Abstract:
A planar lightwave circuit is provided that includes a substrate and at least one boundary positioned in the substrate and defining a cavity. The boundary is substantially non-transmissive and absorbing for wavelengths of stray light present in the vicinity of the boundary. The boundary possesses substantial symmetry under at least one symmetry group operation.

Description:
FIELD OF THE INVENTION 
     The present invention relates to the fabrication of an optical waveguide device for optical isolation of a light emitter, a scattered light source or a light detector. The invention discloses a versatile resonator structure that is formed using a reflective absorbing boundary around a device and which can be applied substantially independently of the actual geometry of the device to be isolated. 
     BACKGROUND OF THE INVENTION 
     The manipulation of input and output light signals to and from optical fiber transmission lines generally requires that the signals be processed in some fashion, examples of which might include amplification, power splitting or the addition and/or dropping of signals. With the persistent trend towards miniaturization and integration, the optical circuits which best serve these processing functions are more and more being integrated on optical chips as a single module. The resulting optical circuits, which carry channel waveguides as their fundamental light-guiding elements, are generally referred to as planar lightwave circuits or PLCs. Current planar waveguide technology typically prepares a PLC by lithographically patterning light-guiding channels either directly upon (or buried beneath) the surface of a rigid planar substrate, or within a sequence of dielectric films separately deposited on the substrate. 
     In cases where the waveguide channels are formed in direct association with the substrate the substrate composition is usually chosen with a view to taking advantage of its specific electronic or electro-optic properties in addition to its mechanical characteristics. Patterning can be induced by ion exchange or by metallic diffusion. As an example of the latter process, a metallic film that has already been lithographically formed into a channel pattern can be heated to a temperature sufficient to induce a thermal diffusion of metal atoms into the surface region of the substrate (e.g. U.S. Pat. No. 5,749,132 by A. Mahapatra and S. A. Narayanan). In this manner a high-refractive-index light-guiding waveguide pattern can be created close to the surface of the substrate. The guides so formed can then be buried, if desired, by utilizing a second thermal-diffusion patterning process employing a metal that is able to generate a lower-refractive-index covering. 
     In PLCs where the waveguide channels are formed within a sequence of dielectric films deposited on a rigid substrate, the substrate usually plays only a thermal-mechanical roll. For these structures, the simplest situation sees the deposition of a sequence of three films (often referred to respectively as lower cladding (or buffer), core, and upper cladding) utilizing photolithography to pattern the required waveguide and component designs into the core layer. The refractive index of the core composition is chosen to be larger than those of the cladding layers to ensure good optical confinement within the core waveguides. An exposition of this general technology can be found in U.S. Pat. No. 4,902,086 by C. H. Henry et al. 
     In the context of the present invention the term ‘planar lightwave circuit’ (or PLC) should be interpreted to embrace all light-guiding circuits patterned into or onto rigid planar substrates. In particular, it should not be construed as limited to the specific categories examples of which have been described above. 
     In addition to signal processing circuits, which comprise optical network nodes, network termination points, such as light transmitters and light receivers, can also be integrated with other elements on a single PLC chip. Examples of light transmitters that can be so integrated are heterostructure end-emitting lasers, vertical cavity surface-emitting lasers and light emitting diodes. The most commonly used integrated light receivers are different types of photodiodes. For both transmitters and receivers, it is necessary that they be coupled to a single well-defined set of optical modes in the planar lightwave circuit. Generally the optical modes that carry the light signal around the optical chip are confined modes guided by waveguide channels. However, there are other unconfined optical modes (often designated as ‘radiation modes’ or ‘stray light’) present in the chip. These can enter devices on the chip and severely limit their performance. It therefore becomes necessary to design features that are capable of significantly limiting the undesirable access unconfined modes to these devices. 
     Light power in such radiation modes is almost exclusively scattered light, and there are usually several sources of scattered light on the optical chip. For example, there can be substantial power present on an optical chip in radiation modes due to imperfect coupling of a fiber to a planar waveguide at the chip interface. While it is disadvantageous to couple any power from a light signal into radiation modes on the chip, this is difficult to avoid in practice. Signal processing devices on the PLC chip can also be sources of scattered light. These sources are either there by design, such as in some types of variable optical attenuators where redundant light is dumped into the cladding, or they occur because of fabrication imperfections. On active chips, such as in waveguide amplifiers, another source of scattered light is amplified spontaneous emission from the gain material deposited on the chip. At the receiver end of the optical network, scattered light from any the above sources may interfere with the light signal propagating in a waveguide and can cause major signal degradation. 
