## Abstract

We report on modeling, development, and optical characterization of fused silica photonic crystal fiber with germanium doped microinclusion placed in the middle of the core. The fiber is designed to efficiently couple and guide LP_{02} mode. It offers high optical density in the center region, large mode separation, low losses, and small dispersion with relatively flat profile for both LP_{01} and LP_{02} modes in 1-1.6 µm wavelength range. We demonstrate that by changing geometrical and material parameters of the inclusion partially independent tuning of propagation constants of individual modes is possible, what might be found is a variety of potential applications, e.g., in nonlinear optics. We also show that diffraction-limited propagation of LP_{02} mode in free space can be exploited in microscopy or lab-on-a-chip systems, where the proposed fiber can be used for light delivery.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Within the last decades, higher-order modes (HOM) excited in optical fibers have found various applications in different types of photonic devices. The reason behind is their distinct from fundamental mode optical properties such as electric field distributions, complex polarization patterns or waveguide dispersion and propagation characteristics. For instance, HOMs have been exploited in optical communication for space multiplexing and dispersion management, offering an attractive alternative to free-space gratings and prisms [1]. Due to their ability to operate with substantial optical power while providing a means of sensing spectral signature changes they have been widely used in a range of sensor systems [2]. Finally, some of HOMs can also carry orbital angular momentum, what can be used in the future to increase the data capacity of the next generation optical networks [3].

Higher-order modes are often perceived as not suitable for nonlinear optics. The main reason behind is that they have a large effective area what typically corresponds to low local intensities. However, unlike Gaussian-like fundamental mode, where nonlinear distortions are expected to be roughly proportional to the peak intensity of a pulse of light, HOMs require a more precise description to capture the physics of various nonlinear responses. Since a majority of effects relay on an intensity dependent phase-change the profile of the entire mode and not just the peak intensity matters [4]. This opens new possibilities and adds an extra degree of freedom in designing the optical performance of fiber based nonlinear devices [5]. For example, it has been demonstrated recently that HOMs can play an important role in the process of supercontinuum generation [6,7].

Among HOM a particularly interesting one is a “Mexican sombrero” LP_{02} (HE_{12}) mode, with a bright spot in the center surrounded by a single ring. It is linearly polarized, circularly symmetric and has attractive diffractive properties. The standard method to generate LP_{02} mode is either with the use of spatial light modulators [8] or LP_{01}-L_{02} mode converters in the form of long-period gratings [9,10]. Thanks to improved efficiencies [11], new fabrication techniques [12] and new designs [13] of the converters, there has been lately much interest in applications uniquely enabled by LP_{02} mode. For instance, in multi-photon microscopy by using the intensity distribution of an LP_{02} mode instead of conventionally Gaussian-shaped beam profile the resolution of the system can be significantly improved [14]. Dispersion properties of LP_{02} mode permit the nonlinear generation of radiation deeper into the ultraviolet than is possible with the fundamental mode [15] and allow highly efficient generation of Cerenkov radiation [16].

However, for the successful operation of proposed devices the key requirement is propagation stability of LP_{02} mode. Typically in classical step-index and photonic crystal fibers (PCF) LP_{02} mode is degenerated with LP_{21}, having almost equal propagation constant. Therefore non-uniformities in the manufacturing process and micro bendings result in impure mode propagation and intermodal coupling limiting its practical applications. Moreover, since fundamental LP_{01} mode and LP_{02} partially overlap with each other in space it is difficult to selectively excite only HOM.

In this paper, we propose a new design of a fused silica PCF with germanium doped microinclusion embedded in the center of the core. The fiber is dedicated to efficiently couple and support LP_{02} higher-order mode. We show numerical and experimental analysis of the inclusion influence on optical properties of the fiber, taking into account intensity distribution profiles, modal index, losses, dispersion characteristics, power coupling and diffractive properties.

