Losses and Energy Backflows of the Fundamental Core Mode in Solid Core Micro Structured Optical Fibers

Abstract

This work discusses the behavior of transverse energy fluxes of the fundamental core mode of a holey fibers and a photonic band gap fibers when the polarization state of this mode changes. The behavior of the transverse component of the Poynting vector of the fundamental core mode is considered for both linear and elliptical polarization. It is demonstrated that despite the difference in the distribution of the Poynting vector stream lines in the cross section of the fibers for the two polarizations, the leakage loss level is maintained constant due to the forward and reverse energy flows in the radial direction. Differences in the level of leakage losses in different micro structured fibers arise from the vortex structure of the Poynting vector of the fundamental core mode.

1. Introduction

This paper examines the unusual behavior of leakage losses in well-known solid-core micro structured fibers [1]. Here we consider two types of fibers, namely, the holey fiber [2] and the all solid band gap fiber (ASBG fiber) [3]. The cladding of such optical fibers consists of a periodic array of air holes or dielectric rods running their length. Unique optical properties of holey fibers and ASBG fibers are the primary reason for their efficient theoretical and experimental investigations. Such properties of holey fibers as a large mode area [4], extraordinary dispersion values [5] and endlessly single-mode guidance [2] find applications in industry [6], supercontinuum generation [7], sensing [8]. The cladding structure of the holey fibers make them sensitive to fluctuations of the fiber geometry. This can lead to an increase in leakage losses and to a change in the dispersion properties of the holey fibers. In part, this problem can be solved with help of ASBG fibers. In this case, the deviations of fiber geometrical parameters can be reduced by using two types of glass with properly matched thermal and chemical properties. In this work, these fibers have one and two layers of holes—dielectric rods in the cladding. In addition, in the case of the ASBG fiber, selected diameters of the germanate cladding rods are consistent with experimental loss measurements [9]. According to the ARROW model [10], the thickness of the cladding layers and the diameters of the cladding rods are selected in such a way that leakage losses for core modes are minimal. The ARROW model tells us that the cladding rods of ASBG fibers reflect light in an anti-resonant mode, similar to a Fabry-Perot resonator. That is, a direct analogy is traced between the reflection and localization of radiation in a planar waveguide with a cladding made of planar layers with a higher refractive index than that of the core and in the ASBG fiber with a cladding consisting of dielectric rods with a refractive index higher than the surrounding silica glass matrix [11]. On the other hand, in the case of holey fibers, the ARROW model cannot be used to explain the light localization in the core surrounded by layers of air holes. In this case, it is assumed that since the refractive index of the cladding has on average a lower refractive index than the core, a modified principle of total internal reflection can be used to explain the light localization [1].

In this work, we propose to look at the problems of light localization in the above-mentioned micro-structured fibers from a more general point of view, namely, using the so-called “hydrodynamic approach” [12]. In this case, what is considered is not the reflection of electromagnetic fields from individual elements of the fiber cladding, but the behavior of the stream lines of energy flows of the core modes, which is similar to the lines of motion of fluid particles in hydrodynamics. In this case, it is possible to observe complex interference phenomena in the energy fluxes of the core modes with strong light localization in the fiber core, for example, vortices. It is assumed that the structure of vortices in the energy flow of the core modes in the fiber cross section can directly affect the degree of light localization in the fiber core. In work [13], it was shown that singular optics approaches can be applied to energy flows of core modes of solid-core micro-structured fibers such as dividing the Poynting vector of core modes into spin and orbital parts. It is worth noting here that all core modes in holey fibers and ASBG fibers are leaky with a corresponding level of losses [1]. It was also demonstrated in [13] that leakage losses in these fibers are determined mainly by the orbital part of the Poynting vector of the core modes, while the spin part is responsible for the twist of the stream lines of the transverse component of the Poynting vector of the core modes in the fiber cross section.

In order to verify these findings, in this paper we carry out numerical calculations of the stream lines of the transverse component of the Poynting vector of the fundamental core mode, as well as its radial and azimuthal components for the case of elliptical and linear polarizations. It is demonstrated that despite the strong difference in the distribution of the azimuthal components of the Poynting vector of the fundamental core mode, the loss level is determined only by the forward and reverse energy flows of the core mode along the radial direction. In addition, vortices of the transverse component of the Poynting vector of the fundamental core mode are formed in different ways for two different polarizations of the core mode.

