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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 3 — Feb. 2, 2009
  • pp: 1508–1517
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A novel ultra-low loss hollow-core waveguide using subwavelength high-contrast gratings

Ye Zhou, Vadim Karagodsky, Bala Pesala, Forrest G. Sedgwick, and Connie J. Chang-Hasnain  »View Author Affiliations


Optics Express, Vol. 17, Issue 3, pp. 1508-1517 (2009)
http://dx.doi.org/10.1364/OE.17.001508


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Abstract

We propose a novel ultra-low loss single-mode hollow-core waveguide using subwavelength high-contrast grating (HCG). We analyzed and simulated the propagation loss of the waveguide and show it can be as low as 0.006dB/m, three orders of magnitude lower than the lowest loss of the state-of-art chip-scale hollow waveguides. This novel HCG hollow-core waveguide design will serve as a basic building block in many chip-scale integrated photonic circuits enabling system-level applications including optical interconnects, optical delay lines, and optical sensors.

© 2009 Optical Society of America

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

1. H. Ou “Different index contrast silica-on-silicon waveguides by PECVD,” Electron. Lett. 2, 212–213 (2003) [CrossRef]

], 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

2. P. Y. Yu and M. Cardona, Fundamentals of Semiconductors: Physics and Material Properties (Springer, 2001)

]. 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

3. J. N. McMullin, R. Narendra, and C. R. James, “Hollow metallic waveguides in silicon V-grooves,” IEEE Photon. Technol. Lett. 5, 1080–1082 (1993). [CrossRef]

]. 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

5. P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, J. Knight, and P. St. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005). [CrossRef] [PubMed]

]; however, the lowest loss for a chip-based hollow-core waveguide is still high, at ~10 dB/m using DBRs [4

4. Y. Sakurai and F. Koyama, “Control of group delay and chromatic dispersion in tunable hollow waveguide with highly reflective mirrors,” Jpn. J. Appl. Phys. 43, 1091–1093 (2004). [CrossRef]

]. 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

6. C. F. R. Mateus, M. C. Y. Huang, D. Yunfei, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16, 518–520 (2004). [CrossRef]

]. We also showed that its incorporation as the top mirror of a vertical cavity surface emitting laser (VCSEL) [7–9

7. M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nature Photon. 1, 119–122 (2007). [CrossRef]

] as well as a high-Q resonator [10

10. Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16, 17282–17287 (2008). [CrossRef] [PubMed]

].

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

11. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, (Princeton University Press, 1995).

], 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

12. V. Karagodsky, Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, C.A. 94720, Y. Zhou, M.C.Y. Huang, and C. J. Chang-Hasnain are preparing a manuscript to be called “full-wave analysis and design rules for high-contrast gratings.”

]. 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.

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

α(dB/m)=10Llog(PtPi)=10Llog(RN)=10Llog(R)=10tanθDlog(R)
(1)

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

Given that, δ << 1 , log(R)=ln(1δ)ln100.43δ and θ << 1, tan θθ, therefore the waveguide loss can be estimated as

α(dB/m)4.3θDδ

kxD=2πλsinθ·D=;m=1,2,...

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 = /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+tg) and outside the grating (x>D/2+tg). 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.

(Ey)core=mρmcos(kxmx)ei(kz+mkg)z

kxm2+(kz+mkg)2=k02,

where ko 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

(Ey)grating=nAncos(βnx+ϕn)eiΓnz+Bncos(βnx+ψn)eiΓnz

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

(Ey)outside=mτmeikxmxei(kz+mkg)z

The magnetic field can be obtained from the electric field using the relation H=1iωμ×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/kz, 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: tg/λ = 0.258, Λ/λ = 0.423 , η=0.45 and εr =3.62.

4. Design and simulation

Rigorous coupled wave analysis (RCWA) [11

11. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, (Princeton University Press, 1995).

] 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 tg 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. tg 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, tg = 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.

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 3rd mode (2nd lowest odd order mode) is drastically higher, 200 times, than that of the 1st mode.
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.

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.

