1. Introduction

The three-dimensional micro- and nanolithography based on the femtosecond-laser writing and following phosphoric acid etching allows to fabricate microstructured optical waveguides in YAG crystals with high refractive index contrast [1,2]. In these waveguides, the core is a pristine crystal region, while the cladding is composed of the hollow nano- or microchannels regularly arranged like ions in a crystal lattice. Such waveguides demonstrate an excellent light confinement due to huge refractive index modulation in the cladding. This combined technology permits to dramatically reduce mode leakage, the main drawback of the depressed cladding waveguides, and thus to expand their operation from near-IR into mid-IR region. In general, the new 3D lithography paves a way for manufacturing of optical waveguide structures like photonic crystal fibers, which revolutionized the fiber optics science and industry in the past.

In 2019 Ródenas et al. demonstrated the strong mode confinement in such waveguide, however propagation loss was measured to be rather high (around 50 dB/cm at 1.55 µm) [1]. We think that the main loss mechanism is based on a mode scattering on the hollow channel walls. Further, a lower propagation loss was measured in mid-IR (around 0.5 dB/cm at 4 µm) in a waveguide with analogous architecture, but a larger laser core diameter [2]. However, we suggest that this diminishment is due to the scattering decrease with wavelength increase. Unfortunately, no loss data were given for 1.55 µm wavelength region in this paper. Thus, propagation loss reduction in NIR in the hollow channel waveguides is still an unsolved task.

In the current study, we show a way to reduce the scattering loss. We exploited the famous facetted etching due to preferential etching in crystals along dislocation that leads to the development of crystalline facets with low Miller indices. Due to lowered roughness of these facets after etching, we have got low scattering on the facetted waveguide walls. To get the developed system of dislocation, we implemented helical writing of the depressed cladding elementary units along [111] crystallographic axis. Thus, we have got a tubular-like modified region, that is, of almost perfect axial symmetry, oriented along the crystallographic direction [111]. The following phosphoric acid etching of such helix allowed us to fabricate the hollow channel with six walls oriented parallel to {110} crystallographic planes.

2. Experiment

2.1. Waveguide fabrication

At first, we tested the etching of laser modified tracks of the tubular-like form oriented along different crystallographic directions of YAG:Nd lattice, namely [110] and [111]. Before the etching, the crystal was modified with a femtosecond laser operating at 1030 nm under a pulse repetition rate of 15 kHz with a pulse duration of 180 fs. The laser beam was focused to the crystal volume at depth of 200 µm by an objective lens with a numerical aperture NA = 0.65 with the spherical aberration correction collar. The beam was focused along [112] crystallographic axis for both cases of the track direction. A spectroscopic slit, being placed before the objective lens, was used to form a lens-like beam waist (LBW) [3,4]. The average diameter of LBW was nearly 10 µm, while thickness was estimated to be 1 µm. The sample was mounted on a high-precision 3D translator (Aerotech ABL1000). The motion of the translator was coded so that LBW moves along the helical path in the crystal volume. The X-axis was moving with a constant speed along the direction perpendicular to the largest cross-section of LBW, while two other axes were making elliptical movement in the plane of the LBW largest cross-section under the cyclic frequency of 15 Hz (Y-Z plane). The X-axis speed was chosen to set the helix pitch in the range 1-2 µm. The diameter of the tested helixes was in the range of 15-40 µm. The laser polarization was linear and parallel to the X-axis.

Laser inscription with LBW ensures equivalent thickness of the tubular like modification walls, and consequently the high order rotational symmetry [3]. Series of helixes have been written in both crystallographic directions under different pulse energies, helix pitches and diameters. Then crystals were wet etched in a mixture of H₃PO₄:H₂SO₄:H₂O (1:1:2 molar parts) at 110°C. The helixes 3 mm long were completely etched for 1 month forming through hollow channels in the crystals. The selectivity of etching was estimated to be as high as 330 (selectivity definition was done in [5]). However, at the channel entrance, the selectivity was poor up to the channel depth of about 50 µm. This manifested in enlarged channel sizes at its entrance and poor aspect ratio, which is similar to the already observed in [5]. We think that this is due to the enhanced plastic deformation and consequently increased dislocation concentration near the crystal ends. To further investigate the regular part of hollow channels the entrance part was removed by grinding crystal facets by nearly 100 µm after the etching. Typical end views of just written and etched helixes after grinding are shown in Fig. 1. It was found that channel walls are uniformly facetted along the entire channel, and the etched facets coincide with crystallographic planes as shown in Fig. 1.

