Patent Publication Number: US-8526005-B1

Title: System and method for calibrating optical measurement systems that utilize polarization diversity

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     The present application claims the benefit of U.S. Provisional Patent Application Ser. No. 61/297,239, entitled “REAL-TIME POLARIZATION DIVERSITY CALIBRATION TECHNIQUE” filed on Jan. 21, 2010, which is hereby incorporated by reference in its entirety for all purposes. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     Not applicable. 
     BACKGROUND 
     1. Field 
     The subject technology relates generally to optical measurements, and more specifically to systems and methods for calibrating for optical measurement systems that utilize polarization diversity. 
     2. Background 
     A waveplate or retarder is an optical device that alters the polarization state of a light wave travelling through it. A waveplate works by shifting the phase between two perpendicular polarization components of the light wave. A typical waveplate is simply a birefringent crystal with a carefully chosen orientation and thickness. The crystal is cut so that the extraordinary axis or “optic axis” is parallel to the surfaces of the plate. Light polarized along this axis travels through the crystal at a different speed than light with the perpendicular polarization, creating a phase difference. Therefore, one of the two perpendicular polarization components experiences a retardation (e.g., slowdown) with respect to the other component in the waveplate. Such polarization diversity is utilized to propagate information in interferometric measurements and sensing systems. 
     Systems that use polarization diversity to propagate information are sensitive to drift and systematic effects in the birefringence and retardance of the optical components of the system. Calibration is required to characterize and compensate for (e.g., subtract out) these spurious effects. The update rate of calibration is dependent upon the time scales of drift and noise, balanced against sensing requirements. For example, in homodyne metrology where the relative phase of the two polarization components (e.g., in-phase (I) and quadrature (Q) sensing beams) are sensed to determine a position and/or a change in position, the systematic biases in retardance, diattenuation and birefringence need to be calibrated in order to accurately monitor motion to, e.g., a 100 pm level. 
     In conventional calibration methods, a motion is applied to the entire optical system in order to generate the &gt;1 wave phase shift in the I and Q sensing polarizations. This is usually done by a deliberate actuation of a mirror, which also tends to modulate the data beam as well as the sensing beam. For systems with moderate drift and high precision requirements, these calibration procedures need to be performed frequently, thereby disturbing the normal operation (e.g., measurement or sensing) of the system. 
     Accordingly, a need exists in an optical measurement system that utilizes polarization diversity to provide a calibration procedure that can be performed without disturbing the measurement of the system. 
     SUMMARY 
     Systems and methods disclosed here can be used for extracting relevant parameters for calibration in an optical measurement system. In certain aspects of the present disclosure, the calibration can be achieved in real time without disturbing the normal operation (e.g., sensing and/or measurement) in the optical measurement system. 
     According to one aspect of the present disclosure, an optical measurement system utilizing polarization diversity is provided. The system can comprise a light source configured to transmit an incident light beam in a beam direction. The system can further comprise a waveplate having a rotation axis and configured to receive at least a portion of the incident light beam, the received incident light beam causing a first polarization component and a second polarization component to propagate in the waveplate. The waveplate is configured to rotate about the rotation axis to cause an equal common phase shift in the first and second polarization components while maintaining a differential phase shift in the first and second polarization components. 
     According to one aspect of the present disclosure, a method of calibrating an optical measurement system utilizing polarization diversity is disclosed. The method can comprise causing a first light polarization component and a second light polarization component to propagate in a waveplate having a rotation axis, while rotating the waveplate about the rotation axis. The method can further comprise causing an equal common phase shift in the first and second light polarization components while maintaining a differential phase shift in the first and second light polarization components. The method can further comprise sensing a relative phase between the first and second light polarization components. The method can further comprise determining at least one calibration parameter based on the relative phase. 
     According to one aspect of the present disclosure, a waveplate for use in an optical measurement system utilizing polarization diversity is provided. The waveplate can comprise a first plate comprising a first material and configured to receive an incident light beam and propagate therein a first light polarization component and a second light polarization component. The waveplate can further comprise a second plate comprising a second material coupled to the first material and configured to receive the first and second polarization light components from the first plate. The waveplate can have a rotation axis around which the waveplate is configured to be rotated with a minimum optical path difference between the first and second polarization components through the first and second plates. In one preferred aspect, in a retarder, there is a desired optical path difference between the two polarizations, and that difference is preserved. 
     According to one aspect of the present disclosure, an optical measurement system utilizing polarization diversity is provided. The system can comprise a light source configured to transmit an incident light beam. The system can further comprise a waveplate having a rotation axis and configured to receive at least a portion of the incident light beam, the received incident light beam causing a first polarization component and a second polarization component to propagate in the waveplate. The system can further comprise a rotation mechanism coupled to the waveplate and configured to rotate the waveplate about the rotation axis by a predetermined angle to cause an equal common phase shift in the first and second polarization components while maintaining a differential phase shift in the first and second polarization components. The system can comprise photodetectors disposed at an opposite side of the waveplate with respect to the light source and configured to receive a light beam emerging from the waveplate and convert the emerging light beam into electrical signals. The system can further comprise a signal conversion module configured to receive the electrical signals and convert the electrical signals into a digital representation. The system can further comprise a processor configured to receive the digital representation, sense a relative phase between the first and second light polarization components in the emerging light beam, and determine at least one calibration parameter based on the relative phase. The system can further comprise a memory in data communication with the processor and configured to store the at least one calibration parameter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram depicting a polarized light beam travelling along a first direction and incident on a waveplate. 
