Method for manufacturing a robust inhibited-coupling hollow-core thz waveguide and robust inhibited-coupling hollow-core thz waveguide

Abstract

The invention relates to a robust inhibited-coupling hollow-core THz waveguide, comprising: - a microstructured cladding (SCF1) comprising a plurality of tubular patterns (MCF1) distributed in a ring around a core (C1) so as to confine at least one radiation at an optimal wavelength λ op in the core - an outer cladding (GE1) surrounding the microstructured cladding (SCF1), wherein a waveguide formed by the outer cladding and the microstructured cladding is referred to as the initial waveguide; - an inner cladding (GI) arranged between the microstructured cladding and the outer cladding, wherein the tubular patterns are embedded in the inner cladding so as to enable them to be mechanically retained in the waveguide, and wherein an index and a thickness e gi of the inner cladding are optimised according to a diameter of the core and the optimal wavelength λ op , such that a ratio α i / α r between a linear attenuation coefficient of the robust waveguide and a linear attenuation coefficient of the initial waveguide is less than or equal to 5 at the optimal wavelength λ op .

Description

DESCRIPTION Title of the invention: Method for manufacturing a robust THz hollow-core waveguide with inhibited coupling and Robust THz hollow-core waveguide with inhibited coupling Domain

[0001] The invention falls within the field of microstructured hollow-core optical fibers with inhibited coupling. Previous technique:

[0002] Hollow-core microstructured fibers can be separated into two main categories: bandgap guided fibers (HC-PCF-PBG for Hollow-Core; Photonic-Cristal-Fiber; Photonic BandGap in English) and inhibited coupling fibers (HC-PCF-IC for Hollow-Core; Photonic-Cristal-Fiber; Inhibited Coupling in English).

[0003] These types of fibers offer numerous advantages for use in gyroscopes, including low losses, and the theoretical possibility of transferring light from the fiber core to a resonant element in the microstructured cladding, or conversely, from the resonant element to a guiding core. Furthermore, these fibers exhibit light leakage from their core with a specific azimuthal distribution and polarization dependence.

[0004] Light guidance in PCF-PBG fibers is caused by the existence of photonic band gaps created by the periodicity of the microstructured cladding. The guidance of HC-PCF-PBG band gap fibers is achieved in a periodicity defect (the core) whose refractive index m is lower than that of the cladding. HC-PCF-PBG fibers have hollow cores (air cross-section, refractive index m=1). This guidance is therefore different from that of conventional optical fibers or PCF fibers with solid core microstructured cladding, where the guidance and confinement of light in the core is caused by total internal reflection between the core (refractive index m) and the microstructured cladding (which has an average refractive index n2eff lower than m).

[0005] Alternatively, coupling inhibited fibers (or IC fibers) are optical fibers with negatively curved core walls, offering numerous advantages. They allow for low attenuation, polarization retention, relatively large core modes, very low spatial overlap between the core mode and the silica core wall, and coupling efficiency exceeding 90% (see Debord, B. et al. Hollow-Core Fiber Technology: The Rising of “Gas Photonics”. Fibers 2019, 7, 16). In an IC-type HC-PCF hollow-core fiber, since the core and cladding modes have the same effective refractive index, light propagating in the core could theoretically also propagate in the cladding.However, the confinement and guidance of the core mode in these fibers is based on the inhibition (or antiresonance) between the coupling of the core mode and the sheath mode. This inhibition is achieved through the structure of the microstructured sheath and that of the core contour. In other words, the dot product between the electric field of the core mode (|4>core) and that of the sheath mode (|<|)sheath) is very small. The coupling term between these two modes can be reduced by having a small spatial intersection between the fields |4>core) and |<|>sheath) or by a large phase shift between the transverse spatial phase of the core mode and that of the sheath mode. Since this confinement only exists for certain wavelength ranges, discontinuities are observed in the refractive index dispersion curve and therefore in the transmission curve.

[0006] Figure 1A shows a propagation loss curve as a function of wavelength for a typical HC-PCF type IC FE fiber known from the prior art and illustrated in Figure 1B. More precisely, Figure 1B schematically illustrates a cross-section of an FE fiber called HC-PCF-IC-SR-TL because the microstructured cladding GS comprises a plurality of tubular lattice (MT) motifs—typically made of silica—arranged around the hollow core (C) in a single ring (SR) within an outer cladding GE. This arrangement is known per se, in order to achieve the confinement and guidance of radiation at a wavelength op in core C1, the MT tubular motifs have a wall thickness t cf and an index n cf such as with me N.

[0007] In the illustrated example, the wall thickness of the tubular patterns is t cf= 2 m. Figure 1A shows discontinuities in the propagation loss curve. Wavelengths within the Zcf range exhibit very low losses because there is a small spatial intersection between the core and cladding fields, or a large phase shift between the transverse spatial phase of the core mode and that of the cladding mode. Wavelengths within this Zcf range can therefore be guided within the core C over long distances. The curve in Figure 1A illustrates an example of wavelength A o which can be guided by the fiber in Figure 1B. Conversely, wavelengths in the Zcp range have much higher losses because there is significant coupling between cladding modes and the core mode and therefore cannot be efficiently guided.

[0008] In the terahertz (THz) range, commercially available waveguides have a very high linear attenuation coefficient (from 20 dB / m to 14 dB / m in the 0.22–0.33 THz band, from 67 dB / m to 47 dB / m in the 0.5–0.75 THz band, or from 130 dB / m to 88 dB / m in the 0.75–1.1 THz band), which limits propagation beyond a few centimeters. Some large-core waveguides exist, allowing propagation with low losses at the expense of broad multimode guidance. However, in practice, these waveguides are difficult to manufacture because they require etching a micrometric structure along the waveguide, limiting the length of manufactured waveguides to a few tens of centimeters and resulting in a significant manufacturing cost (several tens of thousands of euros).Similarly, inhibited-coupling hollow-core waveguides appear to be a very promising solution for low-loss guidance of terahertz radiation, given that air is the optimal medium for propagating THz waves. However, in practice, the most efficient anti-resonant waveguides are limited to laboratory demonstrations, as they lack a sheath to mechanically support the dielectric tube assembly. These waveguides exhibit fragility and mechanical instability due to the presence of a microstructured sheath providing anti-resonance confinement.