 

Fig. 1. Superimposed pictures of the cross-sections of just written and etched helixes with the helix axes oriented along crystallographic directions [110] (a) and [111] (b). The dark grey area corresponds to the etched region of the crystal. The diameters of the written helixes were 35 µm (a) and 40 µm (b). Dashed lines indicate projections of crystallographic planes to the perpendicular cross-section plane (plane of the Figure). Miller indices for these crystallographic planes are shown in brackets. The etching is illustrated in Visualization 1.

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Cross-section sizes of the etched channels are remarkably larger than the diameters of the written helixes. However, these sizes correlate with the diameters. Namely, they increase with the diameter of the written helix as well as with the input laser pulse energy. In Fig. 2 dependences of the cross-section sizes on the pulse energy for different helix diameters for the channels etched along [110] are shown.

 

Fig. 2. Cross-section sizes d1 (blue points and line) and d2 (red points and line) of the channel etched along [110] crystallographic direction (Fig. 1(a)) against laser pulse energy for different helix diameter: 15 µm (squares), 25 µm (circles) and 35 µm (triangles). The helix pitch was fixed and equaled to 1 µm.

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Based on the above findings, we had fabricated the series of waveguides along [111] crystallographic axis. Two types of architecture were considered, which are waveguides, composed of four crystalline hollow channels possessing C2 rotational symmetry, and waveguides, composed of six crystalline hollow channels possessing C6 rotational symmetry. Firstly, the corresponding number of spirals was written with the technique described above. The four hollow channel waveguides were fabricated by etching at 110°C as described above for the solitary helical tracks. In order to increase the selectivity, the six-channel waveguides were etched at 83°C in a mixture of H₃PO₄:H₂SO₄:H₂O (1:1:2 molar parts) for 39 days [6]. After etching the waveguide ends were ground off by 100 µm to remove conical parts with poor selectivity and polished. Brightfield microscopic pictures of the waveguides, which demonstrate the best mode confinement in its class, are shown in Fig. 3.

 

Fig. 3. The end (a,c) and top (b,d – along the laser beam direction) views of the waveguides, composed of four (a,b, WG1) and six (c,d, WG2) hollow crystalline channels. The written helix diameter and pitch were equaled to 35 µm and 2 µm for WG1, and 7 µm and 2 µm for WG2 correspondingly. The red arrows show the crystallographic direction for the laser beam, used or the inscription. The crystallographic plane of the waveguide ends is noted by purple color.

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2.2. Waveguide characterization

The waveguides were characterized in near-IR at 1550 nm wavelength region and in mid-IR at 3390 nm.

In near-IR region Gaussian beam of the single-frequency tunable laser diode (Thorlabs TLK-L1550) was coupled to a waveguide end by a two-lens system. The lens with focal distances of 20 mm was set in front of waveguide input, and a long focal distance lens was set far off. The coupling of the laser beam was tuned to selectively excite only few low modes by adjusting the distance between the lenses. The output end was imaged by Olympus 10x objective lens at InGaAs sensor of the ArtCam-0016TNIR camera. WG1 was found to be multimode. This is illustrated by a structured pattern in intensity distribution at the waveguide output (Fig. 4). However, since the fundamental mode has the lowest divergency, it was possible to get the image of the fundamental mode under a few mm shift of the imaging lens apart from the crystal end (Fig. 4). In opposite to WG1 it was possible to predominantly excite the fundamental mode in WG2 by proper tuning the two-lens system (Fig. 5). The shape of the intensity distribution was only slightly changing when the imaging lens have been shifted from the WG2 output end. No remarkable differences in the images were observed while linear vertical or horizontal polarization was at the input.