         FIG. 2  is a diagram depicting a waveplate having a special rotation axis according to certain aspects of the present disclosure. 
         FIG. 3  is an exemplary optical measurement system according to certain aspects of the present disclosure. 
         FIG. 4  is a diagram depicting light propagation through a waveplate having a single stack construction according to certain aspects of the present disclosure. 
         FIG. 5  is a diagram depicting light propagation through a waveplate having a two-stack construction according to certain aspects of the present disclosure. 
         FIG. 6  is a flowchart illustrating an exemplary process for calibrating an optical measurement system utilizing polarization diversity. 
         FIG. 7  is a schematic block diagram of an exemplary optical measurement system utilizing polarization diversity and configured to be calibrated according to certain aspects of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a diagram depicting a polarized light beam  10  travelling along a first direction (e.g., z direction  3 ) and incident on a waveplate  100 . The light beam  10  has a linear polarization  11  along a second direction (e.g., along y direction  2 ). The waveplate  100  is assumed to have an ordinary axis (o-axis)  101  and an extraordinary (e-axis)  103 . The waveplate  100  is selected and oriented such that a desired phase shift between the two polarization components of light is achieved. For example, the orientation and thickness of the waveplate  100  is such that a quarter-wave phase shift between the polarization components is realized in an emerging light beam  20 , thereby enabling quadrature phase measurements based on I and Q beams. For example, if the waveplate  100  is a quarter-waveplate, the light component associated with a larger index of refraction is retarded by 90° in phase (a quarter wavelength) with respect to that associated with a smaller index, such that the emerging light beam  20  becomes circularly polarized. In one aspect, the effective retardance needs to be ¼ wave for I and Q signals to be generated. It should be noted that in an example with the Michelson interferometer, the waveplate is ⅛ wave because the beam is double-passed. 
     According to certain aspects of the present disclosure, the waveplate  100  is part of a sensing or measurement system in which the waveplate  100  is inserted in the path of a sensing beam or sensing beams, but not in the path of a data beam. In another example, the waveplate  100  may be inserted in the path of a sensing beam or sensing beams as well as the path of a data beam depending on the data beam. The light beam  10  is linearly polarized, so typically (but not always) the polarization  11  of the light beam  10  is at 45 degrees to the o- and e-axes  101 ,  103  of the uniaxial crystal which comprises the waveplate  100 . Therefore, the single polarization  11  of the light beam  10  may be thought of as being a sum of two polarizations along the o- and e-axes  101 ,  103  of the waveplate  100 , and the light beam  10  having the single linear polarization  11  may be thought of as comprising two light rays, namely an ordinary ray (o-ray) polarized along the o-axis  101  and an extraordinary ray (e-ray) polarized along the e-axis  103 . 
     One of the properties of the waveplate  10  is that it is configured to impart a predetermined relative phase shift between the o- and e-rays. However, for most sensing devices, it is desirable to modulate both o- and e-rays in phase by an equal amount to calibrate the optical system (e.g., the “bias” changes by the same amount, so the difference in phase shifts remains exactly the same). Conventionally, as indicated above, such modulation is achieved by motion of components of the optical system being calibrated. 
       FIG. 2  is a diagram depicting a waveplate  200  having a special rotation axis  211  according to certain aspects of the present disclosure. A small rotation (e.g., shaking or wobbling) around or about the rotation axis  211  causes an equal common phase shift in the orthogonal polarizations (e.g., I and Q) in the emerging light beam  20  while maintaining a differential phase shift (e.g., 90°) necessary for the I and Q sensing. In the illustrated example, the rotation axis  211  is at a certain orientation angle (θ)  213  from an axis (e.g., x-axis) coplanar with the waveplate  200 . 
     The rotation of the waveplate  200  about the rotation axis  211  makes it possible to modulate both o- and e-rays in phase by an equal amount so that the phase difference between the rays remains the same. During a calibration procedure, the waveplate  200  is rotated by a sufficient rotation angle (ω)  215  to generate at least one wave of common phase shift for fitting the curves and extracting the calibration parameters. In certain embodiments, the rotation angle (ω)  215  can be in a range, for example, between 1 and 2 degrees, depending on the wavelength of the light and the index of refraction of the waveplate material. Since the waveplate  200  is not present in the path of the data beam in certain sensing system embodiments, the data beam in such systems is undisturbed. Because the waveplate used to generate the I and Q sensing beams is also used to generate the common phase shift, the calibration can be performed without disturbance to the system under measurement, and so can either be done frequently or continuously in real-time, with no interruption to the data. 
     The existence and utilization of such a special rotation axis of a waveplate for calibration purposes is unexpected in view of the fact that, in general (e.g., except for the case of a special tilt axis), varying indices of refraction for the o- and e-rays cause the rays to refract through the material of the waveplate along different paths, thereby tending to change their relative phase shift. The calibration systems and methods of the present disclosure based on the rotation of a waveplate about a special rotation axis are shown to be applicable in spite of these facts. 
       FIG. 3  is an exemplary optical measurement system  300  according to certain aspects of the present disclosure. In the illustrated example of  FIG. 3 , the sensing system  300  is a Michelson interferometer. However, a person skilled in art shall understand in view of the present disclosure that the systems and methods of the present disclosure can be applied to other types of optical measurement systems based on polarization diversity including, but not limited to, an optical communication system, material properties characterization, and displacement path length metrology. 