## 2. The concept of the LP_{02} mode supporting fiber

In classical step-index and standard PCFs the core has a uniform refractive index distribution and any change in the value of the refractive index of the glass affects all of the modes. However, since HOMs have a different than fundamental mode intensity field distributions a microinclusion placed in the center of the fiber might influence different modes more selectively. As it is shown further, this idea can be successfully used to remove the degeneracy between HOM by increasing the difference in their propagation constants and thus allowing pure propagation of LP_{02}. Moreover, such design can significantly reduce losses, grant efficient light coupling from free space into the LP_{02} mode and give flexibility in the engineering of dispersion properties.

Firstly, an evaluation of the concept was performed on the basis of the numerical investigations. As a test bench and as a reference structure we considered a PCF made from fused silica, which schematic is presented in Fig. 1(a). The cladding of the fiber is filled with air holes of d = 0.92 µm diameter with the lattice constant Λ equal to 3.028 µm. The core is formed by replacement of 7 capillaries with solid rods resulting in the total core diameter of c = 9.35 µm. The geometry of the fiber permits propagation of 4 higher-order modes, up to LP_{21}. As can be seen in Fig. 2(a). where with dashed lines we present modal index (MI) of particular modes, LP_{02} and LP_{21} have very similar propagation constant, especially for 1.55um wavelength. This causes the modes to couple energy to each other and results in impure propagation of LP_{02} mode.

In the same graph with solid lines we mark MI of the proposed fiber having same geometrical parameters and made from the same base glass, but with the 22 mol% germanium doped microinclusion of diameter a = 3 µm embedded in the center of the core (see also Fig. 1(c)). The values in the subscript indicate either the amount of germanium in host glass or in the inclusion. Fiber with inclusion also supports 4 modes, but in this case we observe much stronger wavelength dependence of the MI of particular modes and larger split between them. The presence of the substructure breaks the degeneracy between the HOMs with an effective difference of Δn = 0.003 for 1.55 µm wavelength and thus allows undisturbed propagation of LP_{02}.

To illustrate that the strong influence of the inclusion on optical properties comes mainly from the geometry and not simply from the higher refractive index of the glass in Fig. 2(b) we show the difference in values of MI calculated for various waveguide designs. We compare the MI of the pure fused silica fiber with the Ge doped inclusion located in the center (Fiber_{Ge0%}w/inc_{Ge22%}) with reference fibers which do not have inclusions but are made either from pure silica (Fiber_{Ge0%}) or silica doped with germanium (Fiber_{Ge22%}, see also Fig. 1(b)). As can be seen from dotted lines in the case of fibers with uniform refractive index distribution in the core region the difference in MI between doped and undoped fiber is almost wavelength independent and have very similar values for all modes. This indicates that replacing the fused silica glass with germanium doped fused silica doesn’t modify considerably the shape of effective refractive indexes of any considered modes but only shifts their value by a constant. On the contrary, the change in MI between fiber with and without inclusion (solid lines) varies meaningfully both in wavelength and among modes. As a consequence also the higher-order dispersion effects are enhanced in the proposed design. In Fig. 3(b) we show the group delay profiles calculated for both configurations. Again, for the fiber with the microinclusion in the center we observe the larger variation of values and clearer dependence from the wavelength of propagating light. As it is demonstrated in the next section the modal index of individual modes, dispersion properties and the separation between modes, can be precisely controlled by tuning the geometrical and material properties of the inclusion. This means that the proposed approach is not only useful for guiding of LP_{02} mode, but the flexibility of the design makes it also attractive for many linear and nonlinear applications which need to fulfill phase matching conditions [17]. Phase can be matched either within the same mode for different wavelengths or intermodal since LP_{01} and LP_{02} modes have a large spatial overlap in the center region and therefore can strongly interact with each other.

The strong sensitivity of optical properties of the fiber to the presence of the microinclusion results from the change of the areas of individual modes. In Fig. 4 we show the intensity field distributions of LP_{01} and LP_{02} modes calculated for 1.55 µm wavelength for fused silica fiber with the 22 mol% germanium doped microinclusion located in the center of the core (c,d).