[0009] The invention aims to overcome certain problems of the prior art by means of an optimized THz hollow-core waveguide with inhibited coupling (and a manufacturing method) of this waveguide) which includes, in particular, an inner sheath, positioned between the microstructured sheath and the outer sheath of the waveguide, in which the tubular patterns are interlocked to ensure their mechanical retention within the waveguide. The inner sheath is optimized so that the linear attenuation coefficient of the robust waveguide at the optimal wavelength to be guided is minimally impacted by its presence. Thus, this inner sheath ensures mechanical stability and reinforcement of the waveguide without significantly altering the propagation conditions of the THz electromagnetic wave, and even improves propagation performance in certain aspects (described later). Summary of the invention:

[0010] To this end, an object of the invention is a method for manufacturing a robust THz hollow-core waveguide with inhibited coupling, said method comprising a design step and a material manufacturing step of the robust waveguide thus designed, said method being characterized in that the design step comprises the following steps: - Selection of a so-called optimal wavelength op to be guided in the robust waveguide - Design of a so-called initial THz hollow-core waveguide with inhibited coupling comprising: o a microstructured cladding comprising a plurality of tubular motifs distributed in a ring around a core so as to confine at least one radiation at the optimal wavelength  op in said core o an outer sheath surrounding the microstructured sheath - Design of an inner sheath in a first dielectric or metallic material, intended to be placed between the microstructured sheath and the outer sheath of said initial waveguide designed in the second step, thus forming the robust waveguide in which the tubular patterns are embedded in the inner sheath so as to allow their mechanical retention within the robust waveguide, said inner sheath design step (including a substep of optimizing the index and thickness of the inner sheath in function of core diameter and optimal wavelength op so that a ratio has r / has £ between a linear attenuation coefficient of the robust waveguide and a linear attenuation coefficient of the initial waveguide, either less than or equal to 5 and preferably less than or equal to 2 at the optimal wavelength  op .

[0011] According to one embodiment of the manufacturing process of the invention, the physical manufacturing step comprises: - a first sub-step in the manufacturing of the inner and outer sheaths, - a second sub-step in the manufacture of tubular patterns, and - a third sub-step of assembling the tubular patterns and an assembly formed by the inner sheath and the outer sheath.

[0012] Preferably, in the preceding embodiment, the first manufacturing substep is carried out by additive printing.

[0013] Preferably, in the preceding embodiment, the inner sheath is made in an adhesive, for example a UV adhesive, the third manufacturing substep then being carried out by gluing the inner sheath with the tubular patterns.

[0014] Another object of the invention is a robust THz hollow-core waveguide with inhibited coupling comprising: - a microstructured sheath comprising a plurality of tubular motifs arranged in a ring around a core so as to confine at least one radiation to an optimal wavelength op in said heart - an outer sheath surrounding the microstructured sheath, a waveguide formed by said outer sheath and said microstructured sheath being called the initial waveguide, - an inner sheath made of a first dielectric or metallic material and disposed between the microstructured sheath and the outer sheath, the tubular motifs being embedded in the inner sheath so as to allow their mechanical retention in the robust waveguide, a complex index n gi And a thickness e gi the internal sheath being optimized according to a diameter D c of the heart and the optimal wavelengthop so that a ratio has r The difference in attenuation between a linear coefficient of the robust waveguide and a linear coefficient of the initial waveguide must be less than or equal to 5 at the optimal wavelength. op .

[0015] According to one embodiment of the robust waveguide of the invention, the inner sheath is a ring having a thickness e gi substantially constant in a plane transverse to the robust waveguide and substantially constant longitudinally.

[0016] According to one embodiment of the robust waveguide of the invention, the inner sheath is arranged discontinuously longitudinally, for example in a plurality of sections of the same length L t separated longitudinally by a distance AL Z .

[0017] According to one embodiment of the robust waveguide of the invention, a ratio e gi lr tThe ratio between the thickness of the inner sheath and the radius of the tubular patterns is adapted according to the diameter of the core, an absorption coefficient of the material constituting the tubular patterns, and the guiding wavelength so that the ratio has r lai is less than or equal to 5 at the optimal wavelength  op .

[0018] Preferably, the ratio e gi lr t is between 1 and 2 and preferably between 0.05 and 0.95.

[0019] According to one embodiment of the robust waveguide of the invention, a real part of the index n gi the internal sheath is between 1.4 and 3, preferably between 1.4 and 2 at the optimal wavelength.

[0020] According to one embodiment of the robust waveguide of the invention, an imaginary part of the index n giThe internal sheath thickness is between 0.0001 and 0.015. This range is particularly suitable for guiding THz waves with low attenuation.

[0021] According to one embodiment of the robust waveguide of the invention, a ratio e gi lr t between the thickness of the inner sheath and a radius of the tubular patterns are adapted to allow a reduction of the peaks of strong attenuations of a transmission spectrum of said robust waveguide.

[0022] According to one embodiment of the robust waveguide of the invention, the first dielectric material is silica, glue or plastic.

[0023] According to a preferred embodiment of the robust waveguide of the invention, the microstructured cladding is formed of tubular patterns having a refractive index n t between 1.4 and 2.5, a thickness between 0.2 and 0.3 times the optimal wavelength, and a diameter Dt between 0.3 and 0.9 times the core diameter, the core diameter being between 6 and 20 times the optimal wavelength (preferably between 8 and 12), the inner cladding being made of silica or a first material having a refractive index n gi with a real part between 1.4 and 2.5 at the optimal wavelength and a ratio e gi lr t The thickness of the inner sheath and the radius of the tubular patterns are between 0.05 and 0.75. This embodiment has been identified by the inventors as providing optimal guidance performance in the THz range for both joined and disjointed tubular patterns. The core diameter is preferably between 8 and 12 times the optimal wavelength to allow for better guidance.

[0024] According to one embodiment of the robust waveguide of the invention, a thickness and diameter of the tubular patterns are constant longitudinally and in which the internal sheath is adapted so that a distribution of the tubular patterns in a plane transverse to the robust waveguide varies longitudinally so that the robust waveguide allows, from a first end of the robust waveguide to a second end of the robust waveguide, a transition from a first intensity distribution of a guided fundamental mode to a second intensity distribution of the guided fundamental mode.

[0025] According to one embodiment of the robust waveguide of the invention, a thickness and diameter of the tubular patterns are constant longitudinally and in which the inner sheath is adapted so that a distribution of the tubular patterns in a plane transverse to the robust waveguide varies longitudinally so that the diameter of the core of the waveguide varies longitudinally and so that a cross-section of a fundamental mode undergoes a homothety longitudinally.

