Abstract:
Apparatus for providing Bias-Instability reduction in Fiber Optic Gyroscopes are provided. In one embodiment, an optical circuit for a fiber optic gyroscope having a broadband light source and an optical fiber loop comprises: a PM fiber of length v; an IOC coupled to the PM fiber via a pigtail of length d 1 , a second pigtail with length of d 2 , and a third pigtail with length of d 3 ; a splitter that splits light received from the first pigtail into a first and second beams directed to the second and third pigtails; and a depolarizer circuit coupled to said fiber loop including a first fiber section of length x, a second fiber section of length z, a third fiber section of length w and a fourth fiber section of length y; and wherein the lengths of v, x, y, d 1  and d 3  are proportioned to avoid regions of high bias instability.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application is a continuation of, and claims priority to, U.S. patent application Ser. No. 12/187,932 (hereafter “the &#39;932 Application”), entitled “BIAS-INSTABILITY REDUCTION IN FIBER OPTIC GYROSCOPES” filed on Aug. 7, 2008. 
    
    
     GOVERNMENT LICENSE RIGHTS 
     The U.S. Government may have certain rights in the present invention as provided for by the terms of a Government Contract No. N00030-08-C-0010 with the U.S. Navy. 
    
    
     BACKGROUND OF THE INVENTION 
     In a fiber optic gyroscope, polarization errors result from interference between the primary lightwaves and spurious lightwaves, as well as interference among spurious lightwaves. The weak spurious lightwaves come from polarization cross-couplings at non-ideal fiber splices, in birefringent coil fiber, at junctions between integrated optical circuit (IOC) waveguides and its pigtail fibers, and inside optical components. These spurious lightwaves reaching the photodetector carry erroneous Sagnac phase information because they traveled nonreciprocal paths in the sensing loop. Methods of using Lyot-type fiber depolarizers and tailoring of the depolarizer polarization maintaining (PM) fiber lengths have been suggested to mitigate polarization errors. In prior art polarization error models, the polarization errors are evaluated in the time domain by keeping track of time and phase delays of spurious lightwaves originating from cross-couplings in the optical circuit. Wavelength dependent properties of the optical component, such as polarization dependent loss (PDL), polarization mode dispersion (PMD), etc., are often completely or partially ignored in the model. Specifically, for a depolarized gyroscope using a non-polarization maintaining single-mode (SM) fiber coil, the impact of the SM coil birefringence on the polarization error is empirically taken into account by assuming a broadened light source coherence function. Such simplifications lead to inaccuracies (up to one order of magnitude of deviation) in evaluation of polarization errors. More accurate modeling methods are needed to find optimal design parameters of interferometric fiber optic gyroscopes with reduced polarization error and bias instability. 
     SUMMARY 
     In one embodiment, an optical circuit for a fiber optic gyroscope having a broadband light source and a single mode optical fiber loop comprises: a polarization maintaining (PM) delay management fiber of length v, having a first end and a second end, wherein the first end of the polarization maintaining delay management fiber is coupled a directional coupler that receives light from the broadband light source; an integrated optical circuit (IOC) having the integrated optical circuit coupled to the second end of the polarization maintaining delay management fiber via a first polarization maintaining fiber pigtail of length d 1 ; the integrated optical circuit coupled to a second polarization maintaining fiber pigtail with length of d 2 ; and the integrated optical circuit coupled to a third polarization maintaining fiber pigtail with length of d 3 ; wherein the integrated optic circuit (IOC) comprises a polarization maintaining waveguide device having waveguide length of L IOC  and comprise a polarizing element so that the total effective polarization extinction ratio (PER) of the integrated optic circuit is ε. The circuit further comprises a splitter that splits light received from the first polarization maintaining fiber pigtail into a first beam directed to the second polarization maintaining fiber pigtail and a second beam directed to the third polarization maintaining fiber pigtail; and a depolarizer circuit coupled to said fiber loop, said depolarizer region including a first optical fiber section of length x, a second optical fiber section of length z, a third optical fiber section of length w and a fourth optical fiber section of length y, each comprising polarization maintaining fibers having a beat length of L B ; wherein the second optical fiber section is coupled to the first optical fiber section via a first splice, the first optical fiber section is coupled to a first end of the single mode optical fiber loop via a second splice, the fourth optical fiber section is coupled to the third optical fiber section via a third splice, and the third optical fiber section is coupled to a second end of the single mode optical fiber loop via a fourth splice; and wherein x+z is substantially equal to w+y; wherein the lengths of v, x, y, d 1  and d 3  are proportioned to avoid regions of high bias instability defined by the expressions v=x−(L IOC +y+d 1 +d 3 ), v=2x−(L IOC +y+d 1 +d 3 ), v=3x−(L IOC +y+d 1 +d 3 ), and v=4x+y−(L IOC +y+d 1 +d 3 ). 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Preferred and alternative embodiments of the present invention are described in detail below with reference to the following drawings. 
         FIG. 1  illustrates an optical circuit of a depolarized gyroscope according to an embodiment of the invention; 
         FIG. 2  illustrates an example of a bias versus temperature plot for a gyroscope with polarization errors obtained according to an embodiment of the invention; 
         FIGS. 3   a  and  3   b  illustrate two coil coherence functions and their associated wavelength dependent differential group delays (DGDs) in accordance with an embodiment of the present invention; 
         FIG. 4  illustrates a comparison of bias thermal instabilities under temperature ramping for two gyroscopes having identical optical components except for the coil (coil-I and II, respectively) according to an embodiment of the present invention; 
         FIG. 5  illustrates an example of calculation of relative gyroscope bias fluctuation amplitude as a function of the 2×2 coupler polarization dependent splitting ratio (PDSR); 
         FIG. 6  illustrates an example of calculation of relative gyroscope bias fluctuation amplitude as a function of the cross-coupling at splice  32  of  FIG. 1 ; 
         FIG. 7  illustrates an example of calculation of polarization error induced relative bias fluctuation amplitude as a function of IOC polarization extinction ratio; 
         FIG. 8  illustrates dependence of relative bias fluctuation amplitude on the length v, of PM fiber  15  of  FIG. 1  for a depolarizer design according to an embodiment; 
         FIGS. 9   a  and  9   b  illustrate a first example of contour plots of relative bias fluctuation amplitude as a function of PM fiber lengths, x and v; 
         FIGS. 10   a  and  10   b  illustrate a second example of contour plots of relative bias fluctuation amplitude as a function of fiber lengths, x and w; and 
         FIG. 11  illustrates an operating environment in which an embodiment of the invention can be implemented. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 11  illustrates an example of a suitable computing system environment  100  on which an embodiment of the invention may be implemented. The computing system environment  100  is only one example of a suitable computing environment and is not intended to suggest any limitation as to the scope of use or functionality of embodiments of the invention. Neither should the computing environment  100  be interpreted as having any dependency or requirement relating to any one or combination of components illustrated in the exemplary operating environment  100 . 
     Embodiments of the invention are operational with numerous other general-purpose or special-purpose computing-system environments or configurations. Examples of well-known computing systems, environments, and/or configurations that may be suitable for use with embodiments of the invention include, but are not limited to, personal computers, server computers, hand-held or laptop devices, multiprocessor systems, microprocessor-based systems, set-top boxes, programmable consumer electronics, network PCs, minicomputers, mainframe computers, distributed-computing environments that include any of the above systems or devices, and the like. 
     Embodiments of the invention may be described in the general context of computer-executable instructions, such as program modules, executed by a computer. Generally, program modules include routines, programs, objects, components, data structures, and the like that perform particular tasks or implement particular abstract data types. Embodiments of the invention may also be practiced in distributed-computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed-computing environment, program modules may be located in both local- and remote-computer storage media including memory storage devices. 
     The operating environment illustrated in  FIG. 11  typically includes at least some form of computer readable media. Computer readable media can be any available media that can be accessed by one or more components of such operating environment. By way of example, and not limitation, computer readable media may comprise computer storage media and communication media. Computer storage media includes volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by one or more components of such operating environment. Communication media typically embody computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, RF, infrared and other wireless media. Combinations of the any of the above should also be included within the scope of computer readable media. 
     With reference to  FIG. 11 , an exemplary system for implementing an embodiment of the invention includes a computing device, such as computing device  100 . In its most basic configuration, computing device  100  typically includes at least one processing unit  103  and memory  104 . 
     Depending on the exact configuration and type of computing device, memory  104  may be volatile (such as random-access memory (RAM)), non-volatile (such as read-only memory (ROM), flash memory, etc.) or some combination of the two. This most basic configuration is illustrated in  FIG. 11  by dashed line  106 . 
     Additionally, device  100  may have additional features/functionality. For example, device  100  may also include additional storage (removable and/or non-removable) including, but not limited to, magnetic or optical disks or tape. Such additional storage is illustrated in  FIG. 11  by removable storage  108  and non-removable storage  110 . Computer storage media includes volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer-readable instructions, data structures, program modules or other data. Memory  104 , removable storage  108  and non-removable storage  110  are all examples of computer storage media. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by device  100 . Any such computer storage media may be part of device  100 . 
     Device  100  may also contain communications connection(s)  112  that allow the device to communicate with other devices. Communications connection(s)  112  is an example of communication media. Communication media typically embodies computer-readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, radio-frequency (RF), infrared and other wireless media. The term computer-readable media as used herein includes both storage media and communication media. 
     Device  100  may also have input device(s)  114  such as keyboard, mouse, pen, voice-input device, touch-input device, etc. Output device(s)  116  such as a display, speakers, printer, etc. may also be included. 
     An embodiment of the present invention pertains to interferometric fiber optic gyroscopes (IFOGs), and particularly to depolarized IFOGs using non-polarization-maintaining SM fiber in the sensing coil. More particularly, an embodiment of the present invention pertains to the reduction of bias instability originated from polarization errors. Embodiments include both analytical/computational and empirical methods of testing and designing such gyroscopes. 
     The optical circuit  1  of a depolarized gyroscope is shown in  FIG. 1 . Depending on the embodiment, the circuit  1  may be computationally modeled using computer-executable instructions or may be a tangible device. A broadband light source  11  generates a lightwave {right arrow over (E)} 0  that is composed of many waves with different wavelengths and polarization states at the splice  30 . In the general case, the light emitted by broadband light source  11  may be described as a summation of many optical waves, i.e. {right arrow over (E)} 0 =Σ{right arrow over (E)} 0 (λ,|p           ), where λ is wavelength (the spectrum of which ranges from λ 1  to λ 1 ) and |p          is a vector on the Poincaré sphere representing a particular polarization state. The phases of these waves with different wavelengths and polarization states may be uncorrelated. For a perfect unpolarized light source, the light intensity associated with all orthogonal polarization states will be identical. When the unpolarized light passes through an optical component with polarization-dependent loss (PDL), the lightwaves of orthogonal polarization states experience different losses and acquire different intensity at the output of the component.
     A strict theoretical description of a broadband light source may require taking into account all the wavelength components and polarization states. However, for simplicity of the following discussion and without loss of generality, it may be assumed that the light source only contains two orthogonal polarization states for each wavelength component arriving at splice  30  in  FIG. 1 . The electric field of the broadband light source  11  can thus be expressed by a Jones vector 
                       E   0     ⁡     (   λ   )       =       (           E     0   ⁢   x             (   λ   )               E     0   ⁢   y             (   λ   )           )     =         E     0   ⁢   x       ⁡     (   λ   )       ⁢       (         1               t     11   ⁢   y       ⁡     (   λ   )             )     .                 (   1   )               
Here we may assume the PDL experienced by the light source before point  30  is along the Y axis (i.e., the polarization state that experiences more loss is along the axis orthogonal to the plane of paper in  FIG. 1 ). The amplitude transmission coefficient t 11y  (λ) quantifies the degree of polarization or PDL of light emitted by the light source  11  when arriving at splice  30 .
 
