As is known in the art, high data rate optical fiber communication is achieved by subdividing the transmission spectrum of the fiber into many separate channels and then transmitting data rates compatible with electronic devices on each channel. This process is known as wavelength division multiplexing (WDM) and requires the use of optical filters, known as wavelength division multiplexers, in order to combine the lower data rate signals at the fiber input and to separate the data signals at the output. The current state of the art in WDM technology is what is commonly referred to as an arrayed waveguide grating (AWG). AWGs exhibit a competitive advantage over traditional thin film filters and bulk gratings since the AWGs are integrated on a chip, and are thus considerably smaller, more stable, and provide a greater degree of functionality.
In spite of the advantages listed above, AWGs have some limitations. Primarily, AWGs occupy large chip areas (>>10 cm2) and provide only moderate spectral efficiency. Integrated resonant devices such as ring resonators and photonic bandgap (PBG) or standing wave resonators occupy much smaller areas (<<10−3 cm2) and hold potential for greatly increased spectral efficiency. Resonators are essential optical devices for many other important optical components including lasers and nonlinear switches. A resonant cavity is characterized by its modal volume V and its quality factor Q where Q is a dimensionless lifetime, the time for the mode energy to decay by e−2π. Many useful devices require cavities that exhibit a very large Q and small volume V.
Three dimensional photonic bandgap (PBG) structures with singular defect sites have been shown to exhibit infinite quality factors while occupying small modal volumes. Similarly, the theoretical Qs of ring resonators can be very high. However, on account of the difficulty in fabricating the complicated geometries associated with three-dimensional PBG structures and ring resonators, considerable interest has been directed toward the simple one-dimensional case. Yet, upon reducing the dimensionality, PBG structures have been shown to radiate, which results in vastly diminished Qs, thereby limiting the devices' utility and effectiveness.
In view of the foregoing it would be desirable to provide integrated optic resonators that either minimize or totally eliminate radiation in one-dimensional PBG structures. It would be further desirable that the integrated optic resonators are manufacturable by planar fabrication techniques.
With the foregoing background in mind, it is an object of the present invention to provide a device including a plurality of waveguides wherein adjacent waveguides are mode-matched to each other through adjustments of the waveguides' core permittivities and the waveguides' cladding permittivities in order to minimize junction radiation and provide an optical cavity having a large quality factor Q and small modal volume V.
The invention will be better understood by reference to the following more detailed description and accompanying drawings in which:
One-dimensional integrated PBG structures are typically formed by imposing reflecting boundaries at λ/4 separations or multiples of λ/2 thereof. In the center of the device, a defect or shift in the structure is introduced of length λ/4 or multiples of λ/2 thereof. Such structures are not limited to quarter-wave layers with half-wave defects but may be formed in a multitude of geometries. Initially, such structures were formed from gratings etched into the waveguides. More recently, such structures have been formed by etching holes into the waveguide cores. In either case, the disruption of the guide results in substantial radiation.
Referring now to
Maintaining a small device size while ensuring a substantially large Q is achieved by the present invention by eliminating radiation at the junction interfaces while still maintaining substantial reflection. When a waveguide mode encounters an interface, there are normally radiation losses. However, if the guided mode in one section can be expressed purely as a linear combination of the forward and backward guided modes of the other section, there will be reflections without scattering or radiation.