[0026] According to one embodiment of the robust waveguide of the invention, the thickness e gi is greater than or equal to 0.01 xr t , with r t a range of tubular patterns. Brief description of the figures:

[0027] Other features, details and advantages of the invention will become apparent from the description provided with reference to the accompanying drawings given by way of example, which represent, respectively:

[0028] [Fig.1 A], a propagation loss curve as a function of wavelength of a typical HC-PCF type IC fiber known from the prior art, illustrated in [Fig.1 B]

[0029] [Fig. 2], a schematic illustration of a cross-section of a robust waveguide according to the invention,

[0030] [Fig. 3A], a schematic illustration of one embodiment of the robust waveguide of the invention

[0031] [Fig. 3B], the evolution of the linear attenuation coefficient of the fundamental HE11 mode as a function of the radiation frequency, for three different waveguides,

[0032] [Fig.3C], the transmission spectrum of two waveguides, including a robust waveguide according to an embodiment of the invention;

[0033] [Fig. 3D], the evolution of the attenuation of the fundamental mode HEn as a function of the radiation frequency, for three different waveguides, including two robust waveguides according to the invention,

[0034] [Fig. 4], a schematic illustration of a cross-section of the robust waveguide according to the MR4-MR8 configurations of the invention

[0035] [Fig. 5], two different graphs tracing the evolution of the linear attenuation coefficient of the fundamental mode in the hollow core of the Ref3, MR5 and MR5 configurations,

[0036] [Fig. 6], an embodiment of the invention in which the robust waveguide is adapted to perform, from a first end of the robust waveguide to a second end of the robust waveguide, a transition between a first transverse intensity distribution of a guided fundamental mode towards a second transverse intensity distribution of the guided fundamental mode,

[0037] [Fig.7], an embodiment of the invention in which the robust waveguide is adapted to achieve, from a first end of the robust waveguide to a second end of the robust waveguide, a longitudinal variation of the core diameter,

[0038] [Fig.8], a graph showing the evolution of the value of the linear attenuation coefficient of the robust waveguide in Figure 7, as a function of the core diameter.

[0039] In the figures, unless otherwise indicated, the elements are not to scale and identical references designate identical elements. Detailed description:

[0040] Figure 2 schematically illustrates a cross-section ST of a robust waveguide 1 according to the invention.

[0041] We begin by describing the general structure of the robust waveguide 1 before detailing the different embodiments and their performance. Finally, we will describe a manufacturing process according to the invention for obtaining the robust waveguide 1 of the invention.

[0042] This robust waveguide 1 is a hollow-core terahertz (THz) waveguide with inhibited coupling which includes a microstructured cladding GS comprising a plurality of tubular MT motifs (also referred to as MT capillaries hereafter) which can be joined or disjoined, an outer GE cladding and an inner Gl cladding.

[0043] More specifically, the MT tubular patterns of the GS microstructured cladding are arranged in a ring around the C core in such a way as to confine at least one radiation at an optimal wavelength within the C core. opin the THz range. The tubular patterns have a diameter (i.e., a maximum dimension in the transverse plane of the waveguide) D t = 2r t and a wall thickness e t As mentioned previously, the guidance of a wavelength  op predetermined is made possible by an appropriate choice of wall thickness e t MT tubular patterns. The thickness e t is related to the material used to make the MT tubular patterns, more precisely the complex refractive index of the material. This material is, for example, silica or a polymer-type material.

[0044] By "THz waveguide", it is meant here that the waveguide of the invention allows the guidance of at least one radiation having the optimal wavelength op with a frequency between 0.1 THz and 30 THz.

[0045] By way of non-limiting example, the robust waveguide 1 illustrated in the embodiment of Figure 2 comprises eight MT tubular patterns that are disjoint from one another. As is known per se, the distribution of the MT tubular patterns in the microstructured cladding GS in the transverse plane of the waveguide determines the azimuthal distribution of the leakage field of the guided radiation in the core. It is understood that the invention is not limited to this specific example and can be implemented with a different number of MT tubular patterns, of more or less complex topology, which may be joined or disjoint (see, for example, Figure 4 described below). An exhaustive description of the various possible arrangements of the tubular patterns, their shapes, and their effects on the guidance would fall outside the scope of the present invention.

[0046] The outer GE sheath surrounds the microstructured GS sheath in such a way as to protect the latter. This outer GE sheath is typically a dielectric material, such as silica or a polymer. A waveguide formed solely by the outer sheath and the microstructured sheath is subsequently referred to as the "initial waveguide." This initial waveguide is therefore identical to the one described in Figure 1B.

[0047] Unlike the initial waveguide, the robust waveguide of the invention comprises an inner sheath Gl disposed between the microstructured sheath GS and the outer sheath GE, and in which the tubular motifs MT are interlocked so as to allow their mechanical retention within the robust waveguide. This inner sheath Gl is made of a first dielectric or metallic material of complex index n gi and takes the form of a layer of thickness e giarranged in a ring around the C core and attached to the GE outer sheath.

[0048] According to a preferred embodiment, the inner sheath is deposited on the initial waveguide via a "glue gun" for example.

[0049] Preferably, in order to best control the guiding conditions along the entire waveguide, the inner cladding has a thickness of e gi substantially constant in a plane transverse to the robust waveguide, and substantially constant longitudinally (i.e., along the length of the waveguide). Thus, in this embodiment, the cross-sectional area ST of the waveguide 1 is constant longitudinally.

[0050] Alternatively, according to another embodiment illustrated in Figure 3A, the inner sheath Gl is arranged discontinuously along its longitudinal axis, for example, as a plurality of sections of the same length L separated longitudinally by a distance AL. This embodiment has the advantage of being more easily fabricated. Indeed, it is then possible to simply encircle the tubular motifs so as to hold them fixed in the waveguide with several ring sections that form the inner sheath sections. As a non-limiting example, L = 1 cm and AL = 10 cm. Alternatively, the sections are not of the same length and / or are not separated by the same distance.

[0051] According to a variant of the embodiment of Figure 3A, some of the longitudinally distributed sections have different internal sheath materials.

