Abstract:
A method and apparatus are disclosed for simulating planar waveguides having a rectangular cross-section using a beam propagation method and design tool based on the FFT-BPM. Constraining the beam propagation method to planar waveguides having rectangular cross-sections significantly reduces the computational complexity and increases accuracy. The beam propagation method can be performed entirely in the angular spectrum domain. The constrained shape of the waveguide allows the structure to be accurately specified by its width and center-to-center arm spacing, thereby avoiding transverse spatial quantization.

Description:
FIELD OF THE INVENTION 
     The present invention relates to beam propagation method and, more particularly, to a method and apparatus for designing planar lightwave circuits using beam propagation methods. 
     BACKGROUND OF THE INVENTION 
     Planar lightwave circuits are well-suited for mass production of optical filters and switches. Planar lightwave circuits, such as step-index waveguides, typically consist of a substrate, a uniform lower cladding, a core that varies discretely in two dimensions, and a uniform upper cladding. Beam propagation methods (BPM) are utilized to investigate lightwave propagation through simulated planar lightwave circuits. For a review of general beam propagation methods, see, for example, K. Okamoto, Fundamentals of Optical Waveguides, Chapter 7, Academic Press (2000), incorporated by reference herein. The two most popular beam propagation methods used in the design of planar lightwave circuits are the split-step Fourier Transform beam propagation method, often referred to as the Fast Fourier Transform (FFT)-BPM, and the finite-difference beam propagation method (FD-BPM). 
     Generally, the FFT-BPM and FD-BPM both approximate a planar waveguide structure by plotting the index of refraction as a function of the spatial coordinates using a spatial grid. Thus, spatial quantization errors are introduced along the waveguide boundaries between the core and the cladding. In addition, the FFT-BPM and FD-BPM both permit simulation of planar lightwave circuits having an arbitrary index distribution, i.e., an arbitrary core cross section. In addition, the FFT-BPM continuously translates between the spatial domain and angular spectrum domain using Fourier transform techniques. Thus, these beam propagation methods have significant processing speed and memory capacity requirements. A need therefore exists for beam propagation methods with reduced computational complexity and spatial quantization errors. 
     SUMMARY OF THE INVENTION 
     Generally, a novel beam propagation method and design tool are disclosed that are based on the FFT-BPM. The present invention provides a beam propagation method that is constrained to planar waveguides having a rectangular cross-section, resulting in significantly reduced computational complexity and better accuracy. Among other benefits, the beam propagation method can be performed entirely in the angular spectrum domain, without translating back and forth to the spatial domain. In addition, the constrained shape of the waveguide allows the structure to be accurately specified by its width and center-to-center arm spacing, thereby avoiding transverse spatial quantization. 
     A more complete understanding of the present invention, as well as further features and advantages of the present invention, will be obtained by reference to the following detailed description and drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a schematic block diagram of an illustrative planar waveguide design tool in accordance with the present invention; 
     FIG. 2 is a flow chart describing an exemplary implementation of the planar waveguide design process of FIG. 1; and 
     FIGS. 3A and 3B, collectively, illustrate exemplary pseudo-code for the planar waveguide design process of FIG.  2 . 
    
    
     DETAILED DESCRIPTION 
     The present invention provides a new beam propagation method based on the FFT-BPM. The present invention recognizes that most planar waveguides have a rectangular cross-section. Thus, the present invention provides a beam propagation method that is constrained to rectangular cross-sections, resulting in significantly reduced computational complexity and better accuracy. 
     Conventional FFT-BPM Techniques 
     For a detailed discussion of FFT-BPM techniques, see, for example, K. Okamoto, Fundamentals of Optical Waveguides, Chapter 7, Academic Press (2000), incorporated by reference herein above. Generally, FFT-BPM techniques typically treat field distortion in the spatial domain due to refractive index variation and plane wave propagation in the angular domain separately for each small step in the propagation direction, z. For each step Δz, the complex field u(x,y), where x and y are the transverse dimensions in the plane of the circuit, is initially multiplied by: 
     
       
         exp[jkΔzΔn(x,y)/n].  (1)  
       
