1. Introduction

The ability to generate long optical delays with low intrinsic loss is useful for a wide range of applications, including optical signal processors, RF filtering, optical buffers, and optical sensing. Optical fibers have been used for these applications with advantages such as ultra-low loss, dispersion and nonlinearity, and an exceedingly large bandwidth. However, they are bulky, heavy, and lack manufacturing scalability. Lithographically defined, chip-scale waveguides have been reported in SiO₂/Si and III–V material systems. They are desirable because they are compact, light-weight, and can be integrated with other optoelectronic devices. The lowest reported loss achieved to date in chip-based waveguides is on the order of 1 dB/m [1], three to four orders of magnitude higher than that of optical fibers. This loss is unacceptably high for most applications requiring 0.01 dB/m. The fundamental reasons for the high losses are: direct band-edge absorption, free carrier absorption, and absorption due to interaction with optical phonons [2]. In addition, these devices are expected to have high nonlinearity and dispersion.

Hollow-core waveguides (HW) are highly promising for achieving fiber-like ultra-low loss, nonlinearity and dispersion because of the elimination of the core material. There have been advances in hollow-core waveguides, ranging from waveguides using a metallic shell, to ones using a distributed Bragg reflector (DBR), to ones with photonic crystals (PhCs), etc [3–5]. The basic principle is to guide the optical beam propagating through air by multiple reflections at the cladding mirrors. A hollow-core PhC optical fiber has shown an extremely low loss of ~0.001 dB/m [5]; however, the lowest loss for a chip-based hollow-core waveguide is still high, at ~10 dB/m using DBRs [4]. The major loss of the waveguide comes from insufficient reflectivity of the cladding DBR mirrors. Ultrahigh reflectivity is essential to achieve ultra-low loss hollow waveguides. Recently, we proposed a novel subwavelength high-index-contrast grating (HCG) as a broadband reflector with very high reflectivity for surface-normal incident light [6]. We also showed that its incorporation as the top mirror of a vertical cavity surface emitting laser (VCSEL) [7–9] as well as a high-Q resonator [10].

In this paper, we propose a novel ultra-low loss hollow-core waveguide structure using HCGs as the high reflectivity cladding to reflect light at a small glancing angle. We show that HCGs with periodicity parallel to the direction of propagation can confine light in the waveguide. Instead of causing the backward wave reflection normally expected from traditional periodic structures, the HCG forms a high reflectivity glancing incidence mirror for the guided wave. At the first glance, this structure may seem quite similar to a photonic crystal slab waveguide (PhC-SW) [11], but with only one crystal period in each direction. However, the guidance principle in a HCG waveguide structure is quite different than a traditional photonic crystal slab waveguide. It is well known that the light in a PhC-SW is confined to the core due to the photonic band gap arising from constructive interference of distributed reflections from each periodic layer. In fact, the concept of photonic bandgap stems from the distributed reflections from multiple periods and hence, one single period can never provide enough reflection or, in this case, guidance. In the case of a high-contrast grating hollow core waveguide, the confinement is due to the constructive interference of multiple grating harmonics in a sub-wavelength periodic structure [12]. This is a totally unexplored concept in guided-wave optics: propagation parallel to the direction of periodicity of a periodic structure using just one layer of HCG on each side to provide lateral confinement. We show an HCG hollow-core slab waveguide design with an exceedingly low propagation loss (>0.01 dB/m) using both rigorous coupled wave analysis (RCWA) and finite-difference time-domain (FDTD) numerical simulation. In addition, we show a potential 2D design with loss estimated to be less than 0.01 dB/m.

2. High contrast grating and hollow-core waveguide

 

Fig. 1. (a) Schematic of high contrast grating (HCG). High index gratings (blue) are surrounded by low index material, typically air. The incident angle, θ, is measured from the plane of the grating. High reflectivity in a sub-wavelength grating (Λ < λ) can be achieved by proper choice of grating parameters (b) Schematic of a 1D HCG hollow-core slab waveguide structure consisting of two reflecting HCGs. (c) Ray optics model for guided mode in a hollow-core slab waveguide. Two reflective surfaces extend infinitely in the y-z plane and light propagates in the z direction. The spacing D between the planes forms the core of the waveguide. Light within the core can be expressed via a plane wave expansion, where the plane waves within the core are characterized by the wave vector k.

