1. Field of the Invention
The field of the invention may relate to methods for designing grating patterns with predetermined reflectivity spectra and the fabrication of optical gratings from such designs.
2. Background
In optics, and in particular in integrated optics, it is often desired to have a reflector with a predetermined reflectivity spectrum. One way to realize such a reflector is by designing a grating on a waveguide such that the reflectivity of the grating matches that of the required reflectivity spectrum. A conventional binary superimposed grating (BSG) is a convenient and elegant device for achieving such reflectors. The reflectivity spectrum of the BSG is tailor-designed for a particular application such as highly-selective filter or a tunable laser. In such applications, it is important to achieve high reflectivity at accurately predetermined wavelengths.
The design of the conventional BSG starts with defining a function which is a superposition of spatial sine functions having spatial frequencies equal to the spatial frequencies that the grating is being designed to reflect. The phases and amplitudes of the sine functions are chosen for the best performance of the grating, in a manner that is well-known to those of skill in the art. Hence, the defined function is:
                                          ψ            ⁡                          (              x              )                                =                                    ∑                              n                =                1                            N                        ⁢                                          A                n                            ⁢                              sin                ⁡                                  (                                                                                    k                        n                                            ⁢                      x                                        +                                          φ                      n                                                        )                                                                    ,                            (        1        )            
where kn are the spatial frequencies corresponding to the wavelengths that the grating is being designed to reflect, and An and φn are the amplitudes and phases, respectively, of the sine functions that are selected for the best performance of the grating. FIG. 1(a) illustrates a short section of an example of ψ(x).
Since the continuous profile described by Eq. (1) is difficult to achieve at optical-wavelength scales using conventional methods of micro and nano-fabrication, the design methods of the conventional BSG starts with converting ψ(x) to a binary, or two-level, profile, such as is shown in FIG. 1(b). This is conventionally done by applying a threshold function to ψ(x). Furthermore, since the minimum feature size that can be fabricated is predetermined for any particular technology chosen to fabricate the BSG, the pattern is further distorted to accommodate this constraint, as is shown in FIG. 1(c). The pattern shown in FIG. 1(c) is the pattern that is fabricated as a grating in a waveguide to achieve the reflectivity spectrum defined by the spatial frequencies kn.
The shortcomings of the conventional BSG as described above stem from two factors: (1) the distortion of the exposed pattern due to the minimum feature size available by use of any particular chosen technology and (2) from the fact that the feature size varies along the waveguide. The first factor contributes to the shift of the reflected spectral lines and line broadening, and is introduced at the design stage. The second factor contributes to line broadening and shift due to aspect-ratio dependent etching, in which narrow features typically etch slower than wider features in a diffusion-limited etch process. This results in a variable etch depth and consequently inconsistent effective index of the etched regions, which compromise device performance.