     The sensing system  300  comprises a light source (e.g., a laser)  310 , a beam splitter  321 , a waveplate  323 , first and second mirrors  325 ,  327 , and a sensor  329 . The light source, which can be a laser, transmits an incident light beam  301  towards the waveplate  323 . The beam splitter  321  splits the incident light beam  301  into a first split light beam  303  and a second split light beam  307 . The first split light beam  303  passes through the waveplate  323  and emerges therefrom as an emerging light beam  305 . The emerging light beam  305  and the second split light beam  307  reflect from the first and second mirrors  325 ,  327 , respectively, and travel back to the beam splitter  321  where they are combined and sent to the sensor  329  as a combined light beam  309 . The sensor  329  uses the o- and e-ray polarizations of the combined light beam  309  to sense how much the first mirror  325  and/or the second mirror  327  has moved, for example. By rotating the waveplate  323  around a special rotation axis as discussed above, the sensing system  300  can be calibrated without disturbing the measurement thereof. 
     With reference to  FIG. 2  for ease of illustration without any intent to limit the scope of the disclosure in any way, various systems and methods of determining the orientation angle ( 0 )  213  by which the special rotation axis  211  is rotated from an axis (e.g., x-axis  201 ) of the waveplate  200  are now described. The orientation angle ( 0 )  213  may vary depending on the materials used in the waveplate  200  (e.g., calcite) and the wavelength of the sensing light. Stated in another way, the orientation angle (θ)  213  may vary depending on the index of refraction of the material, which is a function of the wavelength of light and the material itself. The orientation angle (θ)  213  may also vary depending on the manner in which the waveplate  200  is constructed (e.g., single or multiple stacks). As the waveplate  200  is rotated or tilted by a rotation angle (ω)  215 , the waveplate  200  appears thicker to the light beam travelling in the waveplate  200 , which causes larger phase shifts in the polarization light components (e.g., the n- and e-rays) travelling therein. However, as long as the waveplate  200  is rotated about the special rotation axis  211 , the phase shift for the o- and e-rays is the same, so the difference in the o- and e-ray phases remains the same. 
     There are, in general, two major categories of waveplates—multi-order and zero-order (0-order). A multi-order waveplate is a single plate of a birefringent material. A zero-order waveplate is made of two plates which have a specific thickness difference equal to the desired retardance of the waveplate and in which the o- and e-axes are oriented opposite to each other (i.e., the o-axis in one plate is parallel to the e-axis in the other plate, and vice versa). Zero-order waveplates are less sensitive to angle of incidence, and are accurate for a much larger range of wavelengths, which are the main advantages of such waveplates. 
       FIG. 4  is a diagram depicting light propagation through a waveplate  400  having a single stack construction according to certain aspects of the present disclosure. The waveplate  400  is assumed to comprise of a single optical material (e.g., calcite). It is further assumed that a monochromatic incident light beam  410  of linearly polarized light is incident on the waveplate  400  at an angle of incidence φ  413 . The incident light beam  410  can be described then as a combination of an extraordinary ray (e-ray) and an ordinary ray (o-ray) by the projection of the polarization of the incident light beam  410  on an extraordinary axis (e-axis) and an ordinary axis (o-axis) 90° to the extraordinary axis in the plane of the waveplate  400 . Typically, the polarization of the incident light beam  410  is oriented such that equal amounts of light are in the e- and o-rays (e.g., the polarization is oriented 45° to the e- and o-axes), but that is not required. In fact, the calibration methodology of the present disclosure allows for an imbalance in the amounts of light in the e- and o-rays. In one aspect, there may be substantially no constraints on the magnitude of φ  413 ; it is a parameter for optical design purposes (e.g., to reject ghost reflections, etc.). The magnitude of φ  413 , however, cannot be greater than Brewster&#39;s angle; otherwise no light will go through the plate. 
     It is further assumed that a rotation axis (not shown) about or around which the waveplate  400  is tilted or rotated for calibration purposes is at an angle θ (not shown) relative to an axis (see, e.g., the x-axis  201  of  FIG. 2 ). The angle θ, also referred hereinafter as a “wobble angle”, defines the axis about which small rotations cause no change in the retardation of the waveplate, but provide a common phase shift for the polarization components (e.g., the e- and o-rays) travelling through the waveplate  400  for use in calibration. 
     In one aspect, the angle θ is determined by defining an equation for an optical path difference (OPD) as a function of θ. The OPD corresponds to a difference between optical path lengths (OPLs) through the waveplate  400  between the e-ray parallel to the e-axis and the o-ray parallel to the o-axis, which is normal to the e-axis. In one aspect, the derivative of the OPD is minimized with respect to ω ( 215 ). In one aspect, the OPD needs to be consistent with the retardation (i.e., the retardation phase is equal to 2 times pi times OPD divided by wavelength). One method of determining the angle θ involves iterating the angle θ ( 213 ), taking small steps in ω ( 215 ), calculating the delta OPD, and selecting the angle θ ( 213 ) where the delta OPD is the smallest. An analytical formula for ∂OPD/∂ω (i.e., the derivative) can be derived and minimized by a method such as a simplex method, a trust-region method, Newton&#39;s method, or a line search method. These are examples, and other methods may also be used. 
     Firstly, general equations for light passing through a tilted parallel waveplate  400  are derived. These equations are used later when deriving the physical path lengths of light travelling through the waveplate  400  after solving for the angles of incidence. In  FIG. 4 , n refers to the refractive index of the optical material comprising the waveplate  400 , t refers to the thickness of the waveplate  400 , d refers to a physical path length of a light beam  415  within the waveplate  400 , and d′ refers to a physical distance coaxial to the light beam  415  between entrance and exit points on the parallel waveplate  400  (e.g., the thickness of the missing air). 
     For this configuration, the following relationships exist: 
             d   =     t       1   -       (       sin   ⁢           ⁢   φ     n     )     2                         φ   ′     =       sin     -   1       ⁡     (       sin   ⁢           ⁢   φ     n     )               d′=t ·cos(φ−φ′)
 
