Patent Publication Number: US-2023152515-A1

Title: Interferometer filters with compensation structure

Description:
CROSS-REFERENCES TO OTHER APPLICATIONS 
     This application is a continuation of U.S. application Ser. No. 17/345,724, filed Jun. 11, 2021; which is a continuation of U.S. application Ser. No. 16/599,526, filed Oct. 11, 2019, now U.S. Pat. No. 11,048,041, issued Jun. 28, 2021; which is a continuation of U.S. application Ser. No. 16/514,832, filed Jul. 17, 2019, now U.S. Pat. No. 10,534,130, issued Jan. 14, 2020; which claims priority to U.S. Provisional Patent Application No. 62/851,039 filed May 21, 2019, U.S. Provisional Patent Application No. 62/851,559 filed May 22, 2019 and to Provisional Patent Application No. 62/853,657 filed May 28, 2019. The disclosures of each are hereby incorporated by reference in entirety for all purposes. 
    
    
     FIELD 
     The described embodiments relate generally to optical filter devices. More particularly, the present embodiments relate to Mach-Zehnder interferometer (MZI) filters that include one or more compensation structures to compensate for variations in manufacturing tolerances and/or temperature variations and/or other perturbations. 
     BACKGROUND 
     Currently, there are a wide variety of devices that utilize optical circuits for communications and/or computations. Many optical circuits rely on one or more optical filter elements to filter out undesirable optical frequencies, so an optical frequency range of interest can be isolated. 
     In some applications, an MZI filter which can include a cascaded MZI filter, may demonstrate the theoretical capability of meeting the system specifications. However, when practical fabrication tolerances of the MZI filter are accounted for, the MZI filter may not be able to meet the system specifications without additional tuning. More specifically, an MZI filter employs two parallel waveguides and fabrication variations in the dimensions of the waveguides can produce undesirable shifts in the frequency response of the filter. This can lead to decreased performance parameters of the filter and/or, the failure to meet specifications and unacceptably high yield loss. 
     To compensate for fabrication variations some applications employ one or more heaters that are used to actively tune the filters using the thermo-optic effect in silicon. However, the use of heaters increases power consumption of the circuit and may not be effective for circuits that operate at cryogenic temperatures. Active tuning as a post-fabrication process is another common approach to mitigating fabrication variation, however active tuning can increase expense, may be dependent on foundry-specific processes, and could be intractable for circuits with numerous filters. Therefore, passive compensation structures for MZI filters that are intrinsically tolerant to perturbations from variations in waveguide dimensions and/or other ambient conditions are desired. 
     SUMMARY 
     In some embodiments, a Mach-Zehnder interferometer (MZI) filter comprises a first waveguide having a first length and extending from a first coupler section to a second coupler section, the first waveguide having a constant first width along the first length. A second waveguide having a second length and extending from the first coupler section to the second coupler section includes a tolerance compensation portion positioned between the first coupler section and the second coupler section. The tolerance compensation portion includes a first compensation section having a second width, a second compensation section having a third width and a third compensation section having a fourth width, wherein the fourth width is greater than the third width and the third width is greater than the second width. A first taper portion is positioned between the first coupler section and the first compensation section and transitions from the first coupler section to the second width. A second taper portion is positioned between the first compensation section and the second compensation section and transitions from the second width to the third width. A third taper portion is positioned between the second compensation section and the third compensation section and transitions from the third width to the fourth width. 
     In some embodiments, the first compensation section has a constant second width, the second compensation section has a constant third width and the third compensation section has a constant fourth width. In various embodiments, the tolerance compensation portion is symmetric and includes a fourth compensation section having the third width and a fifth compensation section having the second width. In some embodiments, the tolerance compensation portion in the second waveguide is a first tolerance compensation portion and the first waveguide includes a second tolerance compensation portion that includes a fourth compensation section having a fifth width, wherein the fifth width is greater than the first width. 
     In some embodiments, the first waveguide and the tolerance compensation portion form components of a tolerance compensation structure that compensates for a variation in a width of the first waveguide and a variation in a width of the second waveguide due to manufacturing tolerances. In various embodiments, the tolerance compensation structure reduces a shift in a frequency response of the MZI filter due to the variation in the width of the first waveguide and the variation in the width of the second waveguide. 
     In some embodiments, a method of fabricating a Mach-Zehnder interferometer (MZI) filter tolerant to manufacturing variations comprises forming a substrate and forming a first waveguide on the substrate, the first waveguide having a first length and a first continuous width along the first length, wherein the first width varies within a first range, and forming a second waveguide on the substrate. The second waveguide includes a manufacturing tolerance compensation portion including a first compensation section having a continuous second width that varies in a second range, a second compensation section having a continuous third width that varies in a third range and a third compensation section having a continuous fourth width that varies in a fourth range, wherein the fourth width is greater than the third width and the third width is greater than the second width. 
     In some embodiments, a first taper portion is positioned between a first coupler section and the first compensation section and transitions from the first coupler section to the second width, and a second taper portion is positioned between the first compensation section and the second compensation section and transitions from the second width to the third width. A third taper portion is positioned between the second compensation section and the third compensation section and transitions from the third width to the fourth width. 
     In some embodiments, the tolerance compensation portion is symmetric and includes a fourth compensation section having the third width and a fifth compensation section having the second width. In various embodiments, the tolerance compensation portion in the second waveguide is a first tolerance compensation portion and the first waveguide includes a second tolerance compensation portion that includes a fourth compensation section having a fifth width, wherein the fifth width is greater than the first width. 
     In some embodiments, the manufacturing tolerance compensation portion reduces a shift in a frequency response of the MZI filter caused by the second width varying within the second range, the third width varying within the third range and the fourth width varying within the fourth range. 
     In some embodiments, a Mach-Zehnder interferometer (MZI) filter comprises a first waveguide having a first width extending between a first coupler section and a second coupler section, and a second waveguide extending between the first coupler section and the second coupler section and including a first compensation section having a second width, a second compensation section having a third width and a third compensation section having a fourth width, wherein the fourth width is greater than the third width and the third width is greater than the second width. In various embodiments, the MZI filter further comprises a first taper portion positioned between the first coupler section and the first compensation section and transitioning from the first coupler section to the second width. A second taper portion is positioned between the first compensation section and the second compensation section and transitions from the second width to the third width. A third taper portion is positioned between the second compensation section and the third compensation section and transitions from the third width to the fourth width. 
     In some embodiment, the second waveguide further includes a fourth compensation section having the third width and a fifth compensation section having the second width. In various embodiments, the second waveguide includes a fourth compensation section having the third width and a fifth compensation section having the second width. 
     In some embodiments, a method for making a Mach-Zehnder interferometer (MZI) filter having a compensation section that compensates for a number of perturbations comprises fabricating a first waveguide having a first length and one or more first compensation sections distributed along the first length, wherein each first compensation section of the one or more first compensation sections includes a respective width and length. The method further comprises fabricating a second waveguide having a second length and one or more second compensation sections distributed along the second length, wherein each second compensation section of the one or more second compensation sections includes a respective width and length. Wherein, a sum of the one or more first compensation sections and the one or more second compensation sections is greater than the number of perturbations. 
     In some embodiments, the number of perturbations is selected from a manufacturing tolerance variation in a width of each of the first and the second waveguides, a manufacturing tolerance variation in a thickness of each of the first and the second waveguides and a temperature variation in each of the first and the second waveguides. 
     In some embodiments, a method for making a Mach-Zehnder interferometer (MZI) filter comprises fabricating a first waveguide having a first length and a first continuous width, and fabricating a second waveguide having a second length and a plurality of widths along the second waveguide, wherein the first and the second waveguides simultaneously satisfy: 
     
       
         
           
             
               
                 
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     wherein:
 
m=an integral multiple;
 
λ 0 =wavelength of light in first and second arms;
 
L 1 =reference length of first arm;
 
λ 0 =central wavelength of light in first and second arms;
 
L i =length of i th  portion of second arm;
 
κ i =L i /L 1 ;
 
ν FSR =free spectral range;
 
c=speed of light;
 
X 1 =waveguide width; and
 
X 2 =waveguide thickness.
 