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The schematic of a HCG structure is shown in Fig. 1(a). High index gratings (blue) are surrounded by low index material, typically air or oxide. The gratings are typically made of semiconductors and have a high refractive index >3. There are three physical parameters which control the reflectivity of the grating: period (Λ), thickness (t_g), and duty cycle (η). The thickness and period are subwavelength. Duty cycle is defined as the ratio of the width of the high index material to Λ. The incident angle of the light, θ, is measured from the plane of the grating. When a light beam is incident on a periodic grating, the light is reflected and transmitted into multiple diffraction orders. However, when the period of the grating is less than the wavelength (Λ < λ), all higher order modes are evanescent in the air expect the zeroth order mode. When the grating parameters are optimally designed, destructive interference between the directly transmitted wave and the Bragg transmitted wave leads to extremely high reflectivity. Due to the high index contrast between the gratings and their surroundings broadband high reflectivity is possible. In our previous study, we showed that when light is incident on the grating in surface-normal condition, i.e. θ=90°, the grating exhibits a very broadband high reflectivity [6]. In this study, we choose the grating parameters such that the high reflectivity is achieved at glancing angles.

In this work, we use HCG as a hollow-core waveguide cladding with extremely high reflectivity. The schematic of a 1D HCG-HW slab waveguide structure is shown in Fig. 1(b). It consists of two reflecting HCGs that are periodic in the z direction and infinite in y directions. The two HCGs are spaced a distance D apart. In this study, we consider the input light is launched in z direction with TE polarization (electric field parallel with the grating fingers). Light in the core of the waveguide can be expanded into a series of plane waves bouncing between the reflecting HCG planes at different angles. Due to the high reflectivity of the HCG, a simple ray-optics model may be used to estimate the propagation properties of the HCG-HW as shown in Fig. 1(c). A ray which propagates through a HW of size D at angle θ travels a distance D/ tan θ per bounce. At each bounce, there is a loss of δ=1-R, where R is the reflectivity of the HCG at angle θ. Assuming there is no material loss in the hollow waveguide core (air), the optical propagation loss per unit length over distance L is

where N is the number of bounces in distance L and N = L tan θ/D.

Given that, δ << 1 , $\log \left(R\right)=\frac{\ln \left(1-\delta \right)}{\ln 10}\approx -0.43\delta$ and θ << 1, tan θ ≈ θ, therefore the waveguide loss can be estimated as

For a propagating mode, the round-trip phase shift incurred by the wave from crossing the waveguide in the transverse direction is a multiple of 2π. A discrete set of angles satisfies this condition and forms the modes of the waveguide. Provided that there is no significant phase shift associated with the reflections, the modal angles are

Thus the modes of a HW are determined only by D. Each mode can be described in terms of its characteristic angle θ. For small θ and large D, θ ≅ sin θ = mλ/2D and the propagating loss is inversely proportional to the square of D.

3. Analytical treatment of high-contrast grating hollow-core waveguides

The purpose of this section is to present an analytic formulation for propagating modes in the waveguide and to compare this formulation with the ray-optics approximation. Due to the symmetry of the HCG waveguide, the modes can be classified as even and odd modes. The solution methodology is as follows. Assume the position of the hollow core center is x=0. First, we decompose the waveguide into three regions: inside the core (x<D/2), the grating region (D/2<x<D/2+t_g) and outside the grating (x>D/2+t_g). The electric and magnetic fields for a TE mode are written using Maxwell’s equations in each of the regions. By matching the boundary conditions at the interface, we arrive at a set of homogeneous equations relating the field amplitudes in each region. Axial propagation constant and the mode profiles can be solved by looking for solutions when the determinant of the homogeneous system becomes zero.