     The light fields reaching the photodetector  14  can be modeled using a Jones-matrix description of the gyro optical circuit  1 . Element  12  can be any kind of directional coupler. Here, a 2×2 coupler is used as an example, which may include polarization maintaining (PM) fiber leads. There is normally a rotation angle α between the input light PDL axis and the PM fiber pass axis. The Jones matrix of such an angle rotation at splice  30  can be expressed as 
     
       
         
           
             
               
                 
                   
                     R 
                     30 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                         
                       
                       ) 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     If light emitted by the light source  11  experiences wavelength dependent PDL, the rotation angle α(λ) may be a function of wavelength λ. For simplicity, we implicitly assume the wavelength dependence of every parameter and omit the λ in the following expressions. 
     The propagation of light through the directional coupler  12 , including its fiber leads, can be described by a Jones matrix H in . This matrix takes into account the polarization dependent splitting ratio, PDL, cross-couplings and phase delays incurred by the coupler  12  as light propagates from splice  30  to splice  31 . 
     Element  15  may be a piece of PM fiber that connects the coupler  12  and an integrated optical circuit (IOC)  16 . Element  15  has a Jones matrix of 
                       D   15     =     (         1       0           0         ⅇ       -   ⅈ     ⁢           ⁢     d   15               )       ,           (   3   )               
where d 15  is the relative phase delay of the slow axis with respect to the fast axis of the PM fiber  15 . The IOC pigtails  33 ,  36  and  38  may be PM fibers. They can similarly be described by delay matrices
 
     
       
         
           
             
               
                 
                   
                     
                       D 
                       j 
                     
                     = 
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   ⅈ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   d 
                                   j 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                       
                   
                   ⁢ 
                   
                     j 
                     = 
                     33 
                   
                   , 
                   36 
                   , 
                   38. 
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     The splices,  31  and  32 , between PM fibers, and the connection points,  34 ,  35  and  37 , between IOC pigtail fibers and the IOC waveguides, are not perfect. They introduce small polarization cross-couplings, which can be described by Jones matrices 
     
       
         
           
             
               
                 
                   
                     
                       K 
                       j 
                     
                     = 
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             
                               k 
                               j 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 k 
                                 j 
                               
                             
                           
                           
                             1 
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                       
                   
                   ⁢ 
                   
                     j 
                     = 
                     31 
                   
                   , 
                   32 
                   , 
                   34 
                   , 
                   35 
                   , 
                   37. 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     For light propagation from splice  30  to splice  34 , the total Jones matrix G is 
                   G   =       (           G   11           G   12               G   21           G   22           )     =       K   34     ⁢     D   33     ⁢     K   32     ⁢     D   15     ⁢     K   31     ⁢     H   in     ⁢       R   30     .                 (   6   )               
The electric field at the input point  34  of the IOC waveguide is therefore
 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             A 
                             x 
                           
                         
                       
                       
                         
                           
                             A 
                             y 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
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                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 E 
                                 
                                   0 
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                     = 
                     
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                                 11 
                               
                               ⁢ 
                               
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                                   ⁢ 
                                   x 
                                 
                               
                             
                           
                           
                             + 
                           
                           
                             
                               
                                 G 
                                 12 
                               
                               ⁢ 
                               
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                                   y 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 G 
                                 21 
                               
                               ⁢ 
                               
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                                   0 
                                   ⁢ 
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                             + 
                           
                           
                             
                               
                                 G 
                                 22 
                               
                               ⁢ 
                               
                                 E 
                                 
                                   0 
                                   ⁢ 
                                   y 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Next, a Jones matrix 
                   (         M       N           L       J         )           
is used to represent the transmission of light through the sensing loop  10  in the clockwise direction, without taking into account the IOC phase modulation and its polarization extinction ratio ε. For the depolarized gyroscope shown in  FIG. 1 , the clockwise transmission matrix is
 
                           (         M       N           L       J         )     =       D   41     ⁢     K   37     ⁢     D   38     ⁢     K   25     ⁢     D   26     ⁢     R   27     ⁢     D   28     ⁢     R   29     ⁢     C   10     ⁢     R   24     ⁢     D   23     ⁢     R   22     ⁢     D   21     ⁢     K   20     ⁢     D   36     ⁢     K   35     ⁢       D   40     .                 (   8   )               
Here, D 40  and D 41  are delay matrices for the IOC waveguide  40  and  41 . Delay matrices D 21 , D 23 , D 26 , D 28  are for segments of PM fibers that constitute the depolarizer  200 , which contains an upper section  210  and a lower section  220 . The imperfect splices  20  and  25  with small angle misalignments are represented by matrices K 20  and K 25 . The two 45° angle splices  22  and  27  between the two pair of depolarizer PM fiber segments  21 ,  23  and  26 ,  28  are represented by matrices R 22  and R 27 . Once can also include rotation matrices R 24  and R 29  for the splices  24  and  29  between the depolarizer sections  210 ,  220  and the fiber leads  101 ,  102  of the single-mode sensing coil  10 .
 
     The Jones matrix for propagation of light in the counter-clockwise direction is the transpose of the clockwise counterpart, i.e. 
                     (         M       L           N       J         )     .           
Again, the IOC phase modulation and polarization property are not taken into account in this matrix.
 
     The integrated optical circuit (IOC)  16  may be composed of a polarizing element  18  (alternatively, the polarizing element itself could be a waveguide), a beam splitter/combiner  17  and two or more electrodes  19  for phase-modulation of the light beams passing through the IOC waveguides  40  and  41 . This phase modulation function of IOC is used for gyro bias modulation, which typically changes the phase of light polarized along the pass axis (the horizontal x-axis is assumed here) with a period equal to loop transit time τ. If a clockwise propagating lightwave experiences a modulation phase φ U (t) at the upper waveguide, it experiences a different modulation phase φ U (t+τ) at the lower waveguide. During the same time, the counter-clockwise wave can experience a modulation phase of φ L (t) at the lower waveguide and φ L (t+τ) at the upper waveguide. For the situation of push-pull square-wave modulation, one can simply assume φ U (t)=φ L (t+τ)=φ B  and φ L (t)=φ U (t+τ)=−φ B  without loss of generality. In this case, the matrix that takes into account the IOC bias modulation and polarization extinction ratio for clockwise light propagation is 
                     (           ⅇ     ⅈϕ   B           0           0       ɛ         )     ⁢               (         M       N           L       J         )     ⁢     (           ⅇ     ⅈϕ   B           0           0       ɛ         )       ,               (   9   )               
and that for the counter clockwise propagation is
 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
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                               B 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
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                           ɛ 
                         
                       
                     
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                   ⁢ 
                   
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                           M 
                         
                         
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                     ) 
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           
                             
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                             0 
                           
                           
                             ɛ 
                           
                         
                       
                       ) 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     The combined electric fields of CW and CCW light at IOC output  34  are therefore determined by the following equation: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             B 
                             x 
                           
                         
                       
                       
                         
                           
                             B 
                             y 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       
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                       ⁢ 
                       
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                       ⁢ 
                       
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                       ⁢ 
                       
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                                   B 
                                 
                               
                             
                             
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                       ⁢ 
                       
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                                 y 
                               
                             
                           
                         
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                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Taking into account the Sagnac phase 2φ R  induced by the rotation sensing and expanding the above equation, yields: 
     
       
         
           
             
               
                 
                   
                     B 
                     x 
                   
                   = 
                   
                     
                       
                         
                           A 
                           x 
                         
                         ⁡ 
                         
                           ( 
                           
                             
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                                   ⁢ 
                                   
                                       
                                   
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                                   ⁢ 
                                   
                                       
                                   
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                                     B 
                                   
                                 
                               
                             
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                           y 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
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                                     ⁢ 
                                     
                                         
                                     
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                                       ϕ 
                                       R 
                                     
                                   
                                   + 
                                   
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                                     ⁢ 
                                     
                                         
                                     
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                               N 
                             
                             + 
                             
                               
                                 ⅇ 
                                 
                                   
                                     
                                       - 
                                       i 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ϕ 
                                       R 
                                     
                                   
                                   - 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ϕ 
                                       B 
                                     
                                   
                                 
                               
                               ⁢ 
                               L 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     B 
                     y 
                   
                   = 
                   
                     
                       ɛ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           A 
                           x 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 ⅇ 
                                 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ϕ 
                                       R 
                                     
                                   
                                   + 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ϕ 
                                       B 
                                     
                                   
                                 
                               
                               ⁢ 
                               L 
                             
                             + 
                             
                               
                                 ⅇ 
                                 
                                   
                                     
                                       - 
                                       i 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ϕ 
                                       R 
                                     
                                   
                                   - 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ϕ 
                                       B 
                                     
                                   
                                 
                               
                               ⁢ 
                               N 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         ɛ 
                         2 
                       
                       ⁢ 
                       
                         
                           A 
                           y 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ⅇ 
                               
                                 i 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ϕ 
                                   R 
                                 
                               
                             
                             + 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   i 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ϕ 
                                   R 
                                 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         J 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     To obtain the electric field at the photodetector  14 , matrix F is used to describe the optic path from point  34  to the photodetector  14 . It is: 
                   F   =       (           F   11           F   12               F   21           F   22           )     =       T   14     ⁢     R   13     ⁢     H   out     ⁢     K   31   T     ⁢     D   15     ⁢     K   32   T     ⁢     D   33     ⁢       K   34   T     .                 (   14   )               
Here K 32   T , K 33   T , K 34   T  are the transposes of matrices K 32 , K 33 , K 34  respectively H out  is the coupler matrix for light transmission from splice  31  to photodetector  14 . Matrix
 
                     R   13     =     (           cos   ⁢           ⁢   σ             -   sin     ⁢           ⁢   σ               sin   ⁢           ⁢   σ           cos   ⁢           ⁢   σ           )             (   15   )               
takes into account the mismatch angle of the PM fiber axis with respect to the axis of photodetector polarization-dependent responsivity, which is accounted for by matrix
 
     
       
         
           
             
               
                 
                   
                     T 
                     14 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             1 
                           
                         
                         
                           
                             
                               t 
                               
                                 14 
                                 y 
                               
                             
                           
                         
                       
                       ) 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     With the matrix F, the electric field at the photodetector  14  can be expressed as 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             E 
                             x 
                           
                         
                       
                       
                         
                           
                             E 
                             y 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       F 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 B 
                                 x 
                               
                             
                           
                           
                             
                               
                                 B 
                                 y 
                               
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   F 
                                   11 
                                 
                                 ⁢ 
                                 
                                   B 
                                   x 
                                 
                               
                               + 
                               
                                 
                                   F 
                                   12 
                                 
                                 ⁢ 
                                 
                                   B 
                                   y 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   F 
                                   21 
                                 
                                 ⁢ 
                                 
                                   B 
                                   x 
                                 
                               
                               + 
                               
                                 
                                   F 
                                   22 
                                 
                                 ⁢ 
                                 
                                   B 
                                   y 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     The power of the horizontally polarized light at the photodetector  14  is 
     
       
         
           
             
               
                 
                   
                     I 
                     x 
                   
                   = 
                   
                     
                       
                         E 
                         x 
                       
                       ⁢ 
                       
                         E 
                         x 
                         * 
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             
                               
                                 F 
                                 11 
                               
                               ⁢ 
                               
                                 B 
                                 x 
                               
                             
                             + 
                             
                               
                                 F 
                                 12 
                               
                               ⁢ 
                               
                                 B 
                                 y 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   F 
                                   11 
                                 