     In some embodiments, the second waveguide has a first compensation section having a second width, a second compensation section having a third width and a third compensation section having a fourth width, wherein the fourth width is greater than the third width and the third width is greater than the second width. In various embodiments the second waveguide further includes a first taper portion positioned between a first coupler section and the first compensation section and transitioning from the first coupler section to the second width. A second taper portion is positioned between the first compensation section and the second compensation section and transitions from the second width to the third width. A third taper portion is positioned between the second compensation section and the third compensation section and transitions from the third width to the fourth width. 
     To better understand the nature and advantages of the present disclosure, reference should be made to the following description and the accompanying figures. It is to be understood, however, that each of the figures is provided for the purpose of illustration only and is not intended as a definition of the limits of the scope of the present disclosure. Also, as a general rule, and unless it is evident to the contrary from the description, where elements in different figures use identical reference numbers, the elements are generally either identical or at least similar in function or purpose. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1 A  illustrates a simplified plan view of an example Mach-Zehnder interferometer filter including a passive compensation structure, according to embodiments of the disclosure; 
         FIG.  1 B  illustrates a simplified plan view of an example Mach-Zehnder interferometer filter including two passive compensation structures, according to embodiments of the disclosure; 
         FIG.  1 C  illustrates a simplified plan view of an example Mach-Zehnder interferometer filter including a passive compensation structure, according to embodiments of the disclosure; 
         FIG.  2    illustrates a single stage of a three-waveguide cascaded third order MZI based filter, according to embodiments of the disclosure; 
         FIG.  3    illustrates an incoherently cascaded third-order MZI filter having four stages, according to embodiments of the disclosure; 
         FIG.  4    illustrates effective index parameters as a function of waveguide width and height for a silicon-on-insulator waveguide, according to embodiments of the disclosure; 
         FIGS.  5 A and  5 B  illustrate a plotted derivative, according to embodiments of the disclosure; 
         FIG.  6    illustrates standard deviations for waveguides and couplers, according to embodiments of the disclosure; 
         FIG.  7    illustrates the statistical behavior of four-stage cascaded third-order MZI filter without mitigation mechanisms, according to embodiments of the disclosure; 
         FIG.  8    illustrates designs to minimize susceptibility to fabrication errors, according to embodiments of the disclosure; 
         FIG.  9    illustrates the statistical distribution of cascaded third-order MZI&#39;s with asymmetric arm widths, according to embodiments of the disclosure; 
         FIG.  10    illustrates the statistical distribution of cascaded third-order MZI&#39;s in the absence of coupler variations with respect to fabrication uncertainties, according to embodiments of the disclosure; 
         FIG.  11    illustrates the statistical distribution of MZI properties for three waveguide widths, according to embodiments of the disclosure; 
         FIG.  12    illustrates the statistical distribution of an MZI filter with four waveguide widths, according to embodiments of the disclosure; 
         FIG.  13    illustrates fabrication tolerance achieved using asymmetric widths as well as heights, according to embodiments of the disclosure; 
         FIGS.  14 A- 14 D  illustrate unconventional cross-sections that are compatible with CMOS-foundry processes, according to embodiments of the disclosure; 
         FIG.  15    illustrates the performance of a filter, according to embodiments of the disclosure; 
         FIG.  16    illustrates performance of a filter, according to embodiments of the disclosure; 
         FIG.  17    illustrates performance of the filter, according to embodiments of the disclosure; 
         FIG.  18    illustrates asymmetric widths where width and height variations are independent, according to embodiments of the disclosure; 
         FIG.  19    illustrates a filter having asymmetric widths where width and height variations are independent and each stage is correlated, according to embodiments of the disclosure; 
         FIG.  20    illustrates an embodiment where width and height variations of every stage are correlated, according to embodiments of the disclosure; 
         FIG.  21    illustrates an embodiment where width and height variations are independent but are correlated for all stages, according to embodiments of the disclosure; 
         FIG.  22    illustrates an embodiment where width and height variations are independent but are correlated for all stages, according to embodiments of the disclosure; 
         FIG.  23    illustrates the performance of a filter, according to embodiments of the disclosure; 
         FIG.  24    illustrates yield percentage, according to embodiments of the disclosure; 
         FIG.  25    illustrates a simplified plan view of an example Mach-Zehnder interferometer switch including a passive compensation structure, according to embodiments of the disclosure; and 
         FIG.  26    illustrates a simplified plan view of an example Mach-Zehnder interferometer switch including a passive compensation structure, according to embodiments of the disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     Some embodiments of the present disclosure relate to a passive compensation structure for a Mach-Zehnder interferometer (MZI) filter that improves the filter&#39;s ability to accommodate changes in manufacturing tolerances and/or other perturbations. While the present disclosure can be useful for a wide variety of configurations, some embodiments of the disclosure are particularly useful for cascaded MZI filters that are fabricated using silicon-based structures, as described in more detail below. 
     For example, in some embodiments, an MZI filter includes a pair of waveguides that extend between a first and a second coupler section. The first waveguide has a first continuous width along its length. The second waveguide includes a tolerance compensation portion positioned between the first and the second coupler sections. The tolerance compensation portion includes multiple waveguide sections, each having a different width, as explained in more detail below. The compensation portion can reduce a shift in frequency response of the MZI filter that can be caused by various perturbations, including variations in manufacturing widths of the waveguides, manufacturing variations in thicknesses of the waveguides and variations in temperature. In further embodiments the compensation structure can be designed to reduce a shift in frequency response of the MZI filter that can be caused by myriad perturbations while meeting a resonance requirement, as described in more detail below. 
     In one example the tolerance compensation portion includes waveguide sections having three different widths, however other embodiments may have a lesser number or a greater number of widths. In this example, the tolerance compensation portion includes a first compensation portion having a second width, a second compensation portion having a third width and a third compensation portion having a fourth width, wherein the fourth width is greater than the third width and the third width is greater than the second width. 
     In another example the first waveguide can also have a compensation portion including multiple waveguide sections, each having different waveguide widths. In further examples, the compensation structure can be designed to compensate for a particular number of system perturbations by having a quantity of waveguide widths that is greater than the number of perturbations. In one embodiment the resonance requirement and a number of system perturbations can be accommodated by designing the compensation structure to have at least one more waveguide width than the number of system perturbations. For example in one embodiment a MZI filter can be designed to have insensitivity to width variations and to have a resonance at 1.55 um by having a compensation structure with three different widths, while a compensation structure having two different widths may be used to compensate for width variations only. In further examples, the degree to which the compensation structure can compensate for a particular set of perturbations can be improved by increasing the total number of different waveguide widths, as also described below. 
     In some embodiments, lengths and widths of the compensation structure can be determined using one or more compensation equations. More specifically, the first and the second waveguides of the MZI filter simultaneously satisfy: 
     
       
         
           
             
               
                 
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     wherein:
 
m=an integral multiple;
 
λ 0 =wavelength of light in first and second arms;
 
L 1 =reference length of first arm;
 
L i =length of i th  portion of second arm;
 
κ i =L i /L 1 ;
 
ν FSR =free spectral range;
 
c=speed of light;
 
X 1 =waveguide width; and
 
X 2 =waveguide thickness.
 
     In order to better appreciate the features and aspects of the present disclosure, further context for the disclosure is provided in the following section by discussing one particular implementation of an MZI filter that includes a passive compensation structure, according to embodiments of the disclosure. These embodiments are for explanatory purposes only and other embodiments may be employed in other MZI-based filter devices. In some instances, embodiments of the disclosure are particularly well suited for use with quantum computing circuits because of the intractability of using thermo-optic tuning for these applications. 
       FIG.  1 A  illustrates a simplified plan view of an example Mach-Zehnder interferometer filter  100  including a passive compensation structure  102 , according to an embodiment of the disclosure. As shown in  FIG.  1   , MZI filter  100  includes a first waveguide  104  having a first length  106  and extending from a first coupler section  108  to a second coupler section  110 . First waveguide  104  has a constant first width  112  along first length  106 . A second waveguide  114  includes a compensation portion  116  positioned between first coupler section  108  and second coupler section  110 . Compensation portion  116  includes a first compensation section  118  having a second width  120 , a second compensation section  122  having a third width  124  and a third compensation section  126  having a fourth width  128 . In some embodiments, fourth width  128  is greater than third width  124  and the third width is greater than second width  120 . In some embodiments, the width and length of each compensation portion can be determined using one or more compensation equations, as described in more detail below. 
     In some embodiments, compensation portion  116  is symmetric along second waveguide  114  and further includes a fourth compensation section  130  having third width  124  and a fifth compensation section  132  having second width  120 . In further embodiments, compensation structure  102  may also include a compensation portion positioned within first waveguide  104 , as described in more detail below. 
     In various embodiments, one or more taper portions can be positioned in-between each compensation section to transition between different waveguide widths. More specifically, in some embodiments, a first taper portion  134  is positioned between first coupler section  108  and first compensation section  118  and transitions to second width  120 . A second taper portion  136  can be positioned between first compensation section  118  and second compensation section  122  and transitions from second width  120  to third width  124 . A third taper portion  138  can be positioned between second compensation section  122  and third compensation section  126  and transitions from third width  124  to fourth width  128 . Similarly, a fourth taper portion  140  can be positioned between third compensation section  126  and fourth compensation section  130  and transitions from fourth width  128  to third width  124 . A fifth taper portion  142  can be positioned between fourth compensation section  130  and fifth compensation section  132  and transitions between third width  124  and second width  120 . A sixth taper portion  144  can be positioned between fifth compensation section  132  and second coupler section  110  and can transition from second waveguide width  120 . In some embodiments, first waveguide  104  can also include one or more taper portions to transition widths between first coupler section  108  to first waveguide  104  and from the first waveguide to second coupler section  110 . 
     In some embodiments, each compensation section  118 ,  122 ,  126 ,  130 ,  132  of compensation portion  116  may have a substantially constant width. More specifically, in some embodiments, first compensation section  118  has a constant second width  120 , second compensation section  122  has a constant third width  124 , third compensation section  126  has a constant fourth width  128 , fourth compensation section  130  has a constant third width  124  and fifth compensation section  132  has a constant second width  120 . 
     In some embodiments, each compensation section can have a particular length, as determined by one or more compensation equations, described in more detail below. First compensation section  118  can have a second length  146 , second compensation section  122  can have a third length  148 , third compensation section  126  can have a fourth length  150 , fourth compensation section  130  can have a fifth length  152  and fifth compensation section  132  can have a sixth length  154 . 
     In some embodiments, first length  106  of first waveguide  104 , length of each compensation section  118 ,  122 ,  126 ,  130  and  132 , first width  112  of first waveguide  104  and widths  120 ,  124 ,  128 ,  124 ,  120  of each respective compensation section  118 ,  122 ,  126 ,  130  and  132  of compensation structure  102  can be determined using one or more compensation equations. More specifically, the first and the second waveguides of MZI filter  100  simultaneously satisfy: 
     
       
         
           
             
               
                 
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     wherein:
 
m=an integral multiple;
 
λ 0 =wavelength of light in first and second arms;
 
L 1 =reference length of first arm;
 
λ 0 =central wavelength of light in first and second arms;
 
L i =length of i th  portion of second arm;
 
κ i =L i /L 1 ;
 
ν FSR =free spectral range;
 
c=speed of light;
 
X 1 =waveguide width; and
 
X 2 =waveguide thickness.
 
     For example, in one embodiment, compensation equations can be used to define a compensation structure for a pump-rejection filter for a quantum computer having the following parameters: 
     (i) 120 dB of pump rejection at wavelength λ 0 =1.55 μm;
 
(ii) 25 mdB of signal loss; and
 
(iii) A free-spectral range (FSR) of 2.4 THz.
 