As mentioned earlier, when the light is incident on a periodic grating, it excites multiple diffraction orders. Hence, inside the core region, the solution can be represented as an infinite set of plane waves differing by a Bragg wave number.

where the Bragg wave number k_g= 2π/Λ, k_xm and k_z are the lateral and axial wave numbers respectively related by the expression

where k_o is the free space wavenumber.

In the grating region, the field is given by the sum of infinite number of harmonics in both the high index and low index regions. These solutions are of the form

Periodic boundary conditions are applied to relate the wave numbers in the grating region (β_n, Γ_n) to the bloch wavenumber k_z. The electric field outside the waveguide region is similar to that inside the core comprising of multiple-diffraction orders

The magnetic field can be obtained from the electric field using the relation $H=\frac{1}{\mathrm{i\omega \mu }}\nabla \times E$. The relation between the field amplitudes can be obtained by matching the boundary conditions for both the electric and magnetic fields at the boundaries. This yields a system of homogenous equations. For this system to have a non-zero solution, it must be singular (i.e. determinant of the coefficient matrix is zero). Each such singularity corresponds to a guided mode.

The simplest way to describe each guided mode is by representing it with a characteristic angle θ = arctan (k _x0/k_z, which can be interpreted as the angle of incidence of the wave upon the grating. The high reflectivity of the grating walls leads to an intuitive conclusion that the relation between the incidence angle θ and the waveguide spacing D can be approximated using a ray optics approach, namely sin θ = λM/2D, wherein M=1,2,… is the mode number. This conclusion can be validated by Fig. 2(a), which shows a comparison between the θ - D relation derived from the analytical solution described in this section and the ray optics approximation described above. The symmetric cosine modes correspond to even values of M while the anti-symmetric sine modes correspond to odd values of M. The comparison between the mode profiles obtained analytically to the ray optics approximation is presented in Fig. 2(b).

 

Fig. 2: Excellent agreement is obtained between ray optics and full wave analysis. (a) Relation between the waveguide spacing D and the angle of incidence θ, obtained by the analytical solution is compared with the ray optics approximation. (b) Symmetric mode profile of the fundamental mode obtained by the analytical solution for D =15 um and compared to the ray optics approximation. The data for both figures corresponds to: t_g/λ = 0.258, Λ/λ = 0.423 , η=0.45 and ε_r =3.6².

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4. Design and simulation

Rigorous coupled wave analysis (RCWA) [11] is used to obtain the HCG reflectivity spectra at different incident angle θ, and loss is calculated using Eq. (1). Fig. 3(a) shows a waveguide loss contour plot (dB/m) vs. HCG parameters Λ and s, for a HW with a core size D of 15 μm (corresponding to θ=3°) at λ = 1.55 μm with t_g fixed at 410 nm. The loss of 0.1dB/m and 0.01dB/m are shown by the black and white contours, respectively. Fig. 3(b) shows loss contour plot vs. t_g and s for the case when Λ is fixed at 665 nm. Once again the loss of 0.1dB/m and 0.01dB/m are shown by the black and white contours, respectively. These figures show reasonable tolerance of variations is obtained to accommodate fabrication imperfection.

One fascinating aspect of HCG is the ability to design a large spectral width despite the stringent reflectivity requirement. The following grating parameters are used to achieve a broad spectral width: Λ = 730 nm, t_g = 1.04 μm, η = 65%, high index grating n= 3.6 and low index air n=1. Fig. 3(c) shows the fundamental mode propagation loss of a 15 μm core HCG-HW as a function of wavelength. Large spectral widths of 130 nm and 80 nm can be achieved for loss requirements of 0.1dB/m and 0.01dB/m, respectively.

 

Fig. 3. Calculation of waveguide loss α (dB/m) at 1.55 um for a 15-μm core slab HCG-HW as a function of (a) grating period and semiconductor width (b) thickness and semiconductor width (c) wavelength. 0.1dB/m and 0.01dB/m lines are labeled in the plots.