                                 ⁢ 
                                 
                                   B 
                                   x 
                                 
                               
                               + 
                               
                                 
                                   F 
                                   12 
                                 
                                 ⁢ 
                                 
                                   B 
                                   y 
                                 
                               
                             
                             ) 
                           
                           * 
                         
                       
                       = 
                       
                         
                           
                             
                                
                               
                                 F 
                                 11 
                               
                                
                             
                             2 
                           
                           ⁢ 
                           
                             B 
                             x 
                           
                           ⁢ 
                           
                             B 
                             x 
                             * 
                           
                         
                         + 
                         
                           
                             
                                
                               
                                 F 
                                 12 
                               
                                
                             
                             2 
                           
                           ⁢ 
                           
                             B 
                             y 
                           
                           ⁢ 
                           
                             B 
                             y 
                             * 
                           
                         
                         + 
                         
                           
                             F 
                             11 
                           
                           ⁢ 
                           
                             F 
                             12 
                             * 
                           
                           ⁢ 
                           
                             B 
                             x 
                           
                           ⁢ 
                           
                             B 
                             y 
                             * 
                           
                         
                         + 
                         
                           
                             F 
                             12 
                           
                           ⁢ 
                           
                             F 
                             11 
                             * 
                           
                           ⁢ 
                           
                             B 
                             y 
                           
                           ⁢ 
                           
                             B 
                             x 
                             * 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     Substituting (12) and (13) into (18) yields: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             I 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             
                               ϕ 
                               B 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               
                                  
                                 
                                   F 
                                   11 
                                 
                                  
                               
                               2 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           
                                              
                                             
                                               
                                                 A 
                                                 x 
                                               
                                               ⁢ 
                                               M 
                                             
                                              
                                           
                                           2 
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             1 
                                             + 
                                             
                                               cos 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   
                                                     2 
                                                     ⁢ 
                                                     
                                                       ϕ 
                                                       R 
                                                     
                                                   
                                                   + 
                                                   
                                                     4 
                                                     ⁢ 
                                                     
                                                       ϕ 
                                                       B 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         2 
                                         ⁢ 
                                         ɛ 
                                         ⁢ 
                                         
                                            
                                           
                                             
                                               A 
                                               x 
                                             
                                             ⁢ 
                                             
                                               A 
                                               y 
                                               * 
                                             
                                           
                                            
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               
                                                 
                                                   
                                                      
                                                     
                                                       MN 
                                                       * 
                                                     
                                                      
                                                   
                                                   ⁢ 
                                                   
                                                     ( 
                                                     
                                                       
                                                         cos 
                                                         ⁡ 
                                                         
                                                           ( 
                                                           
                                                             
                                                             ϕ 
                                                             
                                                             M 
                                                             - 
                                                             N 
                                                             
                                                             
                                                             + 
                                                             
                                                             ϕ 
                                                             
                                                             Ax 
                                                             - 
                                                             Ay 
                                                             
                                                             
                                                             + 
                                                             
                                                             ϕ 
                                                             B 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                     
                                                   
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     
                                                       cos 
                                                       ⁢ 
                                                       
                                                         ( 
                                                         
                                                           
                                                             ϕ 
                                                             
                                                             M 
                                                             - 
                                                             N 
                                                             
                                                           
                                                           + 
                                                           
                                                             ϕ 
                                                             
                                                             Ax 
                                                             - 
                                                             Ay 
                                                             
                                                           
                                                           - 
                                                           
                                                             2 
                                                             ⁢ 
                                                             
                                                             ϕ 
                                                             R 
                                                             
                                                           
                                                           - 
                                                           
                                                             3 
                                                             ⁢ 
                                                             
                                                             ϕ 
                                                             B 
                                                             
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                     ) 
                                                   
                                                   + 
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     
                                                       
                                                         
                                                            
                                                           
                                                             ML 
                                                             * 
                                                           
                                                            
                                                         
                                                         ⁢ 
                                                         
                                                           ( 
                                                           
                                                             
                                                             cos 
                                                             ⁡ 
                                                             
                                                             ( 
                                                             
                                                             
                                                             ϕ 
                                                             
                                                             M 
                                                             - 
                                                             L 
                                                             
                                                             
                                                             + 
                                                             
                                                             ϕ 
                                                             
                                                             Ax 
                                                             - 
                                                             Ay 
                                                             
                                                             
                                                             - 
                                                             
                                                             ϕ 
                                                             B 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                           
                                                         
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         
                                                           cos 
                                                           ⁢ 
                                                           
                                                             ( 
                                                             
                                                             
                                                             ϕ 
                                                             
                                                             M 
                                                             - 
                                                             L 
                                                             
                                                             
                                                             + 
                                                             
                                                             ϕ 
                                                             
                                                             Ax 
                                                             - 
                                                             Ay 
                                                             
                                                             
                                                             + 
                                                             
                                                             2 
                                                             ⁢ 
                                                             
                                                             ϕ 
                                                             R 
                                                             
                                                             
                                                             - 
                                                             
                                                             3 
                                                             ⁢ 
                                                             
                                                             ϕ 
                                                             B 
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ɛ 
                                         2 
                                       
                                       ⁢ 
                                       
                                         
                                            
                                           
                                             A 
                                             y 
                                           
                                            
                                         
                                         2 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                                
                                               N 
                                                
                                             
                                             2 
                                           
                                           + 
                                           
                                             
                                                
                                               L 
                                                
                                             
                                             2 
                                           
                                           + 
                                           
                                             2 
                                             ⁢ 
                                             
                                                
                                               
                                                 NL 
                                                 * 
                                               
                                                
                                             
                                             ⁢ 
                                             
                                               cos 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   
                                                     ϕ 
                                                     
                                                       N 
                                                       - 
                                                       L 
                                                     
                                                   
                                                   + 
                                                   
                                                     2 
                                                     ⁢ 
                                                     
                                                       ϕ 
                                                       R 
                                                     
                                                   
                                                   + 
                                                   
                                                     2 
                                                     ⁢ 
                                                     
                                                       ϕ 
                                                       B 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           + 
                           
                             
 
                           
                           ⁢ 
                           
                             
                               
                                  
                                 
                                   F 
                                   12 
                                 
                                  
                               
                               2 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         
                                           ɛ 
                                           2 
                                         
                                         ⁢ 
                                         
                                           
                                              
                                             
                                               A 
                                               x 
                                             
                                              
                                           
                                           2 
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               
                                                  
                                                 N 
                                                  
                                               
                                               2 
                                             
                                             + 
                                             
                                               
                                                  
                                                 L 
                                                  
                                               
                                               2 
                                             
                                             + 
                                             
                                               2 
                                               ⁢ 
                                               
                                                  
                                                 
                                                   NL 
                                                   * 
                                                 
                                                  
                                               
                                               ⁢ 
                                               
                                                 cos 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       ϕ 
                                                       
                                                         N 
                                                         - 
                                                         L 
                                                       
                                                     
                                                     - 
                                                     
                                                       2 
                                                       ⁢ 
                                                       
                                                         ϕ 
                                                         R 
                                                       
                                                     
                                                     - 
                                                     
                                                       2 
                                                       ⁢ 
                                                       
                                                         ϕ 
                                                         B 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                         ɛ 
                                         3 
                                       
                                       ⁢ 
                                       
                                          
                                         
                                           
                                             A 
                                             x 
                                           
                                           ⁢ 
                                           
                                             A 
                                             y 
                                             * 
                                           
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     
                                                       ( 
                                                       
                                                         
                                                           cos 
                                                           ⁡ 
                                                           
                                                             ( 
                                                             
                                                             
                                                             ϕ 
                                                             
                                                             L 
                                                             - 
                                                             J 
                                                             
                                                             
                                                             + 
                                                             
                                                             ϕ 
                                                             
                                                             Ax 
                                                             - 
                                                             Ay 
                                                             
                                                             
                                                             + 
                                                             
                                                             ϕ 
                                                             B 
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                       
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             cos 
                                                             ⁢ 
                                                             
                                                             ( 
                                                             
                                                             
                                                             ϕ 
                                                             
                                                             L 
                                                             - 
                                                             J 
                                                             
                                                             
                                                             + 
                                                             
                                                             ϕ 
                                                             
                                                             Ax 
                                                             - 
                                                             Ay 
                                                             
                                                             
                                                             + 
                                                             
                                                             2 
                                                             ⁢ 
                                                             
                                                             ϕ 
                                                             R 
                                                             
                                                             
                                                             + 
                                                             
                                                             ϕ 
                                                             B 
                                                             
                                                             
                                                             ) 
                                                             
                                                           
                                                           ) 
                                                         
                                                         ⁢ 
                                                         
                                                            
                                                           
                                                             LJ 
                                                             * 
                                                           
                                                            
                                                         
                                                       
                                                       + 
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   
                                                     
                                                       ( 
                                                       
                                                         
                                                           cos 
                                                           ⁡ 
                                                           
                                                             ( 
                                                             
                                                             
                                                             ϕ 
                                                             
                                                             N 
                                                             - 
                                                             J 
                                                             
                                                             
                                                             + 
                                                             
                                                             ϕ 
                                                             
                                                             Ax 
                                                             - 
                                                             Ay 
                                                             
                                                             
                                                             - 
                                                             
                                                             ϕ 
                                                             B 
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                       
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           cos 
                                                           ⁢ 
                                                           
                                                             ( 
                                                             
                                                             
                                                             ϕ 
                                                             
                                                             N 
                                                             - 
                                                             J 
                                                             
                                                             
                                                             + 
                                                             
                                                             ϕ 
                                                             
                                                             Ax 
                                                             - 
                                                             Ay 
                                                             
                                                             
                                                             - 
                                                             
                                                             2 
                                                             ⁢ 
                                                             
                                                             ϕ 
                                                             R 
                                                             
                                                             
                                                             - 
                                                             
                                                             ϕ 
                                                             B 
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         ) 
                                                       
                                                       ⁢ 
                                                       
                                                          
                                                         
                                                           NJ 
                                                           * 
                                                         
                                                          
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           + 
                           
                             
 
                           
                           ⁢ 
                           
                              
                             
                               
                                 F 
                                 11 
                               
                               ⁢ 
                               
                                 F 
                                 12 
                                 * 
                               
                             
                              
                           
                         
                       
                     
                   
                   
                     
                       
                         [ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   ɛ 
                                   ⁢ 
                                   
                                     
                                        
                                       
                                         A 
                                         x 
                                       
                                        
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                      
                                                     
                                                       MN 
                                                       * 
                                                     
                                                      
                                                   
                                                   ⁢ 
                                                   
                                                     ( 
                                                     
                                                       
                                                         cos 
                                                         ⁡ 
                                                         
                                                           ( 
                                                           
                                                             
                                                             ϕ 
                                                             
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                             
                                                             
                                                             + 
                                                             
                                                             ϕ 
                                                             
                                                             M 
                                                             - 
                                                             N 
                                                             
                                                             
                                                             - 
                                                             
                                                             ϕ 
                                                             B 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                     
                                                   
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     
                                                       cos 
                                                       ⁢ 
                                                       
                                                         ( 
                                                         
                                                           
                                                             ϕ 
                                                             
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                             
                                                           
                                                           + 
                                                           
                                                             ϕ 
                                                             
                                                             M 
                                                             - 
                                                             N 
                                                             
                                                           
                                                           + 
                                                           
                                                             2 
                                                             ⁢ 
                                                             
                                                             ϕ 
                                                             R 
                                                             
                                                           
                                                           - 
                                                           
                                                             3 
                                                             ⁢ 
                                                             
                                                             ϕ 
                                                             B 
                                                             
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                     ) 
                                                   
                                                   + 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                      
                                                     
                                                       ML 
                                                       * 
                                                     
                                                      
                                                   