     In other embodiments other suitable parameters can be defined for an MZI filter, as appreciated by one of skill in the art. 
       FIG.  1 B  illustrates a simplified plan view of an example MZI filter  156  including a passive compensation structure, according to an embodiment of the disclosure. As shown in  FIG.  1 B , MZI filter  156  is similar to MZI filter  100  illustrated in  FIG.  1 A . However, in this embodiment, MZI filter  156  includes a compensation portion positioned within each waveguide arm. More specifically, similar to MZI filter  100 , MZI filter  156  includes compensation portion  116  positioned within second waveguide  114 , however, MZI filter  156  also includes a second compensation portion  158  positioned within first waveguide  160 , as described in more detail below. As appreciated by one of skill in the art with the benefit of this disclosure any combination of compensation portions can be employed in an MZI filter and the compensation portions do not need to be the same, or even have similar widths and/or lengths. As described in more detail below, each compensation portion can be uniquely designed according to the compensation equations. 
     As shown in  FIG.  1 B , first waveguide  160  includes second compensation portion  158  that includes a plurality of compensation sections, each having a width and a length as defined by a set of compensation equations, described in more detail herein. Second compensation portion  158  is positioned between first coupler section  108  and second coupler section  110 . Second compensation portion  158  includes a sixth compensation section  164  having fifth width  174  and seventh length  176 , a seventh compensation section  166  having sixth width  178  and a eighth length  180 , and an eighth compensation section  168  having fifth width  174  and seventh length  176 . As described above with regard to  FIG.  1 A , one or more taper portions can be positioned between waveguide sections of different widths to transition from one width to another width. 
       FIG.  1 C  illustrates a simplified model of an MZI filter  172  illustrating geometrical parameters for a set of compensation equations. As shown in  FIG.  1 C , an MZI filter  172  is shown having two parallel waveguides, each having a particular set of geometric parameters. In general, the phase difference between the two waveguide arms is given by Equation (1). 
       ϕ(ω)= k   1 (ω) L   1 −Σ i=2   n+2   k   i (ω) L   i   (Eq. 1)
 
     In Equation (1), ω is the angular frequency of light, k i (ω) is the wave number corresponding to the i th  waveguide width at angular frequency ω, while L i  refers to the length of the i th  waveguide. Note that L i  could be negative, in which case it would mean that it is located on the other arm. In one example, L 1 , L 2 , L 4  are positive while L 3  is negative, then the two arm lengths are L 1 +L 3  and L 2 +L 4 . The simplest case of this class of structures is when each arm has a different but uniform width. 
     Several constraints may be satisfied by the filter design. Firstly, the pump with central wavelength λ 0  can be situated at a transmission minimum (since this is a pump-rejection filter). Therefore, the left-hand side (LHS) of Equation (1) corresponds an integral multiple m of 2π at the center wavelength λ 0 . Since k i (λ 0 )=2πn i (λ 0 )λ 0   −1 , for Equation (2). In writing down the expression for the transmission function, in some embodiments, it is proportional to sin 2 (Ø/2). In various embodiments Ø/2=mπ, or ϕ=2mπ. 
         mλ   0   =L   1 ( n   1 (λ 0 )−Σ i   n   i (λ 0 )κ i )  (Eq. 2)
 
     In Equation (2), κ i =L i /L 1 . In addition, in some embodiments, it may be desirable for the filter to possess a predetermined free-spectral range (FSR). The free-spectral range can be obtained by setting ϕ(ω 0 +2πν FSR )−ϕ(ω 0 )=±2π. Since the FSR may be smaller than the central angular frequency ω 0 , the various k i  can be expanded in a Taylor series about k i (ω 0 ), where dk i /dω=ν gi   −1 =n gi /c. Here n gi  refers to the group refractive index at the center wavelength λ 0 . This yields Equation (3) for ν FSR . 
     
       
         
           
             
               
                 
                   
                     v 
                     FSR 
                   
                   = 
                   
                     c 
                     
                       
                         L 
                         1 
                       
                       ( 
                       
                         
                           n 
                           
                             g 
                             ⁢ 
                             1 
                           
                         
                         - 
                         
                           
                             Σ 
                             i 
                           
                           ⁢ 
                           
                             n 
                             gi 
                           
                           ⁢ 
                           
                             κ 
                             i 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     3 
                   
                   ) 
                 
               
             
           
         
       
     
     To check the validity of Equation (3), a conventional MZI may be considered having arms of differing lengths L 1 , L 2  but the same widths. This yields Equation (4) for ν FSR . 
     
       
         
           
             
               
                 
                   
                     v 
                     FSR 
                   
                   = 
                   
                     
                       c 
                       
                         
                           n 
                           
                             g 
                             ⁢ 
                             1 
                           
                         
                         ( 
                         
                           
                             L 
                             1 
                           
                           - 
                           
                             L 
                             2 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       c 
                       
                         
                           n 
                           
                             g 
                             ⁢ 
                             1 
                           
                         
                         ⁢ 
                         
                           Δ 
                           ⁢ 
                           L 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
     Next, constraints can be derived that make the system invariant to various sources of perturbation, X j . This can be achieved by setting 
     
       
         
           
             
               
                 ∂ 
                 ϕ 
               
               
                 ∂ 
                 
                   X 
                   j 
                 
               
             
             = 
             0. 
           
         
       
     
     A generic approach can be used in which N+1 waveguide widths are used to mitigate N sources of perturbation. In addition, the resonant wavelength λ c  (defined as the location of the transmission minimum in this case) can be made invariant to perturbations as shown in Equation (5). 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         n 
                         1 
                       
                     
                     
                       ∂ 
                       
                         X 
                         j 
                       
                     
                   
                   = 
                   
                     
                       Σ 
                       i 
                     
                     ⁢ 
                     
                       κ 
                       i 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           n 
                           i 
                         
                       
                       
                         ∂ 
                         
                           X 
                           j 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     5 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation (5) is generally valid for various sources of perturbation. For example, X 1 ≡w, where w is waveguide width and X 2 ≡h, where h is waveguide thickness. Additional sources of perturbation can be defined, i.e. X 3 ≡T, where T is temperature, etc. Each source of variation represents an additional linear equation with unknowns κ i  for a given set of w i . 
     While Equation (5) adjusts the resonant wavelength λ c  (the wavelength at which a transmission minimum is present) to be invariant to perturbation, it does not make the shape of the transmission curve near the minimum invariant. In some embodiments, this condition can be imposed by setting the derivative of ∂ 2 ϕ/∂ω∂X j  to be constant. 
     In some embodiments, an additional condition can be imposed to mitigate N different sources of variation yielding Equation (6). 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         2 
                       
                       
                         n 
                         1 
                       
                     
                     
                       
                         ∂ 
                         
                           X 
                           j 
                         
                       
                       ⁢ 
                       
                         ∂ 
                         ω 
                       
                     
                   
                   = 
                   
                     
                       Σ 
                       i 
                     
                     ⁢ 
                     
                       κ 
                       i 
                     
                     ⁢ 
                     
                       
                         
                           ∂ 
                           2 
                         
                         
                           n 
                           i 
                         
                       
                       
                         
                           ∂ 
                           
                             X 
                             j 
                           
                         
                         ⁢ 
                         
                           ∂ 
                           ω 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation (6) also represents a set of linear equations with unknowns κ i  for a given set of w i . In Equation (6), the order of derivatives is swapped for the sake of convenience since ∂n i /∂w is readily obtained from the effective-index dispersion of waveguides. Furthermore, Equations (2)-(3) can be reduced to a single equation with unknowns κ i  by dividing Equation (2) by Equation (3) as shown below in Equations (7a) and (7b). 
     
       
         
           
             
               
                 
                   
                     
                       m 
                       ⁢ 
                       
                         λ 
                         0 
                       
                     
                     
                       c 
                       / 
                       FSR 
                     
                   
                   = 
                   
                     γ 
                     = 
                     
                       
                         ( 
                         
                           
                             
                               n 
                               1 
                             
                             ( 
                             
                               λ 
                               0 
                             
                             ) 
                           
                           - 
                           
                             
                               Σ 
                               i 
                             
                             ⁢ 
                             
                               
                                 n 
                                 i 
                               
                               ( 
                               
                                 λ 
                                 0 
                               
                               ) 
                             
                             ⁢ 
                             
                               κ 
                               i 
                             
                           
                         
                         ) 
                       
                       
                         ( 
                         
                           
                             n 
                             
                               g 
                               ⁢ 
                               1 
                             
                           
                           - 
                           
                             
                               Σ 
                               i 
                             
                             ⁢ 
                             
                               n 
                               
                                 g 
                                 ⁢ 
                                 i 
                               
                             
                             ⁢ 
                             
                               κ 
                               i 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     
                       Eq 
                       . 
                           
                       7 
                     
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       γ 
                       ⁢ 
                       
                         n 
                         
                           g 
                           ⁢ 
                           1 
                         
                       
                     
                     - 
                     
                       n 
                       1 
                     
                   
                   = 
                   
                     
                       Σκ 
                       i 
                     
                     ( 
                     
                       
                         γ 
                         ⁢ 
                         
                           n 
                           
                             g 
                             ⁢ 
                             i 
                           
                         
                       
                       - 
                       
                         n 
                         i 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   
                     
                       Eq 
                       . 
                           
                       7 
                     
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     Equations (5), (6) and (7b) represent a set of 2N+1 linear equations in κ i  for 2N sources of perturbation or constraints. If Equation (6) is ignored, then there are N+1 linear equations in κ i . Thus for a predefined set of N+1 waveguide widths, a solution is yielded by obtaining N+1 values of κ i . Since the various partial derivatives enumerated above are real, a solution to the above problem is generated. Negative values of κ i  are permitted since they represent that section being present in the ‘other’ arm. Thus, the above problem can therefore be cast into a form MX=B as shown below in Equation (8). 
     