     Assuming that the waveplate  400  is a unixial crystal waveplate (e.g., quartz) with ordinary and extraordinary refractive indices n o  and n e , respectively, the polarization states of the incident light beam  410  can be decomposed into a first polarization component parallel to the o-axis and a second polarization component parallel to the e-axis. Because of the tilt or rotation of the waveplate  400 , however, the angle of incidence varies for the e-axis, but not for the o-axis due to the symmetry of the uniaxial crystal. Hence, the following relationships:
 
φ o =φ
 
φ e =2 arcsin(sin φ/2*cos θ),
 
where φ o  is the incidence angle with respect to the ordinary axis and is independent of rotational angle, and φ e  is the incidence angle with respect to the extraordinary axis and requires a coordinate transformation, and contains the dependence on the orientation or “wobble angle” θ.
 
     Solving through Snell&#39;s Law and the index ellipsoid, φ′ e  and φ′ o , the angles of the e- and o-rays after entering the waveplate  400 , are given by: 
     
       
         
           
             
               n 
               e 
               ′ 
             
             = 
             
               1 
               
                 
                   
                     
                       
                         ( 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             φ 
                             e 
                           
                         
                         ) 
                       
                       2 
                     
                     
                       n 
                       o 
                       2 
                     
                   
                   + 
                   
                     
                       
                         ( 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             φ 
                             e 
                           
                         
                         ) 
                       
                       2 
                     
                     
                       n 
                       o 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               φ 
               o 
               ′ 
             
             = 
             
               arcsin 
               ⁡ 
               
                 ( 
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     φ 
                   
                   
                     n 
                     o 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               φ 
               e 
               ′ 
             
             = 
             
               
                 sin 
                 
                   - 
                   1 
                 
               
               ⁡ 
               
                 ( 
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     φ 
                   
                   
                     n 
                     e 
                     ′ 
                   
                 
                 ) 
               
             
           
         
       
     
     The physical path length of the rays through the plate can be calculated as follows: 
               d   o     =     t       1   -       (       sin   ⁢           ⁢   φ       n   o       )     2                         d   e     =     t       1   -       (       sin   ⁢           ⁢     φ   e         n   e   ′       )     2                 
The missing air for each ray can be calculated as follows:
 
 d′   o   =t ·cos(φ−φ′ o )
 
 d′   e   =t ·cos(φ−φ′ e )
 
The optical path length for each ray can be calculated as follows:
 
OPL o   =n   o   ·d   o  
 
OPL e   =n   e   ·d   e  
 
The optical path difference, with corrections due to the fact that the “missing air” is different for the two rays, can then be calculated. In other words, because the rays exit at different points on the surface of the waveplate, they will need to travel different distances to a reference plane.
 
OPD=(OPL o   −d′   o )−(OPL e   −d′   e )
 
     Now, the value of the wobble angle θ can be found that allows the desired optical path difference (OPD) between the beams to remain constant while the OPL for both polarizations is increased with increasing φ. The equation to be solved is: 
                   ∂     OPD   ⁡     (   θ   )           ∂   ω       ⁢     |     ω   =   0         =   0         
The above expression can be used to find the value θ ( 213 ) for which the derivative of the OPD with respect to ω ( 215 ) is zero, evaluated at ω ( 215 )=0. This can be done either by analytically deriving the derivative expression and solving for θ ( 213 ) for which it equals 0, or by doing finite differences (essentially a numerical derivative) of the OPD equation around ω=0 for many values of θ ( 213 ) to find the value of θ ( 213 ) for which the numerical derivative is 0.
 
