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Abstract
In this paper, we present a theoretical and experimental analysis on the properties of rectangular core fibers. We investigate the single-mode regime and the bending properties with respect to the aspect ratio of the rectangular core. In comparison to a standard step-index single-mode fiber, we show that a passive single-mode rectangular core fiber with an aspect ratio of ten is able to transport two times more power without the occurrence of nonlinear effects. For actively doped fibers, the threshold of nonlinear effects can be raised by up to 15%. For experimental verification a rectangular core fiber with an aspect ratio of three is manufactured and analysed. Here, the results between the numerical simulation and the experimental analysis are in agreement.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Single-mode fiber lasers are used in many applications due to their outstanding beam quality, high efficiency and high practicability. The use of different active dopants, like ytterbium, erbium or thulium opens up a wide range of usable wavelengths to address various applications. Fundamental limitations to the output power of single-mode fiber lasers are, for example, stimulated Brillouin- or Raman-scattering [1]. For single-mode fibers, which have a normalized frequency of V < 2.405, the threshold of these nonlinear effects can typically be increased by using a lower numerical aperture and, following the single-mode condition of optical fibers, a higher core area. This way, the mode-field area is increased and thus the optical intensity is reduced. However, fibers with low numerical apertures typically suffer from higher bending sensitivity and therefore higher bending losses.
To overcome these limitations, Large-Mode-Area Fibers with V > 2.405 can be used. These fibers are typically no truly-single-mode fibers, but the single-mode operation can be maintained due to mode filtering of the higher-order modes, due to e.g. fiber bending [2]. However, the output power of these fiber lasers can still be limited due to nonlinear effects or due to transversal mode instabilities [3].
Aside from step-index fibers there are many other types of fibers that can be used within fiber laser systems. These types of fibers are e.g. Large-Pitch fibers [4] or distributed mode filtering rod fibers [5], both of which typically can only be used within a rod-type configuration. Additionally, hollow-core photonic-bandgap fibers can be used [6], but these fibers cannot be used for the amplification or generation of a laser beam.
Due to the presented fundamental limitations of cylindrical symmetric fibers, we examine a different approach for the scaling of the output power of fiber lasers, namely the change of the geometry of the fiber core. Within this article, we present an analysis of step-index single-mode fibers with a rectangular fiber core. We are identifying the single-mode regime of the fiber with different aspect ratios and are simulating the corresponding core area, mode-field area and guidable mode power without the occurrence of nonlinear effects. To avoid transversal mode instabilities within the fiber’s core, it has to be truly single-mode. Since the bending sensitivity of a fiber is an important aspect for the later practical use within an application, we perform an analysis about the bending losses with respect to the bending radius and the aspect ratio of the rectangular core. Afterwards, a first prototype of a rectangular core fiber will be presented, experimentally analyzed and compared to the simulation results. By using this approach, we want to investigate the functionality of a new type of step-index fiber for high power applications. This could either be the use of the fiber as a transport fiber, meaning that the size of the core is not a critical parameter, or the use of the fiber within an amplifier systems, where a big fiber core can be beneficial for the absorption of the pump radiation.
2. Theoretical background and simulation
The eigenmodes in cylindrically symmetric fibers are known as the $L{P_{lm}}$ modes. They define an orthogonal system of modes, which all have a well-defined effective refractive index $ {n_{eff,lm}}$. Each mode is guided within the fiber core, if its effective refractive index fulfills the condition [7]
with the refractive index of the fiber’s core ${n_{Core}}$ and the refractive index of the fiber’s cladding ${n_{Clad}}$. Following this condition, a fiber will be defined as a single-mode fiber at a chosen wavelength, when the above condition is only fulfilled for one, two times degenerate eigenmode. Following that, a fiber with more than one, two times degenerate mode, will be defined as a multi-mode fiber at a defined wavelength.For the simulation of the rectangular core fibers, a finite-elements method is used to solve the full-vectorial wave equation for the calculation of the effective refractive indices and the transversal profiles of all guided modes [8]. For the definition of the single-mode-regime, the condition (1) is used.