     
     where n is the cladding index, Δn(x,y) equals n(x,y)−n, and k equals 2πn/(wavelength). Then u(x,y) is Fourier transformed to get the angular spectrum U(k x , k y ). For a discussion of angular spectrums, see, e.g., J. Goodman, Introduction to Fourier Optics, McGraw-Hill (1968). U(k x , k y ) is then multiplied by: 
     
       
         exp( jk   z   Δz )=exp( j {square root over ( k   2   −k   x   2   −k   y   2 )} Δz ).  (2)  
       
     
     to represent the propagation of the lightwave in free space. The result is then inverse Fourier transformed, z is advanced by Δz, and the cycle repeats until the end of the propagation region. 
     While conventional FFT beam propagation methods continuously translate between the spatial domain and angular spectrum domain using Fourier transform techniques, the present invention recognizes that equation (1) can be Fourier transformed and convolved with the angular spectrum. As discussed further below, the beam propagation method of the present invention speeds up the computation for step-index waveguides and waveguide arrays, where it is easy to Fourier transform equation (1), and where equation (1) is periodic, respectively. In addition, the beam propagation method of the present invention avoids transverse spatial quantization. Thus, it is not necessary to keep an angular spectrum much larger than k, because fields decay exponentially outside of this range. The beam propagation method of the present invention avoids quantization errors by staying in the angular domain (there is some quantization of the impulses in the angular domain, but these are averaged out over many impulses in each step; when simulating non-periodic structures, i.e., when α is chosen to be so large that there is no mutual coupling, then even this quantization does not matter). 
     SINC Beam Propagation Method 
     FIG. 1 is a schematic block diagram of an illustrative planar waveguide design tool  100  in accordance with the present invention. As shown in FIG. 1, the planar waveguide design tool  100  includes certain hardware components, such as a processor  110 , a data storage device  130 , and one or more optional communications ports  130 . The processor  110  can be linked to each of the other listed elements, either by means of a shared data bus, or dedicated connections, as shown in FIG.  1 . The communications port(s)  130  allow(s) the planar waveguide design tool  100  to communicate with other devices over a network (not shown). 
     The data storage device  120  is operable to store one or more instructions, discussed further below in conjunction with FIG. 2, which the processor  110  is operable to retrieve, interpret and execute in accordance with the present invention. As shown in FIG. 1, the data storage device  120  includes a planar waveguide design process  200  incorporating features of the present invention. Generally, the planar waveguide design process  200  simulates any planar lightwave circuit having a rectangular cross-section and investigates lightwave propagation through the simulated device. 
     As is known in the art, the methods discussed herein may be distributed as an article of manufacture that itself comprises a computer readable medium having computer readable code means embodied thereon. The computer readable program code means is operable, in conjunction with a computer system, to carry out all or some of the steps to perform the methods or create the apparatuses discussed herein. The computer readable medium may be a recordable medium (e.g., floppy disks, hard drives, compact disks, or memory cards) or may be a transmission medium (e.g., a network comprising fiber-optics, the world-wide web, cables, or a wireless channel using time-division multiple access, code-division multiple access, or other radio-frequency channel). Any medium known or developed that can store information suitable for use with a computer system may be used. The computer-readable code means is any mechanism for allowing a computer to read instructions and data, such as magnetic variations on a magnetic media or height variations on the surface of a compact disk. 
     Data storage device  120  will configure the processor  110  to implement the methods, steps, and functions disclosed herein. The data storage device  120  could be distributed or local and the processor could be distributed or singular. The data storage device  120  could be implemented as an electrical, magnetic or optical memory, or any combination of these or other types of storage devices. It should be noted that, although data storage device  120  is shown separately from other elements of the planar waveguide design tool  100 , this is not necessarily the case for all applications. Moreover, the term “data storage device” should be construed broadly enough to encompass any information able to be read from or written to an address in the addressable space accessed by processor  110 . With this definition, information on a network is still within data storage device  120  of the planar waveguide design tool  100  because the processor  100  can retrieve the information from the network. 
     FIG. 2 is a flow chart describing an exemplary planar waveguide design process  200  incorporating features and functions of the present invention. As previously indicated, the planar waveguide design process  200  simulates any planar lightwave circuit having a rectangular cross-section and investigates lightwave propagation through the simulated device. Initially, the planar waveguide design process  200  finds the complex starting field u(x,y) during step  210 . 
     Since the present invention considers step-index waveguides, waveguide arrays and other optical devices having a rectangular cross-section, equation (1) can be modified to be: 
     