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

15. H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength”, Appl. Phys. Lett. 80, 416–418, (2002). [CrossRef]

]. 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

16. K. Jackson, S. Newton, B. Moslehi, M. Tur, C. Cutler, J. Goodman, and H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Tech. 33, 193–204, (1985). [CrossRef]

].

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.

kx=ksinθ=2πsinθ/λ,kxDx=;m=1,2,...
ky=ksinφ=2πsinφ/λ,kyDy=;n=1,2,...

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, tg = 460 nm and η = 46%. The bottom HCG sits on 2.5 µm SiO2 and its dimensions are Λ = 570 nm, tg = 390 nm and η = 46%, whereas for left and right HCGs, Λ = 622 nm, tg = 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

17. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

]. 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 Dx and Dy, respectively.

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

18. B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. mat. 4, 207–210 (2005). [CrossRef]

].

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.

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 neff = kz/ko. 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 Ey(y) is then multiplied by the field profile Ey(x) obtained using analytical solution to get the full transverse field profile Ey(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%.

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.

References and Links

1.

H. Ou “Different index contrast silica-on-silicon waveguides by PECVD,” Electron. Lett. 2, 212–213 (2003) [CrossRef]

2.

P. Y. Yu and M. Cardona, Fundamentals of Semiconductors: Physics and Material Properties (Springer, 2001)

3.

J. N. McMullin, R. Narendra, and C. R. James, “Hollow metallic waveguides in silicon V-grooves,” IEEE Photon. Technol. Lett. 5, 1080–1082 (1993). [CrossRef]

4.

Y. Sakurai and F. Koyama, “Control of group delay and chromatic dispersion in tunable hollow waveguide with highly reflective mirrors,” Jpn. J. Appl. Phys. 43, 1091–1093 (2004). [CrossRef]

5.

P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, J. Knight, and P. St. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005). [CrossRef] [PubMed]

6.

C. F. R. Mateus, M. C. Y. Huang, D. Yunfei, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16, 518–520 (2004). [CrossRef]

7.

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nature Photon. 1, 119–122 (2007). [CrossRef]

8.

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nature Photon. 2, 180–184 (2008). [CrossRef]

9.

Y. Zhou, M. C. Y. Huang, and C. J. Chang-Hasnain, “Tunable VCSEL with ultra-thin high contrast grating for high-speed tuning,” Opt. Express 16, 14221–14226 (2008). [CrossRef] [PubMed]

10.

Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16, 17282–17287 (2008). [CrossRef] [PubMed]

11.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, (Princeton University Press, 1995).

12.

V. Karagodsky, Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, C.A. 94720, Y. Zhou, M.C.Y. Huang, and C. J. Chang-Hasnain are preparing a manuscript to be called “full-wave analysis and design rules for high-contrast gratings.”

13.

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981). [CrossRef]

14.

Y. Zhou, M. C. Y. Huang, and C. J. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high-contrast grating,” IEEE Photon. Technol. Lett. 20, 434–436 (2008). [CrossRef]

15.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength”, Appl. Phys. Lett. 80, 416–418, (2002). [CrossRef]

16.

K. Jackson, S. Newton, B. Moslehi, M. Tur, C. Cutler, J. Goodman, and H. J. Shaw, “Optical fiber delay-line signal processing,” IEEE Trans. Microwave Theory Tech. 33, 193–204, (1985). [CrossRef]

17.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

18.

B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. mat. 4, 207–210 (2005). [CrossRef]

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(230.7370) Optical devices : Waveguides

ToC Category:
Diffraction and Gratings

History
Original Manuscript: November 7, 2008
Revised Manuscript: December 21, 2008
Manuscript Accepted: December 30, 2008
Published: January 26, 2009

Citation
Ye Zhou, Vadim Karagodsky, Bala Pesala, Forrest G. Sedgwick, and Connie J. Chang-Hasnain, "A novel ultra-low loss hollow-core waveguide using subwavelength high-contrast gratings," Opt. Express 17, 1508-1517 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-3-1508


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