Another important aspect is the bending sensitivity of the fiber. A high bending loss can result in a restriction of the practicability of the fiber and therefore limit the number of applications. For the simulation of the fiber bending we apply a linear variation to the refractive index profile of the fiber [9], which will then be used as an approximation of the bent waveguide. The propagation of the electric field is computed using the Wide-Angle Beam-Propagation-Method (WA-BPM) and for the discretization a finite differences approach is used [8].
3. Single-mode conditions of rectangular core fibers
In order to simulate the influence of the core geometry on the mode parameters, the aspect ratio of the core and the core area are varied. An exemplary normalized refractive index profile and the cross section of a rectangular core fiber are shown in Fig. 1. Within the simulation a resolution of 0.2 µm in x and y direction is used. The numerical aperture of the core will be kept constant at 0.06, and the cladding diameter will be kept constant at 100 µm, which is in the area of typical diameters for a single-mode fibers. The vacuum wavelength $\lambda $ of the calculated modes is 1064 nm.
To find the set of parameters, at which condition (1) is fulfilled for a rectangular core fiber, the effective refractive index of the first higher order mode is calculated with respect to the core area and the aspect ratio. The core area for a chosen aspect ratio, at which the effective refractive index of the higher order mode is exactly the refractive index of the cladding, is seen as the biggest core area at which the fiber is still a single-mode fiber.
The effective refractive indices of the $L{P_{11}}$-like higher order modes with respect to the aspect ratio and the core area are shown in Fig. 2. For effective refractive indices ${\textrm{n}_{eff}}$ above the refractive index of the cladding $ {\textrm{n}_{Cl}}$, the fiber is within the multi-mode regime, and for effective refractive indices below the fiber is within the single-mode regime. It can be observed that the intersection of effective refractive indices and the refractive index of the cladding is at lower core areas for higher aspect ratios, which means that the core area has to be kept smaller for higher aspect ratios to still enable a truly single-mode operation. Since the simulation can only be carried out at discrete core areas, no continuous analysis can be performed. For an additional improvement of the comparability of the calculated data, a linear fit between the calculated nearest points to the refractive index of the cladding is performed. This way, the exact intersection of the calculated refractive index and the refractive of the cladding can be found and used for the later analysis of the fiber parameters.
Fig. 2. Effective refractive indices of the $L{P_{11}}$-like higher order modes with respect to the aspect ratio and the core area
4. Modal characteristics of rectangular core fibers
Since the thresholds of many nonlinear effects scale with the intensity, the threshold of nonlinear effects does not only depend on the power, but also on the transversal shape of the laser mode. Therefore, a laser mode with a low peak intensity ${I_{Peak}}$ at a defined integrated laser power ${P_{Mode}}$ is beneficial for increasing the nonlinear thresholds. To compare the suitability of the calculated rectangular fiber core modes with different aspect ratios for a high power transport and the generation and amplification of high power signals, the maximum power of a laser mode ${P_{Mode}}$ at which a well-defined peak intensity threshold ${I_{Threshold}}$ is not exceeded is calculated. In the following, the normalized mode power is defined as the calculated data, normalized by the data of a cylindrically symmetric round core fiber with a numerical aperture of 0.06 and V=2.405.
Following this, the core area, the mode field area and the resulting normalized mode power with respect to the aspect ratio of single-mode rectangular core fibers are shown in Fig. 3. Here, the interpolated points at which the effective refractive index is exactly the refractive index of the cladding are shown. The data shows that the core area has to become smaller for higher aspect ratios, for the enabling of the truly single-mode operation. Additionally, the difference in the core area becomes smaller for higher aspect ratios. While a local minimum of the normalized mode power can be found at an aspect ratio of 2, it becomes bigger than the initial value for aspect ratios bigger than 4. The mode field area shows a similar characteristic. Following the characteristic of the mode field area and the core area, the benefit of rectangular core fibers could increase for higher aspect ratios.