     It therefore becomes necessary to devise a structure that can be used for optical isolation. Such a structure focuses on isolating a specific PLC device (such as the receiver) from light in radiation modes, but it can also be structured to isolate individual sources of scattered light from the rest of the chip. One common method of isolation in this context is the use of deep air trenches, geometrically positioned in a manner that can optimally intercept stray light that is propagating substantially parallel to a waveguide and redirect it away from the sensitive locations (see, for example, Pat. No. WO02097491 by D. Kitcher et al.). Another method is the introduction of light-absorbing regions to severely attenuate, rather than redirect, problematic radiation modes. In addition to the careful positioning of absorbing regions, the efficiency of stray light capture can be improved by decorating their shapes with protruding or notched facets to facilitate a more efficient coupling of scattered light into these lossy regions (see, for example, Pat. Nos. EP0883000 by T. S. Hoekstra, and WO03007034 by I. E. Day et al.). The structure by Day et al. is illustrated in FIG. 1. a , and comprises a waveguide 101, and light absorbing doped regions 102. The above references are directed solely to intercept stray light that propagates substantially parallel to the waveguide and therefore they rely on the proximity of the absorbing regions to the waveguide for efficient stray light absorption. Another approach to optical isolation is a monomode spatial optical filter (U.S. Pat. No. 5,093,884 by Gidon et al.), which is illustrated in FIG. 1. b ), and comprises a curved waveguide section 103 and a light absorber 104 with an irregular sawtooth pattern and a geometrically asymmetrical shape with respect to the waveguide axis. 
     SUMMARY OF THE INVENTION 
     The present invention greatly improves upon the efficiency of capture (and subsequent attenuation) of stray light that is achievable by the conventional devices. The invention does so by introducing the concept of absorbing resonant cavities, in which scattered light is coupled into resonator chambers bounded by reflective absorbing surfaces. More specifically, the invention is directed to a resonant cavity bounded by a reflective absorbing boundary around the device to be isolated. The cavity has at least one opening to allow for a light signal to couple into or out of the device. The boundary of the cavity is non-transmitting as well as partially reflecting and absorbing for wavelengths of the scattered light. Light that is not directed at the device, either directly, or indirectly by use of a waveguide, is coupled into one or more cavity modes. Light in the cavity modes is either reflected out of the cavity or is substantially attenuated before arriving at the device. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1.   a . Schematic of a prior art optical isolator based on doped dielectric regions near a waveguide. 
         FIG. 1.   b . Schematic of a prior art monomode spatial optical filter based on absorbers near a curved waveguide. 
       FIG.  2 . Schematic of a straight optical isolator cavity. 
       FIG.  3 . Schematic cross-sectional side view of an optical isolator cavity. 
         FIG. 4.   a . Schematic of an optically isolated PIN detector. 
         FIG. 4.   b . Schematic cross-sectional side view of an optically isolator cavity around a PIN detector. 
       FIG.  5 . Schematic of light ray propagation in a straight optical isolator cavity. 
         FIG. 6.   a . A histogram of light ray attenuation for a straight optical isolator cavity. 
         FIG. 6.   b . A plot of light ray attenuation for a straight optical isolator cavity versus ray exit angle. 
         FIG. 7.   a . Schematic of a concave optical isolator cavity with curved boundaries with mirror symmetry. 
         FIG. 7.   b . Schematic of a concave optical isolator cavity with perpendicular boundaries with mirror and rotational symmetries. 
         FIG. 7.   c . Schematic of a concave optical isolator cavity with arbitrary straight boundaries with rotational symmetry. 
         FIG. 7.   d . Schematic of an optical isolator cavity with arbitrary asymmetric boundaries. 
         FIG. 8.   a . Schematic of an sawtooth type optical isolator cavity comprising a series of identical concave cavities. 
         FIG. 8.   b . Schematic of an optical isolator cavity comprising a series of concave and straight cavities. 
         FIG. 8.   c . Schematic of an optical isolator cavity comprising a symmetric cavity and additional asymmetric cavity boundaries. 
       FIG.  9 . Schematic of an optical isolator sawtooth cavity around a PIN detector. 
         FIG. 10.   a . A histogram of light ray attenuation for a sawtooth optical isolator cavity. 
         FIG. 10.   b . A plot of light ray attenuation for a sawtooth optical isolator cavity versus ray exit angle. 
         FIG. 11. A  plot of attenuation efficiency versus cavity length for a straight, sawtooth and perfectly absorbing optical isolator cavity. 
         FIG. 12.   a . Schematic of a straight optical isolator cavity around an optical tap monitor. 
         FIG. 12.   b . Schematic of a sawtooth optical isolator cavity around an optical tap monitor. 
         FIG. 13.   a . A histogram of light ray attenuation for a straight optical isolator cavity around an optical tap monitor. 
         FIG. 13.   a . A histogram of light ray attenuation for a sawtooth optical isolator cavity around an optical tap monitor. 
     