       
         
           
             
               
                 
                   M 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               γ 
                               ⁢ 
                               
                                 n 
                                 
                                   g 
                                   ⁢ 
                                   2 
                                 
                               
                             
                             - 
                             
                               n 
                               2 
                             
                           
                         
                         
                           
                             
                               γ 
                               ⁢ 
                               
                                 n 
                                 
                                   g 
                                   ⁢ 
                                   3 
                                 
                               
                             
                             - 
                             
                               n 
                               3 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               γ 
                               ⁢ 
                               
                                 n 
                                 
                                   g 
                                   ⁡ 
                                   ( 
                                   
                                     N 
                                     + 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               n 
                               
                                 N 
                                 + 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 2 
                               
                             
                             
                               ∂ 
                               
                                 X 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 3 
                               
                             
                             
                               ∂ 
                               
                                 X 
                                 1 
                               
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 
                                   N 
                                   + 
                                   2 
                                 
                               
                             
                             
                               ∂ 
                               
                                 X 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           … 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 2 
                               
                             
                             
                               ∂ 
                               
                                 X 
                                 K 
                               
                             
                           
                         
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 3 
                               
                             
                             
                               ∂ 
                               
                                 X 
                                 K 
                               
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 
                                   N 
                                   + 
                                   2 
                                 
                               
                             
                             
                               ∂ 
                               
                                 X 
                                 K 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 ∂ 
                                 2 
                               
                               
                                 n 
                                 2 
                               
                             
                             
                               
                                 ∂ 
                                 ω 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   X 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∂ 
                                 2 
                               
                               
                                 n 
                                 3 
                               
                             
                             
                               
                                 ∂ 
                                 ω 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   X 
                                   1 
                                 
                               
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               
                                 ∂ 
                                 2 
                               
                               
                                 n 
                                 
                                   N 
                                   + 
                                   2 
                                 
                               
                             
                             
                               
                                 ∂ 
                                 ω 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   X 
                                   1 
                                 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           … 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               
                                 ∂ 
                                 2 
                               
                               
                                 n 
                                 2 
                               
                             
                             
                               
                                 ∂ 
                                 ω 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   X 
                                   K 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∂ 
                                 2 
                               
                               
                                 n 
                                 3 
                               
                             
                             
                               
                                 ∂ 
                                 ω 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   X 
                                   K 
                                 
                               
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               
                                 ∂ 
                                 2 
                               
                               
                                 n 
                                 
                                   N 
                                   + 
                                   2 
                                 
                               
                             
                             
                               
                                 ∂ 
                                 ω 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   X 
                                   K 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     8 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             X 
             = 
             
               [ 
               
                 
                   
                     
                       κ 
                       2 
                     
                   
                 
                 
                   
                     
                       κ 
                       3 
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       κ 
                       
                         N 
                         + 
                         2 
                       
                     
                   
                 
               
               ] 
             
           
         
       
       
         
           
             B 
             = 
             
               [ 
               
                 
                   
                     
                       
                         γ 
                         ⁢ 
                         
                           n 
                           
                             g 
                             ⁢ 
                             1 
                           
                         
                       
                       - 
                       
                         n 
                         1 
                       
                     
                   
                 
                 
                   
                     
                       
                         ∂ 
                         
                           n 
                           1 
                         
                       
                       
                         ∂ 
                         
                           X 
                           1 
                         
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         ∂ 
                         
                           n 
                           1 
                         
                       
                       
                         ∂ 
                         
                           X 
                           K 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           ∂ 
                           2 
                         
                         
                           n 
                           1 
                         
                       
                       
                         
                           ∂ 
                           ω 
                         
                         ⁢ 
                         
                           ∂ 
                           
                             X 
                             1 
                           
                         
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         
                           ∂ 
                           2 
                         
                         
                           n 
                           1 
                         
                       
                       
                         
                           ∂ 
                           ω 
                         
                         ⁢ 
                         
                           ∂ 
                           
                             X 
                             K 
                           
                         
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
       FIG.  2    illustrates a single stage of a three-waveguide Cascaded third order MZI based filter  200  using a solution to Equation (8). Each stage can be incoherently cascaded as shown in  FIG.  3    that illustrates an incoherently cascaded third-order MZI filter  300  having four stages  305 ,  310 ,  315 ,  320 . 
     In some embodiments, it may be considered that the above set of equations do not consider loss or extinction ratio thus it may be possible that the obtained lengths from the above set of constraints violate the parameters of the extinction ratio. 
     In some embodiments, the use of more or less than N+1 waveguide widths can be used. In either case, the problem is modified to an optimization problem, i.e. a solution to min(MX−B) is desirable. 
     In some embodiments, the transitions in waveguide widths may not considered because the waveguide widths may be marginally different and therefore the transition lengths between these may not be relatively large, approximately 1 micron, in one embodiment. This can be relatively smaller than the length of one of the arms, for example approximately 100 microns, in one embodiment. 
     The discussion above disclosed an approach to make the MZI&#39;s tolerant to sources of perturbation. The next section discloses a design process including an approach to test the statistical performance of an MZI device. 
     The first step is to define the geometry of the device and obtain refractive indices of waveguides as functions of w, h, T . . . and other variables for various angular frequencies co. In some embodiments, this can be accomplished using commercial mode solvers. Upon obtaining this information, it can be stored in the form of look-up tables. To simplify storing the spectral dependencies, the refractive index data can be fit as follows and the coefficients n, ∂n/∂w, ∂ 2 n/∂ω 2  can be stored yielding Equation (9). 
     
       
         
           
             
               
                 
                   
                     n 
                     ⁡ 
                     ( 
                     
                       ω 
                       , 
                       
                         
                           X 
                           1 
                         
                         ⁢ 
                         … 
                         ⁢ 
                            
                         
                           X 
                           N 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       n 
                       ⁡ 
                       ( 
                       
                         
                           ω 
                           0 
                         
                         , 
                         
                           
                             X 
                             1 
                           
                           ⁢ 
                           … 
                           ⁢ 
                              
                           
                             X 
                             N 
                           
                         
                       
                       ) 
                     
                     + 
                     
                       
                         
                           ∂ 
                           
                             n 
                             ⁡ 
                             ( 
                             
                               
                                 X 
                                 1 
                               
                               , 
                               
                                 … 
                                 ⁢ 
                                    
                                 
                                   X 
                                   n 
                                 
                               
                             
                             ) 
                           
                         
                         
                           ∂ 
                           ω 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           ω 
                           - 
                           
                             ω 
                             0 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           
                             ∂ 
                             2 
                           
                           
                             n 
                             ⁡ 
                             ( 
                             
                               
                                 X 
                                 1 
                               
                               , 
                               
                                 … 
                                 ⁢ 
                                    
                                 
                                   X 
                                   N 
                                 
                               
                             
                             ) 
                           
                         
                         
                           ∂ 
                           
                             ω 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             ω 
                             - 
                             
                               ω 
                               0 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation (8) can then be solved to obtain various ratios κ i . If an exact solution cannot be obtained, variation of the central resonant wavelength Δλ c  can be minimized for given standard deviations in perturbation sources σ X     j    according to Equation (10). 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                       λ 
                       c 
                     
                   
                   = 
                   
                     
                       λ 
                       0 
                     
                     ⁢ 
                     
                       
                         
                           Σ 
                           j 
                         
                         ⁢ 
                         
                           σ 
                           
                             X 
                             j 
                           
                         
                         ⁢ 
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               
                                 ∂ 
                                 
                                   n 
                                   1 
                                 
                               
                               
                                 ∂ 
                                 
                                   X 
                                   j 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   - 
                                   ∑ 
                                 
                                 
                                   
                                     i 
                                     = 
                                     2 
                                   
                                 
                                 
                                   
                                     N 
                                     + 
                                     2 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   κ 
                                   i 
                                 
                                 ⁢ 
                                 
                                   
                                     ∂ 
                                     
                                       n 
                                       i 
                                     
                                   
                                   
                                     ∂ 
                                     
                                       X 
                                       j 
                                     
                                   
                                 
                               
                             
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                       
                         
                           n 
                           
                             g 
                             ⁢ 
                             1 
                           
                         
                         - 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               2 
                             
                             
                               N 
                               + 
                               2 
                             
                           
                           
                             
                               κ 
                               i 
                             
                             ⁢ 
                             
                               n 
                               gi 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     10 
                   
                   ) 
                 
               
             
           
         
       
     
     The value of L 1  can be determined using Equation (3). The second MZI in the third-order MZI will can possess L′ 1 =2L 1  but the same values of κ i . Using the obtained values of L i , the values of t 1 , t 2 , t 3  may be optimized as well as a number of stages N to meet the specifications of extinction ratio, transmission loss and extinction bandwidth. In some embodiments, extinction bandwidth (BW) may be larger than the central wavelength shift Δλ c , e.g. BW&gt;&gt;Δλ c . A Monte-Carlo analysis of the system can be performed by repeating a relatively large number (R N ) of random simulations. The sampling can be conducted with knowledge of correlations in a representative fabrication process. In some embodiments, the process can be repeated until a favorable yield is obtained. 
     In some embodiments, numerical methods can be used to develop a MZI filter. The output of a filter can obtained using transfer matrices. A cascaded third-order filter can include directional couplers and the propagation of light in the two arms. A filter can be defined to be third-order when two asymmetric MZI&#39;s of differential length ΔL and 2ΔL are cascaded coherently. The transfer matrices for directional couplers and MZI arms are shown in Equation (11). 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       c 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             t 
                           
                           
                             
                               - 
                               jK 
                             
                           
                         
                         
                           
                             
                               - 
                               jK 
                             
                           
                           
                             t 
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       M 
                       
                         p 
                         ⁢ 
                         m 
                       
                     
                     = 
                     
                       
                         α 
                         
                           1 
                           / 
                           2 
                         
                       
                       [ 
                       
                         
                           
                             
                               
                                 e 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     ϕ 
                                     m 
                                   
                                 
                               
                               ⁢ 
                               
                                 α 
                                 m 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     11 
                   
                   ) 
                 
               
             
           
         
       
     
     In Equation (11), |t| 2  is the transmission coefficient of the directional coupler. Notably, K=√{square root over (1−|t| 2 )} while Ø r  (r=1, 2) corresponds to the differential phase in each of the two asymmetric MZI&#39;s that constitute a cascaded third-order filter. α m =e −rα′ΔL/2  correspond to the additional losses that accrue due to the differential length in each MZI, while α=e −rα′L/2  is the common absorption experienced by the nominal length L of the MZI arms. For the general multi-waveguide case, L=min(L 1 , Σ i κ i L 1 ) and ΔL=|L 1 −Σ i κ i L 1 |. Note that α′ is the absorption coefficient in units of 1/meter. 
     Upon utilizing the above transfer matrices the following expressions for the elements H mk  of the overall transfer matrix of the cascaded third-order filter was obtained. A single third-order filter can be defined by three couplers with corresponding parameters t 1 , t 2 , t 3  and two phase and absorption terms Ø r , α r , where r=1, 2 as shown in Equations (12a), (12b), (12c) and (12d). 
         H   11 (ω)=α[− K   1 (ω)( t   2 (ω) K   3 (ω)+α 2   K   2 (ω) t   3 (ω) e   −jϕ     2     (ω) )−α 1   t   1 (ω) e   −jϕ     1     (ω) ( K   2 (ω) K   3 (ω)−α 2   t (ω) t   3 (ω) e   −jϕ     2     (ω) )]  (Eq. 12a)
 