 

Fig. 4. Fabry–Pérot fringers of the laser power transmitted through WG1. The blue line was recorded when the output end was imagined on the detector (full power in the upper insert image was detected), and the red line corresponds to the shift of the imaging lens from the waveguide end (the fundamental mode shown in the low insert image was detected). The dashed line denotes the boundary of the waveguide core.

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Fig. 5. Fabry–Pérot fringers of the laser power transmitted through WG2. Intensity distribution at the waveguide output is shown in the insert.

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Mode composition was calculated with COMSOL Multiphysics. It was found that WG1 and WG2 possess 8 and 11 modes correspondingly with leakage loss lower than 0.5 dB/cm.

The propagation loss was measured with the Fabry–Pérot technique [7]. While the wavelength of the laser diode was scanned, the oscillation of the transmitted light power was detected by InGaAs photodiode which was set instead of the camera (Figs. 4, 5). A diaphragm with the tunable aperture was placed in front of the photodiode to ensure that only the guided modes were detected. Good repeatability of Fabry–Pérot fringes in relatively wide spectral range with only small amplitude modulation for both waveguides pointed out on the prevailing excitation of the fundamental mode. The few abrupt jumps in the transmittance are due to mode hopping in the tunable laser and do not fluence in the results. The combined reflection-loss coefficient $\tilde{R}$ takes a maximum of 0.075 at 1552 nm for WG1 under the condition that all excited modes were detected (Fig. 4). Calculations made according to the procedure proved in [7] give the upper limit of the propagation loss to be as low as 1.4 dB/cm. However, when we detected power transmitted in the fundamental mode (Fig. 4), we found that the reflection-loss coefficient maximum increased up to 0.079, and the calculated upper limit of propagation loss of the fundamental mode was reduced to 0.7 dB/cm. The analogous procedure applied to WG2 gives the upper limit of the propagation loss of the fundamental mode as low as 0.5 dB/cm.

The propagation loss was found to be independent of the input beam polarization.

In mid-IR region few-mode the He-Ne laser beam was coupled to a waveguide end by a ZnSe lens with the focal length of 30 mm. The output waveguide end was imaged by a Ge lens with the focal length of 10 mm onto a sensor of pyro-crystalline array camera (PiroCam-III, Electrophysics). The corresponding near-field images are shown in Fig. 6.

 

Fig. 6. The intensity distribution at the output of WG1 (a) and WG2(b) at 3390 nm.

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3. Discussions

The most interesting results of the tubular-like track etching are the faceting of the walls of hollow channels, and the dependency of the etching geometry on the crystallographic direction of the track. Similar etching behavior was not reported elsewhere. The reason of the observed etching regime is obviously in the track geometry that defines prevailed directions of etching.

It is known that the high rate selective chemical wet etching is due to penetration of acid solution along dislocations [8]. We think that the laser modified tube-like region is etched first, as it contains laser modified bonds and nanotracks [9], and high density of dislocations [10]. Then etching of crystal region around the tubular-like track runs along dislocations, which glided in planes which are perpendicular or nearly perpendicular to the tube axis. When helix is written along [110] direction, the plane (110), perpendicular to helix axis, coincides with one of the main planes of dislocation gliding (Fig. 1(a)). There are two main Burger vectors in this plane, [111] and $[11\bar{1}]$, which give rise to two directions of dislocations, $[11\bar{2}]$ and $[112]$ correspondingly. Etching in these directions forms hollow channel walls oriented along $(11\bar{2})$ and (112) crystallographic planes. There is a secondary gliding system with the same gliding plane (110), but another Burger vector [110]. This system generates dislocations directed along [001]. Etching along these dislocations produces (001) crystallographic plane. So as dislocations in the secondary gliding system are less developed than in the main one, the etching rate in [001] direction is lower than in $[11\bar{2}]$ and [112] directions (Fig. 1(a)) that resulting in d2 < d1 (Fig. 1(a), Fig. 2).