    
    
     DETAILED DESCRIPTION 
     It is worthy to note that any reference herein to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the invention. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment. 
     Embodiment 1 of the invention is illustrated in FIG.  2 . The invention comprises a cavity  203  defined by at least one absorbing reflective boundaries  204 . The cavity  203  has at least one open port  205  for a waveguide  201  to carry an optical signal through the cavity  203 . An optical device  202  may be inside the cavity  203 . A light signal can be input using the waveguide  201  at one of the ports  205 . Scattered light is also present in the proximity of the ports  205  and can enter the cavity  203 . The inner edges  207  of the regions  204  as well as the outer edges  208  of the boundaries  204  are substantially non-transmissive for wavelengths of the scattered light present. The outer edges  208  function as optical isolators for the device  202  to protect it from scattered light that would be directly incident on the device  202  if the boundaries  208  was absent. The cavity  203  defined by the inner edges  207  serves as an absorber for scattered light that enters the ports  205 . Scattered light inside the cavity  203  is reflected and absorbed each time a light ray hits an inner edge  207 . The two main effects of optical isolation are twofold. First, the power density of the scattered light near the device  202  is substantially decreased as compared to the case when the boundaries  204  are absent. Second, any stray light entering a port  205  will be substantially attenuated by the time the light exits the cavity  203  via another port  205 . 
       FIG. 3  shows a schematic cross-section of this embodiment of the light absorbing cavity. The device  301  is embedded in a cladding material  302 , which in turn is positioned on a silicon wafer  303 . The absorbing reflective regions are created by etching trenches  304  through the cladding material  302  substantially approaching the silicon wafer  303 . In the next processing step, one ore more metal layers  305  are deposited to cover the walls of the trenches  304 . Optionally, some of the metal can subsequently be removed to expose other parts of the PLC. The absorbing reflective regions are now defined by the metal layers on the outer walls  306  of the trenches  304  and the metal layers on the inner walls  307  of the trenches  304 . Scattered light in the cladding that is incident on the metal layers  305  will be partially reflected and partially absorbed by the metal and thus the device  301  will be optically isolated from scattered light. 
     Embodiment 2 of the invention is illustrated in  FIGS. 4.   a  and  b . The structure comprises straight cavity  401  defined by an absorbing reflective boundary  402 . Inside the cavity  401 , a PIN detector  403  is fabricated that detects light directed towards it by the waveguide  404 . A substantial portion of the scattered light that enters through the cavity opening  405  will be absorbed inside the cavity  401  before it reaches the PIN detector  403 .  FIG. 4.   b  shows a cross-sectional side view of the structure taken along the waveguide  404 . The waveguide  404  is embedded in a cladding material  406 , which in turn is positioned on top of a silicon wafer  407 . A trench  408 , whose cross-section is triangular, is etched through the cladding material  406  and the waveguide  404  and a metal layer is deposited on the angled side of the trench  408  to form the turning mirror  409 . Light guided in the waveguide  404  enters the trench  408  via the waveguide termination  410 , is reflected from the turning mirror  409  toward the PIN detector  403 . The absorbing reflective boundary for optical isolation can be defined in the same manner as illustrated in FIG.  3 . The resulting structure in cross-section comprises the trench  411  and the metal layer  412  deposited inside the trench  411 . 
     To evaluate the optical isolation efficiency of the straight absorbing cavity, we calculate the attenuation for scattered light rays originating at the cavity opening  405  and detected by the PIN detector using a ray tracing model. With reference to  FIG. 5 , the cavity  501  is defined by its straight boundaries  502  and has length L and width W. Scattered light rays  503  are assumed to originate at N equidistant points  504  at the cavity opening  505  substantially covering the entire opening. From each originating point  504 , we follow M light rays  503 . Each light ray and the cavity boundaries  504  enclose a propagation angle a which is randomly chosen from the range [−90°, 90°]. The light rays are partially reflected and partially absorbed whenever they hit a cavity boundary  504 . The portion R of the light power that is reflected is related to the angle of incidence θ as follows: 
       R   =                n   ⁢           ⁢   cos   ⁢           ⁢   θ     -     n   a           n   ⁢           ⁢   cos   ⁢           ⁢   θ     +     n   a              2         
 