As a reference, we present the mode distribution for a fiber with similar geometrical parameters but without inclusion (a,b). Non-uniform refractive index distribution in the core region modifies significantly the intensity profiles of both considered modes. Due to the incorporation of the defect the mode area of the LP_{02} changes from 757 µm^{2} to 92 µm^{2}. The bright spot in mode profile occupies the very center part of the inclusion, while the surrounding ring fits the fabricated from the undoped fused silica glass core. On the contrary, the effective area of the fundamental mode is reduced from 51 µm^{2} to 12 µm^{2}, but the mode fills mainly the microinclusion and not its surrounding. As can be seen in Fig. 2(a) the size of the overlap between the inclusion and the high intensity part of the modes has a direct impact on the strength of the MI change. The MI of the LP_{01} shifts more significantly towards higher values than it is in the case of LP_{02} mode. Such behavior can be expected since the Ge doped area has the highest refractive index in the fiber, larger by 0.033 than the index of pure fused silica glass for 1.55 µm wavelength. Moreover, as we show in section 3 the size and the shape of intensity distribution profiles of LP01 and LP02 modes have also substantial influence on propagation losses.

Finally, the LP_{02} mode is a so-called Bessel-like beam. Although it is only an approximation to Bessel beams as true Bessel beams require infinite spatial extent, LP_{02} mode still provides some of its interesting properties. For example it has partially diffraction resistant propagation in free space [18]. Since smaller diffraction in free-space means higher optical density, this feature can be exploited in lab-on a chip systems, where the fiber is used for light delivery offering a better signal to noise ratio and therefore larger sensitivity. Moreover, in many imaging techniques the resolution that can be achieved is highly dependent on the intensity distribution that is being used. Since circularly symmetric higher order modes can be focused to tighter spot than the Gaussian-shaped beams they might be an interesting alternative offering better performance of existing devices [14]. In Figs. 5(a) and 5(b) we present angular distribution of light intensity emerging from the output of the proposed fiber simulated for 1.55 µm wavelength for LP_{01} and LP_{02} mode respectively. White lines correspond to the drop in intensity by a factor of e^{2}. The calculated numerical aperture (NA) for fundamental mode is twice larger than the one of the HOM. According to simulations, this ratio scales with the geometrical parameters of the fiber always offering a more diffraction resistant propagation of LP_{02} mode.

## 3. Tailoring the optical properties of the fiber

The flexibility of the proposed design allows tailoring of the fiber optical properties in order to fit to certain tasks. In this section we show that depending from the application requirements effective mode area and the MI can be partially independently tuned among the modes. In the further analysis we will focus only on LP_{01} and LP_{02}, which are attractive for having the highest intensities in the center of the core and which can be simultaneously coupled into the fiber. The optimization is done by changing the Ge concentration and the radius of the microinclusion. In the presented numerical analysis the refractive index of heavily germanium doped fused silica is calculated on the basis of the Sellmeier dispersion equation [19]:

_{i}, SCi, GBi, and GCi are the Sellmeier coefficients for the SiO

_{2}and GeO

_{2}glasses, respectively. X corresponds to GeO2 concentration in mol%. The influence of the amount Ge in the microinclusion on MI is shown in Fig. 5(a) (solid lines). For reference we also present the mode refractive index for uniformly doped PCF fiber (dotted lines). Since the fundamental and higher-order mode have different intensity distributions in the core region, the presence of the microinclusion affects them dissimilarly. The difference in MI between fundamental and LP

_{02}mode varies between 0.031 and 0.011, only when the doping is not uniform. This feature is essential for applications where the intermodal phase matching is crucial.