         H   12 (ω)=α[− jt   1 (ω)( K   3 (ω) t   2 (ω)+α 2   K   2 (ω) t   3 (ω) e   −jϕ     2     (ω) )+ jα   1 (ω) K   i (ω) e   −jϕ     1     (ω) (ω)( K   2 (ω) K   3 (ω)−α 2   t (ω) t   3 (ω) e   −jϕ     2     (ω) )]  (Eq. 12b)
 
         H   21 (ω)=α[ t   1   −jK   1 (ω)( t   2 (ω) t   3 (ω)+α 2   K   2 (ω) K   3 (ω) e   −jϕ     2     (ω) )− jα   1   t   1 (ω) e   −jϕ     1     (ω) (ω)( K   2 (ω) t   3 (ω)−α 2   t   2 (ω) K   3 (ω) e   −jϕ     2     (ω) )]  (Eq. 12c)
 
         H   22 (ω)=α[ t   1 (ω)( t   2 (ω) t   3 (ω)+α 2   K   2 (ω) K   3 (ω) e   −jϕ     2     (ω) )+ jα   1   k   1 (ω) e   −jϕ     1     (ω) (ω)( K   2 (ω) t   3 (ω)+α 2   K   3 (ω) t   2 (ω) e   −jϕ     2     (ω) )]  (Eq. 12d)
 
     The validity Equations (12a)-(12d) can be shown by verifying that |H qp (ω)| 2 +|H pp (ω)| 2 =1 for q, p=1, 2 under conditions of no loss (i.e. α′=0). This relates to the conservation of energy. The transmission loss and pump-rejection ratios can be calculated in Equations (13a) and (13b), respectively. 
     
       
         
           
             
               
                 
                   
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     In these embodiments the waveguides considered are silicon-on-insulator (SOI) strip waveguides, however other embodiments can use different configurations. The material dispersion can be based on the Palik model at room temperature. The dispersion of the effective index can be fit according to Equation (9). In this embodiment the center wavelength λ 0 =2πc/ω 0 =1.55 μm. The obtained coefficients are plotted in  FIG.  4    showing the effective index parameters as a function of waveguide width and height for a silicon-on-insulator waveguide. The obtained effective index is fit to Equation (9). 
     A full parameter sweep of the refractive index over angular frequency ω, waveguide width (ω) and thickness (h) is performed. In  FIGS.  5 A- 5 B , the derivatives are plotted with respect to w, h of the effective index at the wavelength λ 0 =1.55 μm. 
     In  FIGS.  5 A and  5 B , the derivative ∂n/∂w is plotted. The value is invariant with thickness but changes dramatically with width. This indicates that waveguide width variations can be mitigated using this approach. On the other hand, while ∂n/∂h does vary with width, it only does so mildly; it varies more with regard to thickness. The magnitude of change is about four times larger than ∂n/∂w. In some embodiments, standard deviations σ h  of the thickness tend be smaller than those of the widths (see Table. 1 showing parameters of simulations), which reduces their impact. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Parameters used for simulations 
               
            
           
           
               
               
            
               
                 Parameter 
                 Value 
               
               
                   
               
               
                 Standard deviation of  waveguide width (σ ω ) 
                 3 nm 5   
               
               
                 Standard deviation of  waveguide height (σ h ) 
                 0.5 nm 6   
               
               
                 Material index 
                 Palik (from Lumerical) 
               
               
                 Temperature (T) 
                 300K 
               
               
                 Absorption coefficient (α′) 
                 0.3 dBcm −1  or 7.5 m −1   
               
               
                 Transmission coefficients  (|t 1 (ω 0 )| 2 , |t 2 (ω 0) | 2 , |t 3 (ω 0 )| 2 ) 
                 0.5, 0.75, 0.93 
               
               
                 Number of stages 
                 4 
               
               
                 Pump, signal and idler distributions 
                 Gaussian with 5 GHz e −2  bandwidth 
               
               
                 Correlations 
                 ω, h for each third-order MZI  stage uncorrelated. 
               
               
                   
               
            
           
         
       
     
     Due to the relative invariance of ∂n/∂h, with respect to w, the strategy of using multiple waveguide widths to mitigate variation in this parameter may not be very efficacious for particular applications. In principle, a solution is possible but the lengths of arms obtained turn out to be in the range of centimeters which can be too large for some applications. Therefore, in some applications that may benefit from small filter sizes, it would be beneficial to reduce the values of σ h . 
     The coupling coefficients of the directional couplers can be determined by obtaining the even and odd modes of the coupled waveguide system. The coupling length can then be determined according to Equation (14). 
     
       
         
           
             
               
                 
                   
                     
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                       m 
                     
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                     ω 
                     ) 
                   
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                     sin 
                        
                     [ 
                     
                       
                         
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                           ⁢ 
                           
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     The statistical performance of standard cascaded third-order filters is examined to estimate the yield for such devices. In this approach a Monte-Carlo calculation was employed. Waveguide widths and thicknesses were chosen at random and their effective indices are obtained from the previously generated look-up tables. Similarly, the effective super-mode indices of the couplers are obtained. The coupling coefficients are then calculated using Equation (14) and the parameters from Table 1 are used. The standard deviations for waveguide width σ w =3 nm and thickness σ h =0.5 nm are plotted in  FIG.  6    showing statistical distributions of effective index n eff . 
     In this embodiment the entire dispersion curve has been shifted. The distribution of effective indices is slightly asymmetric. Therefore, in assuming a 3 nanometer waveguide width standard deviation and 0.5 nanometer standard deviation in thickness, this example evaluates variations more germane to die-to-die or intra-die variations. Therefore, a relevant parameter may be the critical dimension uniformity (CDU). 
     The overall performance for a N=4 stage, incoherently cascaded third-order filter can then be obtained. The design described above had the goal of meeting the specifications for a pump rejection filter, that can be, in one example, 120 dB of rejection and 50 mdB of loss. However, from  FIG.  7    that shows the statistical behavior of cascaded third-order MZI&#39;s without mitigation mechanisms, it can be seen that the mean rejection ratio has shifted to approximately 60 dB and the mean absorption coefficient has shifted to approximately 1800 mdB, however these may have different values in other embodiments. 
     In one embodiment, a fabrication tolerant MZI design uses asymmetric widths for each MZI arm. In this particular embodiment it is desired to mitigate variations to both thickness (h) and width (w), so the quantity in Equation (10) is minimized. The results are plotted in  FIG.  8    that illustrates designs to minimize susceptibility to fabrication errors. The minimization procedure yields a value of Δλ c ≈700 pm at various values of κ for varying values of w 1  and h=220 nm. Incidentally, the minimization yields ∂n 1 /∂w−∂n 2 /∂w=0, while being at the mercy of σ h |∂n 1 /∂h−∂n 2 /∂h|. Therefore, in some embodiments, σ h  should be reduced. 
     As shown in  FIG.  9   , the statistical distribution of cascaded third-order MZI&#39;s with asymmetric widths w 1 =500 nm and w 2 =540 nm, h=220 nm are illustrated. In some embodiments, this value can be reduced by increasing the height of the waveguides. For instance at h=245.5 nm, Δλ c ≈580 pm. However, it also appears that using thicker waveguides in some embodiments causes the transmission loss to increase due to dispersion. Therefore, over engineering this aspect of the system may not be worthwhile for some embodiments. Using such a configuration,  FIG.  9    illustrates the performance for N=4 incoherently cascaded third-order filters. An improvement in performance compared to that depicted in  FIG.  7    is evident with the mean rejection ratio shifting to 110 dB and mean loss shifting to 188 mdB. 
     Furthermore, if coupler variations with respect to fabrication uncertainties (simply referred to as coupler variations henceforth) are ignored, then the performance is shown in  FIG.  10    illustrating the statistical distribution of cascaded third-order MZI&#39;s in the absence of coupler variations with respect to fabrication uncertainties. The rejection ratio shifts to 154 dB, while the transmission loss changes to 165 mdB. In some embodiments, this can indicate that coupler variations predominantly produce vertical movements in the spectral response while the index changes produce mainly horizontal shifts. Horizontal shifts affect both rejection ratio and transmission loss, while vertical shifts predominantly affect rejection ratios. 
     In some embodiments, while using asymmetric arms can make |∂n 1 /∂w−∂n 2 /∂w|=0, it may not correlate to a transmission minimum located at λ 0 =1.55 μm. In the above embodiments, it is fortuitous that for κ+δκ, the above resonance condition is satisfied. Here, δκ is a relatively small amount of adjustment imparted to κ. Therefore, there may be a residual error of −δκ∂ 2   2 /∂w, which is may be undesirable. However, if two additional waveguide widths are used (i.e. w 2 , w 3 ), then some embodiments may have improved results. This is demonstrated in  FIG.  11   , where one of the arms contains two widths of 0.5, 0.66 microns. More specifically,  FIG.  11    illustrates the statistical distribution of MZI properties for three waveguide widths L 1 =22.96 μm, m=58, κ i =[1, 4.2805, −4.6974] and w i =[0.5, 0.56, 0.66] microns. In this embodiment, the design can be constrained to satisfy a condition for transmission minimum at λ 0  (Equation (1)), FSR (Equation (3)) and insensitivity to width variations (Equation (5)). 
     In  FIG.  11   , coupler variations are neglected, building on the results from  FIG.  10   . As can be seen, there is an additional 10 dB improvement in rejection ratio, while an improvement in transmission loss by approximately 70 mdB. While thickness variations may not be mitigated using this approach since ∂n/∂h is not a function of w, the additional constraint of having ∂ 2 Ø∂w∂ω=0 may be included, which yields the performance in  FIG.  12    showing the statistical distribution of an MZI filter with four waveguide widths. L 1 =25.51 microns, κ i =[1, 4.1464, −4.5875, 0.1662] and w i =[0.5, 0.56, 0.66, 0.76] microns. The shape of the distribution appears to change, although improvements in mean values do not appear to occur. 
     In principle, compensation for perturbations in w, h can be simultaneously achieved by choosing arms with different w, h as shown in Equation (15). 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           n 
                           1 
                         