     A typical waveplate has a multiple stack construction in which two plates of a birefringent material are stacked with their axes at 90° rotations with respect to each other and with thicknesses that differ according to a desired retardation. For example, an eighth-wave waveplate can be built using two plates with a thickness difference of Δd=λ/8/Δn, where λ is the free-air wavelength of an incident light beam, and Δn is the difference in the indices of refraction of the material between the o- and e-axes. The two plates can comprise the same material or different materials having different indices of refraction. 
       FIG. 5  is a diagram depicting light propagation through a waveplate  500  having a first plate  500   a  and a second plate  500   b  according to certain aspects of the present disclosure. It is assumed that a monochromatic incident light beam  510  of linearly polarized light is incident on the waveplate  500  at an angle of incidence  513  at a first (air-to-the first plate  500   a ) interface  501 . The incident light beam  510  comprises an extraordinary ray (e-ray) and an ordinary ray (o-ray) by the projection of the polarization of the incident light beam  510  on an extraordinary axis (e-axis) and an ordinary axis (o-axis) normal to the e-axis in the plane of the plate  500   a . It is further assumed that a rotation axis (not shown) around which the waveplate  500  is tilted or rotated for calibration purposes is at an angle θ (not shown) relative to an axis (see, e.g., the x-axis  201  of  FIG. 2 ). The angle θ defines the axis about which small rotations cause no change in the retardation of the waveplate  500 , but provide a common phase shift for the light polarization components (e.g., the e- and o-rays) travelling through the waveplate  500  for use in calibration. 
     In  FIG. 5 , different subscripts will be used to identify separate light polarization components and the different plates  500   a ,  500   b  since the polarization which is oriented with the e-axis in the first plate  500   a , is oriented with the o-axis in the second plate  500   b . Subscripts “a” and “b” are used for the light polarization components once separated within the first plate  500   a . “a” corresponds to an o-ray  515   a  associated with the o-axis in the first plate  500   a , and “b” corresponds to an e-ray  515   b  associated with the e-axis in the first plate  500   a . Subscripts “1” and “2” are used to denote the first and second plates  500   a ,  500   b , respectively. 
     In  FIG. 5 , n o  and n e  refer to ordinary and extraordinary refractive indices of the optical material comprising the first and second plates  500 , respectively; t 1  and t 2  refer to the thicknesses the first and second plates  500 , respectively; d 1a  and d 1b  refer to the physical path lengths of the o- and e-rays  515   a ,  515   b , respectively, within the first plate  500   a ; d 2a  and d 2b  refer to the physical path lengths of e- and o-rays  517   a ,  517   b , respectively, within the second plate  500   b ; and d a ′ and d b ′ refer to physical distances coaxial to the two rays “a” and “b” between an entrance point  505  and respective exit points  509   a ,  509   b  on the waveplate  500  (e.g., the thicknesses of the missing air). 
     The equations for angles of incidence and distances the o- and e-rays travelling within the first plate  500   a  are given by: 
     
       
         
           
             
               n 
               
                 e 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
               
               ′ 
             
             = 
             
               1 
               
                 
                   
                     
                       
                         ( 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             φ 
                             e 
                           
                         
                         ) 
                       
                       2 
                     
                     
                       n 
                       o 
                       2 
                     
                   
                   + 
                   
                     
                       
                         ( 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             φ 
                             e 
                           
                         
                         ) 
                       
                       2 
                     
                     
                       n 
                       o 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               φ 
               
                 1 
                 ⁢ 
                 a 
               
             
             = 
             
               arcsin 
               ⁡ 
               
                 ( 
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     φ 
                   
                   
                     n 
                     o 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               φ 
               
                 1 
                 ⁢ 
                 b 
               
             
             = 
             
               
                 sin 
                 
                   - 
                   1 
                 
               
               ⁡ 
               
                 ( 
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     φ 
                   
                   
                     n 
                     
                       e 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                     ′ 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               d 
               
                 1 
                 ⁢ 
                 a 
               
             
             = 
             
               
                 t 
                 1 
               
               
                 
                   1 
                   - 
                   
                     
                       ( 
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             φ 
                             o 
                           
                         
                         
                           n 
                           o 
                         
                       
                       ) 
                     
                     2 
                   
                 
               
             
           
         
       
       
         
           
             
               d 
               
                 1 
                 ⁢ 
                 b 
               
             
             = 
             
               
                 t 
                 1 
               
               
                 
                   1 
                   - 
                   
                     
                       ( 
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             φ 
                             e 
                           
                         
                         
                           n 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           ′ 
                         
                       
                       ) 
                     
                     2 
                   
                 
               
             
           
         
       
     
     Next, angles of incidence of the o- and e-rays  515   a ,  515   b  at a second (e.g., the first plate  500   a -to-the second plate  500   b ) interface  502  are determined. As with the first interface  501 , the angle of incidence for the o-ray  517   a  at the second interface  501   b  is simply the angle of incidence, while the angle of incidence for the e-ray  517   b  contains a dependence on θ. 
     Since the value of the variable θ is fixed with the decision to place a particular orientation for the first plate  500   a , the effective rotation angle of the second plate  500   b  is 90°−θ. That changes the function of the internal incidence angle along the e-axis in the second waveplate  500   b  to:
 
