Abstract:
Embodiments are described that simultaneously block glare arising from the reflection of polarized sunlight and can also allow a polarized liquid crystal display (LCD) to be clearly viewed. Embodiments may be useful for use as polarized sunglasses. Embodiments operate by using a unique property of a circular polarizer. One side of the circular polarizer substantially blocks the transmission of incident polarized light whereas the other side allows substantially complete transmission of incident polarized light. Using these opposite behaviors, polarizing sunglasses can be constructed in which the top portion uses a first side of the circular polarizer to block the polarized glare and the lower portion uses a second side of the circular polarizer, opposite from the first side, to view the LCD. Because two different polarizers are used, these sunglasses are called dual-polarization sunglasses to differentiate them from the traditional linear polarizer sunglasses

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims the benefit of U.S. Provisional Patent Application Ser. No. 61/686,071, filed on May 29, 2012, the entire content of which is hereby incorporated by reference in its entirety. 
     
    
     BACKGROUND 
       [0002]    1. Field of the Invention 
         [0003]    This invention relates to polarized sunglasses that can be used to simultaneously block the glare arising from unpolarized sunlight reflected from a planar surface, while allowing for the reading of liquid crystal displays (LCDs) without the need for substantial movement of the polarized sunglasses. 
         [0004]    2. Description of Related Art 
         [0005]    Polarized sunglasses have been available for many years, in which polarized lenses have been used to filter out undesirable reflections, e.g., glare, and to reduce eye strain and fatigue. In addition, because polarizing sunglasses are tinted or appear to be tinted, the transmitted intensity is reduced, thereby allowing further reduction in eye fatigue. The use of linear polarizing glasses has until recently been satisfactory for use in most environments. 
         [0006]    Polarized sunglasses are widely used by automobile drivers to reduce the glare from road reflections and atmospheric scattering (e.g., sky polarization). Commercial polarized sunglasses are linear vertical polarizers that take advantage of the Brewster reflection angle. Unpolarized light from the sun contains electromagnetic (“EM”) field components at substantially all orientation angles relative to the direction of propagation. Upon reflection of unpolarized light from a planar reflection surface at the Brewster angle, the component perpendicular to the plane of the reflection surface vanishes and only the component parallel to the plane of the reflection surface remains. Polarized sunglasses are constructed using linear vertical polarizers to view the reflected light with the result that the perpendicular component of the reflected light is blocked by the linear vertical polarizer and the intensity of the glare is dramatically reduced 
         [0007]    However, liquid crystal displays (LCDs) are a new type of display that has appeared in recent years and are now ubiquitous. For example, automobile dashboards have evolved from using a single, relatively simple dashboard display to using numerous LCD displays that provide information to the driver such as time, tuning station for the radio, CD player, etc. This transformation to the usage of numerous displays has been highly criticized by regulatory automobile organizations. The National Highway Traffic Safety Administration has criticized the increasing number of so-called dash-apps. Their sheer number has become overwhelming and has led to distractions and reduced driving safety. Indeed, N.H.T.S.A. even regards navigation as interfering inherently with a driver&#39;s ability to safely control the vehicle. The agency points out that 17 percent, or nearly 900,000, of police-reported accidents in the United States in 2010 were because of driver distraction. Among their recommendations is that any in-dash operation which requires the driver to look away from the road for more than two seconds be disabled because of the distance traveled. In two seconds a vehicle moving at a speed of 60 mph travels a distance of 176 feet. 
         [0008]    Often a linear polarizer is incorporated into the design of an LCD display, which causes light emitted from the LCD display to be linearly polarized, [usually] linear +45° polarized (L+45P). If the driver views the display while he or she is driving, the LCD display may be blocked by the linear polarized sunglasses and artifacts such as dark patches may appear on the display. In order to view the LCD display, many drivers may lift their polarized sunglasses so that the polarized sunglasses are not in their line of sight to the LCD display. The result is that for several seconds when driving in a glare environment, the driver is distracted from viewing the road, which exacerbates the dangerous and life threatening conditions noted by the N.H.T.S.A. 
         [0009]    One solution to this problem has been to divide the sunglass lens so that the upper portion of the lens is linearly polarized and blocks glare, whereas the lower portion of the lens is left unpolarized. A drawback of this solution is that for comfortable viewing, the tint and the optical transmission of the lower part of the sunglass using a non polarizing material must be substantially identical to the tint and transmission in the upper polarizing element. 
         [0010]    Therefore, a need exists to provide sunglasses that simultaneously reduce glare and allow an LCD display to be viewed, with substantially identical tint and transmission properties throughout the lens in order to provide a comfortable and safer driving environment. 