where n is the refractive index of the cladding material and n a  is the complex refractive index of the material comprising the cavity boundaries at each reflection. The portion of the light that is not reflected is absorbed by the cavity walls. By following light rays, we can calculate the attenuation for each light ray as they arrive at the PIN detector that is positioned at the cavity end  505 .
 
     We plot the results of the model in  FIG. 6  for parameter values L=1 mm and W=0.05 mm, N=20, M=100, n=1.444 corresponding to silica and n a =4.04+3.82i corresponding to titanium at 1.55 μm wavelength. The histogram in  FIG. 6.   a  shows the distribution of light power attenuation in the range [0 dB, 60 dB] for light rays detected by the PIN detector. The graph illustrates that out of the 2000 light rays followed during the simulation about 10% reaches the detector directly with small or no attenuation. These are the light rays that propagate through the cavity substantially parallel to the cavity walls and are detected by the PIN detector after a few or no reflections. The rest of the rays are substantially attenuated. 
     The light attenuation inside the cavity results in a narrowing of the scattered light beam exiting at the output port of the cavity into the PIN detector. To quantify the beam divergence, we approximate analytically the angular dependence of the light intensity at the output port. The propagation angle β enclosed by a light ray and the cavity boundaries at the output port is β=±α, depending on whether the number of reflections suffered by the light ray is even or odd.  FIG. 6.   b  shows the simulated dependence of the attenuation on the propagation angle β. Since the number of reflections suffered by a light ray during its propagation through the cavity is approximately |L tan β/W|, the total attenuation (in dB) as a function of the propagation angle follows the functional form 
         -       20   ⁢   L   ⁢           ⁢   tan   ⁢           ⁢   β       W   ⁢           ⁢   ln   ⁢           ⁢   10         ⁢   ln   ⁢                n   ⁢           ⁢   sin   ⁢           ⁢   β     -     n   a           n   ⁢           ⁢   sin   ⁢           ⁢   β     +     n   a              .         
 
Approximating this expression for small angles, we obtain that the angular intensity distribution of the scattered light beam is 
       exp   ⁡     (       -       4   ⁢   nL              n   a          ⁢   W         ⁢     β   2       )         
 
which is a Gaussian beam with full angular width of 
                n   a          ⁢     W   /   2     ⁢   nL         
 
based on the 1/e 2  points). In this case, the full angular width is about 36°.
 
     A figure of merit that can be used to illustrate how well scattered light is attenuated by the cavity is the attenuation efficiency, defined as p ρ=1−P d  /P, where P d  is the total light power reaching a detector at one end of the cavity and P i  is the light power input at the input port of the cavity. Attenuation efficiency is between zero (meaning no attenuation is achieved) and one (meaning complete attenuation). We first compute the attenuation efficiency for a cavity whose walls are perfectly absorbing and non-reflecting. The perfectly absorbing cavity only allows light through in angle subtended by the end of the cavity at the originating point of the light ray. The efficiency for the perfectly absorbing cavity of width W and length L can be straightforwardly calculated as 
         ρ   p     =       1   -       1     π   ⁢           ⁢   W       ⁢       ∫       -   W     /   2       W   /   2       ⁢           ⁢       ⅆ   y     ⁢           ⁢   arctan   ⁢           ⁢     WL       L   2     -       W   2     /   4     +     y   2                 =         2   π     ⁢   arctan   ⁢           ⁢     L   W       -       L     π   ⁢           ⁢   W       ⁢   ln   ⁢           ⁢       L   2         W   2     +     L   2                   
 
We will use this as an ideal but non-physical benchmark to evaluate the attenuation efficiency of resonant cavities. For the straight cavity whose characteristics are plotted in  FIG. 6 , ρ=0.86, while for a perfectly absorbing cavity of the same dimensions ρ p =0.984.
 