                       
                       
                         ∂ 
                         w 
                       
                     
                     
                       | 
                       
                         
                           w 
                           1 
                         
                         , 
                         
                           h 
                           1 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           
                             n 
                             2 
                           
                         
                         
                           ∂ 
                           w 
                         
                       
                       
                         | 
                         
                           
                             w 
                             2 
                           
                           , 
                           
                             h 
                             2 
                           
                         
                       
                         
                       
                         
                           ∂ 
                           
                             n 
                             1 
                           
                         
                         
                           ∂ 
                           h 
                         
                       
                       
                         | 
                         
                           
                             w 
                             1 
                           
                           , 
                           
                             h 
                             1 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             n 
                             2 
                           
                         
                         
                           ∂ 
                           h 
                         
                       
                       
                         | 
                         
                           
                             w 
                             2 
                           
                           , 
                           
                             h 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     15 
                   
                   ) 
                 
               
             
           
         
       
     
     This results in a value of Δλ c =26 pm. The results are plotted in  FIG.  13    illustrating fabrication tolerance achieved using asymmetric widths as well as heights. w 1 =500 nm, w 2 =535 nm, h 1 =220 nanometers and h 2 =245 nanometers. The average pump rejection shifts to 174 dB and the average loss is 125 mdB, which is smaller compared to the case when only asymmetric widths without coupler variations ( FIG.  10   ) are considered. Here, too the effect of coupler variations have been ignored. The marginal increase in absorption relative to  FIGS.  11    and  12  is that the constraint of fixing λ c  is not satisfied. In some embodiments, waveguide geometries that effectively enable different heights (such as rib waveguides) can be used. Furthermore, some embodiments can use both different heights and multiple widths to further improve performance. 
     While obtaining different thicknesses can be challenging in some embodiments, there may be ways to accomplish this by using unconventional cross-sections that are compatible with current CMOS-foundry processes, as shown in  FIGS.  14 A- 14 D . In one example embodiment shown in  FIG.  14 A , a conventional strip waveguide is shown. In  FIG.  14 B , a cross-section which has an additional silicon nitride or silicon layer on top of the SOI strip waveguide that modifies the effective height is shown.  FIG.  14 C  illustrates a rib waveguide and  FIG.  14 D  illustrates a modified rib waveguide showing two other embodiments that offer height changes. 
     The effect of coupler dispersion and insertion loss on the system can now be considered. All the systems are assumed to possess the three-waveguide design from  FIG.  11   , however other embodiments may have other configurations. In the first embodiment illustrated in  FIG.  15   , the coupler dispersion is retained while waveguide loss is reduced to 0.1 dB/cm.  FIG.  15    illustrates performance of the filter when the loss is reduced to 0.1 dB/cm and with couplers robust to fabrication variations but with varying transmission with respect to frequency. 
     With this improvement, the transmission loss has reduced to 93 mdB, while the pump rejection has been minimally altered. On the contrary, when the loss is maintained at 0.3 dB/cm but the couplers are fab-tolerant and also not dispersive, the transmission loss falls below the 50 mdB level as shown in  FIG.  16   .  FIG.  16    illustrates performance of the filter when the loss is 0.3 dB/cm with couplers that are robust to fabrication variations and also with constant transmission with respect to frequency. 
     Thus, in order to meet device specifications, in some embodiments, the couplers may be fab-tolerant and broadband. When the loss is also reduced to 0.1 dB/cm with couplers robust to fabrication and also with constant transmission coefficients with respect to frequency, transmission losses reduce to 28 mdB as can be seen in  FIG.  17   .  FIG.  17    illustrates performance of the filter when loss is reduced to 0.1 dB/cm with couplers robust to fabrication variations and also with constant transmission with respect to frequency. In order to reduce transmission loss values below 25 mdB, the number of stages may be reduced to three, although this may also reduce the mean pump rejection ratio. Therefore, in order to further improve the yield, improved process control or reduced values of σ w , σ h  may be needed. 
     As described above, the variations of width and thickness were treated as independent random variables and each stage was assumed to vary independently. In this section the case when the width and height variations are uncorrelated but all stages are well-correlated is evaluated. When the correlation between each stage increases, the spread in performance increases as shown in  FIG.  18   .  FIG.  18    illustrates asymmetric widths where width and height variations are independent and every stage is correlated with a loss of 0.3 dB/cm. 
     However, in some embodiments, if the couplers are made insensitive to fabrication, then the performance improves as seen in  FIG.  19   .  FIG.  19    shows an embodiment having asymmetric widths where width and height variations are independent and each stage is correlated. Couplers are considered fabrication tolerant and the loss is 0.3 dB/cm. Using three or four waveguide widths, as was the case in  FIGS.  11  and  12   , improves performance even more, bringing elements close to specifications in  FIG.  20   .  FIG.  20    illustrates an embodiment where width and height variations of every stage are correlated. In additional to mitigating coupler variations, one embodiment uses three waveguide widths. The waveguide loss assumed in this embodiment is 0.3 dB/cm. 
     If develop broadband couplers are developed while maintaining loss at 0.3 dB/cm, the performance improvement is line with trends in the previous embodiments, as shown in  FIG.  21   .  FIG.  21    illustrates an embodiment where width and height variations are independent but are correlated for all stages. The structure uses three waveguide widths and fab-tolerant and broadband couplers and the insertion loss is 0.3 dB/cm. As illustrated in  FIG.  22   , the loss is reduced to 0.1 dB/cm, which brings the performance to similar levels as shown in  FIG.  17   . More specifically, even when the correlations are not favorable, the devices have comparable yield.  FIG.  22    illustrates an embodiment where width and height variations are independent but are correlated for all stages. The structure uses three waveguide widths, fab-tolerant and broadband couplers as well as a reduced insertion loss of 0.1 dB/cm. 
     Serial improvements are summarized that can be achieved for various design improvements shown in Table 3. In some embodiments, broadband, fabrication insensitive couplers enable the system to meet performance specifications. In further embodiments, reducing waveguide losses on-chip may help improve the performance and yield. In addition embodiments having σ w &lt;3 nanometers and σ h &lt;0.5 nanometers may be used. 
     Table 3 summarizes different embodiments that may have reduced performance and also identifies various strategies that could potentially address the performance. Each point labelled (i)-(iv) in Table 3 is discussed in more detail below. 
     (i) In some embodiments, the use of asymmetric arm widths may achieve tuning-free operation of cascaded third-order filters. Use of three or four waveguide widths helps achieve pinning the transmission minimum and also compensates ∂ 2 n/∂ω∂ω. 
     (ii) In some embodiments, the use of multi-waveguide sections can mitigate many sources of variation but due to the invariance of ∂n/∂h to w, this approach may need long device lengths to mitigate thickness variations. In principle, using different waveguide heights can also address thickness variation issues, although this may not be a CMOS-foundry compatible process. Some embodiments may use unconventional waveguide geometries to effectively engineer a height difference. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                   
                 Mean Pump 
                   
                   
                 Mean 
                 μ loss  + 
                   
               
               
                   
                 rejection 
                 μ − 
                 ≥120 
                 Transmission 
                 σ loss   
                 ≤25 mdB 
               
               
                 Design 
                 μpump (dB) 
                 σ pump (dB) 
                 dB(%) 
                 loss (mdB) 
                 (mdB) 
                 (%) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 Standard third-order, 4 stage MZI, 
                 60 
                 44 
                 0 
                 1830 
                 3074 
                 0 
               
               
                 0.3 dB/cm 
               
               
                 Asymmetric widths, 0.3 dB/cm 
                 110 
                 92 
                 28 
                 188 
                 280 
                 0 
               
               
                 Asymmetric widths and robust 
                 154 
                 130 
                 87 
                 164 
                 237 
                 0 
               
               
                 couplers, 0.3 dB/cm 
               
               
                 Multiple widths, robust couplers, 
                 163 
                 97.4 
                 99.4 
                 110 
                 155.1 
                 0 
               
               
                 0.3 dB/cm 
               
               
                 Standard third-order, 4 stage MZI, 
                 163 
                 98.2 
                 99.43 
                 44 
                 80 
                 2.2 
               
               
                 0.3 dB/cm 
               
               
                 Multiple widths, robust couplers and 
                 163 
                 143.5 
                 97.1 
                 93 
                 144.1 
                 0 
               
               
                 0.1 dB/cm loss 
               
               
                 Multiple widths, robust and broadband 
                 163 
                 141.5 
                 97.4 
                 28 
                 67 
                 67.3 
               
               
                 couplers and 0.1 dB/cm loss 
               
               
                 Multiple widths, robust couplers and 
                 122 
                 104 
                 55.1 
                 22 
                 56 
                 76.2 
               
               
                 0.1 dB/cm loss, 3 stages 
               
               
                 σ w  = 1 nm, σ h  = 0.25 nm and Multiple 
                 188 
                 175 
                 100 
                 10.5 
                 12.5 
                 99.7 
               
               
                 widths, robust, broadband couplers, 
               
               
                 0.1 dB/cm, 4 stages 
               
               
                 σ w  = 1 nm, σ h  = 0.25 nm and Multiple 
                 141 
                 131 
                 98.5 
                 7.8 
                 8.8 
                 100 
               
               
                 widths, robust, broadband couplers, 
               
               
                 0.1 dB/cm, 3 stages 
               
               
                 Asymmetric widths, correlated stage 
                 111 
                 73.26 
                 36 
                 210 
                 609 
                 0 
               
               
                 variations, 0.3 dB/cm 
               
               
                 Asymmetric widths, robust couplers and 
                 148 
                 102 
                 69 
                 178 
                 329.3 
                 0 
               
               
                 correlated stage variations, 0.3 dB/cm 
               
               
                 Multiple widths, robust couplers and 
                 164 
                 123 
                 78 
                 105 
                 177 
                 0 
               
               
                 correlated stage variations, 0.3 dB/cm 
               
               
                 Multiple widths, robust and broadband 
                 163 
                 120.75 
                 80 
                 44 
                 103 
                 21 
               