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As the HW loss is inversely proportional to D ², it is desirable to design a large core waveguide. However, a large D leads to a large number of modes, which may be detrimental. The HCG-HW offers a unique advantage: simultaneously having a large core and maintaining a single polarization and transverse mode. This is because of the HCG reflectivity angular and polarization dependence, thus we can design the HCG to yield higher loss for larger θ, i.e. higher order modes, such that they are preferentially filtered out. Figure 4 plots α vs. incident angle for D=15μm, with the first four modes marked on the curve. The loss for the fundamental mode is as low as 0.0026dB/m. The 2^nd order mode loss is also rather low at 0.057dB/m. However, choosing launching conditions with symmetric transverse field components would excite only the odd order modes, so that the next mode is the 3^rd mode, whose loss is 200 times higher than that of the 1^st mode. With this angular loss dependence, HCG-HW can simultaneously achieve low loss for the fundamental mode and high suppression towards higher order modes.

Next, finite-difference time-domain (FDTD) numerical simulation is used to calculate the propagation loss of the HCG-HW. Figure 5(a) shows the 2D electric field intensity profile along the propagation direction of an HCG-HW with a 15μm core size. In this simulation, TE-polarized light with a mode profile matching the fundamental mode is launched into a 2 mm long HCG-HW. The spectral width of the launched light is about 4 nm. As shown in Fig. 5(a), the light is guided inside the hollow core, and the field intensity outside the HCG-HW is only 10^-8 of the intensity at the center of the hollow core. This agrees with the δ~10^-8 result which is obtained by using RCWA. Due to simulation boundary conditions and the abrupt cutoff of the HCG walls at the start and end of the waveguide, only the middle 0.8 mm is used for calculating the waveguide propagation loss. Figure 5(b) shows the normalized Ey intensity at the center of the hollow core as a function of waveguide length z. Since we are looking for a less than 10^-5 intensity drop over the waveguide length, high frequency noise from the simulation is removed from the data and linear regression is used to extract the loss from the data, as plotted in the red dash line in Fig. 5(b). A waveguide propagation loss of 0.006±0.0024dB/m with 95% fitting confidence bounds is obtained. Currently the 2 mm simulated waveguide length is limited by our computational power. A more accurate loss number can be achieved by simulating longer waveguide length.

 

Fig. 4. Propagation loss of 1D slab HCG-HW vs θ for the first four TE modes in a 15 μm HCG-HW. Due to symmetry, one can avoid launching into the second order mode. Hence the difference between the first and third order modes is the most important for modal screening. In this case, the loss of 3^rd mode (2^nd lowest odd order mode) is drastically higher, 200 times, than that of the 1st mode.

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Fig. 5. (a) Simulated electric field intensity profile of HCG-HW. Color is labeled in log scale. The field intensity outside the HCG-HW is only 10^-8 of the intensity at the center of the hollow core. (b) Normalized electrical field at the center of the HCG-HW as a function of waveguide length. Linear regression is used to fit the curve as plotted in the red dash line. A waveguide propagation loss of 0.006±0.0024dB/m with 95% fitting confidence bounds is obtained.

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While other highly reflective structures, such as DBR or PhC, require multiple layers to achieve a high reflectivity, the HCG structure can achieve a very high light confinement in a HW by using a single layer of cladding grating, Because of this, the field penetration depth in a HCG-HW cladding is significantly reduced compared to DBR or PhC based HW. In addition, as shown in the FDTD simulation results, the field intensity inside grating is very low, only 10^-4 to 10^-3 of the intensity at the center of the waveguide core. Using the field profile data from FDTD simulation, we integrate the total field energy inside the hollow-core as well as the total energy inside the HCG cladding layer. The result we get shows that the energy inside HCG layers is only 10^-7 to 10^-6 of the total energy inside waveguide core, a much lower number than the ones in other HW structures.