     Embodiment 3 of the invention is illustrated in FIG.  7 . With reference to  FIG. 7.   a , the invention comprises a cavity  701  defined by at least two absorbing reflective boundary  702  on both sides of the waveguide  703 . In this embodiment, the cavity  701  has a concave shape as viewed from inside. This shape results in a higher attenuation for a light ray entering the cavity than in the case of a straight cavity of Embodiment 1. Most light rays incident on a boundary  702  are not only redirected towards another inside cavity boundary  702  as in the case of a straight cavity but the propagation angle also changes substantially. In fact, a scattered light ray originating at port  704  may be reflected in such a way that it will propagate backwards, towards port  704 . The shape of the cavity forces the light ray to suffer more reflections before it exits the cavity than in a straight cavity and thus it will experience higher attenuation. The cavity may have other concave shapes as well, such as illustrated in  FIGS. 7.   b  and  c .  FIG. 7.   b  shows schematically a cavity  705  defined by two sets of perpendicular absorbing reflective boundary sections  706  and  707 , both of which boundaries enclose an approximately 45° angle with the waveguide  708 . The advantage of this particular embodiment is that it never redirects scattered light that is not propagating substantially parallel to the waveguide  708  to become substantially parallel to the waveguide  708 . However, as illustrated in  FIG. 7.   c , the absorbing reflective boundary sections  709  and  710  may define any angle with each other and with the waveguide  711 . 
     For the purposes of stray light absorption, it is advantageous for the cavity to be symmetric, that is, to possess symmetry under one or more symmetry group operation. Such symmetry group operations are, for instance, reflection with respect to a line (mirror symmetry), reflection or rotation with respect to a point (point symmetry) or linear translation (translational symmetry). The cavities  701  and  705  in  FIGS. 7.   a  and  b  possess mirror symmetry with respect to the optical axes  703  and  708 , respectively. However, the resonant cavity may possess a mirror symmetry with respect to arbitrary line enclosing an arbitrary angle with the optical axis of the waveguide. One example is cavity  705  in  FIG. 7.   b , where a symmetry axis is perpendicular to the optical waveguide axis  708 . To put it another way, the waveguide need not follow the symmetry axis of the cavity for the cavity to effectively isolate the region optically. The cavities  705  and  712  in  FIGS. 7.   b  and  c  possess rotational symmetry as well, because there is a specific non-zero angle (i.e. 180°) by which the cavity can be rotated around a pre-selected point and the rotated image coincides with the original. This angle can be an arbitrary angle, not necessarily 180°. The symmetry operations may be combined in series to obtain additional symmetries. 
     It is also understood that slight variations in design or fabrication that destroy perfect symmetry for the cavity will not substantially change the optical isolation properties of the resonant cavity. However, even though for the purposes of absorbing stray light in the cavity, it is advisable for the cavity to be substantially symmetric, the cavity need not possess any symmetry to be functional if the objective is to block light directly incident on the device to be protected. This is the case, for instance, for the cavity directly surrounding the detector in the optical tap monitor depicted in FIG.  4 . An asymmetric version of the cavity  713  protecting the detector  714  is shown in  FIG. 7.   d.    
     Embodiment 4 of the invention is illustrated in FIG.  8 . In this embodiment, absorbing reflective trenches of the previous embodiments are combined in series to increase the effectiveness of attenuation. With reference to  FIG. 8.   a , the sawtooth type cavity  801  comprises a series of identical cavities  802  shown in  FIG. 7.   b . Scattered light entering at port  803  is well attenuated by the time it exist the cavity at port  804 . One can take any combination of single cavities with any parameters to construct longer, more efficient cavities, as illustrated in  FIG. 8.   b , where the cavity  805  comprises a series of straight and concave cavities. An irregular series of single cavities may improve the efficiency of attenuation. As mentioned above, it is advantageous for a light absorbing cavity to possess some symmetry.  FIG. 8.   c  shows schematically a cavity  806  which consists of a series of cavities, but in this case only one cavity  807  of those comprising cavity  806  is symmetric. Other sections of the cavity boundary may serve a different purpose, such as redirecting light rays in a different direction, or connecting cavities. 