φ e2 =2 arcsin(sin φ/2*sin θ)
 
while the function for φ o  remains unchanged. Thus the functions for the lengths of the optical paths in the second plate  500   b  are given by:
 
     
       
         
           
             
               n 
               
                 e 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 2 
               
               ′ 
             
             = 
             
               1 
               
                 
                   
                     
                       
                         ( 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             φ 
                             
                               e 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                         ) 
                       
                       2 
                     
                     
                       n 
                       o 
                       2 
                     
                   
                   + 
                   
                     
                       
                         ( 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             φ 
                             
                               e 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                         ) 
                       
                       2 
                     
                     
                       n 
                       o 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               d 
               
                 2 
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     Because of the varying geometries and angles of refraction changes, the two rays “a” and “b” exit the second plate  500   b  at different points on the exterior surface of the second plate  500   b . As they do, they then travel different path lengths through the air. This differential path, also referred to as the differential missing air path, is determined. Referring to  FIG. 5  again, the differential path can be determined by calculating the distance, d b ′, d b ′ parallel to the in-air beam path from the entrance point  505  of the first waveplate  500   a  to the exit point  509   a ,  509   b  of the second waveplate  500   b  for each ray. The results are given by:
 
 d   a   ′=d   1a  cos [φ−φ 1a   ]+d   2a  cos [φ−φ 2a ]
 
 d   b   ′=d   1b  cos [φ−φ 1b   ]+d   2b  cos [φ−φ 2b ]
 
     It should be noted that the missing air paths are d a ′ and d b ′. Missing air can be understood in the context of the light travelling through the air to the plate. It leaves the plate and travels through the air along a path parallel to the incident light. The distance between those two planes in which the light has been travelling in glass can be considered as the missing air. In one aspect, the missing air can be utilized to determine the difference in the paths because the two rays come out of the glass at different locations. 
     Now the effect of tilt angle on optical path length (OPL) can be calculated for each polarization path:
 
OPL a   =d   1a   ·n   o   +d   2a   ·n′   a2  
 
OPL b   =d   1b   ·n′   a1   +d   2b   ·n   o  
 
     using the effective index of refractions that is defined above. 
     Finally, the optical path difference between the different polarization paths can be calculated by subtracting the two optical path lengths, with corrections for the fact that the two rays “a” and “b” exit at the different exit points  509   a ,  509   b  due to the different angles of refraction.
 
OPD=(OPL a   −d′   a )−(OPL b   −d′   b ),
 
The corrections account for the fact that the beams  520   a ,  520   b  exit at two different points.
 
     Now, the value of the wobble angle θ can be found that allows the desired optical path difference (OPD) between the beams to remain constant while the OPL for both polarizations is increased with increasing φ. The equation to be solved is: 
                   ∂     OPD   ⁡     (   θ   )           ∂   ω       ⁢     |     ω   =   0         =   0         
The above expression can be used to find the value θ ( 213 ) for which the derivative of the OPD with respect to ω ( 215 ) is zero, evaluated at ω ( 215 )=0. This can be done either by analytically deriving the derivative expression and solving for θ ( 213 ) for which it equals 0, or by doing finite differences (essentially a numerical derivative) of the OPD equation around ω=0 for many values of θ ( 213 ) to find the value of θ ( 213 ) for which the numerical derivative is 0.
 