2. Energy Flows of the Fundamental Core Mode in Holey Fibers

Let us consider the behavior of the transverse component of the Poynting vector of the fundamental core mode (HE₁₁) for a holey fiber with one and two layers of air holes in the cladding, as well as the separate distributions of the radial and azimuthal components of the Poynting vector of this mode ${\overrightarrow{P}}_{transv}\left(r,\phi \right)=P_r\overrightarrow{r}+P\overrightarrow{\phi }$ in the fiber cross-section. All further calculations are performed in Comsol Multiphysics for the elliptically and linearly polarized ($\overrightarrow{E}=\left\{0;E_y\right\}$) fundamental core mode. The triangular mesh was used for calculations with size of $\lambda /\left(5n\right)$, where n is the refractive index of the material.

All calculations are performed at a practically important wavelength of λ = 1.06 µm (refractive index of silica glass is 1.44968). The ratio of the diameter of the air hole to the pitch is equal to d/Λ = 0.12, which should ensure that the fiber in question is single-mode [2]. The pitch size is Λ = 14 µm (Figure 1).

Leakage losses in these fibers are determined by the radial component of the Poynting vector $P_r$. Its distribution and the stream lines of the transverse component of the Poynting vector for the elliptical polarization of the fundamental core mode are shown in Figure 2a. The loss value for the fiber with one layer of air holes in the cladding was 2162 dB/m. The great losses in this case are determined by the outflowing energy of the fundamental core mode moving along almost radially directed lines that flow around air holes in the fiber cladding. The distribution of the azimuthal component of the Poynting vector of the fundamental core mode is shown in Figure 2b. In order to quantitatively characterize the movement of the fundamental mode energy in the radial and azimuthal directions, we calculate the average value of the $P_r$ and $P_{\phi }$ components over the fiber cross section by taking the following integrals $P_r^{average}={\displaystyle \int P_r}dS/S$ and $P_{\phi }^{average}={\displaystyle \int P_{\phi }}dS/S$, where S is an area of the fiber cross section. For the case shown in Figure 2 these values are equal to $P_r^{average}=0.5{\mathrm{W}/\mathrm{m}}^2$ and $P_{\phi }^{average}=3\times {10}^{-3}{\mathrm{W}/\mathrm{m}}^2$, respectively.

Similar distributions of the Poynting vector components for the case of linear polarization are shown in Figure 3. From Figure 3a it is clear that the process of the fundamental core mode energy leakage in the radial direction has not changed qualitatively for linear polarization. Therefore, both the magnitude of the losses and the value of the integral $P_r^{average}={\displaystyle \int P_r}dS/S\simeq 0.5{\mathrm{W}/\mathrm{m}}^2$ for linear polarization remained the same. On the other hand, the distribution of the azimuthal component of the Poynting vector of the fundamental core mode (Figure 3b) has qualitatively changed and it has a strict discrete rotational symmetry. In this case, a periodic change in the sign of the azimuthal component leads to the integral value of $P_{\phi }^{average}=2.8\times {10}^{-7}{\mathrm{W}/\mathrm{m}}^2$. According to the findings in the work [13], the spin part of the Poynting vector has a small but finite value precisely because of the mode energy of leakage from the fiber core.

For a holey fiber with two layers of air holes in the cladding, the behavior of the radial component of the Poynting vector does not fundamentally change (Figure 4), namely, the leaking energy of the fundamental core mode also flows around the holes of the second layer. That is why the dependence of losses in such fibers on a wavelength is non-resonant [1]. For simplicity purposes, here we present the distributions of the Poynting vector and stream lines only for the linear polarization of the fundamental core mode.

The values of the integrals for the averaged values of the radial and azimuthal components of the Poynting vector have the values of $P_r^{average}=0.13{\mathrm{W}/\mathrm{m}}^2$ and $P_{\phi }^{average}=2.7\times {10}^{-7}{\mathrm{W}/\mathrm{m}}^2$ for linear polarization. In the case of elliptical polarization, these quantities have the following values of $P_r^{average}=0.13{\mathrm{W}/\mathrm{m}}^2$ and $P_{\phi }^{average}=2.9\times {10}^{-3}{\mathrm{W}/\mathrm{m}}^2$. The radial part of the average Poynting vector decreased due to the addition of a new layer of air holes in the cladding (reduction of leakage losses), and the value of $P_{\phi }^{average}$ decreases as the cross-section area of the holey fiber with a discrete rotational symmetry of the arrangement of air holes and the symmetric distribution of $P_{\phi }$ increases, leading to a decrease in the value of the integral $P_{\phi }^{average}={\displaystyle \int P_{\phi }}dS/S$. The value of losses when adding a new layer of air holes in the cladding was 324 dB/m.