[0052] According to a first embodiment, the inner sheath Gl is a separate layer from the outer sheath GE, made of a first dielectric material distinct from that of the outer sheath. Alternatively, according to a second embodiment, the inner sheath Gl is a continuation of the outer sheath GE, and is therefore made of the same material as the latter. Generally, in all embodiments of the invention, no portion of the tubular patterns MT is embedded in the outer sheath GE, and, by convention, the boundary between the outer sheath GE and the inner sheath Gl is defined by the end of the tubular patterns most radially eccentric with respect to the core center.

[0053] The inner cladding Gl allows for the fixed positioning of each MT tubular pattern—both in the transverse plane of the waveguide and longitudinally—thus ensuring mechanical stability and robust reinforcement of the waveguide compared to the original waveguide. However, the presence of the inner cladding alters the complex effective index perceived by the radiation within the waveguide. robust waveform. Also, it is necessary that the different parameters (complex index n) gi and thickness e gi ) of the inner cladding are optimized so that the linear attenuation coefficient at the optimal wavelength of the robust waveguide is minimally impacted by the presence of the inner cladding.

[0054] Through numerous simulations and experiments, the inventors determined that, for a predetermined torque of diameter D c of the heart and optimal wavelength op, it was possible to optimize the index n gi and the thickness e gi of the internal sheath Gl so that a ratio a r / has £ between a linear attenuation coefficient of the robust waveguide and a linear attenuation coefficient of the initial waveguide is less than or equal to 5 (and preferably less than or equal to 2) at the optimal wavelength  op Each of these parameters n gi summer gi influence the linear attenuation coefficient a t of the robust waveguide and are dependent on the various parameters of the initial waveguide and in particular on the various parameters of the MT tubular patterns (e.g. wall thickness, diameter, distribution in the transverse plane, number of patterns) which determine the optimal wavelength  op of guidance and the associated linear attenuation coefficient. However, importantly, there is a pair n gi , e gi) optimal for each couple (D c , op ).

[0055] The thickness e gi The thickness of the inner sheath Gl must be sufficient to mechanically support the MT tubular patterns. Although the minimum thickness required to support the MT tubular patterns depends on the material of the inner sheath and the size of the MT tubular patterns, through several experiments, the inventors determined that a thickness e gi greater than 0.01 xr t allowed the mechanical support of the MT tubular patterns. Also, preferably, the thickness e gi of the internal sheath Gl is greater than 0.01 xr t .

[0056] Figure 3B shows the evolution of the linear attenuation coefficient of the fundamental mode HEn as a function of radiation frequency, for three different waveguides, according to three curves C1, C2, and C3. The three curves C1, C2, and C3 are simulation results. Curve C1 corresponds to the evolution of the linear attenuation coefficient of the fundamental mode HEu in an initial waveguide comprising 10 silica-sealed MT capillaries of diameter D t = 2.6 mm and wall thickness 134 pm forming a hollow core in air of diameter D c = 6 mm and with an external silica sheath. The complex refractive index of the silica at the optimal wavelength op = 300 |im (f op= I THz) is nt = 1.954 - 0.0047 i (corresponding to an absorption coefficient of 849 dB / m). Curve C2 corresponds to the evolution of the linear attenuation coefficient of the fundamental mode HE11 in a robust waveguide according to the invention formed from the initial waveguide of curve C1 and comprising an internal silica sheath Gl of thickness e gi = r t This configuration will be referred to as MR1 ​​hereafter. Finally, curve C3 corresponds to the evolution of the linear attenuation coefficient of the fundamental mode HE11 in a robust waveguide according to the invention, formed from the initial waveguide of curve C1 and comprising an internal sheath Gl of thickness e gi = r t and in a first material of index n gi= 1.5 - 0.01 * i over the studied frequency range. This configuration will be called MR2 hereafter. The imaginary part of the index implies that the first material used in the waveguide of curve C3 is absorbing for the optimal wavelength λ op .

[0057] Figure 3B shows that the three curves C1, C2, and C3 are virtually identical and indistinguishable from one another, indicating no significant variation in the linear attenuation coefficient introduced by the inner cladding. Thus, the introduction of the inner cladding, even when it is in an initial absorbing material for the optimal wavelength, op does not significantly impact the propagation conditions of the THz electromagnetic wave when the parameters n gi summer gi are chosen appropriately.

[0058] This result is also confirmed by Table 1 below, which summarizes the calculations of the effective neff index. HE11 and the linear attenuation coefficient a HE11 of the fundamental mode HEn for different configurations Ref1, Ref2, MR1-MR3. More precisely, the effective index and linear attenuation coefficient Ref2 are obtained for the initial waveguide of curve C1 in Figure 3B. Similarly, the effective index and linear attenuation coefficient Ref1 are obtained for the initial waveguide of curve C1 in Figure 3B, but without the external silica sheath. The MR1 and MR2 configurations have been described in Figure 3B, and the MR3 configuration corresponds to a robust waveguide according to the invention, identical to that of configuration MR2 but with an internal sheath made of a first dielectric material having an index n ni such q 1 ue n ni *• = 3 - 0.01 * i. Table 1

[0059] These simulation results do not illustrate any significant variation in these values ​​by the introduction of the inner sheath in the MR1-MR3 configurations of the invention compared to the Ref1 and Ref2 configurations known in the prior art, even when the sheath has a real part of the index n gi very high as in the MR3 configuration or high absorption as in the MR2 and MR3 configurations.

[0060] Figure 3C shows the transmission spectrum of a waveguide according to an embodiment MR2' of the invention (curve C2'). More specifically, the waveguide of embodiment MR2' of the invention has a length of 30 cm and comprises 10 MT silica-sealed capillaries of diameter D t = 2.6 mm and wall thickness 127 pm forming a hollow core in air of diameter D c= 6 mm and with an outer silica sheath. The complex refractive index of silica at the optimal wavelength op = 300 [im (f op = 1 THz) is n t = 1.954 - 0.0047i (corresponding to an absorption coefficient of 849 dB / m). In embodiment MR2', the inner sheath was formed by applying glue using a glue gun, without any special precautions. This layer of glue provides mechanical support for the waveguide.

[0061] For comparison, the CT curve corresponds to the spectrum of an initial waveguide identical to the waveguide of embodiment MR2', except that it does not include an internal sheath.

[0062] Curve C2' shows that the waveguide spectrum of embodiment MR2' is characterized by different transmission windows induced by the antiresonance mechanism. Comparison of curves CT and C2' demonstrates that there is a very small variation in the transmission spectrum induced by the addition of the inner sheath, which is made here of a highly absorbing material. These results demonstrate that adding an internal sheath results in a negligible increase in waveguide attenuation.