                                                   ⁢ 
                                                   
                                                     ( 
                                                     
                                                       cos 
                                                       ⁡ 
                                                       
                                                         ( 
                                                         
                                                           
                                                             ϕ 
                                                             
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                             
                                                           
                                                           + 
                                                           
                                                             ϕ 
                                                             
                                                             M 
                                                             - 
                                                             L 
                                                             
                                                           
                                                           + 
                                                           
                                                             ϕ 
                                                             B 
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     
                                                       + 
                                                       cos 
                                                     
                                                     ⁢ 
                                                     
                                                       ( 
                                                       
                                                         
                                                           ϕ 
                                                           
                                                             
                                                               
                                                             
                                                             
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                             
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           
                                                             M 
                                                             - 
                                                             L 
                                                           
                                                         
                                                         - 
                                                         
                                                           2 
                                                           ⁢ 
                                                           
                                                             ϕ 
                                                             R 
                                                           
                                                         
                                                         - 
                                                         
                                                           3 
                                                           ⁢ 
                                                           
                                                             ϕ 
                                                             B 
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     ɛ 
                                     2 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         A 
                                         y 
                                       
                                       ⁢ 
                                       
                                         A 
                                         x 
                                         * 
                                       
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     
                                                       
                                                          
                                                         N 
                                                          
                                                       
                                                       2 
                                                     
                                                     ⁢ 
                                                     
                                                       cos 
                                                       ⁡ 
                                                       
                                                         ( 
                                                         
                                                           
                                                             ϕ 
                                                             
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                             
                                                           
                                                           + 
                                                           
                                                             ϕ 
                                                             
                                                             Ax 
                                                             - 
                                                             Ay 
                                                             
                                                           
                                                           - 
                                                           
                                                             2 
                                                             ⁢ 
                                                             
                                                             ϕ 
                                                             R 
                                                             
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                   + 
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     
                                                       
                                                          
                                                         L 
                                                          
                                                       
                                                       2 
                                                     
                                                     ⁢ 
                                                     
                                                       cos 
                                                       ⁡ 
                                                       
                                                         ( 
                                                         
                                                           
                                                             ϕ 
                                                             
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                             
                                                           
                                                           + 
                                                           
                                                             ϕ 
                                                             
                                                             Ax 
                                                             - 
                                                             Ay 
                                                             
                                                           
                                                           + 
                                                           
                                                             2 
                                                             ⁢ 
                                                             
                                                             ϕ 
                                                             R 
                                                             
                                                           
                                                           + 
                                                           
                                                             2 
                                                             ⁢ 
                                                             
                                                             ϕ 
                                                             B 
                                                             
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                   + 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                      
                                                     
                                                       NL 
                                                       * 
                                                     
                                                      
                                                   
                                                   ⁢ 
                                                   
                                                     ( 
                                                     
                                                       
                                                         cos 
                                                         ⁡ 
                                                         
                                                           ( 
                                                           
                                                             
                                                             ϕ 
                                                             
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                             
                                                             
                                                             + 
                                                             
                                                             ϕ 
                                                             
                                                             Ax 
                                                             - 
                                                             Ay 
                                                             
                                                             
                                                             - 
                                                             
                                                             ϕ 
                                                             
                                                             N 
                                                             - 
                                                             L 
                                                             
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                     
                                                   
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     cos 
                                                     ⁢ 
                                                     
                                                       ( 
                                                       
                                                         
                                                           ϕ 
                                                           
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           
                                                             Ax 
                                                             - 
                                                             Ay 
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           
                                                             N 
                                                             - 
                                                             L 
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     ɛ 
                                     2 
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         A 
                                         x 
                                       
                                       ⁢ 
                                       
                                         A 
                                         y 
                                         * 
                                       
                                     
                                      
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       MJ 
                                       * 
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     cos 
                                                     ⁡ 
                                                     
                                                       ( 
                                                       
                                                         
                                                           ϕ 
                                                           
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           
                                                             M 
                                                             - 
                                                             J 
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           
                                                             Ax 
                                                             - 
                                                             Ay 
                                                           
                                                         
                                                         + 
                                                         
                                                           2 
                                                           ⁢ 
                                                           
                                                             ϕ 
                                                             B 
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   + 
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     cos 
                                                     ⁢ 
                                                     
                                                       ( 
                                                       
                                                         
                                                           ϕ 
                                                           
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           
                                                             M 
                                                             - 
                                                             J 
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           
                                                             Ax 
                                                             - 
                                                             Ay 
                                                           
                                                         
                                                         - 
                                                         
                                                           2 
                                                           ⁢ 
                                                           
                                                             ϕ 
                                                             B 
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   + 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     cos 
                                                     ⁡ 
                                                     
                                                       ( 
                                                       
                                                         
                                                           ϕ 
                                                           
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           
                                                             M 
                                                             - 
                                                             J 
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           
                                                             Ax 
                                                             - 
                                                             Ay 
                                                           
                                                         
                                                         + 
                                                         
                                                           2 
                                                           ⁢ 
                                                           
                                                             ϕ 
                                                             R 
                                                           
                                                         
                                                         + 
                                                         
                                                           2 
                                                           ⁢ 
                                                           
                                                             ϕ 
                                                             B 
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   + 
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   cos 
                                                   ⁡ 
                                                   
                                                     ( 
                                                     
                                                       
                                                         ϕ 
                                                         
                                                           
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                           
                                                           - 
                                                           12 
                                                         
                                                       
                                                       + 
                                                       
                                                         ϕ 
                                                         
                                                           M 
                                                           - 
                                                           J 
                                                         
                                                       
                                                       + 
                                                       
                                                         ϕ 
                                                         
                                                           Ax 
                                                           - 
                                                           Ay 
                                                         
                                                       
                                                       - 
                                                       
                                                         2 
                                                         ⁢ 
                                                         
                                                           ϕ 
                                                           R 
                                                         
                                                       
                                                       - 
                                                       
                                                         2 
                                                         ⁢ 
                                                         
                                                           ϕ 
                                                           B 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   ɛ 
                                   3 
                                 
                                 ⁢ 
                                 
                                   
                                      
                                     
                                       A 
                                       y 
                                     
                                      
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                    
                                                   
                                                     NJ 
                                                     * 
                                                   
                                                    
                                                 
                                                 ⁢ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       cos 
                                                       ⁡ 
                                                       
                                                         ( 
                                                         
                                                           
                                                             ϕ 
                                                             
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                             
                                                           
                                                           + 
                                                           
                                                             ϕ 
                                                             
                                                             N 
                                                             - 
                                                             J 
                                                             
                                                           
                                                           + 
                                                           
                                                             ϕ 
                                                             B 
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                     + 
                                                   
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   
                                                     cos 
                                                     ⁢ 
                                                     
                                                       ( 
                                                       
                                                         
                                                           ϕ 
                                                           
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           
                                                             N 
                                                             - 
                                                             J 
                                                           
                                                         
                                                         + 
                                                         
                                                           2 
                                                           ⁢ 
                                                           
                                                             ϕ 
                                                             R 
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           B 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   ) 
                                                 
                                                 + 
                                               
                                             
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                    
                                                   
                                                     LJ 
                                                     * 
                                                   
                                                    
                                                 
                                                 ⁢ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       cos 
                                                       ⁡ 
                                                       
                                                         ( 
                                                         
                                                           
                                                             ϕ 
                                                             
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                             
                                                           
                                                           + 
                                                           
                                                             ϕ 
                                                             
                                                             L 
                                                             - 
                                                             J 
                                                             
                                                           
                                                           - 
                                                           
                                                             ϕ 
                                                             B 
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                     + 
                                                   
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   cos 
                                                   ⁢ 
                                                   
                                                     ( 
                                                     
                                                       
                                                         ϕ 
                                                         
                                                           
                                                               
                                                           
                                                           
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                           
                                                         
                                                       
                                                       + 
                                                       
                                                         ϕ 
                                                         
                                                           L 
                                                           - 
                                                           J 
                                                         
                                                       
                                                       - 
                                                       
                                                         2 
                                                         ⁢ 
                                                         
                                                           ϕ 
                                                           R 
                                                         
                                                       
                                                       - 
                                                       
                                                         ϕ 
                                                         B 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     In equation 19, φ M-N , φ M-L , φ N-L , φ L-J  and φ Ax-Ay  express the phase differences of the complex numbers M and N; M and L; N and L; L and J; A x  and A y . Similarly, term φ F11-12  represents the phase differences of complex numbers F 11  and F 12 . 
     Demodulation of the above push-pull square-wave bias modulated signal is accomplished by calculating the differences of the light intensity in the neighboring time slots of loop transit time τ. Since the φ B  flips the sign in the neighboring period, τ, the demodulated signal is calculated by 
     
       
         
           
             
               
                 
                   
                     I 
                     x_dm 
                   
                   = 
                   
                     
                       
                         I 
                         x 
                       
                       ⁡ 
                       
                         ( 
                         
                           - 
                           
                             ϕ 
                             B 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         I 
                         x 
                       
                       ⁡ 
                       
                         ( 
                         
                           ϕ 
                           B 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     For the primary signal (i.e., the first term in Eq. (19)), the demodulated signal is 
     
       
         
           
             
               
                 
                   
                     I 
                     
                       main_x 
                       ⁢ 
                       _dm 
                     
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         
                            
                           
                             F 
                             11 
                           
                            
                         
                         2 
                       
                       ⁢ 
                       
                         
                            
                           
                             
                               A 
                               x 
                             
                             ⁢ 
                             M 
                           
                            
                         
                         2 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     ϕ 
                                     R 
                                   
                                 
                                 - 
                                 
                                   4 
                                   ⁢ 
                                   
                                     ϕ 
                                     B 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     ϕ 
                                     R 
                                   
                                 
                                 + 
                                 
                                   4 
                                   ⁢ 
                                   
                                     ϕ 
                                     B 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       4 
                       ⁢ 
                       
                         
                            
                           
                             F 
                             11 
                           
                            
                         
                         2 
                       
                       ⁢ 
                       
                         
                            
                           
                             
                               A 
                               x 
                             
                             ⁢ 
                             M 
                           
                            
                         
                         2 
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                               ϕ 
                               R 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           4 
                           ⁢ 
                           
                             ϕ 
                             B 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Demodulation of other terms in (19) yields two major types of polarization errors. One type is called amplitude type error, which is proportional to IOC polarization extinction ratio ε. The other type is called intensity type error, which is proportional to ε 2 . Higher order terms are extremely small, and therefore are neglected for purposes of simplicity in the following discussion. Performing demodulation calculation similar to those in Equations (20) and (21), the amplitude-type polarization error light intensity along the x-axis is: 
     
       
         
           
             
               
                 
                   
                     I 
                     
                       amp_x 
                       ⁢ 
                       _dm 
                     
                   
                   = 
                   
                       
                     
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ɛ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                              
                             
                               F 
                               11 
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                            
                           
                             
                               A 
                               x 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               A 
                               y 
                               * 
                             
                           
                            
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                        
                                       
                                         MN 
                                         * 
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       ϕ 
                                                       
                                                         M 
                                                         - 
                                                         N 
                                                       
                                                     
                                                     + 
                                                     
                                                       ϕ 
                                                       
                                                         Ax 
                                                         - 
                                                         Ay 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               ⁢ 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     ϕ 
                                                     B 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 - 
                                                 
                                                   sin 
                                                   ⁡ 
                                                   
                                                     ( 
                                                     
                                                       
                                                         ϕ 
                                                         
                                                           M 
                                                           - 
                                                           N 
                                                         
                                                       
                                                       + 
                                                       
                                                         ϕ 
                                                         
                                                           Ax 
                                                           - 
                                                           Ay 
                                                         