       FIG. 6  is a flowchart illustrating an exemplary process  600  for calibrating an optical measurement system utilizing polarization diversity. The process  600  begins at start state  601  and proceeds to operation  610  in which a waveplate having a special rotation axis is provided. In certain embodiments, the waveplate comprises a single plate of a birefringent material (e.g., calcite). In some embodiments, the waveplate comprises a stack of a first plate of a first birefringent material crystal and a second plate of a second birefringent material. In one aspect, the two plates used in a zero-order waveplate must have a very specific differential thickness, although an individual thickness can be as much as desired. The waveplate produces a known retardation in the light passing therethrough, where the retardation depends on the thickness(es) of the one or more plates and indices of refraction of the one or more birefringent materials. 
     In certain embodiments, the operation  610  includes determining an orientation direction of the rotation axis. As described with respect to  FIGS. 4 and 5 , this can be achieved by first defining an equation for an optical path difference (OPD) between the first and second polarization components through the waveplate as a function of an orientation angle (e.g., θ  213  of  FIG. 2 ) for the rotation axis, and then minimizing the derivative of OPD with respect to ω ( 215 ). 
     The process  600  proceeds to operation  620  in which a first light polarization component and a second light polarization component (e.g., the “a” and “b” rays in  FIG. 5 ) propagate in the waveplate while the waveplate is rotated about the rotation axis. The operation  620  can involve directing an incident light beam (e.g.,  10  of  FIG. 1 ,  301  of  FIG. 3 ,  410  of  FIG. 4 ,  510  of  FIG. 5 ) towards the waveplate. In the illustrated embodiment of  FIG. 3 , the incident light  301  is split into the first split light beam  303  and the second light beam  307 , where the first split light beam  303  is then incident on the waveplate  321 . 
     The process  600  proceeds to operation  630  in which an equal common shift in the first and second light polarization components while a differential phase shift (e.g., a quarter-wave Shift) is maintained in the first and second light polarization components. As described above, this is a consequence of rotating the waveplate about the special rotation axis. 
     The process  600  proceeds to operation  640  in which a relative phase between the first and second light polarization components is sensed. In certain embodiments, the operation  640  involves a phase detector receiving a light beam emerging from the waveplate with the first and second light polarization components contained therein. 
     The process  600  proceeds to operation  650  in which at least one calibration parameter is determined based on the relative phase between the first and second light polarization components. In certain embodiments, the at least one calibration parameter includes a polarization shift between the first and second light polarization components caused by one or more spurious effects such as drift and systematic effects in the birefringence and retardance of the optical components of the optical system. Typically, there may be several systematic errors/biases which cause measurement errors. Additionally, these systematic errors may have a tendency to drift over time, degrading the calibration. An aspect of the subject technology can calibrate these systematic errors. In certain embodiments, the operation  650  includes fitting a curve with values representative of the relative phase between the first and second light polarization components as a function of a rotation angle (e.g., ω  215  of  FIG. 2 ) by which the waveplate is rotated about a rotation axis (e.g.,  211  of  FIG. 2 ), and extracting calibration parameter(s) from the curve. 
       FIG. 7  is a schematic block diagram of an exemplary optical measurement system utilizing polarization diversity and configured to be calibrated according to certain aspects of the present disclosure. The system  700  includes a control/analysis unit  701 , a light source  750  (e.g., a laser), a waveplate  770  having a rotation axis  772 , a rotation mechanism  780  (e.g., a motor) mechanically coupled to the waveplate  770 , and photodetectors  760  (e.g., photodiodes). In the illustrated example of  FIG. 7 , the control/analysis unit  701  includes a processor  702 , which can be a desktop computer or a laptop computer. The processor  702  is capable of communication with a laser control module  706  and a motor control module  708  through a bus  709  or other structures or devices. It should be understood that communication means other than buses can be utilized with the disclosed configurations. 
     The processor  702  may include a general-purpose processor or a specific-purpose processor for executing instructions and may further include an internal memory  719 , such as a volatile or non-volatile memory, for storing data and/or instructions for software programs. The instructions, which may be stored in a memory  710  and/or  719 , may be executed by the processor  702  to control and manage access to the various networks, as well as provide other communication and processing functions. The instructions may also include instructions executed by the processor  702  for various user interface devices, such as a display  712  and a keyboard or keypad (not shown). 
     The processor  702  may be implemented using software, hardware, or a combination of both. By way of example, the processor  702  may be implemented with one or more processors. A processor may be a general-purpose microprocessor, a microcontroller, a Digital Signal Processor (DSP), an Application Specific Integrated Circuit (ASIC), a Field Programmable Gate Array (FPGA), a Programmable Logic Device (PLD), a controller, a state machine, gated logic, discrete hardware components, or any other suitable device that can perform calculations or other manipulations of information. 
     A machine-readable medium (e.g.,  719 ,  710 ) that stores software for control, analysis and other processing functions can be one or more machine-readable media. Software shall be construed broadly to mean instructions, data, or any combination thereof, whether referred to as software, firmware, middleware, microcode, hardware description language, or otherwise. Instructions may include code (e.g., in source code format, binary code format, executable code format, or any other suitable format of code). 
     Machine-readable media may include storage integrated into a processing system, such as might be the case with an ASIC. Machine-readable media (e.g.,  710 ) may also include storage external to a processing system, such as a Random Access Memory (RAM), a flash memory, a Read Only Memory (ROM), a Programmable Read-Only Memory (PROM), an Erasable PROM (EPROM), registers, a hard disk, a removable disk, a CD-ROM, a DVD, or any other suitable storage device. In addition, machine-readable media may include a transmission line or a carrier wave that encodes a data signal. Those skilled in the art will recognize how best to implement the described functionality for the processing system  702 . According to one aspect of the disclosure, a machine-readable medium is a computer-readable medium or computer-readable storage medium encoded or stored with instructions and is a computing element, which defines structural and functional interrelationships between the instructions and the rest of the system, which permit the instructions&#39; functionality to be realized. Instructions can be, for example, a computer program including code. 