[0063] Figure 3D illustrates the effect of the inner cladding material on the absorption of the robust waveguide. More specifically, Figure 3D shows the evolution of the linear attenuation coefficient of the fundamental HEu mode as a function of the radiation frequency, for three different waveguides, according to three curves C3', C4', and C5', which are simulation results.

[0064] Curve C5' corresponds to the evolution of the linear coefficient of the fundamental mode HEu of the initial waveguide in Figure 3C. Similarly, curve C4' corresponds to the evolution of the linear coefficient of the fundamental mode HEu of the robust waveguide according to the MR2' embodiment detailed in Figure 3C with a highly absorbing silica inner sheath (n gi = 1.954 - i*10' 2 Finally, curve C3' corresponds to the evolution of the linear coefficient of the fundamental mode HEu of a robust waveguide according to an embodiment identical to embodiment MR2' detailed in Figure 3C, except that the inner sheath is not made of silica but of a highly absorbing polymer-type material (n gi = 1.5 - i*10 -2 ).

[0065] The ordinate of curves C3', C4', and C5' represents the attenuation, which is given here as a function of the imaginary part of the effective refractive index of the waveguide. Furthermore, the abscissa of curves C3', C4', and C5' is given as a function of the silica absorption, i.e., the imaginary part of the refractive index of the material of the tubular patterns (lm(nt)). This amounts to modeling the waveguide at different frequencies THz, since the refractive index (real part) of silica is constant in this range (n r = 1.954). The attenuation of the inner sheath at different frequencies in the THz range is illustrated by the points in the figure, i.e., Im(n gi ) = -4.6 . 10“ 4 at 0.1 THz; Im(n gi ) = -1.6. 10" 3 at 0.3 THz, Im(n gi ) = -2.8. 10“ 3 at 0.6 THz, Im(n gi ) = — 4.6. 10 -3 at 1.0 THz and Im(n gi ) = — 1.2 .10 -2 at 2.5 THz.

[0066] Finally, Figure 3D shows the curve C6' which traces the evolution of the ratio a r the ratio between a linear attenuation coefficient of robust waveguides and a linear attenuation coefficient of the initial waveguide

[0067] Figure 3D illustrates firstly that the C3' and C4' curves coincide and that there is a reduction in the attenuation of the HEu mode during a reduction of the absorption of the tube material, down to a plateau value. Figure 3D also demonstrates that, for a high absorption coefficient of the tubular patterns (lm(nt) > 10' 4 - either in the THz range), the addition of an internal silica or highly absorbent polymer sheath has no significant impact on the HEn fundamental mode attenuation. For reference, the ratio — is 2.30

[0068] Thus, through these results and other simulations, the inventors determined that a ratio a r / has£ less than or equal to 5 (and preferably less than or equal to 2) at the optimal wavelength op was possible in the terahertz range for an inner sheath having a material such that the imaginary part of the index n gi of the internal sheath is between 0.0001 and 0.015. This characteristic is applicable in the case where the sheath includes joined and disjointed tubular patterns.

[0069] Since previous results were obtained with joined tubular patterns, the focus is now on confirming that the inner sheath produces the same effects in a robust waveguide with disjointed tubular patterns. Indeed, it is known that reducing the diameter of the capillaries so that they are disjointed reduces the linear attenuation of the fundamental mode and, under certain conditions, significantly increases the attenuation of higher-order modes, thus enabling effective single-mode propagation.

[0070] The simulation results presented in Table 2 below detail, for 6 different configurations Ref3, MR4-MR8, the calculations of the linear attenuation coefficient a HE11 of the fundamental mode HEN, of the ratio a TE01 / has HE11 between the linear attenuation coefficient of the TE01 mode and that of the fundamental mode HEn, and the difference Aa = a r HE11 - has i HE11between the linear attenuation coefficient of the fundamental mode HEu of each MR4-MR8 configuration with that of the reference configuration Ref3. Table 2

[0071] The reference configuration Ref3 corresponds to an initial waveguide comprising 6 disjointed MT silica capillaries with a wall thickness e t = 100 m, and a diameter D t = 2.8 mm, and with a core diameter of D c = 4 mm with an outer sheath of refractive index 1. This initial waveguide is suitable for guiding an optimal frequency of 600 GHz. Indeed, through simulations, the inventors determined that an optimal value — = 0.7 of the diameter of the D cThe MT capillary ratio relative to the core diameter allows for an increase in the attenuation of the TE01 mode by a factor of approximately 40 compared to the attenuation of the HEn mode. This enables near-single-mode waveguide guidance. It is important to note that in the case of a silica waveguide, the variation in the attenuation of the fundamental mode as a function of the core diameter D c is relatively low. In the case of a waveguide composed of 8 capillaries, the diameter of the capillaries has little influence on the attenuation of the HE11 mode, which allows for the creation of waveguides with a ratio — ~ 0.4. Of

[0072] Figure 4 schematically illustrates a cross-section of the robust waveguide according to the MR4-MR8 configurations. The MR4 configuration corresponds to a robust waveguide according to the invention formed from the initial waveguide of the Ref3 configuration and with an inner silica sheath with an overlap between the inner sheath Gl and the MT tubular patterns (corresponding to the ratio r t THE gi which is worth 0.25. The MR5 configuration corresponds to a robust waveguide according to the invention formed from the initial waveguide of the Ref3 configuration and with an internal sheath in a first dielectric material of index n gi = 1.5 - 0.01 * i and with an overlap between the inner sheath Gl and the tubular patterns MT which is 0.25. The MR6 configuration is identical to the MR5 configuration but the overlap between the inner sheath Gl and the tubular patterns MT is 0.1. The MR7 configuration is identical to the MR5 configuration but the overlap between The internal sheath Gl and the tubular patterns MT are 0.05. Finally, the MR8 configuration is identical to the MR5 configuration but the overlap between the internal sheath Gl and the tubular patterns MT is 0.01.

[0073] Table 2 illustrates the possibility of adding an internal sheath around the silica capillaries, composed of a first absorbing dielectric material (silica or a first dielectric material having a complex refractive index n). gi = 1.5 - 0.01 * i) without significantly altering the attenuation a HE11of the fundamental mode propagated in the hollow core formed by 6 disjoint capillaries. Indeed, the addition of the inner sheath results in an increase Ac in the attenuation of the HEu fundamental mode of each MR4-MR8 configuration compared to that of the Ref3 configuration of less than or equal to 0.218 dB / m. It is noted that the use of a silica inner sheath (MR4 configuration) allows for an increase Ac half that of using an inner sheath made of a first dielectric material having a complex refractive index n gi = 1.5 - 0.01 * i (MR5 configuration).