                                                       
                                                       - 
                                                       
                                                         2 
                                                         ⁢ 
                                                         
                                                           ϕ 
                                                           R 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ⁢ 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     3 
                                                     ⁢ 
                                                     
                                                       ϕ 
                                                       B 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                      
                                     
                                       ML 
                                       * 
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               - 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       ϕ 
                                                       
                                                         M 
                                                         - 
                                                         L 
                                                       
                                                     
                                                     + 
                                                     
                                                       ϕ 
                                                       
                                                         Ax 
                                                         - 
                                                         Ay 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             ⁢ 
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   ϕ 
                                                   B 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               + 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       ϕ 
                                                       
                                                         M 
                                                         - 
                                                         L 
                                                       
                                                     
                                                     + 
                                                     
                                                       ϕ 
                                                       
                                                         Ax 
                                                         - 
                                                         Ay 
                                                       
                                                     
                                                     + 
                                                     
                                                       2 
                                                       ⁢ 
                                                       
                                                         ϕ 
                                                         R 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             ⁢ 
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   3 
                                                   ⁢ 
                                                   
                                                     ϕ 
                                                     B 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
 
                       
                       ⁢ 
                       
                         4 
                         ⁢ 
                         ɛ 
                         ⁢ 
                         
                           
                              
                             
                               
                                 F 
                                 11 
                               
                               ⁢ 
                               
                                 F 
                                 12 
                                 * 
                               
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                           
                              
                             
                               A 
                               x 
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                          
                                         
                                           MN 
                                           * 
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   - 
                                                   
                                                     sin 
                                                     ⁡ 
                                                     
                                                       ( 
                                                       
                                                         
                                                           ϕ 
                                                           
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           
                                                             M 
                                                             - 
                                                             N 
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 ⁢ 
                                                 
                                                   sin 
                                                   ⁡ 
                                                   
                                                     ( 
                                                     
                                                       ϕ 
                                                       B 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   + 
                                                   
                                                     sin 
                                                     ⁡ 
                                                     
                                                       ( 
                                                       
                                                         
                                                           ϕ 
                                                           
                                                             
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                             
                                                             - 
                                                             12 
                                                           
                                                         
                                                         + 
                                                         
                                                           ϕ 
                                                           
                                                             M 
                                                             - 
                                                             N 
                                                           
                                                         
                                                         + 
                                                         
                                                           2 
                                                           ⁢ 
                                                           
                                                             ϕ 
                                                             R 
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 ⁢ 
                                                 
                                                   sin 
                                                   ⁡ 
                                                   
                                                     ( 
                                                     
                                                       3 
                                                       ⁢ 
                                                       
                                                         ϕ 
                                                         B 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                        
                                       
                                         ML 
                                         * 
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       ϕ 
                                                       
                                                         
                                                           F 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           11 
                                                         
                                                         - 
                                                         12 
                                                       
                                                     
                                                     + 
                                                     
                                                       ϕ 
                                                       
                                                         M 
                                                         - 
                                                         L 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               ⁢ 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     ϕ 
                                                     B 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 - 
                                                 
                                                   sin 
                                                   ⁡ 
                                                   
                                                     ( 
                                                     
                                                       
                                                         ϕ 
                                                         
                                                           
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             11 
                                                           
                                                           - 
                                                           12 
                                                         
                                                       
                                                       + 
                                                       
                                                         ϕ 
                                                         
                                                           M 
                                                           - 
                                                           L 
                                                         
                                                       
                                                       - 
                                                       
                                                         2 
                                                         ⁢ 
                                                         
                                                           ϕ 
                                                           R 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ⁢ 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     3 
                                                     ⁢ 
                                                     
                                                       ϕ 
                                                       B 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     The intensity-type polarization error light intensity along the x-axis is 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   I 
                                   
                                     
                                       int 
                                       ⁢ 
                                       _x 
                                     
                                     ⁢ 
                                     _dm 
                                   
                                 
                                 = 
                                 
                                   
                                     4 
                                     ⁢ 
                                     
                                       ɛ 
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                          
                                         
                                           F 
                                           11 
                                         
                                          
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                          
                                         
                                           A 
                                           y 
                                         
                                          
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                        
                                       
                                         NL 
                                         * 
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             ϕ 
                                             
                                               N 
                                               - 
                                               L 
                                             
                                           
                                           + 
                                           
                                             2 
                                             ⁢ 
                                             
                                               ϕ 
                                               R 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           2 
                                           ⁢ 
                                           
                                             ϕ 
                                             B 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   - 
                                   
                                     4 
                                     ⁢ 
                                     
                                       ɛ 
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                          
                                         
                                           F 
                                           12 
                                         
                                          
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                          
                                         
                                           A 
                                           x 
                                         
                                          
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                        
                                       
                                         NL 
                                         * 
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             ϕ 
                                             
                                               N 
                                               - 
                                               L 
                                             
                                           
                                           - 
                                           
                                             2 
                                             ⁢ 
                                             
                                               ϕ 
                                               R 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           2 
                                           ⁢ 
                                           
                                             ϕ 
                                             B 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                 
                               
                               ⁢ 
                               
                                   
                               
                                 
                             
                             ⁢ 
                             4 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ɛ 
                               2 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                                
                               
                                 
                                   F 
                                   11 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   F 
                                   12 
                                   * 
                                 
                               
                                
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                                
                               
                                 
                                   A 
                                   y 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   A 
                                   x 
                                   * 
                                 
                               
                                
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           
                                              
                                             N 
                                              
                                           
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               ϕ 
                                               
                                                 
                                                   F 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   11 
                                                 
                                                 - 
                                                 12 
                                               
                                             
                                             + 
                                             
                                               ϕ 
                                               
                                                 Ax 
                                                 - 
                                                 Ay 
                                               
                                             
                                             - 
                                             
                                               2 
                                               ⁢ 
                                               
                                                 ϕ 
                                                 R 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             2 
                                             ⁢ 
                                             
                                               ϕ 
                                               B 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         + 
                                         
                                           
                                              
                                             L 
                                              
                                           
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               ϕ 
                                               
                                                 
                                                   F 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   11 
                                                 
                                                 - 
                                                 12 
                                               
                                             
                                             + 
                                             
                                               ϕ 
                                               
                                                 Ax 
                                                 - 
                                                 Ay 
                                               
                                             
                                             - 
                                             
                                               2 
                                               ⁢ 
                                               
                                                 ϕ 
                                                 R 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             2 
                                             ⁢ 
                                             
                                               ϕ 
                                               B 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           + 
                           
                             
                                 
                                 
                             
                             ⁢ 
                             8 
                             ⁢ 
                             
                               ɛ 
                               2 
                             
                             ⁢ 
                             
                                
                               
                                 
                                   F 
                                   11 
                                 
                                 ⁢ 
                                 
                                   F 
                                   12 
                                   * 
                                 
                               
                                
                             
                             ⁢ 
                             
                                
                               
                                 
                                   A 
                                   x 
                                 
                                 ⁢ 
                                 
                                   A 
                                   y 
                                   * 
                                 
                               
                                
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                                
                               
                                 MJ 
                                 * 
                               
                                
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   
                                     ϕ 
                                     R 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               cos 
                               ( 
                               
                                 
                                   ϕ 
                                   
                                     
                                       F 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       11 
                                     
                                     - 
                                     12 
                                   
                                 
                                 + 
                               
                                 
                             
                             ⁢ 
                             
                               ϕ 
                               
                                 M 
                                 - 
                                 J 
                               
                             
                           
                           + 
                         
                           
                       
                       ⁢ 
                       
                         ϕ 
                         
                           Ax 
                           - 
                           Ay 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                             ϕ 
                             B 
                           
                         
                         ) 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     In a similar manner, we can obtain the demodulated main signal along the y-axis 
     
       
         
           
             
               
                 
                   
                     I 
                     
                       main_y 
                       ⁢ 
                       _dm 
                     
                   
                   = 
                   
                     4 
                     ⁢ 
                     
                       
                          
                         
                           F 
                           21 
                         
                          
                       
                       2 
                     
                     ⁢ 
                     
                       
                          
                         
                           
                             A 
                             x 
                           
                           ⁢ 
                           M 
                         
                          
                       
                       2 
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                             ϕ 
                             R 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             4 
                             ⁢ 
                             
                               ϕ 
                               B 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     The amplitude-type polarization error intensity along the y-axis is 
     
       
         
           
             
               
                 
                   
                     I 
                     
                       amp_y 
                       ⁢ 
                       _dm 
                     
                   
                   = 
                   
                       
                     
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ɛ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                              
                             
                               F 
                               21 
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                            
                           
                             
                               A 
                               x 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               A 
                               y 
                               * 
                             
                           
                            
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                        
                                       
                                         MN 
                                         * 
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       ϕ 
                                                       
                                                         M 
                                                         - 
                                                         N 
                                                       
                                                     
                                                     + 
                                                     
                                                       ϕ 
                                                       
                                                         Ax 
                                                         - 
                                                         Ay 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               ⁢ 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     ϕ 
                                                     B 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 - 
                                                 
                                                   sin 
                                                   ⁡ 
                                                   
                                                     ( 
                                                     
                                                       
                                                         ϕ 
                                                         
                                                           M 
                                                           - 
                                                           N 
                                                         
                                                       
                                                       + 
                                                       
                                                         ϕ 
                                                         
                                                           Ax 
                                                           - 
                                                           Ay 
                                                         
                                                       
                                                       - 
                                                       
                                                         2 
                                                         ⁢ 
                                                         
                                                           ϕ 
                                                           R 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ⁢ 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     3 
                                                     ⁢ 
                                                     
                                                       ϕ 
                                                       B 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                      
                                     
                                       ML 
                                       * 
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               - 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       ϕ 
                                                       
                                                         M 
                                                         - 
                                                         L 
                                                       
                                                     
                                                     + 
                                                     
                                                       ϕ 
                                                       
                                                         Ax 
                                                         - 
                                                         Ay 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             ⁢ 
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   ϕ 
                                                   B 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               + 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       ϕ 
                                                       
                                                         M 
                                                         - 
                                                         L 
                                                       
                                                     
                                                     + 
                                                     
                                                       ϕ 
                                                       
                                                         Ax 
                                                         - 
                                                         Ay 
                                                       
                                                     
                                                     + 
                                                     
                                                       2 
                                                       ⁢ 
                                                       
                                                         ϕ 
                                                         R 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             ⁢ 
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   3 
                                                   ⁢ 
                                                   
                                                     ϕ 
                                                     B 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
 
                       
                       ⁢ 
                       
                         4 
                         ⁢ 
                         ɛ 
                         ⁢ 
                         
                            
                           
                             
                               F 
                               21 
                             
                             ⁢ 
                             
                               F 
                               22 
                               * 
                             
                           
                            
                         
                         ⁢ 
                         
                           
                              
                             
                               A 
                               x 
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                        
                                       
                                         MN 
                                         * 
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 - 
                                                 
                                                   sin 
                                                   ⁡ 
                                                   
                                                     ( 
                                                     
                                                       
                                                         ϕ 
                                                         
                                                           
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             21 
                                                           
                                                           - 
                                                           22 
                                                         
                                                       
                                                       + 
                                                       
                                                         ϕ 
                                                         
                                                           M 
                                                           - 
                                                           N 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ⁢ 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     ϕ 
                                                     B 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 + 
                                                 
                                                   sin 
                                                   ⁡ 
                                                   
                                                     ( 
                                                     
                                                       
                                                         ϕ 
                                                         
                                                           
                                                             F 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             21 
                                                           
                                                           - 
                                                           22 
                                                         
                                                       
                                                       + 
                                                       
                                                         ϕ 
                                                         
                                                           M 
                                                           - 
                                                           N 
                                                         