     Embodiment 5 of the invention is illustrated in FIG.  9 . This structure is a cavity  901  which surrounds a PIN detector  902  and a waveguide  903  guiding a light signal towards the detector  902 . The cavity  901  contains a sawtooth cavity  904  of Embodiment 4. The cavity  901  is an improvement over the straight cavity of Embodiment 2. To estimate the degree of improvement, we calculate the attenuation of scattered light rays using a ray tracing method, illustrated previously in  FIG. 5  with the same parameters. We assume that the concave cavities  902  are identical and are bounded on each side by two straight boundary sections  905  and  906  that are perpendicular to each other and enclose a 45° angle with the waveguide  904 . The smallest extent of the cavity  901  perpendicular to the waveguide  904  is 50 μm and the length of the cavity  901  is also 1 mm as in the case of the analysis for Embodiment 2. The largest extent of the cavity  901  perpendicular to the waveguide  904  is 150 μm. 
     The results of the simulation are shown in FIG.  10 . The histogram in  FIG. 10.   a  shows the distribution of light power attenuation in the range [0 dB, 60 dB] for light rays detected by the PIN detector. The graph illustrates that out of the 2000 light rays followed during the simulation less than 2% reaches the detector directly with no attenuation. These are the light rays that propagate through the cavity substantially parallel to the cavity boundaries and suffer no reflections before being directly detected by the PIN detector. The rest of the rays that hit the cavity boundaries at least once suffer a large number of reflections due to the concave nature of the resonators comprising the cavity. These rays are attenuated by at least 29.9 dB by the time they are detected by the PIN detector. The graph in FIG  10 . b  shows the angular dependence of the attenuation. Only the very narrow range of rays with exit angles between ±1.5° are not attenuated by the cavity. 
     To further demonstrate the optical isolation efficiency of resonant cavities, we plot in  FIG. 11  a graph of the attenuation efficiency ρ versus the length of the cavities for a straight cavity of Embodiment 2, for a sawtooth cavity of Embodiment 5 as well as for a perfectly absorbing cavity for comparison. The parameters (other than their length) of the cavities are the same as for the ones whose simulation results are illustrated in  FIGS. 6 and 10 . The graph shows that even the straight cavity has a high attenuation efficiency of larger than 0.7 for lengths larger than 0.4 mm and that the attenuation efficiency of the sawtooth cavity is almost identical to that of the perfectly absorbing cavity for lengths larger than 0.2 mm. 
     Embodiment 6 of the invention is illustrated in FIG.  12 . With reference to  FIG. 12.   a , the structure comprises a cavity  1201  defined by straight boundaries  1202 . The cavity surrounds a signal waveguide  1203 , an optical tapping device  1204 , a tap waveguide  1205  and a PIN detector  1206 . An optical signal passes through the waveguide  1203  from input port  1207  and most of it propagates through to output port  1208 . Some of the signal power is tapped of for monitoring purposes at the optical tap  1204  and is sent to the tap waveguide  1205  to the PIN detector  1206 . This optical monitoring device is protected from scattered light that would be directly incident on the PIN detector area  1206 . However, scattered light can enter through the open input port  1207  and the output port  1208 . The cavity  1201  surrounding the optical tap monitor structure attenuates scattered light before this light arrives at the PIN detector  1206 . This cavity  1201  is an improvement over previous embodiments The cavity  1201  comprises at least two straight cavities  1209  and  1210  whose axes are at an angle φ with each other. The scattered light beam that enters port  1207  will be a Gaussian beam with a narrow angular waist by the time it arrives at the tap device  1204 . Due to the angular offset of the cavity  1210 , only a small portion of this beam will enter cavity  1210 , thus improving the overall attenuation efficiency of the cavity  1201 . By increasing the angle φ, the attenuation efficiency can be further improved. 
       FIG. 12.   b  illustrates Embodiment 7 of the invention. This structure is an improvement over Embodiment 6 in that the straight boundaries  1202  is now replaced by sawtooth boundaries  1211  which results in a higher attenuation efficiency. 
     To evaluate the performance of these types of cavities, we model light propagation in the cavities using a ray tracing method described above, with scattered light rays entering the cavities both from the input and the output ports. The results of the simulation are displayed in FIG.  13 . The histograms in  FIGS. 13.   a  and  b  show the distribution of power attenuation for 2000 light rays detected by the detector that originate at the input and output ports of a cavity with straight and sawtooth boundaries, respectively. While most light rays are well attenuated for the straight cavity, about 1.5% of them reach the detector with about a 1 dB attenuation. The sawtooth cavity achieves an even better performance: even for the strongest ray reaching the detector, the attenuation is over 35 dB. The attenuation efficiency for the straight cavity is 0.98 while that of the sawtooth cavity is a near perfect 0.99999975.