3. Energy Flows of the Fundamental Core Mode in All Solid Band Gap Fibers

Let us now consider an ASBG fiber with rods in the cladding instead of holes filled with an air. The refractive index contrast between the cladding rod and the surrounding silica glass matrix is $\Delta n=0.028$. The geometric parameters of the ASBG fiber are the same as for the holey fiber discussed in the previous section. This is an exact replica of the ASBG fiber that was used in the experiments in [9]. The geometric parameters of the ASBG fiber and the refractive index contrast between the cladding rods and the silica glass matrix were selected to provide a minimum of losses in the transmission band near a wavelength of 1.06 µm [9].

Figure 5a shows the distribution of the radial component of the Poynting vector of the fundamental core mode $P_r$, the stream lines of the transverse component of the Poynting vector ${\overrightarrow{P}}_{transv}$ and the distribution of the azimuthal component of $P_{\phi }$ for an ASBG fiber for elliptical polarization. First of all, it is clear that the transverse component of the Poynting vector behaves in a completely differently than in holey fibers. An increase in the refractive index contrast in the cladding rods compared to the refractive index of the surrounding silica glass matrix leads to negative values of the radial component of the Poynting vector of the fundamental core mode as was already shown in [12]. The losses of the fundamental core mode in this ASBG fiber at a wavelength of 1.06 µm were 48 dB/m for both polarizations.

The azimuthal component of the Poynting vector of the fundamental core mode also exhibits vortex behavior (Figure 5b). The center of the vortices of the radial and azimuthal components of the Poynting vector lies inside the cladding rods. The averaged values of the radial and azimuthal components of the Poynting vector for elliptical polarization are equal to $P_r^{average}=9.4\times {10}^{-3}{\mathrm{W}/\mathrm{m}}^2$ and $P_{\phi }^{average}=2.5\times {10}^{-2}{\mathrm{W}/\mathrm{m}}^2$, respectively.

In the case of linear polarization of the fundamental core mode of the ASBG fiber, the stream lines of the transverse component of the Poynting vector change greatly (Figure 6a).

The vortex centers for this component are now located outside the cladding rods, and the cladding rods themselves transmit the maximum energy flow of the core mode in the radial direction. The azimuthal component of the Poynting vector of the fundamental core mode has the same symmetrical structure as that of holey fiber (Figure 3b and Figure 6b). The averaged values of the radial and azimuthal components for linear polarization are equal to $P_r^{average}=9.4\times {10}^{-3}{\mathrm{W}/\mathrm{m}}^2$ and $P_{\phi }^{average}=3\times {10}^{-8}{\mathrm{W}/\mathrm{m}}^2$, respectively.

If you draw a circle with a radius of 12.8 μm passing immediately behind the cladding rods, then by plotting the distributions of $P_r$ along it for both polarizations, you can clearly see that the losses in both cases should indeed coincide (Figure 7).

Thus, the decrease in leakage losses in the case of ASBG fiber compared to holey fiber can be explained by the presence of energy—flow—reversing dynamics [14] in the fundamental core mode for the radial component $P_r$, which is not related to the polarization of the core mode.

A similar behavior can be observed for an ASBG fiber with two layers of germanate rods in the cladding. In the case of elliptical polarization of the fundamental core mode, the distribution of the radial component of the Poynting vector of the fundamental mode $P_r$, the stream lines of the transverse component of this vector ${\overrightarrow{P}}_{transv}$ and its azimuthal component $P_{\phi }$ are shown in Figure 8. The highest values of mode energy fluxes in the radial and reverse directions are observed in the rods of the first layer (Figure 8a). A similar picture is observed for the movement of the mode energy flow in the azimuthal direction (Figure 8b). Reverse core mode energy flows in the radial direction in the second layer of cladding rods are observed in every second rod. As will be shown below, they make the main contribution to reducing leakage losses for the fundamental core mode of the ASBG fiber. The averaged values of the radial and azimuthal components for elliptical polarization are equal to $P_r^{average}=1.5\times {10}^{-4}{\mathrm{W}/\mathrm{m}}^2$ and $P_{\phi }^{average}=2.3\times {10}^{-3}{\mathrm{W}/\mathrm{m}}^2$, respectively. Losses in the ASBG fiber with two layers of germanate rods in the cladding for both types of polarizations were equal to 0.7 dB/m at a wavelength of 1.06 µm.