                                                       
                                                       + 
                                                       
                                                         2 
                                                         ⁢ 
                                                         
                                                           ϕ 
                                                           R 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ⁢ 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     3 
                                                     ⁢ 
                                                     
                                                       ϕ 
                                                       B 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                      
                                     
                                       ML 
                                       * 
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   
                                                     ϕ 
                                                     
                                                       
                                                         F 
                                                         ⁢ 
                                                         
                                                             
                                                         
                                                         ⁢ 
                                                         21 
                                                       
                                                       - 
                                                       22 
                                                     
                                                   
                                                   + 
                                                   
                                                     ϕ 
                                                     
                                                       M 
                                                       - 
                                                       L 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             ⁢ 
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   ϕ 
                                                   B 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               - 
                                               
                                                 sin 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       ϕ 
                                                       
                                                         
                                                           F 
                                                           ⁢ 
                                                           
                                                               
                                                           
                                                           ⁢ 
                                                           21 
                                                         
                                                         - 
                                                         22 
                                                       
                                                     
                                                     + 
                                                     
                                                       ϕ 
                                                       
                                                         M 
                                                         - 
                                                         L 
                                                       
                                                     
                                                     - 
                                                     
                                                       2 
                                                       ⁢ 
                                                       
                                                         ϕ 
                                                         R 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             ⁢ 
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   3 
                                                   ⁢ 
                                                   
                                                     ϕ 
                                                     B 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     The intensity-type polarization error along the y-axis is 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   I 
                                   
                                     
                                       int 
                                       ⁢ 
                                       _y 
                                     
                                     ⁢ 
                                     _dm 
                                   
                                 
                                 = 
                                 
                                   
                                     4 
                                     ⁢ 
                                     
                                       ɛ 
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                          
                                         
                                           F 
                                           21 
                                         
                                          
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                          
                                         
                                           A 
                                           y 
                                         
                                          
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                        
                                       
                                         NL 
                                         * 
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             ϕ 
                                             
                                               N 
                                               - 
                                               L 
                                             
                                           
                                           + 
                                           
                                             2 
                                             ⁢ 
                                             
                                               ϕ 
                                               R 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           2 
                                           ⁢ 
                                           
                                             ϕ 
                                             B 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   - 
                                   
                                     4 
                                     ⁢ 
                                     
                                       ɛ 
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                          
                                         
                                           F 
                                           22 
                                         
                                          
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                          
                                         
                                           A 
                                           x 
                                         
                                          
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                        
                                       
                                         NL 
                                         * 
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             ϕ 
                                             
                                               N 
                                               - 
                                               L 
                                             
                                           
                                           - 
                                           
                                             2 
                                             ⁢ 
                                             
                                               ϕ 
                                               R 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           2 
                                           ⁢ 
                                           
                                             ϕ 
                                             B 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                 
                               
                               ⁢ 
                               
                                   
                               
                                 
                             
                             ⁢ 
                             4 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ɛ 
                               2 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                                
                               
                                 
                                   F 
                                   21 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   F 
                                   22 
                                   * 
                                 
                               
                                
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                                
                               
                                 
                                   A 
                                   y 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   A 
                                   x 
                                   * 
                                 
                               
                                
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           
                                              
                                             N 
                                              
                                           
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               ϕ 
                                               
                                                 
                                                   F 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   21 
                                                 
                                                 - 
                                                 22 
                                               
                                             
                                             + 
                                             
                                               ϕ 
                                               
                                                 Ax 
                                                 - 
                                                 Ay 
                                               
                                             
                                             - 
                                             
                                               2 
                                               ⁢ 
                                               
                                                 ϕ 
                                                 R 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             2 
                                             ⁢ 
                                             
                                               ϕ 
                                               B 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         + 
                                         
                                           
                                              
                                             L 
                                              
                                           
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               ϕ 
                                               
                                                 
                                                   F 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   21 
                                                 
                                                 - 
                                                 22 
                                               
                                             
                                             + 
                                             
                                               ϕ 
                                               
                                                 Ax 
                                                 - 
                                                 Ay 
                                               
                                             
                                             + 
                                             
                                               2 
                                               ⁢ 
                                               
                                                 ϕ 
                                                 R 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             2 
                                             ⁢ 
                                             
                                               ϕ 
                                               B 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           + 
                           
                             
                                 
                                 
                             
                             ⁢ 
                             8 
                             ⁢ 
                             
                               ɛ 
                               2 
                             
                             ⁢ 
                             
                                
                               
                                 
                                   F 
                                   21 
                                 
                                 ⁢ 
                                 
                                   F 
                                   22 
                                   * 
                                 
                               
                                
                             
                             ⁢ 
                             
                                
                               
                                 
                                   A 
                                   x 
                                 
                                 ⁢ 
                                 
                                   A 
                                   y 
                                   * 
                                 
                               
                                
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                                
                               
                                 MJ 
                                 * 
                               
                                
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   
                                     ϕ 
                                     R 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               cos 
                               ( 
                               
                                 
                                   ϕ 
                                   
                                     
                                       F 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       21 
                                     
                                     - 
                                     22 
                                   
                                 
                                 + 
                               
                                 
                             
                             ⁢ 
                             
                               ϕ 
                               
                                 M 
                                 - 
                                 J 
                               
                             
                           
                           + 
                         
                           
                       
                       ⁢ 
                       
                         ϕ 
                         
                           Ax 
                           - 
                           Ay 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         
                           ϕ 
                           B 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     The electric fields, A x  and A y , in expressions from (21) to (26) can be replaced by the incoming broadband light source fields, E 0x  and E 0y . After applying Eq. (7), the intensity |A x | 2 , |A y | 2  and A x A* y  can be expanded into the following form: 
     
       
         
           
             
               
                 
                   
                     
                        
                       
                         A 
                         x 
                       
                        
                     
                     2 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               G 
                               11 
                             
                             ⁢ 
                             
                               E 
                               
                                 0 
                                 ⁢ 
                                 x 
                               
                             
                           
                           + 
                           
                             
                               G 
                               12 
                             
                             ⁢ 
                             
                               E 
                               
                                 0 
                                 ⁢ 
                                 y 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 G 
                                 11 
                               
                               ⁢ 
                               
                                 E 
                                 
                                   0 
                                   ⁢ 
                                   x 
                                 
                               
                             
                             + 
                             
                               
                                 G 
                                 12 
                               
                               ⁢ 
                               
                                 E 
                                 
                                   0 
                                   ⁢ 
                                   y 
                                 
                               
                             
                           
                           ) 
                         
                         * 
                       
                     
                     = 
                     
                       
                         
                           
                              
                             
                               G 
                               11 
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                           
                              
                             
                               E 
                               
                                 0 
                                 ⁢ 
                                 x 
                               
                             
                              
                           
                           2 
                         
                       
                       + 
                       
                         
                           
                              
                             
                               G 
                               12 
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                           
                              
                             
                               E 
                               
                                 0 
                                 ⁢ 
                                 y 
                               
                             
                              
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   
                     
                        
                       
                         A 
                         y 
                       
                        
                     
                     2 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               G 
                               21 
                             
                             ⁢ 
                             
                               E 
                               
                                 0 
                                 ⁢ 
                                 x 
                               
                             
                           
                           + 
                           
                             
                               G 
                               22 
                             
                             ⁢ 
                             
                               E 
                               
                                 0 
                                 ⁢ 
                                 y 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 G 
                                 21 
                               
                               ⁢ 
                               
                                 E 
                                 
                                   0 
                                   ⁢ 
                                   x 
                                 
                               
                             
                             + 
                             
                               
                                 G 
                                 22 
                               
                               ⁢ 
                               
                                 E 
                                 
                                   0 
                                   ⁢ 
                                   y 
                                 
                               
                             
                           
                           ) 
                         
                         * 
                       
                     
                     = 
                     
                       
                         
                           
                              
                             
                               G 
                               21 
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                           
                              
                             
                               E 
                               
                                 0 
                                 ⁢ 
                                 x 
                               
                             
                              
                           
                           2 
                         
                       
                       + 
                       
                         
                           
                              
                             
                               G 
                               22 
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                           
                              
                             
                               E 
                               
                                 0 
                                 ⁢ 
                                 y 
                               
                             
                              
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     In the above equations, the terms that have phase difference of E 0x  and E 0y  components can be ignored because they are uncorrelated and average to zero over time. 
     Summing the intensities along both x- and y-axes, and using equations, E 0x (λ)=√{square root over (S(λ))}, E 0y (λ)=t 11y √{square root over (S(λ))} implied in Eq. (1) for light with intensity S(λ) at wavelength λ, the main signal light intensity becomes 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       main_dm 
                     
                     ⁡ 
                     
                       ( 
                       λ 
                       ) 
                     
                   
                   = 
                   
                     4 
                     ⁢ 
                     
                       S 
                       ⁡ 
                       
                         ( 
                         λ 
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                              
                             
                               F 
                               11 
                             
                              
                           
                           2 
                         
                         + 
                         
                           
                              
                             
                               F 
                               21 
                             
                              
                           
                           2 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                              
                             
                               G 
                               11 
                             
                              
                           
                           2 
                         
                         + 
                         
                           
                             
                                
                               
                                 G 
                                 12 
                               
                                
                             
                             2 
                           
                           ⁢ 
                           
                             
                                
                               
                                 t 
                                 
                                   11 
                                   ⁢ 
                                   y 
                                 
                               
                                
                             
                             2 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                          
                         M 
                          
                       
                       2 
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                             ϕ 
                             R 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             4 
                             ⁢ 
                             
                               ϕ 
                               B 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     The total main signal intensity is obtained through wavelength integration over the entire light source spectrum. From (30), the scale factor to convert light intensity to rate in unit of deg/hr is obtained, 
                   SF   =       1           4   ⁢       ∫     λ   1       λ   2       ⁢       S   ⁡     (   λ   )       ⁢     (              F   11          2     +            F   21          2       )     ⁢     (              G   11          2     +                                    G   12          2     ⁢            t     11   ⁢   y            2       )     ⁢          M        2     ⁢     sin   ⁡     (     4   ⁢     ϕ   B       )       ⁢           ⁢     ⅆ   λ               ·         λ   _     ⁢   c       2   ⁢   π   ⁢           ⁢     L   coil     ⁢     D   coil         ·     180   π     ·   3600.             (   31   )               
where L coil  and D coil  are the sensing coil  10  fiber length and average coil diameter, and  λ  is the light source mean wavelength. The scale factor is used to convert spurious light intensities to bias error with unit of deg/hr.
 