For a fabricated waveguide structure, it has been long recognized that the scattering loss from interference roughness and fabrication imperfection can degrade the waveguide performance and introduce scattering loss. Scattering loss is can be estimated using Rayleigh scattering cross-section as α(dB/m) = 4.3(S_inc/P_tot)σ_RρS, where S_inc is the incident Poynting vector at the vicinity of the scattering centers, P_tot is the total power carried through the waveguide and ρ _S is the surface density of the scattering centers, and σ _R is the Rayleigh scattering cross-section. The scattering loss for HCG-HW is expected to be very low because the power near and inside the reflecting HCG is almost zero (only 10^-7 to 10^-6 of the total power), which means that S _inc is extremely small. In addition, the small penetration depth of HCG also provides less chances for scattering centers on HCG surface to interact with the field, which also results in a lower scattering loss. In our previous experimental study on HCGs with surface normal incident angle [14], we show that HCG structure can be extremely robust against fabrication imperfection.

In data communication, nonlinearity can severely degrade signals in both analog and digital system-level applications of low loss waveguides. HCG-HW can significantly reduce nonlinear effects in data transmission, since the optical power is tightly confined in “linear” air, which means that a very small fraction of the optical field interacts with the grating material. From FDTD simulation results, the field intensity located in the high-index solid region is only 10^-7 to 10^-6 of the intensity in the core of the waveguide. Therefore, we estimate that the overall reduction in the effective nonlinear coefficient per unit length is ~10⁶ to 10⁷ lower than in regular silicon waveguides, which means the HCG-HW can handle a higher power over a longer distance, thereby dramatically increasing the overall system performance.

The dispersion of a waveguide consists of material dispersion and waveguide dispersion. For a chip-scale silicon waveguide, the dispersion is dominated by the material dispersion, which is as high as 1000 ps/nm/km [15]. However, for HCG-HW, from the FDTD simulation result, the field intensity located in the high-index silicon region is only 10^-7 to 10^-6 of the intensity in the core of the waveguide. Therefore, the material dispersion contribution is less than 10^-3 ps/nm/km, which can be ignored. The dispersion relationship of the HCG-HW can be calculated by ray optics as well as our analytical solution. For the fundamental mode in a HCG-HW with D=15 μm at 1.55 μm, we get the dispersion parameter d of 5.8 ps/nm/km using ray optics estimation and 8.2 ps/nm/km using the analytical solution. This dispersion number indicates that the proposed HCG-HW can support a > 2 THz RF modulation over a length of 5 m [16].

For optical delay line applications, temperature variations will lead to index changes in conventional waveguides, and hence result in degradation of delay phase precision. Given that the optical field penetration into the HCG is only 10^-7 to 10^-6, the HCG-HW will be highly robust against temperature variation. Assuming a typical index change coefficient of 10^-4 per degree in the semiconductor part of HCG, the change of group velocity can be calculated based on the analytical formulation. The result shows that a change of 80°C over 150 m would translate into 50 fs time delay precision or 0.05% of 2π in phase shift for a 10 GHz signal, an extremely high 10^-8/°C total phase delay precision.

5. 2D HCG hollow-core waveguide

2D confined HCG-HW can be designed using a rectangular waveguide structure as shown in Fig. 6(a). RCWA and ray optics are used to optimize the dimensions of the HCG cladding, as shown in Fig. 6(b). Compared to just one characteristic angle in the 1D HCG-HW, the 2D HCG-HW cladding grating has two incident characteristic angles, θ and φ, which are the angle between incident beam and y-z plane and the angle between incident beam and x-z plane, respectively. For specific θ and φ, high reflectivity can be obtained by optimizing the grating dimensions using RCWA. Figure 6(c) shows a round trip ray trace looking along the z axis direction.