[0074] Furthermore, Table 2 highlights an optimal value for the recovery e gi / r t capillaries with an internal sheath of 0.25 allowing for attenuation c HF11of the fundamental mode of 0.326 dB / m for an internal silica sheath or of 0.390 dB / m for a first highly absorbing dielectric material having a complex refractive index n gi = 1.5 - 0.01 * i.

[0075] The results in Table 2 are confirmed by Figure 5, which presents two different graphs showing the evolution of the linear attenuation coefficient of the fundamental mode in the hollow core of the Ref3, MR5, and MR5' configurations. The MR5' configuration is identical to the MR5 configuration according to the invention, except that the first dielectric material of the inner sheath does not exhibit absorption (complex refractive index n). gi = 1.5).

[0076] More specifically, in the lower graph of Figure 5, curve C6 represents the evolution of the linear attenuation coefficient of the fundamental mode HEn as a function of the radiation frequency in the Ref3 configuration. Curve C4 represents the evolution of the linear attenuation coefficient of the fundamental mode HEu in the MR5 configuration, and curve C5 represents the evolution of the coefficient linear attenuation of the fundamental mode HEn in the MR5' configuration. Firstly, it is noted that curves C4 and C5 coincide, implying that adding an absorption coefficient to the inner cladding has a very small influence on the value of the fundamental mode attenuation coefficient. This result demonstrates the possibility of using absorbing materials for the cladding. Furthermore, the addition of the absorbing inner cladding eliminates spurious peaks of high attenuation present in the attenuation spectrum of the waveguide without a visible cladding between 0.45 GHz and 0.55 GHz in curve C6, but not in curves C4 and C5.

[0077] Furthermore, curve C7 in the top graph of Figure 5 represents the evolution of the ratio a r / has £ depending on the frequency of the radiation, with a r the linear attenuation coefficient of the fundamental mode HEn in the MR5 configuration and a tThe linear attenuation coefficient of the fundamental mode HEn in the Ref3 configuration. The top graph of Figure 5 illustrates that adding the inner cladding results in a slight increase, less than a factor of 2 across most of the spectrum, in the linear attenuation coefficient compared to the Ref3 configuration without the inner cladding. For frequencies above 0.8 GHz, a reduction in the linear attenuation coefficient is even observed thanks to the addition of the inner cladding. Similarly, as seen in the bottom graph of Figure 5, a reduction in the linear attenuation coefficient is observed at certain frequencies between 0.45 GHz and 0.55 GHz thanks to the addition of the inner cladding, which helps to suppress spurious peaks of high attenuation present in this spectral range.

[0078] The MR4, MR5, and MR5' configurations according to the invention are preferred configurations for guiding radiation at the optimal wavelength corresponding to the 600 GHz frequency in order to efficiently guide over the 560 GHz–620 GHz range (referenced as BTS in Figure 5 below), which corresponds to the emission range of many THz sources. They were developed following numerous simulations and optimizations, particularly regarding the choice of the anti-resonant transmission window and the minimization of the amplitude of spurious peaks related to the thickness t c of the capillary wall. The diameter of the capillaries D t is optimized by calculating the attenuation spectrum of the HE11 mode in (depending on the diameter of the core) c These results show that the ratio D t / D c = 0.70 allows for the best attenuation spectrum (low attenuation and moderate amplitude spurious peaks in the transmission band).

[0079] Furthermore, the core diameter influences the intensity of coupling to the cladding modes. For example, in the preferred MR4, MR5, and MR5' configurations according to the invention, the attenuation spectrum of an initial waveguide with a 4 mm diameter core exhibits lower attenuation peaks than for a 3 mm diameter core. Since the peak amplitude is relatively low, at a level acceptable for propagating THz waves, a diameter of 4 mm was chosen as the minimum value. A larger diameter further reduces the peaks and the attenuation coefficient of the HE11 mode. However, this leads to a convergence of the effective indices of the HE11 mode and higher-order modes, which can make it difficult to excite only the HE11 mode.

[0080] As illustrated in Figure 5 below, the waveguides of the preferred configurations MR4, MR5, and MR5' allow the fundamental mode to propagate with a linear attenuation of less than 0.5 dB / m over the considered BTS frequency range. Furthermore, they enable confinement of the HE11 mode with a diameter of 2 mm at half maximum height (in power) and approximately 2.5 mm at 1 / e, at 600 GHz.

[0081] More generally, in order to obtain highly satisfactory waveguide performance with a linear attenuation of less than 0.5 dB / m over the BTS frequency range, the microstructured GS sheath of the robust waveguide is formed of disjoint (or disjoint) tubular patterns with a refractive index n t between 1.4 and 2.5, a thickness between 0.2 and 0.3 times the optimal wavelength, and a diameter D tbetween 0.3 and 0.9 times the core diameter, the core diameter being between 6 and 20 times the optimal wavelength (preferably between 8 and 12), the inner cladding being made of silica or a first material having a refractive index n gi with a real part between 1.4 and 2.5 at the optimal wavelength and a ratio e gi lr t between the thickness of the inner sheath and a radius of the tubular patterns being preferably between 0.05 and 0.75.

[0082] Thus, the invention consists of adding an internal sheath Gl around the hollow core of a THz waveguide with inhibited coupling, without significantly altering the propagation conditions of the fundamental mode within the hollow core. This innovation offers the following advantages: - It ensures mechanical stability and robust mechanical reinforcement of the waveguide - It helps to protect the core of the guide from external contaminants (e.g., dust) - It allows the use of absorbent materials to make the sheath (e.g. glue, supports or guides made by 3D printing, plastic)

[0083] In the illustrations of Figures 2 and 4, the MT tubular motifs have a circular cross-section (i.e., circular cylindrical tubes), but it is understood that the invention is not limited to this embodiment. The tubular motifs may, for example, be nested structures, that is, an interlocking of different concentric tubes with progressively smaller diameters. Alternatively, these tubular motifs may be elliptical tubes, with the major axis of the ellipses oriented radially towards the center of the fiber, or any other shape known to those skilled in the art.