The importance of microinclusion for stable propagation of LP_{02} mode can be also seen in the study of waveguide losses in the system (see Fig. 6(b)). In standard PCF this HOM has typically larger waveguide losses than the fundamental mode and suffers more significantly from bending or glass impurities [17]. The reason behind is the lower confinement of the electric field in the center of the waveguide and because of this larger effective mode area. This effect can be clearly seen also in the case of our reference fiber with uniform refractive index distribution in the core (dashed lines). The uniform doping of the fused silica glass reduces the losses, however only by a few percent. On the contrary, the microinclusion having a 12% Ge concentration decreases the waveguide losses of LP_{02} by 2 orders of magnitude (solid lines). Same conclusions are valid for the dependence of mode effective area on the amount of germanium presented in the inset of Fig. 5(b). Due to the presence of the 12% doped microinclusion the mode area of the LP_{02} mode is significantly smaller.

In the view of the fact that the dispersion properties of the fiber are one of the key factors in applications like e.g. supercontinuum generation in Fig. 7 we have presented the influence of the Ge concentration in the inclusion on dispersion shape. As it is demonstrated the flatness and zero dispersion wavelength (ZDW) can be tuned to a high degree in both fundamental and LP_{02} mode. In general it can be concluded, that for low Ge atoms concentrations in the microinclusion we obtain a flat normal dispersion for LP_{02} mode while with the increase of doping level the dispersion goes into anomalous with ZDW shift toward longer wavelengths.

The optical properties of the fiber can be also tuned by changing the geometrical parameters of the microinclusion. In Figs. 8(a) and 8(b) we present the influence of the inclusion radius on MI and effective mode area calculated for 22 mol% germanium doped microinclusion. Depending from the diameter the difference in MI between LP_{01} and LP_{02} modes varies nonuniformly between 0.019 to 0.006 in the considered range. Also the effective mode area changes significantly and nonmonotonically with the size of the defect, offering a large degree of freedom in the design process [20].

Especially interesting situation occurs for radius equal to 1.5 µm when there is the largest diversity in mode areas between fundamental and HOM. Since it might pronounce coupling of the LP_{02} over the fundamental we have studied this case further. In Fig. 9 we present calculated power efficiencies of coupling light into selected modes guided in considered optical fiber with inclusion with different objective lenses and for various distances from the focus. The estimations are done on the basis of overlap integral formalism which measures the fraction of electromagnetic fields that overlaps between the two field profiles in the forward direction [17].

_{02}than LP

_{01}reaching 0.55 when using objective with numerical aperture equal 0.2. Moreover, the ratio of power carried in both modes can be further tuned by changing the distance of fiber from the focus of the lens. It is an important feature in applications which requires to simultaneously lunch equal powers in both modes [22]. LP

_{02}and LP

_{01}are the only modes in the fiber which can be efficiently excited when the optical axis of the fiber is aligned with the optical axis of the objective.

## 4. Fabrication and characterization of the PCF with microinclusion in the core

On the basis of the presented numerical analysis, using the stack and draw technique we have developed a fiber dedicated to couple and guide LP_{02} higher-order mode. The fiber is made from pure fused silica glass. The geometrical parameters of the PCF are the same as described in the section 2 of the manuscript which are: d = 0.92 µm, Λ = 3.028 µm, c = 9.35 µm. In the center of the core we have embedded a 22 mol% germanium doped microinclusion with diameter equal to 3 µm. According to the data presented in Figs. 8(b) and 9, such width allows to couple LP_{02} with higher efficiency than LP_{01} mode when a lens with low NA is used. The SEM image of the fabricated structure is shown in Fig. 10(a).

The optical performance of the developed fiber was verified in the series of experiments. The fiber effectively supports 3 modes: LP_{01}, LP_{11} and LP_{02}. The LP_{21} mode has too strong attenuation to be observed. Using an input beam aligned along the optical axis of the fiber we have further limited the amount of exited mods only to LP_{01} and LP_{02}. As predicted in the numerical analysis (see Fig. 9) when using the objective with NA equal 0.45 and by varying the distance from the focus we were able to selectively couple the light either to LP_{01} or LP_{02}. In Figs. 10(b) and 10(c) we present their near-field images recorded using 1/2” CCD NIR (1460-1600nm) analog camera (Edmund Optics) and the corresponding mode intensity profiles calculated in numerical simulations.