               
                 couplers and correlated stage variations, 
               
               
                 0.3 dB/cm 
               
               
                 Multiple widths, robust and broadband 
                 163 
                 120 
                 78 
                 28 
                 95 
                 79 
               
               
                 couplers and correlated stage variations, 
               
               
                 0.1 dB/cm 
               
               
                 σ w  = 1 nm, σ h  = 0.25 nm and Multiple 
                 187 
                 163 
                 99.2 
                 10.5 
                 13.85 
                 99.4 
               
               
                 widths, robust, broadband couplers, 
               
               
                 0.1 dB/cm, correlated 
               
               
                 σ w  = 1 nm, σ h  = 0.25 nm and Multiple 
                 141 
                 124 
                 91 
                 7.8 
                 10.38 
                 99.6 
               
               
                 widths, robust, broadband couplers, 
               
               
                 0.1 dB/cm, 3 stages 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
               
                 Problem 
                 Reason 
                 Value 
                 Strategy 
               
               
                   
               
             
            
               
                 (i) Resonance shift 
                   
                   
                 Multiple waveguide widths 
               
               
                 (ii) Sensitivity  to height 
                 Equal  heights 
                 62 mdB  
                 Si—SiO2—Si, Si—SiN—Si  waveguides 
               
               
                 (iii) Transmission loss 
                 Dispersion 
                   
                 Broadband couplers 
               
               
                   
                 in couplers 
                   
                   
               
               
                 (iv) Bandwidth of filter 
                 Roll-off 
                 2 nm at 
                 Reduced waveguide loss 
               
               
                   
                   
                 −150 dB 
                 Additional stages or  alternate architectures 
               
               
                   
               
            
           
         
       
     
     As shown in  FIG.  23   , in some embodiments, reducing σ w  to 1 nanometer and σ h  to 0.25 nanometer from 3 and 0.5 nanometer respectively enables the specifications to comfortably meet the goals. 
     (iii) In some embodiments, the role of coupler dispersion and variations with fabrication may be important. Designing couplers that are more broadband and insensitive to fabrication variations may be needed to make a filter robust to perturbations. 
     (iv) In some embodiments, to meet specifications, loss may reach approximately 0.1 dB/cm. This may enable specifications to be exceeded by adding further cascaded third-order MZI filter stages. In further embodiments, using three stages may meet rejection ratio targets while keeping losses below the 25 mdB level. 
     (v) In some embodiments, further improvement of fabrication tolerances to σ w &lt;&lt;3 nm and σ h &lt;&lt;0.5 nanometer may improve the mean pump rejection to 188 dB and average loss to 10.54 mdB for a four stage cascaded third-order MZI as is seen in  FIGS.  23 A and  23 B . 
       FIG.  24    illustrates yield percentage of loss ≤25 mdB and rejection ratio ≥120 db. Variations are correlated, with insertion loss of 0.1 dB/cm, broadband and fab-tolerant couplers and multiple waveguide width arms. 
     Although MZI filter  100  (see  FIG.  1   ) is described and illustrated as one particular type of MZI-based photonic device, a person of skill in the art with the benefit of this disclosure will appreciate that compensation structures as described above are suitable for use with myriad other MZI-based photonic devices. For example, in some embodiments the MZI passive compensation structures disclosed herein can be implemented in MZI-based photonic switching devices, as described in more detail below. 
       FIGS.  25  and  26    show example MZI-based photonic switches  2500  and  2600 , respectively, that include one or more variable phase-shifters and can also include one or more compensation structures. Photonic switches  2500  and  2600  are similar to MZI filter  100  (see  FIG.  1   ), each having two parallel waveguides ( 2510   a ,  2510   b  in  FIG.  25 , and  2610     a  and  2610   b  in  FIG.  26   ), however photonic switches  2500  and  2600  each include one or more phase shifters ( 2505   a ,  2505   b ,  2505   c  in  FIGS.  25  and  2605    in  FIG.  26   ) disposed in one or more waveguides of each photonic switch. Phase-shifters ( 2505   a ,  2505   b ,  2505   c  in  FIGS.  25  and  2605    in  FIG.  26   ) can be implemented a number of ways in integrated photonic circuits and can provide control over the relative phases imparted to the optical field in each waveguide. In some embodiments, variable phase-shifters can be implemented using thermo-optical switches. 
     In some embodiments thermo-optical switches can use resistive elements fabricated on a surface of the photonic device. Employing the thermo-optical effect in these devices can provide a change of the refractive index n by raising the temperature of the waveguide by an amount of the order of 10 −5  K. One of skill in the art having had the benefit of this disclosure will understand that any effect that changes the refractive index of a portion of the waveguide can be used to generate a variable, electrically tunable, phase shift. For example, some embodiments can use beam splitters based on any material that supports an electro-optic effect. In some embodiments so-called χ (2)  and χ (3)  materials can be used such as, for example, lithium niobate, BBO, KTP, BTO, and the like and even doped semiconductors such as silicon, germanium, and the like. 
     In some embodiments, switches with variable transmissivity and arbitrary phase relationships between output ports can also be achieved by combining directional couplings (e.g., directional couplings  2515   a ,  2515   b  in  FIGS.  25  and  2615     a ,  2615   b  in  FIG.  26   ), and one or more variable phase-shifters (e.g., phase-shifters  2505   a ,  2505   b ,  2505   c  in  FIGS.  25  and  2605    in  FIG.  26   ) within each photonic switch. Accordingly, complete (e.g., analog or digital) control over the relative phase and amplitude of the two output ports can be achieved by varying the phases imparted by phase shifters ( 2505   a ,  2505   b ,  2505   c  in  FIGS.  25  and  2605    in  FIG.  26   ).  FIG.  26    illustrates a slightly simpler example of a MZI-based photonic switch that allows for variable transmissivity between ports  2620   a  and  2620   b  by varying a phase imparted by phase shifter  2605 . 
     In some embodiments one or more compensation structures can be implemented within MZI-based photonic switches  2500 , 2600  using compensation equations similar to those described above with regard to MZI filter  100  (see  FIG.  1   ). More specifically, the compensation equations can be used to determine a width and a length of each compensation portion that can be used to reduce a shift in frequency response caused by various perturbations, including variations in manufacturing widths of the waveguides, manufacturing variations in thicknesses of the waveguides and variations in temperature. Similar to the compensation structures described for MZI filter  100  (see  FIG.  1   ), compensation structures can be employed in one or more waveguides ( 2510   a ,  2510   b  in  FIG.  25 , and  2610     a  and  2610   b  in  FIG.  26   ), and each compensation structure can each have a quantity of waveguide widths that is greater than the number of perturbations, however the governing equations may be different for an MZI-based photonic switch embodiment, as described in more detail below. 
     The phase relationship in an MZI-based photonic switch embodiment may be described as follows. The first two terms can be the same as MZI filter  100  (see  FIG.  1   ), however a third term corresponding to a sum of various index changes, Δn j , weighted by various overlap integrals Γ j , can be added, as described by Equations (16) and (17). 
     
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             m 
                           
                           + 
                           1 
                         
                         ) 
                       
                       ⁢ 
                       
                         λ 
                         0 
                       
                     
                     2 
                   
                   = 
                   
                     
                       
                         
                           n 
                           1 
                         
                         ( 
                         
                           ω 
                           0 
                         
                         ) 
                       
                       ⁢ 
                       
                         L 
                         1 
                       
                     
                     - 
                     
                       
                         Σ 
                         i 
                       
                       ⁢ 
                       
                         κ 
                         i 
                       
                       ⁢ 
                       
                         L 
                         1 
                       
                       ⁢ 
                       
                         
                           n 
                           i 
                         
                         ( 
                         
                           ω 
                           0 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         Σ 
                         j 
                       
                       ⁢ 
                       
                         
                           Γ 
                           j 
                         
                         ( 
                         
                           ω 
                           0 
                         
                         ) 
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         
                           n 
                           j 
                         
                         ( 
                         
                           ω 
                           0 
                         
                         ) 
                       
                       ⁢ 
                       
                         L 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     16 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   	 
                   
                     
                       v 
                       
                         F 
                         ⁢ 
                         S 
                         ⁢ 
                         R 
                       
                     
                     = 
                     
                       c 
                       
                         
                           L 
                           1 
                         
                         ( 
                         
                           
                             n 
                             
                               g 
                               ⁢ 
                               1 
                             
                           
                           - 
                           
                             
                               Σ 
                               i 
                             
                             ⁢ 
                             
                               n 
                               
                                 g 
                                 ⁢ 
                                 i 
                               
                             
                             ⁢ 
                             
                               κ 
                               i 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     17 
                   
                   ) 
                 
               
             
           
         
       
     
     The corresponding compensation equation for the case of an MZI-based photonic switch which requires invariance to width can be described by Equation (18). 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         ϕ 
                         j 
                       
                     
                     
                       ∂ 
                       w 
                     
                   
                   = 
                   
                     
                       
                         L 
                         1 
                       
                       ( 
                       
                         
                           
                             ∂ 
                             
                               n 
                               1 
                             
                           
                           
                             ∂ 
                             w 
                           
                         
                         - 
                         
                           
                             
                               Σ 
                               i 
                             
                             ⁢ 
                             
                               κ 
                               i 
                             
                             ⁢ 
                             
                               ∂ 
                               
                                 n 
                                 i 
                               
                             
                           
                           
                             ∂ 
                             w 
                           
                         
                         + 
                         
                           
                             
                               
                                 Σ 
                                 j 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   Γ 
                                   j 
                                 
                               
                             
                             
                               ∂ 
                               w 
                             
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                             n 
                             j 
                           
                         
                       
                       ) 
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     18 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation (18) can be reduced to Equation (19). 
     
       
         
           
             
               
                 
                   
                     
                       
                         Σ 
                         i 
                       
                       ⁢ 
                       
                         κ 
                         i 
                       
                       ⁢ 
                       
                         ∂ 
                         
                           n 
                           i 
                         
                       
                     
                     
                       ∂ 
                       w 
                     
                   
                   = 
                   
                     
                       
                         Σ 
                         j 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             Γ 
                             j 
                           
                         
                         
                           ∂ 
                           w 
                         
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         n 
                         j 
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           n 
                           1 
                         
                       
                       
                         ∂ 
                         w 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     19 
                   
                   ) 
                 
               
             
           
         
       
     
     In some embodiments, Equations (20) through (22) can be used to account for compensation of higher-order derivatives. 
     