[0084] Figure 6 illustrates an embodiment of the invention in which the robust waveguide is adapted to perform, from a first end E1 of the robust waveguide to a second end E2 of the robust waveguide, a transition from a first transverse intensity distribution I1 of a guided fundamental mode to a second transverse intensity distribution I2 of the guided fundamental mode. For this purpose, the thickness and diameter of the tubular MT patterns and the thickness e giof the inner cladding are constant longitudinally. Furthermore, the inner cladding is adapted so that the distribution of tubular patterns in a plane transverse to the robust waveguide varies longitudinally, allowing the robust waveguide to transition the transverse intensity distribution. Indeed, the position of the tubular patterns in the transverse plane of the waveguide determines the transverse intensity distribution, and the inner cladding determines the position of the tubular patterns in the transverse plane of the waveguide. Thus, the inner cladding allows for a high degree of modularity in the longitudinal distribution of the position of the tubular patterns which allows the longitudinal control of the transverse intensity distribution.

[0085] By way of non-limiting example, the waveguide in Figure 6 is based on the MR4 configuration according to the invention, modified to allow a transition from a first transverse intensity distribution of Gaussian frequency in a circular core 4 mm in diameter to a second transverse intensity distribution of substantially elliptical frequency in a rectangular core measuring 5.65 mm x 2 mm (or vice versa). This shape transition results in a slight, but very acceptable, increase in the linear attenuation coefficient from 0.326 dB / m at 0.6 THz.

[0086] This type of robust "transition" waveguide provides an effective solution to problems related to differences in THz beam shape between two structures, such as injecting a THz beam from a THz horn antenna into a circular beam or into a waveguide with a circular core. Indeed, this rectangular core shape is particularly well-suited for coupling a waveguide to a horn antenna of a THz source.

[0087] Figure 7 illustrates an embodiment of the invention in which the robust waveguide is adapted to achieve, from a first end E1 of the robust waveguide to a second end E2 of the robust waveguide, a variation in the core diameter. This variation is made possible by the inner cladding, which determines the position of the MT tubular patterns in the transverse plane, these patterns determining the core diameter. Thus, by longitudinally varying the position of the MT tubular patterns in the transverse plane of the waveguide while ensuring that the intensity distribution of the fundamental mode remains constant longitudinally, the inner cladding allows a longitudinal variation of the core diameter. To ensure this variation of the core diameter only, it is necessary that the thickness and diameter of the tubular patterns be constant longitudinally. It is also preferable that the thickness e giof the internal sheath is constant longitudinally.

[0088] This type of robust "transition" waveguide provides an effective solution to problems related to beam diameter differences. THz in free space or confined in a structure, or between two structures, such as, for example, the injection of a THz beam propagated in free space into a waveguide whose core is smaller than the diameter of the beam.

[0089] By way of non-limiting example, in the illustration of Figure 7, the waveguide of Figure 7 is based on the MR5 configuration according to the invention, modified to allow an increase in the core diameter from 4 mm to 9.3 mm without significantly increasing the linear attenuation coefficient of the fundamental mode HEn.

[0090] This result is illustrated in Figure 8, where curve C8 plots the evolution of the linear attenuation coefficient of the robust waveguide in Figure 7 as a function of the core diameter. For comparison, curve C9 plots the evolution of the linear attenuation coefficient as a function of the core diameter of the initial waveguide in the MR5 configuration (i.e., the robust waveguide in the MR5 configuration without the inner cladding), modified to allow an increase in the core diameter from 4 mm to 9.3 mm.

[0091] Figure 8 allows us to observe two critical aspects of the implementation of Figure 7.

[0092] Firstly, on curve C8, we note that reducing the core diameter from 4 mm to 9.3 mm does not increase the linear attenuation coefficient of the HE11 fundamental mode and even reduces it: it goes from a value of less than 0.3 dB / m for a core diameter of 4 mm to a value of less than 0.07 dB / m for a diameter of 9.3 mm.

[0093] Furthermore, by comparing curves C8 and C9, we observe that the introduction of the inner sheath allows for better confinement of THz waves at the optimal frequency of 600 GHz, as evidenced by a linear attenuation coefficient for curve C8 that is lower than that of curve C9. This latter result clearly illustrates that, under certain conditions, the inner sheath improves the propagation conditions of THz radiation in the robust waveguide.

[0094] Another object of the invention is a method for manufacturing the robust waveguide 1 according to the invention. This manufacturing method comprises a design step and a physical fabrication step of the robust waveguide thus designed.

[0095] The design phase includes a first step A of selecting the optimal wavelength op to be guided in the robust waveguide. In step B, the initial waveguide is designed, comprising: - the microstructured cladding GS comprising the MT tubular motifs arranged in a ring around the core C so as to confine radiation to the optimal wavelength  op in the heart - the GE outer sheath surrounding the GS microstructured sheath

[0096] Finally, the design phase includes a final step C for designing the inner sheath Gl, intended to be placed between the microstructured sheath and the outer sheath of the initial waveguide designed in step B. This conceptually forms the robust waveguide in which the tubular patterns are embedded within the inner sheath in such a way as to ensure their mechanical retention within the robust waveguide. The design step of the inner sheath Gl includes a substep for optimizing the index n gi and the thickness e gi of the inner sheath as a function of the core diameter D c and the optimal wavelength  op so that the ratio has r / has £ the ratio between the linear attenuation coefficient of the robust waveguide and the linear attenuation coefficient of the initial waveguide is less than or equal to 5 (preferably less than or equal to 2) at the optimal wavelength  op .

[0097] Furthermore, preferably, the physical manufacturing step of the manufacturing process of the invention comprises a first substep for manufacturing the inner and outer sheaths. This first substep can be carried out by any method known to those skilled in the art, either separately for the inner and outer sheaths or simultaneously for both. Preferably, it is carried out using an additive manufacturing technique (3D printing) to limit costs, with the inner sheath being "printed" onto the outer sheath.

[0098] The material fabrication stage includes a second sub-stage for manufacturing the tubular patterns. Again, this second sub-stage can be carried out by any method known to those skilled in the art. For example, the tubular patterns are manufactured using techniques employed to produce photonic crystal fibers or conventional telecommunications optics. given the similarity of dimensions and materials between the tubular patterns of the waveguides according to the invention and the cores of these conventional optical fibers.