To determine the attenuation we have used a standard cutback technique. Surprisingly, fabricated fiber has lower attenuation for LP_{02} mode (0.84 dB/m) than for the fundamental (2.18 dB/m) for 1.55 µm wavelength. We investigated this effect further and made additional simulations in which took into account material losses caused by germanium doping of the microinclusion and indeed we obtained similar behavior. Under the assumption that the complex refractive index equals 1.47713 + 0.001i the calculated values are 0.3 dB/m and 2.49 dB/m for LP_{02} and LP_{01} modes respectively. Apparently, strong confinement of fundamental Gaussian mode into Ge doped micronclusion and uniform filling of its area results also in higher overall attenuation.

We have also verified the diffractive properties of modes emerging from the fiber end. The NA of fundamental LP_{01} mode measured for 1.55 µm wavelength is more than twice wider than the one of the LP_{02} (0.24 vs 0.11) what corresponds well with the data presented in Fig. 5 (0.24 and 0.09 respectively). In Fig. 11(a) we show camera recorded far field projection images measured 9 mm away from the fiber output and the profiles predicted by numerical calculations under the assumption that both output beams carry the same amount of power. Both in the theory and experiment the LP_{02} is narrower than LP_{01} and if coupled with the same efficiency has a 4-times higher intensity peak. Despite the fact that the first diffraction ring of LP_{02} is hardly visible the mode preserved partially diffraction resistant propagation in free space, similar to Bessel-like beams. This is a useful feature since the ring will not create an unwanted background e.g. in sensors or imaging applications.

Finally, we measured the dispersion characteristic of fabricated fiber (11b dashed lines). We used white-light interferometry in an equal-path Mach-Zender setup. The light source was a supercontinuum laser with an average output power of 105mW. W detection in VIS and NIR ranges was done using two spectrometers OceanOptics Red Tide USB650 and Avantes AvaSpec-NIR256-1.7. Based on the SEM photo the dispersion we have calculated the dispersion also numerically (solid lines). The data are in good agreement both for fundamental, as well as HOMs. LP_{01} and LP_{02} mode exhibit a relatively flat dispersion profile (−60 to −17 ps/nm/km and −17 to 81 ps/nm/km respectively) in 1-1.6 µm wavelength range fundamental mode is all-normal dispersion while the zero dispersion wavelength for LP_{02} equals 1.087 µm.

## 5. Summary

In this paper we proposed a new design of a PCF dedicated specially to efficiently excite LP_{02} mode and to enable its pure propagation. By placing a germanium doped microinclusion in the center of the core we break the degeneracy between the LP_{02} and the LP_{21} modes with effective difference Δn = 0.003 for 1.55 µm wavelength. We demonstrate that the change of geometrical and material parameters of the inclusion allows partially independent tuning of effective refractive index of fundamental and LP_{02} HOM what potentially allows adjusting the phase matching conditions for nonlinear applications. Moreover, thanks to the optimized mode effective areas of LP_{01} and LP_{02} the design allows the precise control coupling efficiencies between them. The presented optical properties of fabricated fiber confirm the correctness of the concept. The prototype allows lower attenuation for LP_{02} (0.84 dB/m) mode than for LP_{01} (2.18 dB/m) and relatively flat dispersion profile for both fundamental (−60 to −17 ps/nm/km) and LP_{02} mode (−17 to 81 ps/nm/km) in 1-1.6 µm wavelength range. It also offers diffraction resistant propagation in free space, devoid of unwanted background.

## Funding

Foundation for Polish Science Team Programme co-financed by the European Union under the European Regional Development Fund (TEAM TECH /2016-1/1).

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