     The total amplitude type error, taking into account both x- and y-axes, is 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       amp_dm 
                     
                     ⁡ 
                     
                       ( 
                       λ 
                       ) 
                     
                   
                   = 
                   
                     4 
                     ⁢ 
                     ɛ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       S 
                       ⁡ 
                       
                         ( 
                         λ 
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                              
                             
                               F 
                               11 
                             
                              
                           
                           2 
                         
                         + 
                         
                           
                              
                             
                               F 
                               21 
                             
                              
                           
                           2 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                        
                       
                         
                           G 
                           11 
                         
                         ⁢ 
                         
                           G 
                           21 
                           * 
                         
                       
                        
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                              
                             
                               MN 
                               * 
                             
                              
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ϕ 
                                   
                                     M 
                                     - 
                                     N 
                                   
                                 
                                 + 
                                 
                                   ϕ 
                                   
                                     
                                       G 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                     - 
                                     21 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           
                              
                             
                               ML 
                               * 
                             
                              
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ϕ 
                                   
                                     M 
                                     - 
                                     L 
                                   
                                 
                                 + 
                                 
                                   ϕ 
                                   
                                     
                                       G 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                     - 
                                     21 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                         
                       
                         
                           ( 
                           
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   ϕ 
                                   B 
                                 
                                 ) 
                               
                             
                             - 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   3 
                                   ⁢ 
                                   
                                     ϕ 
                                     B 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                         + 
                         
                           4 
                           ⁢ 
                           ɛ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             S 
                             ⁡ 
                             
                               ( 
                               λ 
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                    
                                   
                                     F 
                                     11 
                                   
                                    
                                 
                                 2 
                               
                               + 
                               
                                 
                                    
                                   
                                     F 
                                     21 
                                   
                                    
                                 
                                 2 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                              
                             
                               
                                 G 
                                 12 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 G 
                                 22 
                                 * 
                               
                             
                              
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                                
                               
                                 t 
                                 
                                   11 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   y 
                                 
                               
                                
                             
                             2 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                    
                                   
                                     MN 
                                     * 
                                   
                                    
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       ϕ 
                                       
                                         M 
                                         - 
                                         N 
                                       
                                     
                                     + 
                                     
                                       ϕ 
                                       
                                         
                                           G 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                         
                                         - 
                                         22 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                     
                                   
                                     
                                        
                                       
                                         ML 
                                         * 
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             ϕ 
                                             
                                               M 
                                               - 
                                               L 
                                             
                                           
                                           + 
                                           
                                             ϕ 
                                             
                                               
                                                 G 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 1 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 2 
                                               
                                               - 
                                               22 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     sin 
                                     ( 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ϕ 
                                       B 
                                     
                                     ) 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 ( 
                                 
                                   sin 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       3 
                                       ⁢ 
                                       
                                         ϕ 
                                         B 
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                               - 
                               
                                 4 
                                 ⁢ 
                                 ɛ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   S 
                                   ⁡ 
                                   
                                     ( 
                                     λ 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     
                                       F 
                                       11 
                                     
                                     ⁢ 
                                     
                                       F 
                                       12 
                                       * 
                                     
                                   
                                    
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                          
                                         
                                           G 
                                           11 
                                         
                                          
                                       
                                       2 
                                     
                                     + 
                                     
                                       
                                         
                                            
                                           
                                             G 
                                             12 
                                           
                                            
                                         
                                         2 
                                       
                                       ⁢ 
                                       
                                         
                                            
                                           
                                             t 
                                             
                                               11 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               y 
                                             
                                           
                                            
                                         
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                          
                                         
                                           MN 
                                           * 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               ϕ 
                                               
                                                 
                                                   F 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   11 
                                                 
                                                 - 
                                                 12 
                                               
                                             
                                             + 
                                             
                                               ϕ 
                                               
                                                 M 
                                                 - 
                                                 N 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                          
                                         
                                           ML 
                                           * 
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               ϕ 
                                               
                                                 
                                                   F 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   11 
                                                 
                                                 - 
                                                 12 
                                               
                                             
                                             + 
                                             
                                               ϕ 
                                               
                                                 M 
                                                 - 
                                                 L 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           ϕ 
                                           B 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           3 
                                           ⁢ 
                                           
                                             ϕ 
                                             B 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 4 
                                 ⁢ 
                                 ɛ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   S 
                                   ⁡ 
                                   
                                     ( 
                                     λ 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     
                                       F 
                                       21 
                                     
                                     ⁢ 
                                     
                                       F 
                                       22 
                                       * 
                                     
                                   
                                    
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                          
                                         
                                           G 
                                           11 
                                         
                                          
                                       
                                       2 
                                     
                                     + 
                                     
                                       
                                         
                                            
                                           
                                             G 
                                             12 
                                           
                                            
                                         
                                         2 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                            
                                           
                                             t 
                                             
                                               11 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               y 
                                             
                                           
                                            
                                         
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                          
                                         
                                           MN 
                                           * 
                                         
                                          
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ( 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           
                                             ϕ 
                                             
                                               
                                                 F 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 21 
                                               
                                               - 
                                               22 
                                             
                                           
                                           + 
                                           
                                             ϕ 
                                             
                                               M 
                                               - 
                                               N 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                          
                                         
                                           ML 
                                           * 
                                         
                                          
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ( 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           
                                             ϕ 
                                             
                                               
                                                 F 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 21 
                                               
                                               - 
                                               22 
                                             
                                           
                                           + 
                                           
                                             ϕ 
                                             
                                               M 
                                               - 
                                               L 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           ϕ 
                                           B 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           3 
                                           ⁢ 
                                           
                                             ϕ 
                                             B 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     The total intensity-type error is 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       
                         int 
                         ⁢ 
                         _dm 
                       
                     
                     ⁡ 
                     
                       ( 
                       λ 
                       ) 
                     
                   
                   = 
                   
                     
                         
                         
                     
                     ⁢ 
                     4 
                     ⁢ 
                     
                       ɛ 
                       2 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       S 
                       ⁡ 
                       
                         ( 
                         λ 
                         ) 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                              
                             
                               F 
                               11 
                             
                              
                           
                           2 
                         
                         + 
                         
                           
                               
                             
                               
                                  
                                 
                                   F 
                                   21 
                                 
                                  
                               
                               2 
                             
                             ) 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                    
                                   
                                     G 
                                     21 
                                   
                                    
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                      
                                     
                                       G 
                                       22 
                                     
                                      
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                      
                                     
                                       t 
                                       
                                         11 
                                         ⁢ 
                                         y 
                                       
                                     
                                      
                                   
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                              
                             
                               NL 
                               * 
                             
                              
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 ϕ 
                                 
                                   N 
                                   - 
                                   L 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             sin 
                             ( 
                             
                                 
                             
                             ⁢ 
                             
                               2 
                               ⁢ 
                               
                                 ϕ 
                                 B 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           4 
                           ⁢ 
                           
                             ɛ 
                             2 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             S 
                             ⁡ 
                             
                               ( 
                               λ 
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                    
                                   
                                     F 
                                     12 
                                   
                                    
                                 
                                 2 
                               
                               + 
                               
                                 
                                    
                                   
                                     F 
                                     22 
                                   
                                    
                                 
                                 2 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                    
                                   
                                     G 
                                     11 
                                   
                                    
                                 
                                 2 
                               
                               + 
                             
                               
                           
                           ⁢ 
                           
                               
                             
                               
                                 
                                    
                                   
                                     G 
                                     12 
                                   
                                    
                                 
                                 2 
                               
                               ⁢ 
                               
                                 
                                    
                                   
                                     t 
                                     
                                       11 
                                       ⁢ 
                                       y 
                                     
                                   
                                    
                                 
                                 2 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                              
                             
                               NL 
                               * 
                             
                              
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 ϕ 
                                 
                                   N 
                                   - 
                                   L 
                                 
                               
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     The total error rate originated from both amplitude- and intensity-type polarization errors may thus be: 
     
       
         
           
             
               
                 
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     Integration over wavelength from λ 1  to λ 2  covers the broadband light source wavelength range. Data characterizing the total error rate and/or other error information may be displayed to output device  116 . 
     The approach described above predicts a bias offset when there are polarization errors. Since the polarization errors are the result of interference of waves propagating along different respective optical paths, relative changes of optical paths will cause the corresponding polarization error to fluctuate. The said “optical path” depends on the polarization state of the light. For a PM fiber, there are two polarization modes. One mode is polarized along the so-called “slow polarization axis” in the fiber cross-section. It acquires more phase delays than the other polarization mode which is polarized along the orthogonal “fast polarization axis”. The slow mode effectively travels along a longer optical path than the fast mode. To quantify the phase difference associated with the fast and slow modes in a PM fiber, it is advantageous to express the lengths of fiber in terms of “beat length” (L B ), such that L B =λ/Δn, where λ is the light wavelength, and Δn is the difference in effective mode refractive index between the fast and the slow polarization modes. Typically, L B  ranges from 1 mm to 4 mm. The phase difference between slow and fast light after propagating over fiber length of L B  is 2π. Typically, the fiber beat length L B  increases slightly when temperature increases (i.e., the temperature is ramped). Changing the PM fiber temperature is equivalent to changing the phase difference of the slow and fast light waves, and is one of the effective ways to excite polarization error induced bias fluctuations. For example, if the PM fiber  15  is assumed to have a length of v=10000 L B , and the beat length L B  of the PM fiber has a temperature sensitivity of 500 ppm/° C., the relative phase change of the fast and slow light will undergo five cycles per ° C., i.e. 10π/° C.  FIG. 2  illustrates a plot of such relative bias versus temperature change for a gyroscope with polarization errors using an approach according to an embodiment of the invention. Since absolute bias depends on specific gyroscope design, only relative bias amplitude values in arbitrary linear unit are used throughout the document without lost of significances. To perform a real temperature ramp measurement, the gyro is put in a temperature chamber under constant rotation rate, e.g. the earth rotation rate. The temperature is gradually increased while rates are continuously read from the gyro. If the gyro under test has substantial polarization errors, the rates will exhibit sinusoidal oscillation patterns versus temperature. The temperature period of the oscillation is a good indicator of the phase delay involved in the polarization error. Measuring bias fluctuation under temperature ramping of selected sections of the optical circuit is an effective way to identify the cause of the polarization error and allow a testing user to replace or otherwise modify inferior portions in the circuit with more optimally configured portions. Additionally, based on the above-described analytical approach, an embodiment includes analytically simulating the bias fluctuations under temperature ramping. The temperature ramping test and methods according to embodiments of the invention are powerful tools in gyroscope polarization error investigations. 
     In the calculation method developed above, the wavelength dependent Jones matrix of the single mode non-polarization-maintaining fiber coil is used. The matrix can actually be measured directly using a polarization-mode-dispersion (PMD) characterization instrument. The coil matrix can also be modeled by treating the coil as a concatenation of many random birefringent elements. With the availability of the coil Jones matrix, the coherence function and the wavelength dependent differential group delay (DGD) of the coil can be determined.  FIG. 3   a  shows two coil coherence functions. Coil-I is represented by a solid line and coil-II is represented by a dashed line. These coil coherence functions, or autocorrelation functions, characterize interference fringe contrast as a function of delay length of broadband lightwaves passing through the coil. At delay length of 2 mm, coil-I exhibits much smaller fringe contrast (−30 dB) compared to coil-II (−18 dB).  FIG. 3   b  shows the wavelength dependent DGDs and the averaged DGDs of the corresponding coil-I and coil-II. Coil-I has an average DGD value of 0.85 ps, much smaller than that of coil-II (dashed line, average DGD=1.81 ps). In other words, coil-I has smaller internal birefringence than coil-II. In the design of depolarized gyroscopes, fabricating coils with low birefringence is advantageous in lowering polarization errors.  FIG. 4  shows a comparison of relative bias fluctuation amplitude under temperature ramping for two gyroscopes having identical design, except for their respective coils, coil-I and II.  FIG. 4  illustrates that the gyroscope using coil-II (dashed lines with circle marks) exhibits significantly larger bias instability than that using coil-I (solid line). 
     The coupler  12  in the optical circuit also plays an important role in controlling polarization errors. As mentioned earlier, the Jones matrix of the coupler shall take into account the polarization dependent splitting ratio (PDSR) and PDL. For light propagating from point  30  to point  31 , the x- and y-polarized light have different amplitude transmission coefficient s x  and s y . There are also cross-couplings between x- and y-polarized light that can be quantified by parameter k 12 . After taking into account the phase delays, d 121 , d 122  of the input and output pigtails  121  and  122  of the coupler, the Jones matrix of the coupler  12  can be written as 
                     H   in     =         (         1       0           0         ⅇ     -     id   122               )     ⁢     (         1         k   12               -     k   12           1         )     ⁢     (           s   x         0           0         s   y           )     ⁢     (         1       0           0         ⅇ     -     id   121               )       ≈       s   x     ⁡     (         1           k   12     ⁢     ⅇ     -     id   121                       -     k   12       ⁢     ⅇ     -     id   122                   (       s   y       s   x       )     ⁢     ⅇ     -     i   ⁡     (       d   121     +     d   122       )                   )       ≈     (         1           k   12     ⁢     ⅇ     -     id   121                       -     k   12       ⁢     ⅇ     -     id   122                   ⅇ     -     i   ⁡     (       d   121     +     d   122       )           ⁢       10       L   PDSR     /   10                 )               (   35   )               
In the above equation L PDSR  is the polarization splitting ratio difference in decibel units. Other PDL are omitted for simplification. Since the common loss and phase experienced by both x- and y-polarized light are not important in the final result of the polarization error calculation, the factor of s x  in the expression of (35) may be omitted in the last step. Similarly, for propagation of light from point  31  to photodetector splice  13 , the coupler matrix can be written as
 