     The light control module  706  may be a hardware module or a software module or a combination of both (e.g., a firmware) and may contain hardware components and/or control programs that are configured to control the light source  750 , which needs to be a narrow band laser in a preferred embodiment. The light control module  706  is configured to send one or more control signals to the light source  750  via an output port  724 , thereby causing the light source  750  to transmit an incident light beam  715  towards the waveplate  770 . In a preferred embodiment, the light must be monochromatic linearly polarized light. In certain embodiments, the light control module  706  is part of and resides in the light source  750 . 
     The motor control module  708  may be a hardware module or a software module or a combination of both (e.g., a firmware) and may contain hardware components and/or control programs that are configured to control the rotation mechanism  780 , which can be any electrically controlled motor including, but not limited to, a server motor or a stepper motor. The motor control module  708  is configured to send one or more control signals (e.g., PWM pulses) to the rotation mechanism  780  via an output port  728 , thereby causing the rotation mechanism  780  to rotate the waveplate  770  about a rotation axis  772  by a predefined rotation angle. The light can be monochromatic linearly polarized light. In certain embodiments, the rotation mechanism may include an encoder that sends information indicative of an angular position of the waveplate  750  to the motor control module  708  and/or the processor  702 . 
     In the illustrated example of  FIG. 7 , the photodetectors  760  (e.g., photodiodes) is disposed at an opposite side of the waveplate  770  with respect to the light source  750 . A light beam  717  emerging from the waveplate  770  is detected by the photodetectors  760 , which converts the detected emerging light beam  717  into electrical signals. As described above, the emerging light beam  717  can contain two light polarization components that have a relative phase therebetween. The electrical signals outputted by the photodetectors  760  are received by a signal conditioning/conversion module  714  via an input port  722 . The signal conditioning/conversion module  714  conditions (e.g., filters and amplifies) the electrical signals and converts (e.g., digitizes) them into a digital representation. The digital representation is then received and processed (e.g., analyzed) by the processor  702  to sense the relative phase between two light polarization components in the emerging light beam  717 . An input/output port may refer to one or more input/output ports. 
     Certain aspects of calibration processes (e.g., process  600 ) for an optical measurement system utilizing polarization diversity can be implemented in a processor (e.g.,  702  of  FIG. 7 ) and a memory (e.g.,  719 ,  710 ). For instance, the operation  640  for sensing a relative phase between the first and second light polarization components and the operation  650  for determining at least one calibration parameter (e.g., a polarization shift caused by spurious effects) may be performed by the processor  702 . Various coefficients and parameters (e.g., rotation angles and curve-fitting parameters) associated with the above sensing and determination and results thereof (calibration parameters) may be stored in the memory  710 ,  719 . Some results, such a fitted curve and calibration parameter extracted therefrom may be displayed on the display  712 . 
     In one aspect, a preferred configuration requires the output sensor to be a polarizing beam splitter (to separate the two polarizations) and two photodetectors. 
     It is understood that the specific order or hierarchy of steps in the processes disclosed is an illustration of exemplary approaches. Based upon design preferences, it is understood that the specific order or hierarchy of steps in the processes may be rearranged. Some of the steps may be performed simultaneously. The accompanying method claims present elements of the various steps in a sample order, and are not meant to be limited to the specific order or hierarchy presented. 
     The previous description is provided to enable any person skilled in the art to practice the various aspects described herein. Various modifications to these aspects will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other aspects. Thus, the claims are not intended to be limited to the aspects shown herein, but is to be accorded the full scope consistent with the language claims, wherein reference to an element in the singular is not intended to mean “one and only one” unless specifically so stated, but rather “one or more.” Unless specifically stated otherwise, the term “some” refers to one or more. Pronouns in the masculine (e.g., his) include the feminine and neuter gender (e.g., her and its) and vice versa. Headings and subheadings, if any, are used for convenience only and do not limit the invention. 
     Terms such as “top,” “bottom,” “front,” “rear” and the like as used in this disclosure should be understood as referring to an arbitrary frame of reference, rather than to the ordinary gravitational frame of reference. Thus, a top surface, a bottom surface, a front surface, and a rear surface may extend upwardly, downwardly, diagonally, or horizontally in a gravitational frame of reference. 
     A phrase such as an “aspect” does not imply that such aspect is essential to the subject technology or that such aspect applies to all configurations of the subject technology. A disclosure relating to an aspect may apply to all configurations, or one or more configurations. A phrase such as an aspect may refer to one or more aspects and vice versa. A phrase such as an “embodiment” does not imply that such embodiment is essential to the subject technology or that such embodiment applies to all configurations of the subject technology. A disclosure relating to an embodiment may apply to all embodiments, or one or more embodiments. A phrase such an embodiment may refer to one or more embodiments and vice versa. 
     The word “exemplary” is used herein to mean “serving as an example or illustration.” Any aspect or design described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other aspects or designs. 
     All structural and functional equivalents to the elements of the various aspects described throughout this disclosure that are known or later come to be known to those of ordinary skill in the art are expressly incorporated herein by reference and are intended to be encompassed by the claims. Moreover, nothing disclosed herein is intended to be dedicated to the public regardless of whether such disclosure is explicitly recited in the claims. No claim element is to be construed under the provisions of 35 U.S.C. §112, sixth paragraph, unless the element is expressly recited using the phrase “means for” or, in the case of a method claim, the element is recited using the phrase “step for.” Furthermore, to the extent that the term “include,” “have,” or the like is used in the description or the claims, such term is intended to be inclusive in a manner similar to the term “comprise” as “comprise” is interpreted when employed as a transitional word in a claim.