For linear polarization of the fundamental core mode (Figure 9), reverse energy flows of the core mode are also observed in every second rod of the second layer of cladding rods. In this case, the patterns of mode energy leakage qualitatively coincide with the mode energy leakage and its reverse energy flows in the radial direction for elliptical polarization (Figure 9a). In the first layer of cladding rods, there are no vortex motions of the core mode energy, in contrast to the case of elliptical polarization. The distribution of the azimuthal component of the Poynting vector, similarly to ASBG fiber and holey fibers, has a strictly symmetrical structure which leads to the value of $P_{\phi }^{average}=2.5\times {10}^{-11}{\mathrm{W}/\mathrm{m}}^2$. The averaged value of the radial component was the same as for the case of elliptical polarization.

In order to make sure that the energy leakage in the radial direction of the fundamental core mode has similar behavior for both polarizations let us draw a circle with a radius of 25 μm passing immediately behind the second layer of the cladding rods and plot the distributions of $P_r$ along it for both polarizations (Figure 10).

The resulting distribution of $P_r$ exactly corresponds to the vortex structure of the radial energy flows and stream lines of the transverse component of the Poynting vector shown in Figure 8 and Figure 9. The transmission peaks of fundamental core mode energy also correspond to every second rods in the second layer of the cladding rods. Negative values of the radial component of the Poynting vector are not observed in Figure 10 because the circle is located a little further from the maximum values of the reverse mode energy flows along the radial coordinate, but the distribution completely repeats the shape of the core mode energy movement in each of the vortices near the cladding rods of the second layer. Thus, the fundamental core mode energy flows out in the same way from the fiber cladding for both mode polarizations.

4. Discussion

The study of radial [14] and azimuthal [15] reverse energy flows in laser beams has yielded very interesting results. In the work [14] it was reported that the approach based on modulation of the internal flows of laser beams represents a nuanced perspective in optical field design. In the above cases of energy flows of leaky fundamental core mode in solid-core micro-structured ASBG fibers, reverse energy flows along the radial direction were also observed. The reason for the occurrence of reverse flow along the radial coordinate is the presence of vortices in the transverse component of the Poynting vector of the fundamental core mode. The vortex structure or the ’skeleton’ of the transverse component of the Poynting vector in the cross-section of the fiber creates a complex pattern of balance between forward and reverse mode energy flows, which is confirmed by integrating the radial component of the core mode energy flow over the fiber cross-section. Vortices of the transverse component of the Poynting vector of the core mode are formed in the region of individual cladding rods, and this is true for ASBG fibers with any number of layers of cladding rods. The reverse motion of the core mode energy flows significantly reduces the total outflow energy flow in the radial direction and therefore reduces leakage losses of the core mode. This is the difference between ASBG fibers and holey fibers, because near the air holes of the holey fibers cladding, the energy flow of the fundamental core mode simply wraps around them without generating reverse energy flows. This leads to significantly higher leakage losses for holey fibers.

A similar phenomenon can be observed for the azimuthal component of the Poynting vector of the fundamental core mode. Azimuthal backflows are observed both for holey optical fibers and ASBG fibers for both polarizations. The distribution of the azimuthal projection of the Poynting vector over the cross section of the micro-structured fibers with reverse energy flows has a strict discrete rotational symmetry for linear polarization. This gives extremely small values to the azimuthal component of the Poynting vector when it is numerically integrated over the cross section of the optical fibers.

Therefore, it can be argued that the phenomena associated with the internal energy flows in core modes of micro-structured optical fibers and related phenomena occurring in vortex beams are very similar to one another. Thus, this may have important practical consequences related to reducing leakage losses in micro-structured solid core fibers with reverse energy flows of core modes along the radial direction.

5. Conclusions

The occurrence of reverse radial energy fluxes of the fundamental core mode in solid core micro structured optical fibers can lead to a significant reduction in leakage losses for these modes. Similar effects can be observed in other types of micro structured optical fibers, including hollow core ones. Radial reverse energy flows of the core modes are characteristic of optical fibers in which the core has a lower refractive index than the cladding elements. Reverse azimuthal energy flows can occur in optical fibers with cladding elements having a lower refractive index than the core. A deeper understanding of the principles of controlling the transverse energy flows the core modes of micro structured optical fibers by changing the geometric structure of their claddings can lead to the creation of new types of optical devices with unique characteristics.