                     H   out     =         (         1       0           0         ⅇ     -     id   123               )     ⁢     (           r   x         0           0         r   y           )     ⁢     (         1         -     k   12                 k   12         1         )     ⁢     (         1       0           0         ⅇ     -     id   122               )       ≈     (             10       L   PDSR     /   10                 -     k   12       ⁢     ⅇ     -     id   122                       k   12     ⁢     ⅇ     -     id   123                 ⅇ     -     i   ⁡     (       d   122     +     d   123       )                 )               (   36   )               
where d 123  is the phase delay caused by the coupler pigtail  123 . Although small PDSR has been assumed in the approximations of expressions, it is not intended to limit the application of the method to use more general forms of H in  and H out  matrices. As illustrated in  FIG. 5 , the relative bias fluctuation amplitude under temperature ramping is found to decrease monotonically as PDSR approaches zero. It is therefore advantageous to use a coupler having small PDSR and PDL to reduce the bias instability.
 
     Most of the spurious light producing polarization errors originates from cross-couplings at fiber splices or in optical components. It is therefore advantageous to reduce these cross-couplings as much as possible.  FIG. 6  shows a dependence of relative bias error as a function of the cross-coupling at splice  32  (i.e., k 32 ). It is obvious that smaller k 32  produces smaller bias. A method, according to an embodiment, to reduce cross-couplings at splices between two PM fibers is to use the identical or similar fiber types. The distortions involved at the joint points are reduced when the two connecting fibers have the same mode field diameter, stress elements and refraction index. 
     In the calculation of polarization error, one can observe that the amplitude-type polarization error is proportional to the IOC polarization extinction ratio (PER), ε, and the intensity type error is proportional to ε 2 . As an example,  FIG. 7  shows a calculation for polarization error induced relative bias instabilities as a function of IOC PER. Indeed, by selecting IOC with smaller PER, bias fluctuation amplitude may be reduced. 
     Design of depolarizer  200  and the PM fiber  15  are effective ways to reduce gyro bias error according to an embodiment. As illustrated in the example of  FIG. 1 , the depolarizer  200  is composed of two sections,  210  and  220 . Section  210  is composed of two pieces of PM fiber,  21  and  23 , with polarization axis oriented about 45° with each other at splice  22 . Section  220  is composed of two pieces of PM fiber,  26  and  28 , with polarization axis oriented about 45° with each other at splice  27 . Section  15  is a piece of PM fiber that generates delays for the slow and fast light before such light enters the IOC  16 . The lengths of PM fibers  15 ,  21 ,  23 ,  26  and  28  are designated in  FIG. 1  as, respectively, v, z, x, y and w. Together with IOC pigtail and waveguide lengths,  33 ,  36 ,  38 ,  40  and  41 , determination of the lengths of PM fibers  15 ,  21 ,  23 ,  26  and  28  are important in the design of depolarized gyroscopes from the standpoint of controlling polarization error. These lengths determine the relative time delays involved in the broadband light source interference. Longer delays cause decrease of interference fringe contrast and less bias fluctuations under temperature changes. By tailoring the lengths of these PM fibers, it is possible to control the polarization error below a certain value. For example,  FIG. 8  shows, in association with gyroscopes employing, respectively, coil-I and coil-II discussed above, a dependence of polarization error induced relative bias fluctuation amplitude on the length of PM fiber  15  in unit of beats for a design of depolarizer  200  according to an embodiment. Selecting length, v, far away from an error peaking region is an effective way to reduce polarization error induced bias instability. 
     In an embodiment, the length, w, is selected to be approximately twice that of length, x (i.e., w=2x). In addition, the physical lengths of the depolarizer section  210  and  220  are kept approximately the same (i.e., z=x+y). Sections  210 ,  220  may be winded onto the coil  10  with the symmetrical points, i.e. points with equal optical length measured from splitting point  17  in IOC, located close to each other to avoid the temperature-fluctuation-induced non-reciprocal phase noise (i.e., the so-called Shupe effect induced noise). To minimize the overall depolarizer length, the length, y, and IOC pigtail lengths,  33 ,  36  and  37  may be selected to be as short as possible. In such an embodiment of the model, the optimal lengths of x and v are determined once the IOC, coupler and coil properties are known. 
     As an example of showing the method to optimize the depolarizer and PM fiber  15  design, the calculated relative bias fluctuation amplitude under temperature ramp is plotted as functions of x and v in the contour plots illustrated in  FIG. 9   a  and  FIG. 9   b .  FIG. 9   a  is plotted for a gyroscope using the coil-I associated with  FIG. 3  that has relatively smaller birefringence.  FIG. 9   b  is plotted for a gyroscope using the coil-II associated with  FIG. 3  that has relatively larger birefringence. In both figures, dark regions have larger bias errors than those in the light regions. Same color codes are used for  FIGS. 9   a  and  9   b  for comparison. Several v-x regions exhibit low bias errors and may thus be preferable. These regions are marked as A, B, C, D and E. They are located between five dark regions with large bias fluctuation amplitudes. From comparison of  FIGS. 9   a  and  9   b , it may be seen that the coil birefringence has significant impact on the gyroscope design. Larger coil birefringence causes minimization of the optimal-design regions and larger background bias errors in the relatively optimal design regions. 
     Still referring to  FIGS. 9   a  and  9   b , the four narrow dark regions, roughly along the southwest-northeast direction, are found to have v-x relations close to: 
                         v   =       ⁢       4   ⁢   x     +   y   -     d     32   -   27                     ⁢       ridge   ⁢           ⁢   between   ⁢           ⁢   A   ⁢           ⁢   and   ⁢           ⁢   B     ,                 v   =       ⁢       3   ⁢   x     -     d     32   -   27                     ⁢       ridge   ⁢           ⁢   between   ⁢           ⁢   B   ⁢           ⁢   and   ⁢           ⁢   C     ,                 v   =       ⁢       2   ⁢   x     -     d     32   -   27                     ⁢       ridge   ⁢           ⁢   between   ⁢           ⁢   C   ⁢           ⁢   and   ⁢           ⁢   D     ,                 v   =       ⁢     x   -     d     32   -   27                     ⁢     ridge   ⁢           ⁢   between   ⁢           ⁢   D   ⁢           ⁢   and   ⁢           ⁢     E   .                     (   37   )               
where d 32-27 =L IOC +y+d 33 +d 38  is the relative phase delay of the fast and slow light of the optical path between  32  and  27 , which includes the IOC waveguide  41 , pigtails  33  and  38 , and the depolarizer PM fiber  26 . These regions of high bias instability are associated with amplitude type polarization errors originating from the interference of the IOC-passed (x-polarized) waves and the IOC-rejected (y-polarized) waves traveling through the sensing loop  10 . A preferable choice of design is often situated in the region A because of its larger low error region providing greater margins against coil uncertainties. However, designs in region A may require longer lengths of depolarizer and PM fibers  15 , which may be of concern for gyro packaging and control of Shupe effect. Regions B, C, D and E are therefore of interest in design choices for reducing the PM fiber lengths.
 
     However, the depolarizer x length cannot be lowered below a certain value. The broader vertically oriented regions on the left sides of  FIGS. 9   a  and  9   b  can undermine bias performance if the length x is too small. This region is associated with polarization errors insensitive to the PM fiber  15  length v. Such errors mostly originate from interference among waves passing through the PM fiber  15  in the same polarization axis. As depolarizer length x increases, this type of error decreases. 
     As another illustration of an embodiment of a depolarized gyroscope, one can optimize the lengths x and w as separate parameters, rather than observing the relation w=2x. In such an embodiment, the length y can be set to a fixed value, and w+y is set equal to the sum of x and z (i.e., z=y+w−x). As such, the depolarizer sections  210  and  220  have the same length. Next, the PM fiber  15  may be first set to a large value. As discussed elsewhere herein, it is known that v-insensitive error can be minimized by setting the length v, to a very large value. The design may now be reduced to the determination of the optimal lengths of x and w. Once optimal x and w is found to minimize the v insensitive polarization error, the PM fiber  15  length v can then be determined to reduce the v sensitive polarization errors. One will see below that w=2x used in an earlier-discussed embodiment is indeed one of the optimal selections of the w-x relationship for better bias stability. 
       FIG. 10   a  and  FIG. 10   b  illustrate plots of relative bias fluctuation amplitude calculation results as functions of x and w using coil-I and coil-II, respectively. There are multiple design regions providing low bias instability. Among them, regions F, G, H, and J are often of greater interest because they offer relatively shorter depolarizer fiber lengths. Region F has an approximate relationship between w and x of w=x/2 or x=2w. Region G has a relation of w=x, which means the two depolarizer PM fiber lengths are close to identical. Region H has a relation of w=2x, which was selected in an above-described embodiment and used in generating  FIGS. 9   a  and  9   b . Regions H and F are of the same type from the point of view of symmetry (i.e., one of the x or w lengths is twice of the other). Region J is a wider region and may have better bias stabilities than those of region F, G and H. 
     An embodiment includes using a temperature-ramping approach, for example, to select optimized values of depolarizer fiber section length w and x substantially in the regions of F, G, H and J shown in  FIGS. 10   a  and  b , to minimize v-insensitive polarization errors while setting v at a substantially long length. In one exemplary embodiment, w is substantially equal to 8000 L B , x is substantially equal to 4000 L B , y is substantially equal to 500 L B , and z substantially equal to 4500 L B . Additionally, the v may be substantially longer than 16500 L B  (region-A in  FIGS. 9   a  and  9   b ), substantially equal to 12500 L B  (region-B in  FIGS. 9   a  and  9   b ), equal to 8000 L B  (region-C in  FIGS. 9   a  and  9   b ), substantially equal to 4000 L B  (region-D in  FIGS. 9   a  and  9   b ), or substantially less than 1000 L B  (region-E in  FIGS. 9   a  and  9   b ). 
     The relative bias amplitude in these regions can be clearly seen by referring back to  FIG. 8 , which is a plot of bias versus PM fiber length, v, when the x=4000 L B , y=500 L B  and w=8000 L B  are selected. Again, the solid line is associated with the coil-I and the dashed line associated with the coil-II. As is illustrated, the designs using larger v-value regions (A, B and C) around v&gt;16500 L B , v=12500 and 8000 L B  have better tolerance to fiber length uncertainties than those using smaller v-values regions (D and E) around v=4000 L B  and v&lt;1000 L B . 
     While preferred embodiments of the invention have been illustrated and described, as noted above, many changes can be made without departing from the spirit and scope of the invention. Accordingly, the scope of the invention is not limited by the disclosure of the preferred embodiment. Instead, the invention should be determined entirely by reference to the claims that follow.