A simple mode-matching proof between a pair of waveguide sections such as any two sections in
(∇t2+μ0εjω2−β2)E=0 (1)
where ∇t2=∇2−∂2/∂z2 denotes the transverse Laplacian. Similarly, for {tilde over (E)},{tilde over (β)},{tilde over (ε)}i in the second waveguide. Since the magnetic and transverse electric fields must be continuous across the junction, the transverse mode profiles ET and {tilde over (E)}T must be at least component-wise proportional if the field solutions are to be composed solely of guided modes. The wave equation for each subsection must be the same, and thus εiω2−β2/μ0={tilde over (ε)}iω2−{tilde over (β)}2/μ0. This implies:
ε1−ε2={tilde over (ε)}1−{tilde over (ε)}2 (2)
This condition is not compatible with the condition of continuity on {circumflex over (n)}·(εiEi) where {circumflex over (n)} is the normal to the waveguide wall, except in the case of εi={tilde over (ε)}i or when the normal component of the Ei is zero. Only TE (transverse electric, i.e. an electric field purely parallel to the waveguide walls) modes may satisfy both of these conditions simultaneously. It remains to be shown that a superposition of these guided modes satisfies the boundary conditions entirely. For this, we must include the magnetic field, whose transverse components are given by Faraday's Law in the TE case:
As a trial solution we consider a superposition of a forward and a backward propagating mode in the left-hand waveguide and a single forward propagating mode in the right-hand waveguide. At the boundary, it is necessary and sufficient that the transverse field profiles be continuous, and thus
ET(1+r)=t·{tilde over (E)}T (4)
HT(1=r)=t·{tilde over (H)}T (5)
where |r|2 and |t|2 are the reflection and transmission coefficients. By applying Eq. (3) to Eq. (5) and solving for r, given from above that the transverse electric-field profiles are proportional, we find that all the boundary conditions are satisfied with the usual reflection coefficient:
with the effective indices
That is, the unique solution of Maxwell's equations consists of forward and backward-propagating modes of the normalized amplitudes 1 and r, respectively in the left-hand guide; and a single forward propagating mode in the right-hand guide of normalized amplitude t=1+r. Therefore when (2) is obeyed, and the excited mode is purely TE, all boundary conditions at the junction are necessarily satisfied by guided-mode solutions and the junction is radiation-free. In two dimensions one can always choose the electric field to be TE polarized. In three dimensions, for cylindrical waveguides, the “azimuthually polarized” TE0m are purely TE: their polarization is everywhere directed along {circumflex over (φ)} (parallel to the walls). Because there are only reflections, the system is effectively one-dimensional and so a quarter-wave stack (thicknesses π/2β and π/2{tilde over (β)}) with a quarter-wave defect can be used to optimally confine light in the axial direction without sacrificing lateral confinement or Q. In fact, the only limitations on the cavity Q will result from the limited number of Bragg layers and the finite extent of the cladding, as well as fabrication imperfections.
A cylindrical cavity with a field propagating in the axial direction is not the only cylindrical geometry that allows for a radiation-free resonator. Alternatively, the Bragg layers may be used to confine a TE mode in the radial direction while total internal reflection is used to confine the mode in the axial direction (e.g.
An FDTD simulation and diagram of a two-dimensional structure that obeys (2) is presented in
T=5.55a, a=0.284λ, ε1=9ε0, ε2=6ε0, and {tilde over (ε)}2=ε0. For the device of
A diagram of an ideal three-dimensional structure with an axially propagating field is presented in
Near perfect high Q structures that more readily lend themselves to fabrication and integration are also highly desirable. One such structure is presented in
Higher order filters may be constructed with any of the aforementioned geometries by simply using a plurality of defect sites.
Active devices may as well be constructed from any of the aforementioned cavity geometries.
Methods for achieving radiation-free and very low radiation optical cavities have been described herein. Importantly, and in contrast to prior designs, low loss cavities are achieved by using four separate permittivities in the cores and claddings of the two waveguide sections that form the resonator building blocks. Principally, this invention allows for the development of very high Q structures and thus well defined filter passbands.
Having described preferred embodiments of the invention it will now become apparent to those of ordinary skill in the art that other embodiments incorporating these concepts may be used. Accordingly, it is submitted that the invention should not be limited to the described embodiments but rather should be limited only by the spirit and scope of the appended claims.
This application claims priority under 35 U.S.C. § 119(e) to provisional patent application Ser. No. 60/335,036 filed Oct. 24, 2001; the disclosure of which is incorporated by reference herein and to provisional patent application Ser. No. 60/332,619 filed Nov. 14, 2001; the disclosure of which is incorporated by reference herein.
This invention was made with government support under Grant Number DMR-9808941 awarded by NSF. The Government has certain rights in the invention.
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