Fig. 3. Core area, mode field area and the normalized mode power with respect to the aspect ratio of single-mode rectangular core fibers
All in all, Fig. 3 shows that the normalized mode power and the mode field area become bigger for higher aspect ratios, which is beneficial for the suppression of nonlinear effects. A similar characteristic can be observed in single-mode round-core fibers. Based on the formula [10]
with the normalized frequency the mode field diameter $MFD$ diverges for ${d_k} \to 0$. For the comparison between the rectangular core fibers and a round fiber, a round fiber with a numerical aperture of 0.06, at a vacuum wavelength of 1064 nm, and a varied core area, will be simulated. In Fig. 4 are the mode-field diameter, the normalized frequency and the normalized mode power for a round single-mode fiber and the calculated rectangular core single-mode fibers at different aspect ratios shown. In the simulation of the round fiber, the core area is varied, while the numerical aperture is kept constant. In principle, both curves show a similar behavior; both curves show a divergent behavior at a specific core area. The fundamental difference is that for the same normalized mode power an up to three times bigger core can be used by using the rectangular core fiber.Fig. 4. Mode field area and normalized mode power of a round-core single-mode fiber and the calculated rectangular core single-mode fibers for different aspect ratios and the corresponding normalized frequency of a round fiber with respect to the core area
While the core area is not a fundamentally limiting value for a passive fiber, it becomes an important value within an actively doped fiber. In a cladding pumped fiber laser, the necessary fiber length to absorb a defined percentage of the pump radiation is inversely proportional to the core area. Following that, a high core area within a fiber laser can be beneficial to increase the threshold ${P_{Th,\; nonlinear}}$ of nonlinear effects, e.g. stimulated Raman-scattering, due to a lower fiber length. The threshold will be approximated using
In Fig. 5 the core area, mode field area and the normalized nonlinear threshold with respect to the aspect ratio of actively doped, single-mode rectangular core fibers is shown. The data is normalized with the data of a cylindrically symmetric round core fiber with a numerical aperture of 0.06 and V=2.405. For aspect ratios higher than 8, the single-mode rectangular core fiber is beneficial for the suppression of nonlinear effects in comparison to a cylindrically symmetric single-mode fiber. At an aspect ratio of 10, the normalized nonlinear threshold is increased by ca. 17%, taking into account the necessary longer fiber length, due to the lower core area. Following the characteristic of the mode field area and the core area, a higher suppression of nonlinear effects can be expected for even higher aspect ratios than 10.Fig. 5. Core area, mode field area and the normalized nonlinear threshold with respect to the aspect ratio of actively doped, single-mode rectangular core fibers
5. Bending characteristics of rectangular core fibers
For many fiber systems, a fiber with a low bending loss is beneficial for a higher practicability of the fiber, especially for highly compact fiber laser systems. Therefore, the bending sensitivity of the simulated single-mode rectangular core fibers at specific bending radii is simulated. For the simulation the model described in section 2 is used. In Fig. 6 the bending loss for simulated rectangular core single-mode fibers for different aspect ratios with respect to the bending radius is shown. For higher aspect ratios and therefore a higher mode field area, the bending sensitivity becomes bigger. This issue could be fixed by using additional structures within the fiber, e.g. a trench or a hole-assistance.
Fig. 6. Bending loss of simulated rectangular core single-mode fibers for different aspect ratios with respect to the bending radius
6. Experimental analysis
For the experimental verification of the results, an Yb-doped, rectangular-core fiber is manufactured by CeramOptec GmbH within ZIM-Project ZF4328102AB6. Figure 7 shows a microscopic measurement of the cleaved fiber facet. During the manufacturing process a deformation of the rectangular core occurred. The dimensions of the fiber core are ca. 4 µm times 11 µm at the widest points, which represents an aspect ratio of ca. 1:2.75. The numerical aperture of the core is 0.1.
For the analysis of the modal characteristics of the rectangular core fiber, a resonator and an ASE-source based on the fiber are set up. This is done to check the influence of the resonator on the modal characteristics, since higher modes could be suppressed. The experiments showed that there is no difference between the modal properties of the two different types of sources. This shows that the used fiber is truly single-mode. In Fig. 8 the measured near-field and far-field profile of the manufactured fiber are shown. The short axis of the rectangular-core fiber has a high numerical aperture, while the longer axis has a lower numerical aperture. The measured numerical apertures are 0.072 and 0.098, which corresponds to the numerical aperture of the fiber’s core, which is ca. 0.1.
