with a counter-clockwise phase wrap of (n × 2π) and an outer vortex with a clockwise phase wrap of (-m × 2π). By choosing the proper Gaussian beam size so as to cover both the inner and outer parts of the phase plate, the far field pattern of the beam will contain several spatially separated lobes due to interference of the wavefronts from the inner and outer vortices, and the number of the lobes is determined by the total charge number (n + m). To achieve the best results, the power incident on the inner vortex phase plate should be equal to the power incident on the outer vortex phase plate. In our experiment, the measured diameter of the incident Gaussian beam on the phase plate is a factor of 2 larger than the diameter of the inner vortex phase plate. This means 40% of the incident beam power is incident on the inner vortex phase plate. In this paper, the (1 + 2) and (1 + 4) concentric vortices, corresponding to 3-lobe and 5-lobe vortices, were investigated. Illustrations of the phase wrap and the microscope images of the fabricated phase plates are shown in Figs. 2(b)-2(e). The phase plates were fabricated on a fused silica wafer. The fused silica wafer was cleaned, coated with photoresist and patterned using optical lithography. Then the wafer was developed and the patterns were transferred from the photoresist to the silica wafer by etching method. The refractive index of the fused silica wafer is 1.436 at 2091 nm, so the total etching depth into the wafer corresponds to 4531 nm for a 2π phase (with 1% etching depth error). Each 2π phase wrap was approximated by 16 different etching depths into the wafer, which corresponding to a calculated 98.7% diffraction efficiency. Since the wafer is not AR coated there will be Fresnel losses at the front and back surface of the wafer that results in a measured ~8% loss. For the transmitted power, the diffraction efficiency of the phase plate was measured to be ~97%. Less than 3% of the total power is measured in the center null of the diffraction beam corresponding to the un-diffracted 0th order Gaussian component. This would be a result of slight etch depth errors or tilt on the phase plate and would contribute to the slight intensity asymmetry on the measured patterns in Fig. 3 compared with the simulated patterns.
Fig. 3 Upper row, the 3-lobe concentric vortex beam. (a)-(c), the simulated beam at the entrance facet of the rod, at the focal point inside the rod and at the exit facet of the rod. (d)-(f) the images of the beam taken by the IR camera at the corresponding locations. Lower row, the 5-lobe concentric vortex beam. (g)-(i), the simulated beam at the entrance facet of the rod, at the focal point inside the rod and at the exit facet of the rod. (j)-(l), the images of the beam taken by the IR camera at the corresponding locations.
4. Results and discussions
In order to ensure the integrity of the concentric vortex beam is not in any way changed by propagating through the rod, the images of the beam pattern were taken at the locations corresponding to the entrance and the exit facets of the rod, as well as the focal point without the presence of the Ho:YAG rod, shown in Fig. 3, with the corresponding simulated beam patterns for comparison. The difference in the optical path length due to the refractive index of the rod (n = 1.807 at 2 μm) was taken into consideration when capturing the images. The images corresponding to the focal point are also shown for reference. For both the 3-lobe and 5-lobe concentric vortex beams, the pattern at the focal point is symmetric, which is inside the rod. Before and after the focal point, the lobes rotate in opposite directions due to the parabolic phase jump carried by the Gaussian beam, or the Gouy phase. The higher the total charge number (n + m), the more “spread out” the beam is and therefore the beam diameter is slightly bigger. It is hard to measure the exact beam size difference between the 3-lobe and the 5-lobe vortices; however, both of them are significantly bigger than the original Gaussian beam. The beam size for both the 3-lobe and 5-lobe vortices are estimated to be approximately 900 μm at both the entrance and exit facets of the rod (the Gaussian beam size at both facets are 2w0 = 300 μm), which means the beams can propagate through the rod without introducing any perturbations from the sidewall of the rod. The measured beams in Fig. 3 look identical to the simulated beams except some slight uneven power distribution among each lobe that may arise from the asymmetry of the collimated Gaussian beam passing through the phase plate.
The images of the output seed through the rod without the pump were also taken, shown in Figs. 4(a) and 4(c). The beam is exactly the same compared with images in free space (shown in Figs. 3(f) and 3(l)), supporting the premise that spatial characteristics of vortex beam are preserved propagating through the rod.
Fig. 4 The images of the concentric vortex beam through the rod amplifier; (a) and (c), with no pump; (b) and (d) with 20 W of incident pump power.
Next the amplification properties of the concentric vortices were investigated. Limited by the available seed laser power, incident seed power of 0.1 W and 0.5 W were used to study the small signal gain and gain saturation effect. Due to the long, 8.5 ms, upper level lifetime of Ho:YAG, this material is good for energy storing and can be potentially used as a power amplifier [23]. Figures 5(a) and 5(b) shows the output seed power versus the incident pump power for 0.1 W incident seed power and 0.5 W incident seed power, respectively. Both the 3-lobe and the 5-lobe vortex beams were measured at each incident seed power. With 0.1 W of seed power and 20 W incident pump power, the output power for 3-lobe vortex and 5-lobe vortex was 0.88 W and 0.49 W, respectively. The corresponding single-pass gain is 8.8 and 4.9. With 0.5 W of seed power and 20 W incident pump power, the output power for 3-lobe vortex and 5-lobe vortex was 2.40 W and 1.83 W, respectively. The corresponding single-pass gain is 4.8 and 3.7. In both cases, the single-pass gain for the 3-lobe vortex beam was higher than the gain for the 5-lobe vortex. The simple explanation for this could be the pump/signal modal overlap. The pump intensity profile was Gaussian. However, the concentric vortex beams contain an intensity minimum at the center that increases in size with the increasing total charge number (n + m). Therefore, as the total charge number (n + m) increases, the modal overlap will decrease.