[0099] Finally, the manufacturing stage includes a third sub-step of assembling the tubular patterns and the assembly formed by the inner and outer sheaths. Preferably, this last sub-step is carried out by bonding, with the inner sheath being made using an adhesive, for example, a UV adhesive. This method offers the clear advantage of ease of implementation.

[0100] The manufacturing process of the invention therefore has the advantage of having a physical manufacturing step that can be easily implemented using techniques well known to those skilled in the art.

Claims

Demands 1. A method for manufacturing a robust (1) THz hollow-core waveguide with inhibited coupling, said method comprising a design step and a material fabrication step of the robust waveguide thus designed, said method being characterized in that the design step comprises the following steps: A. Selection of an optimal wavelength op to be guided in the robust waveguide B. Design of a so-called initial THz hollow-core waveguide with inhibited coupling comprising: - a microstructured sheath (GS) comprising a plurality of tubular motifs (MT) arranged in a ring around a core (C) so as to confine at least one radiation at the optimal wavelength  op in said heart - an outer sheath (GE) surrounding the microstructured sheath (GS) C. Design of an inner sheath (Gl) in a first dielectric or metallic material and intended to be placed between the microstructured sheath and the outer sheath of said initial waveguide designed in step B and thus form the robust waveguide in which the tubular patterns are embedded in the inner sheath so as to allow their mechanical retention in the robust waveguide, said design step of the inner sheath (Gl) comprising a substep of optimization of an index and thickness of the inner sheath as a function of a core diameter and the optimal wavelength λ op so that a ratio has r The difference between a linear attenuation coefficient of the robust waveguide and a linear attenuation coefficient of the initial waveguide is less than or equal to 5 and preferably less than or equal to 2 at the optimal wavelength. op .

2. A manufacturing process according to the preceding claim, wherein the physical manufacturing step comprises: Tl D. a first sub-step in the manufacturing of the inner and outer sheaths, E. a second sub-step of manufacturing the tubular patterns, and F. a third sub-step of assembling the tubular patterns and an assembly formed by the inner sheath and the outer sheath.

3. Manufacturing method according to claim 2, wherein the first manufacturing substep is carried out by additive printing.

4. Manufacturing method according to claim 2 or 3, wherein the inner sheath is made in an adhesive, for example a UV adhesive, the third sub-assembly step then being carried out by gluing the inner sheath with the tubular patterns.

5. Robust (1) THz hollow-core waveguide with inhibited coupling comprising: - a microstructured sheath (GS) comprising a plurality of tubular motifs (MT) arranged in a ring around a core (C) so as to confine at least one radiation to an optimal wavelength op in said heart - an outer sheath (GE) surrounding the microstructured sheath (GS), a waveguide formed by said outer sheath and said microstructured sheath being called the initial waveguide, - an inner sheath (Gl) made of a first dielectric or metallic material and disposed between the microstructured sheath (GS) and the outer sheath (GE), the tubular motifs being embedded in the inner sheath so as to allow their mechanical retention in the robust waveguide, a complex index n gi and a thickness e gi the internal sheath being optimized according to a diameter D c of the heart and the optimal wavelength op so that a ratio has r / has £between a linear attenuation coefficient of the robust waveguide and a linear attenuation coefficient of the initial waveguide, either less than or equal to 5 and preferably less than or equal to 2 at the optimal wavelength  O p.

6. Waveguide according to the preceding claim, wherein the inner sheath is a ring having a thickness e gi substantially constant in a plane transverse to the robust waveguide and substantially constant longitudinally.

7. Waveguide according to claim 5, wherein the internal gain is arranged discontinuously longitudinally, for example in a plurality of sections of the same length L separated longitudinally by a distance AL.

8. Waveguide according to any one of claims 5 to 7, wherein a ratio e gi lr tThe ratio between the thickness of the inner sheath and the radius of the tubular patterns is adapted according to the diameter of the core, an absorption coefficient of the material constituting the tubular patterns (MT), and the guidance wavelength so that the ratio a r / has £ be less than or equal to 5 and preferably less than or equal to 2 at the optimal wavelength op .

9. Waveguide according to the preceding claim, wherein the ratio e gi lr t is between 1 and 2 and preferably between 0.05 and 0.

95.

10. Waveguide according to any one of claims 5 to 9, wherein a real part of the index n gi the internal sheath is between 1.4 and 3, preferably between 1.4 and 2 at the optimal wavelength.

11. Waveguide according to any one of claims 5 to 10, wherein an imaginary part of the index n githe internal sheath is between 0.0001 and 0.

015.

12. Waveguide according to any one of claims 5 to 11, wherein a ratio e gi lr t between the thickness of the inner sheath and a radius of the tubular patterns are adapted to allow a reduction of the peaks of strong attenuations of a transmission spectrum of said robust waveguide.

13. Waveguide according to any one of claims 6 to 12, wherein the first dielectric material is silica, glue or plastic.

14. Waveguide according to any one of claims 6 to 13, wherein the microstructured cladding is formed of tubular patterns having a refractive index n t between 1.4 and 2.5, a thickness between 0.2 and 0.3 times the optimal wavelength, and a diameter D tbetween 0.3 and 0.9 times the core diameter, the core diameter being between 6 and 20 times the optimal wavelength (preferably between 8 and 12), the inner cladding being made of silica or a first material having a refractive index n gi with a real part between 1.4 and 2.5 at the optimal wavelength and a ratio e gi lr t between the thickness of the inner sheath and a radius of the tubular patterns being preferably between 0.05 and 0.

75.

15. Waveguide according to any one of claims 6 to 14, wherein a thickness and diameter of the tubular patterns are constant longitudinally and wherein the inner sheath is adapted so that a distribution of the tubular patterns in a plane transverse to the robust waveguide varies longitudinally so that the robust waveguide allows, from a first end of the robust waveguide to a second end of the robust waveguide, a transition from a first intensity distribution of a guided fundamental mode to a second intensity distribution of the guided fundamental mode.

16. Waveguide according to any one of claims 6 to 14, wherein a thickness and diameter of the tubular patterns are constant longitudinally and wherein the inner sheath is adapted so that a distribution of the tubular patterns in a plane transverse to the robust waveguide varies longitudinally so that the diameter of the core of the waveguide varies longitudinally and so that a cross-section of a fundamental mode undergoes a longitudinal homothety.

17. Waveguide according to any one of claims 6 to 16, wherein the thickness e gi is greater than or equal to 0.01 xr t , with r t a range of tubular patterns.

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