Abstract:
According to embodiments of the invention, the design and fabrication of a binary superimposed grating (BSG) results in better performing devices that may be fabricated using existing technology. The fabrication process includes forming grating features based upon repeating features of the desired superposition function. The design process also relaxes the processing requirement for equivalently performing devices.

Description:
PRIORITY 
     Priority is claimed to U.S. provisional application No. 61/531,783, filed Sep. 7, 2011, the disclosure of which is incorporated herein by reference in its entirety. 
    
    
     BACKGROUND OF THE INVENTION 
     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 k n  are the spatial frequencies corresponding to the wavelengths that the grating is being designed to reflect, and A n  and φ n  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 k n . 
     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. 
     SUMMARY OF THE INVENTION 
     Embodiments of the invention may provide a method of designing a grating pattern with predetermined reflectivity spectrum such that the lines of the grating have uniform width or uniform depth, and their placement is such that the required reflectivity spectrum is achieved. Other embodiments of the invention may provide methods of fabricating gratings from such grating patterns. 
     The fabrication method may include the steps of: 
     1. determining placement for grating features according to a position of repeating features in a desired superposition function; and 
     2. fabricating the grating using a fabrication tool to form the grating features on or as part of a substrate according to the determined placement, wherein the grating features have at least one of a uniform width and a uniform depth 
     Advantages of the improvements will be apparent from the drawings and the description of the preferred embodiment. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In the drawings, wherein like reference numerals refer to similar components: 
         FIG. 1  schematically illustrates steps of creating a grating pattern from a superposition function according to the prior art; and 
         FIG. 2  illustrates an optical waveguide according to an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Embodiments of the current invention may provide new methods for fabrication of BSGs. In a first embodiment, the repeating features are the positions of local minima of the function ψ(x), as shown in  FIG. 1( a ) , may be used to place features of identical widths. The width of the grating features may be determined by the minimum feature size of the particular technology used to fabricate the device. Since the feature placement accuracy of many technologies that may be used to create the grating features is typically much better than the minimum feature size, unlike methods of the prior art, the quality of the BSG that is fabricated may be dependent upon the placement accuracy of the grating features instead of the accuracy of the feature size. For example, in electro-beam (e-beam) lithography, the minimum feature size that can be reliably realized is about 50 nm, limited by the spot size, accelerating voltage, resist thickness, resist type, etc. Feature placement, on the other hand, can be as accurate as 0.1 nm if a high quality beam deflection system is used. 
       FIG. 2  shows an example of an optical waveguide  11  with a grating  13  incorporated into one surface. The grating has grating features  15  which have identical widths and depths, with the placement of the grating features  15  being determined according to the process described above. 
     An additional benefit is that with the approach described above, the demands placed on the lithography in terms of resolution and the minimum grating feature size may be relaxed as compared to the conventional approach. This is because the exposed grating features in the present method can be considerably wider than the minimum grating feature size in the conventional design required to obtain comparable device performance. For example, in the conventional design, it may be necessary to expose grating features as small as 30 nm, and integer multiples thereof, in order to obtain a certain level of device performance. In the present method, 100 nm grating features placed with 1-nm accuracy may provide a considerably better device performance than the conventional design. 
     A lithography tool that may be used to realize structures described herein is a fixed beam moving stage (FBMS) module as part of an electron-beam lithography system, such as those available from Raith USA, Inc., of Ronkonkoma, N.Y. With this fabrication equipment, grating features can be placed with sub-nanometer precision over many millimeters of length, and the placement precision is far better than the minimum grating feature size. Other tools that are capable of creating or fabricating the grating features of the BSG may also be used. 
     The available precision in grating feature placement may yield greater precision in determining the reflected wavelength, which in turn may result in reduced spectral shift and minimized line broadening. Furthermore, since the etched regions may be fabricated having uniform widths for the grating features, the resulting etch depth may be more uniform, which may further improve the line width of the reflected spectrum as compared to the conventional design of BSGs. 
     While the example of a BSG was used to make the discussion more concrete, other approaches may be used to define the position of the features based on the defining superposition function. For example, local maxima may be used instead of local minima. Those skilled in the art will recognize that other features of the superposition function or a different scheme entirely may be used for determining feature placement. 
     The method described herein may allow the high precision of grating feature placement to be exploited. For a given fabrication technology, the grating feature placement precision is typically finer, and depending upon the technology, finer by at least an order of magnitude, than the minimum grating feature size that can be realized. At the same time, in the method described herein, the smallest grating feature may be as small as the minimum width the particular fabrication tool is capable of, which can be relatively large, thereby relaxing fabrication requirements. 
     The foregoing description of the invention illustrates and describes embodiments of the present invention. It is to be understood that the invention is capable of use in various other combinations, modifications, and environments and is capable of changes or modifications within the scope of the inventive concept as expressed herein, commensurate with the above teachings and/or the skill or knowledge of the relevant art. The embodiments described hereinabove are further intended to explain best modes known of practicing the invention and to enable others skilled in the art to utilize the invention in such, or other, embodiments and with the various modifications required by the particular applications or uses of the invention. Accordingly, the description is not intended to limit the invention to the form or application disclosed herein.