When the helix is written along [111] direction, there are no gliding planes that are perpendicular to [111]. However, the gliding planes which are almost perpendicular to [111] belong to $\{ 112\} \left\langle {11\bar{1}} \right\rangle $ gliding system. Three gliding planes of this gliding system, which are equivalent to each other contain three equivalent directions of dislocations. Etching in these directions forms a symmetrical channel with walls oriented along crystallographic planes of {110} family (Fig. 1(b) and Visualization 1).

Two reasons can be responsible for the dominant role of dislocations oriented perpendicular to the helix and gliding in the plane which is perpendicular or almost perpendicular to the helix axis. Firstly, the prevailing multiplication of such dislocations is due to axisymmetric mechanical stress generated in the helix. Secondly, a trivial reason, the shortest path for an acid molecular to penetrate to a given point outside the helix lays in the plane perpendicular to the helix axis, and the etching runs along dislocations, which were already existing in the crystal before laser modification. Although the geometrical factor can completely explain the geometrical peculiarities of the etching, we should take into consideration the first mechanism too, because Fig. 2 shows the experimental data supporting the first proposition. So as the etching distance depends on the laser pulse energy, it is naturally to suggest that laser impact produces plastic deformations outside the exposed zone that leads to the multiplication of dislocation oriented perpendicular to the helix axis, and the density of these dislocations increases with pulse energy increase [10].

Leakage loss of a fundamental mode was calculated with Comsol Multiphysics (Fig. 7). In both waveguides, the calculated leakage losses are predictably increasing with wavelength increase, but still considerably lower than the measured loss. We consider that the main reason of the loss is scattering on the imperfections of the hollow channel walls. Coupling of the fundamental mode with higher order modes and radiation modes takes place due to this scattering. However, propagation loss in our waveguides with facetted channels is two order of magnitude lower than those in the waveguide with the usual oval channels [1]. We explain this fact by lower scattering in our waveguide due to the reduced roughness of the channel walls, because they are parallel to crystallographic planes with low Miller indices [8]. Facetted etching paves the way for obtaining the perfect almost atomic-flat walls under the appropriate choice of etching solution and regimes.

 

Fig. 7. Calculated spectra of the leakage loss in the WG1(black) and WG2(red).

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So as the propagation loss was measured while few lower modes are propagating in the waveguide, the measured values of 0.7 dB/cm and 0.5 dB/cm for WG1 and WG2 correspondingly should be considered as the upper limits of the propagation loss of the fundamental modes. Negligible difference between the loss values cannot allow us to make a conclusion on the preferential waveguide architecture in this concern. However, our modelling predicts that the effective propagation constant difference for modes in WG1 is considerably lower than those for modes in WG2. For example, the ratio of the relative differences between the propagation constants $\Delta n = ({n_{eff}^{L{P_{01}}} - n_{eff}^{LP_{11}}} )/n_{eff}^{L{P_{01}}}$ for LP₀₁ and LP₁₁ modes is $\Delta {n_{^{WG2}}}/\Delta {n_{^{WG1}}} \approx 4$. Thus, it is easier to excite the fundamental mode without excitation higher modes in the six-hollow channel waveguide, than in the four-channel one, which is obviously due to high waveguide symmetry. This is exactly what we observed in the experiments.

4. Conclusion

The direct laser writing of a helical track along one of the main crystallographic axes of YAG:Nd crystal and following wet etching allow to fabricate a faceted hollow channel. The cross-sectional symmetry of the channel depends on the crystallographic direction of the helix axis. A waveguide with hexagonal hollow cladding is fabricated under laser writing along [111] crystallographic axis with the following chemical wet etching. The waveguide efficiently confines light in NIR and mid-IR, the propagation loss of the fundamental mode does not exceed 0.5 dB/cm at 1550 nm.