Fig. 5 The measured output seed power for different concentric vortex beams. (a) Incident seed power of 0.1 W; (b) incident seed power of 0.5 W. Blue for 3-lobe concentric vortex and green for 5-lobe concentric vortex. Solid lines, simulations of ideal cases; Dashed lines, simulations of a 0.3 degree tilted incident pump beam. Dotted line, simulations of 3% Gaussian and 97% concentric vortex.
The simulated gain is also shown in Fig. 5 as solid lines. The experimental data and simulation are in good agreement for the 0.1 W seed case. However, for the case of a 0.5 W seed, the simulation shows approximately 20% higher output seed power compared with the measured seed power. This difference may arise from both experimental and simulation issues. Experimentally, on the pump side, any small misalignment may result in decreased overlap between the pump and seed beams. This will not only degrade the amplified seed beam quality but also decrease the output seed power. The dashed lines in Fig. 5 show the simulations of a misaligned pump beam with a small incident angle of 0.3 degree tilt with respect to the seed beam. Such a small angle will result in an obvious decrease in output seed power. The 0.3 degree tilt is an estimated value for pump beam misalignment in our setup. Meanwhile, on the seed side, the seed is actually a coherent superposition of the desired petal beam and a very small amount of un-diffracted Gaussian beam. This results in a slight amount of mode degradation. Such degradation not only causes diffraction beam asymmetry but also affects the amplified output power. The dotted lines in Fig. 5 show the simulated output power using an input seed with 97% of the power in the petal beam and 3% in a Gaussian beam. Comparing this with the simulation where 100% of the energy is contained in the petal beam (solid line), the amplified output power for the case when the seed contains the pedal beam and Gaussian beam is approximately 2-5% higher in all cases. It’s not surprising because the Gaussian component does have a preferable overlap with the Gaussian pump profile. On the other hand, with the simulation, our model does not include thermal induced effects. As a quasi-3-level material, the gain performance of Ho:YAG suffers from temperature induced laser wavelength re-absorption [22–25]. With only passive cooling, this thermal effect will cause a distinguishable difference between experimental and simulation results, especially with higher seed powers, and this reason contribute majorly to the difference between simulation and experiment for 0.5 W seed.
One thing worth noting is that, at higher pump powers, the gain between the two modes when the seed power is 0.1 W is quite different. However, when the seed power is 0.5 W, the gain difference between the two modes decreases. This suggests that there is gain saturation to some extent for the 3-lobe concentric vortex mode, which is not surprising based on the mode size difference between the two concentric vortex beams. Overall, the concentric vortex beams yield lower mode area intensity (compared with a Gaussian beam of the same power). Therefore, it is possible for the concentric vortex beams to be used as the seed in a power amplifier system and obtain a high energy extraction ratio out of the system. Moreover, comparing the output seed images in Fig. 4 with and without pump power, the seed beam quality maintains well at a high pump power level in both cases, except for some slight distortions in the 5-lobe vortex case. Such distortions mainly arise from input seed asymmetry and pump alignment. The input 5-lobe seed itself shows slight asymmetry and such asymmetry gets intensified with pumping. Also, the pump beam is Gaussian and it is not truly uniformly distributed across the cross-section of the rod, any slight misalignments, including tilt and off-axis incidence, will result in unevenly amplification among the lobes.
5. Conclusion
In conclusion, we demonstrated the amplification of concentric vortex modes through a rod amplifier experimentally and theoretically. Although the pump/signal modal overlap is not optimized, the gain and output power of the seed is rather encouraging. In order to improve the modal overlap, an optimization of the pump beam with the location of the lobes of the concentric vortex beam is necessary. Also, proper thermal management is necessary to power scale the amplifier. As mentioned in [26], the small diameter of the rod reduces surface exchange of the heat with the outer environment; such rods require better thermal contact with a heat sink compared to bulk. Only passive contact cooling is applied in our setup. Active cooling with better thermal contact will be our future work to help with the power scaling in the amplifier. Although the Diffraction Efficiency of the phase plates is high, there is still some residual amount of power in the fundamental mode, <3%, which does get amplified based on the mode overlap with the pump signal. Additional work is currently underway to evaluate the superposition of spatial modes comprised of vortices with different charge numbers and their gain as a function of different pump profiles. In the current setup, the gain is less than 10X and therefore beam degradation due to the coherent interference of the fundamental mode with the concentric optical vortices will remain small. However when scaling up the power level of the amplifier system, it will merit further investigation. On the simulation side, thermal effects will be added to the model to predict the temperature-dependent gain as well as the power limit of the rod amplifier. Combining these numerical tools with the experimental realization of amplified spatial modes may serve as a platform for a number of new applications in propagation, laser machining and sensing.
Acknowledgments
Funding for this research was provided by HEL-JTO and through ONR N00014-141-0264, ARO W911NF-12-1-0536, ARO/MURI W911NF1110297 and HEL-JTO agency through Air Force Office of Scientific Research (AFOSR) on grant FA9550-10-1-0543.
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