In Fig. 9 the measured beam diameter characteristic for the calculation of the beam quality factor ${M^2}$ is shown. As a fit function, the evolution of the beam width of a Gaussian beam is chosen. The measurement leads to $M_x^2 = 1.2$ and $ M_y^2 = 1.1$, which corresponds to a nearly diffraction limited laser beam. Therefore, the fiber can be called a single-mode rectangular-core fiber.
As shown in section 5, the bending characteristics can have an influence on the number of applications a specific fiber can be used for. Therefore, the bending characteristic of the rectangular core fiber is measured here.
The experimental setup is shown in Fig. 10. The pump power is launched into two pieces of the same fiber which are separated by a cladding mode stripper. The first piece of fiber acts as an ASE source, while the second is bent to specific radii to measure the bending losses. Both fiber cleaves have an angle of ca. 8° to avoid lasing. The first cladding mode stripper is used to remove the remaining pump power out of the fiber. Afterwards, the fiber is bent to specific radii, while in all cases a half turn is used. During this bending process, the radiation is coupled from the fiber’s core to the fiber’s cladding. Therefore, a second mode-stripper at the end of the bent fiber is used to remove this radiation. Afterwards the power measurement can be performed.
Fig. 10. Experimental setup for the measurement of the bending characteristic of the manufactured fiber
For the simulation of the bending losses, the measured core geometry shown in Fig. 7 is approximated by a rectangular geometry. In Fig. 11 the measured bending loss with respect to the bending radius and the corresponding simulation results are shown. A high qualitative agreement between the simulation and the experimental results can be observed. Some differences may occur due to the deformation of the rectangular core. While the typical approximation models for the bending losses of a fiber only predict a single rise of the bending losses, as e.g. the formula by Marcuse [11], for this rectangular core fiber another local maximum can be found. The physical origin of this behavior is still under investigation. The here presented experimental and simulation results show a high qualitative accordance. Following that, the measured data shows that the presented ansatz for the simulation of the bending sensitivity of rectangular core fibers is valid. Therefore, it can be expected that for other rectangular core fibers with even higher aspect ratios, as shown in section 5, the bending losses can have a huge impact onto the specific fiber system.
Fig. 11. Measured bending loss with respect to the bending radius and corresponding simulation results for the manufactured fiber
7. Conclusion
We presented a theoretical and experimental analysis on the properties of rectangular core fibers. We investigated the single-mode regime for rectangular core fibers and showed that for higher aspect ratios a lower core area is needed to enable a truly single-mode fiber, while the resulting mode field area becomes bigger for higher aspect ratios. In comparison to a standard step-index single-mode fiber, a passive single-mode rectangular core fiber with an aspect ratio of 10 is able to transport two times more power without the occurrence of nonlinear effects. For actively doped fibers, the threshold of nonlinear effects can be raised by up to 15%. The analysis on the bending properties shows that for increasing aspect ratios an increased bending sensitivity can be expected.
For the experimental analysis, a rectangular core fiber with an aspect ratio of ca. 1:2.75 and a numerical aperture of 0.1 is manufactured by CeramOptec GmbH within ZIM-Project ZF4328102AB6. The near- and far field characteristic and the measured beam quality factor of ca. $M_x^2 = 1.2$ and $ M_y^2 = 1.1$ in the corresponding axes show that the manufactured fiber is a truly single-mode rectangular core fiber. The measured and the simulated bending losses show a high qualitative agreement.
Following our analysis, rectangular core fibers can have beneficial properties in comparison to standard step-index single-mode fibers. We showed that the benefits of rectangular core fibers increase for increasing aspect ratios. Single-mode rectangular core fibers could for example be used in applications, where transversal mode instabilities have to be prevented, which excludes large-mode-area fibers as an alternative.
Funding
Bundesministerium für Wirtschaft und Energie (ZF4328102AB6).
Acknowledgements
We thank CeramOptic GmbH for the manufacturing of the analyzed fiber within the funded project.
Disclosures
The authors declare no conflicts of interest.
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