       
           f ( x,y )[exp( jkΔzΔn/n )−1]+1  (3)  
       
     
     where f(x,y) is a function that is either 0 or 1, and Δn is a constant. Any waveguide distribution can be expressed as a sum of waveguide arrays. Each waveguide array has a width w, period a, and offset (from origin) d. If there is only one waveguide, then a can be a distance at which the coupling between waveguides is negligible. 
     Equation (3) is Fourier transformed during step  220  to get the angular spectrum U(k x , k y ). The angular spectrum U(k x , k y ) is then propagated in free space during step  230  by one step, Δz by multiplying by equation (2). 
     For each set of periodically spaced waveguides we get:                δ        (       k   x     ,     k   y       )       +       [       exp        (     j                 k                 Δ                 z                 Δ                   n   /   n       )       -   1     ]            w   x       a   x              sin        (       k   x            w   x     /   2       )           k   x            w   x     /   2              exp        (     j2                 π          d   x       a   x         )              w   y       a   y              sin        (       k   y            w   y     /   2       )           k   y            w   y     /   2              exp        (     j2                 π          d   y       a   y         )       ×       ∑     p   ,   q            δ        (         k   x     -       2      π                 p       a   x         ,       k   y     -       2      π                 q       a   y           )                   (   4   )                                
     During step  240 , U(k x ,k y ) is convolved with equation (4) (the sinc function) to represent a guiding function of the lightwave in a region of the planar lightwave circuit having a refractive index (instead of Fourier transforming it to translate between the spatial domain and angular spectrum domain as with conventional techniques). The angular spectrum U(k x , k y ) is then advanced during step  250  by one step, Δz. A test is performed during step  260  to determine if the end of the waveguide has been reached. If it is determined during step  260  that the end of the waveguide has not been reached, then program control returns to step  240  and continues in the manner described above. 
     If, however, it is determined during step  260  that the end of the waveguide has been reached, then the inverse fourier transform is performed during step  270  to obtain u(x,y), before program control terminates. 
     To further speed the computation, the sinc functions of equation (4) can be truncated during step  260  after        k   ·     a   26                            
     terms from the center. Because of the truncation, the total power conservation is imperfect. 
     FIGS. 3A and 3B, collectively, illustrate exemplary pseudo-code  300  for the planar waveguide design process  200 . The pseudo-code  300  of FIG. 3 is written as a subroutine in the C programming language. The input arguments are the Fourier transform of the initial field u(x,y); the window-size 2x lim  (which should equal ˜Nπ/k for optimum speed); the step-size Δz; k; the index step Δn/n; and the waveguide width w, center-to-center spacing a, and displacement from the origin d as arrays containing their values for each step of Δz. To simulate non-periodic structures, e.g., evanescent couplers, the two lines in which temp1 and temp2 are assigned are modified. 
     Star Couplers 
     The star coupler, commonly used in waveguide gratings and lenses, consists of two planar arrays of diverging waveguides which have their convergence points on the other&#39;s termination in a planar free-space region. An efficient way to calculate the transmissivity from any waveguide on the one side of the star coupler to any waveguide on the other side of the star coupler is to first perform beam propagation in the waveguide array, starting from when the waveguides are uncoupled to when they reach the free-space region boundary, which has to be done only once, and then use an analytic calculation for the free-space propagation. Total power loss in the BPM is not important since the propagated mode can be renormalized and still provide accurate star-coupler transmissivities. Generally, the planar waveguide design process  200  can be iteratively employed to evaluate the lightwave propagation in the star coupler to ensure the desired power ratio at each output arm. 
     It is to be understood that the embodiments and variations shown and described herein are merely illustrative of the principles of this invention and that various modifications may be implemented by those skilled in the art without departing from the scope and spirit of the invention.