where (m,n) are mode numbers. Similar to the method used in 1D HCG-HW loss calculation, we can use ray optics and RCWA simulation to estimate the loss of 2D rectangular HW-HCG for a different mode (m,n). This mode is quite similar in its “threading through” waveguide nature to the HE modes in fiber. To achieve high reflectivity with all 4 cladding gratings, the dimensions of the top and bottom HCGs are designed to be different from those of the left and right HCGs. For top HCGs, Λ = 674 nm, t_g = 460 nm and η = 46%. The bottom HCG sits on 2.5 µm SiO₂ and its dimensions are Λ = 570 nm, t_g = 390 nm and η = 46%, whereas for left and right HCGs, Λ = 622 nm, t_g = 782 nm, η = 80%. This structure only supports the low loss propagation for one mode and one polarization, in this case, HE (1,1) mode where the majority of E-field components are along y direction. For a 25μm by 25μm hollow-core waveguide with these HCG claddings, we estimate that the loss of the fundamental mode (1,1) is ~0.009 dB/m. To couple light into such a waveguide, a tapered dielectric waveguide or a beam expander can be used to match the fundamental mode and maximize the coupling efficiency [17]. Further optimization of the 2D confined HCG-HW design with even lower loss is still under investigation.

 

Fig. 6. (a) Schematic of a 2D rectangular HCG-HW. (b) Schematic of HCG cladding grating in 2D rectangular HCG-HW. There are two incident characteristic angles θ and φ, which are the angle between incident beam and y-z plane and the angle between incident beam and x-z plane, respectively. (c) A round trip ray trace looking along the z axis direction. The sizes of the hollow-core in x and y direction are D_x and D_y, respectively.

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As mentioned earlier, achieving 2D confinement requires different grating designs for top/bottom and left/right cladding layers which might be difficult to fabricate. Here we propose a novel 2D waveguide that can be fabricated very easily and cost-effectively using 1D waveguides in a planar geometry by simply varying the grating dimensions to achieve lateral confinement. This methodology is similar to what has been referred to as photonic heterostructure [18].

 

Fig. 7. Schematic of the hollow-core waveguide based on photonic heterostructure geometry. Light is confined vertically due to the high-contrast gratings while horizontal confinement is achieved due to effective index difference between core and cladding regions. Effective index method is used to analyze the structure in the lateral direction (y) as shown in the right figure.

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Lateral confinement in a hollow-core waveguide can be achieved by sandwiching the low loss waveguide amid two waveguides with lower effective index in a heterostructure geometry as shown in Fig. 7. Effective index is defined as n_eff = k_z/k_o. In our study, we varied the grating period to change the effective index. Effective index for the core and the cladding regions is calculated using exact analytical formulism described earlier. Once we obtain the effective index for the core and cladding regions, the transverse mode profile can be calculated using the effective index method. In this formulation, we treat each of the regions as an equivalent slab waveguide to obtain the field profile in the y direction. The field profile E_y(y) is then multiplied by the field profile E_y(x) obtained using analytical solution to get the full transverse field profile E_y(x,y). Figure 8 shows the transverse field profile for a heterostructure waveguide with a height of 3.2 um and a width of 4.65um. The grating period in the core and the cladding region is chosen as 0.698 um and 0.3875 um respectively. In this case, we obtain an effective index difference of 1% between the core and the cladding regions.

 

Fig. 8. Transverse intensity profile calculated using effective index method for a heterostructure waveguide with a height of 3.2 um and a height of 10 um. By changing the grating period in core and cladding regions, we obtain an index difference of 1% corresponding to an optical confinement factor of 78%.

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6. Conclusion

We presented an ultra-low loss hollow waveguide structure using high contrast subwavelength grating. A waveguide propagation loss less than 0.01dB/m was obtained by both rigorous coupled wave analysis and finite-difference time-domain numerical simulation. This loss value is three orders of magnitude lower than the lowest loss in the current chip-scale hollow waveguides. We also showed examples of HCG-HW designs with an extremely broad bandwidth (~100nm) and a large fabrication tolerance. In addition, due to the unique angular and polarization dependence of the HCG reflection spectrum, HCG-HW enables single-mode operation with a core that is one order of magnitude larger than conventional waveguides. Finally, HCG-HW can significantly reduce nonlinear effects and dispersion in data transmission, and it is insensitive to temperature variation. With all these desirable attributes, HCG-HW can be an ideal candidate for on-chip optical communication, compact low loss optical delay lines, and interferometric sensors.