       
         
           
             
               
                 
                                    
                   
                     
                       
                         
                           Σ 
                           i 
                         
                         ⁢ 
                         
                           κ 
                           i 
                         
                         ⁢ 
                         
                           
                             ∂ 
                             2 
                           
                           
                             n 
                             i 
                           
                         
                       
                       
                         
                           ∂ 
                           w 
                         
                         ⁢ 
                         
                           ∂ 
                           ω 
                         
                       
                     
                     = 
                     
                       
                         
                           Σ 
                           j 
                         
                         ⁢ 
                         
                           
                             ∂ 
                             
                               ∂ 
                               ω 
                             
                           
                           
                             [ 
                             
                               
                                 
                                   ∂ 
                                   
                                     Γ 
                                     j 
                                   
                                 
                                 
                                   ∂ 
                                   w 
                                 
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                 n 
                                 j 
                               
                             
                             ] 
                           
                         
                       
                       + 
                       
                         
                           
                             ∂ 
                             2 
                           
                           
                             n 
                             1 
                           
                         
                         
                           
                             ∂ 
                             w 
                           
                           ⁢ 
                           
                             ∂ 
                             ω 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     20 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       
                         
                           Σ 
                           i 
                         
                         ⁢ 
                         
                           κ 
                           i 
                         
                         ⁢ 
                         
                           
                             ∂ 
                             2 
                           
                           
                             n 
                             i 
                           
                         
                       
                       
                         ∂ 
                         
                           w 
                           2 
                         
                       
                     
                     = 
                     
                       
                         
                           Σ 
                           j 
                         
                         [ 
                         
                           
                             
                               
                                 ∂ 
                                 2 
                               
                               
                                 Γ 
                                 j 
                               
                             
                             
                               ∂ 
                               
                                 w 
                                 2 
                               
                             
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                             n 
                             j 
                           
                         
                         ] 
                       
                       + 
                       
                         
                           
                             ∂ 
                             2 
                           
                           
                             n 
                             1 
                           
                         
                         
                           ∂ 
                           
                             w 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     21 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             n 
                             1 
                           
                           ( 
                           
                             ω 
                             0 
                           
                           ) 
                         
                         - 
                         
                           
                             Σ 
                             i 
                           
                           ⁢ 
                           
                             
                               n 
                               i 
                             
                             ( 
                             
                               ω 
                               0 
                             
                             ) 
                           
                           ⁢ 
                           
                             κ 
                             i 
                           
                         
                         + 
                         
                           
                             Σ 
                             j 
                           
                           ⁢ 
                           
                             
                               Γ 
                               j 
                             
                             ( 
                             
                               ω 
                               0 
                             
                             ) 
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                             
                               n 
                               j 
                             
                             ( 
                             
                               ω 
                               0 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                     
                       ( 
                       
                         
                           n 
                           
                             g 
                             ⁢ 
                             1 
                           
                         
                         - 
                         
                           
                             Σ 
                             i 
                           
                           ⁢ 
                           
                             n 
                             
                               g 
                               ⁢ 
                               i 
                             
                           
                           ⁢ 
                           
                             κ 
                             i 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     γ 
                     = 
                     
                       
                         
                           ( 
                           
                             m 
                             + 
                             
                               1 
                               2 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           λ 
                           0 
                         
                       
                       
                         c 
                         / 
                         FSR 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     22 
                   
                   ) 
                 
               
             
           
         
       
     
     Generalizing to arbitrary perturbations X k , the set of compensation equations for MX=B can be described by Equations (23) through (25). 
     
       
         
           
             
               
                 
                   M 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               γ 
                               ⁢ 
                               
                                 n 
                                 
                                   g 
                                   ⁢ 
                                   2 
                                 
                               
                             
                             - 
                             
                               n 
                               2 
                             
                           
                         
                         
                           
                             
                               γ 
                               ⁢ 
                               
                                 n 
                                 
                                   g 
                                   ⁢ 
                                   3 
                                 
                               
                             
                             - 
                             
                               n 
                               3 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               γ 
                               ⁢ 
                               
                                 n 
                                 
                                   g 
                                   ⁡ 
                                   ( 
                                   
                                     N 
                                     + 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               n 
                               
                                 N 
                                 + 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 2 
                               
                             
                             
                               ∂ 
                               
                                 X 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 3 
                               
                             
                             
                               ∂ 
                               
                                 X 
                                 1 
                               
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 
                                   N 
                                   + 
                                   2 
                                 
                               
                             
                             
                               ∂ 
                               
                                 X 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           … 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 2 
                               
                             
                             
                               ∂ 
                               
                                 X 
                                 K 
                               
                             
                           
                         
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 3 
                               
                             
                             
                               ∂ 
                               
                                 X 
                                 K 
                               
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 
                                   N 
                                   + 
                                   2 
                                 
                               
                             
                             
                               ∂ 
                               
                                 X 
                                 K 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 2 
                               
                             
                             
                               
                                 ∂ 
                                 ω 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   X 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 3 
                               
                             
                             
                               
                                 ∂ 
                                 ω 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   X 
                                   1 
                                 
                               
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 
                                   N 
                                   + 
                                   2 
                                 
                               
                             
                             
                               
                                 ∂ 
                                 ω 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   X 
                                   1 
                                 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           … 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 2 
                               
                             
                             
                               
                                 ∂ 
                                 ω 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   X 
                                   K 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 3 
                               
                             
                             
                               
                                 ∂ 
                                 ω 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   X 
                                   K 
                                 
                               
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               ∂ 
                               
                                 n 
                                 
                                   N 
                                   + 
                                   2 
                                 
                               
                             
                             
                               
                                 ∂ 
                                 ω 
                               
                               ⁢ 
                               
                                 ∂ 
                                 
                                   X 
                                   K 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     23 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   X 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             κ 
                             2 
                           
                         
                       
                       
                         
                           
                             κ 
                             3 
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             κ 
                             
                               N 
                               + 
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     24 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   B 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               γ 
                               ⁢ 
                               
                                 n 
                                 
                                   g 
                                   ⁢ 
                                   1 
                                 
                               
                             
                             - 
                             
                               n 
                               1 
                             
                             - 
                             
                               
                                 Σ 
                                 j 
                               
                               ⁢ 
                               
                                 
                                   Γ 
                                   j 
                                 
                                 ( 
                                 
                                   ω 
                                   0 
                                 
                                 ) 
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                 
                                   n 
                                   j 
                                 
                                 ( 
                                 
                                   ω 
                                   0 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 ∂ 
                                 
                                   n 
                                   1 
                                 
                               
                               
                                 ∂ 
                                 
                                   X 
                                   1 
                                 
                               
                             
                             + 
                             
                               
                                 Σ 
                                 j 
                               
                               ⁢ 
                               
                                 
                                   ∂ 
                                   
                                     Γ 
                                     j 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     X 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                 n 
                                 j 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               
                                 ∂ 
                                 
                                   n 
                                   1 
                                 
                               
                               
                                 ∂ 
                                 
                                   X 
                                   K 
                                 
                               
                             
                             + 
                             
                               
                                 Σ 
                                 j 
                               
                               ⁢ 
                               
                                 
                                   ∂ 
                                   
                                     Γ 
                                     j 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     X 
                                     k 
                                   
                                 
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                 n 
                                 j 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               
                                 
                                   ∂ 
                                   2 
                                 
                                 
                                   n 
                                   1 
                                 
                               
                               
                                 
                                   ∂ 
                                   ω 
                                 
                                 ⁢ 
                                 
                                   ∂ 
                                   
                                     X 
                                     K 
                                   
                                 
                               
                             
                             + 
                             
                               
                                 
                                   ∂ 
                                   
                                     ∂ 
                                     ω 
                                   
                                 
                                 
                                   Σ 
                                   j 
                                 
                               
                               ⁢ 
                               
                                 
                                   ∂ 
                                   
                                     Γ 
                                     j 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     X 
                                     k 
                                   
                                 
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                 n 
                                 j 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               
                                 
                                   ∂ 
                                   2 
                                 
                                 
                                   n 
                                   1 
                                 
                               
                               
                                 ∂ 
                                 
                                   X 
                                   K 
                                   2 
                                 
                               
                             
                             + 
                             
                               
                                 Σ 
                                 j 
                               
                               ⁢ 
                               
                                 
                                   
                                     ∂ 
                                     2 
                                   
                                   
                                     Γ 
                                     j 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     X 
                                     k 
                                     2 
                                   
                                 
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                 n 
                                 j 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                         
                     25 
                   
                   ) 
                 
               
             
           
         
       
     
     Photonic switches  2500  and  2600  illustrated in  FIGS.  25  and  26   , respectively, and the associated compensation equations are two examples of how compensation structures can be implemented in myriad MZI-based photonic devices. One of skill in the art with the benefit of this disclosure can appreciate that similar compensation structures can be implemented in other MZI-based photonic devices. 
     For simplicity, various components, such as the optical pump circuitry, substrates, cladding, and other components of MZI filter  100  (see  FIG.  1   ) are not shown in the figures. In the foregoing specification, embodiments of the disclosure have been described with reference to numerous specific details that can vary from implementation to implementation. The specification and drawings are, accordingly, to be regarded in an illustrative rather than a restrictive sense. The sole and exclusive indicator of the scope of the disclosure, and what is intended by the applicants to be the scope of the disclosure, is the literal and equivalent scope of the set of claims that issue from this application, in the specific form in which such claims issue, including any subsequent correction. The specific details of particular embodiments can be combined in any suitable manner without departing from the spirit and scope of embodiments of the disclosure. 
     Additionally, spatially relative terms, such as “bottom or “top” and the like can be used to describe an element and/or feature&#39;s relationship to another element(s) and/or feature(s) as, for example, illustrated in the figures. It will be understood that the spatially relative terms are intended to encompass different orientations of the device in use and/or operation in addition to the orientation depicted in the figures. For example, if the device in the figures is turned over, elements described as a “bottom” surface can then be oriented “above” other elements or features. The device can be otherwise oriented (e.g., rotated 90 degrees or at other orientations) and the spatially relative descriptors used herein interpreted accordingly.