       SUMMARY 
       [0011]    Drawbacks of the related art may be addressed using the properties of circular polarizers. Circular polarizers are known in the art as being constructed from a birefringent material having a fast axis and a slow axis, and the thickness of the birefringent material selected such that the phase of an electromagnetic field aligned with the slow axis is retarded by a quarter-wavelength compared to the phase of an electromagnetic field aligned with the fast axis, when both electromagnetic fields pass through the circular polarizer. For this reason, the circular polarizer may also be referred to as a quarter-wave plate (“QWP”). For wavelengths of visible light, approximately 450 nm to approximately 750 nm, the variation in wavelength is relatively negligible and a QWP of a single thickness is adequate for all visible light in accordance with embodiments of the present invention. 
         [0012]    Embodiments in accordance with the present invention provide circular polarizers that are constructed using a combination of two polarizing elements that are joined together: (1) a linear polarizer; and (2) a quarter-waveplate. Light entering the linear polarizer side of the combination emerges as circular polarized light. Light entering the quarter-waveplate side of the combination emerges as linear polarized light. Embodiments in accordance with the present invention provide a polarized sunglass in which the upper portion of the sunglass blocks the glare white the lower portion allows a driver to view an LCD display. This configuration is also referred to herein as a dual-polarized sunglass. The dual-polarized sunglass will allow the driver to wear sunglasses as usual while driving, and simultaneously view the LCD display without removing the sunglasses or otherwise moving the sunglasses relative to the face, by merely moving his or her eyes and/or shifting his or her head, such that a line of sight in a first direction to the road passes through the upper polarizer, and a line of sight in a second direction to the LCD display passes through the lower polarizer. 
         [0013]    Embodiments in accordance with the present invention are also usable with displays that emit circularly polarized light, even though such displays may also be adequately usable with conventional sunglasses. 
         [0014]    The preceding is a simplified summary of embodiments of the disclosure to provide an understanding of some aspects of the disclosure. This summary is neither an extensive nor an exhaustive overview of the disclosure and its various embodiments. It is intended neither to identify key or critical elements of the disclosure nor to delineate the scope of the disclosure but to present selected concepts of the disclosure in a simplified form as an introduction to the more detailed description presented below. As will be appreciated, other embodiments of the disclosure are possible utilizing, alone or in combination, one or more of the features set forth above or described in detail below. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0015]    The above and still further features and advantages of the present invention will become apparent upon consideration of the following detailed description of embodiments thereof, especially when taken in conjunction with the accompanying drawings wherein like reference numerals in the various figures are utilized to designate like components, and wherein: 
           [0016]      FIG. 1  illustrates a cross-sectional view of a portion of a polarized lens for a first incident light, in accordance with an embodiment of the present invention; 
           [0017]      FIG. 2  illustrates polarization properties when an unpolarized light reflects from a surface; 
           [0018]      FIG. 3  illustrates a cross-sectional view of a portion of a polarized lens for a polarized incident light, in accordance with an embodiment of the present invention; 
           [0019]      FIG. 4  illustrates a schematic view of sunglasses in accordance with an embodiment of the present invention; 
           [0020]      FIG. 5  illustrates a schematic view of a lens in accordance with an embodiment of the present invention; 
           [0021]      FIG. 6A  illustrates a cross-sectional view of a polarized lens in accordance with an embodiment of the present invention; 
           [0022]      FIG. 6B  illustrates a cross-sectional view of a polarized lens in accordance with an embodiment of the present invention; and 
           [0023]      FIG. 7  illustrates a perspective view of sunglasses in accordance with an embodiment of the present invention. 
       
    
    
       [0024]    The headings used herein are for organizational purposes only and are not meant to be used to limit the scope of the description or the claims. As used throughout this application, the word “may” is used in a permissive sense (i.e., meaning having the potential to), rather than the mandatory sense (i.e., meaning must). Similarly, the words “include”, “including”, and “includes” mean including but not limited to. To facilitate understanding, like reference numerals have been used, where possible, to designate like elements common to the figures. Optional portions of the figures may be illustrated using dashed or dotted lines, unless the context of usage indicates otherwise. 
       DETAILED DESCRIPTION 
       [0025]    In the following detailed description, numerous specific details are set forth in order to provide a thorough understanding of embodiments or other examples described herein. In some instances, well-known methods, procedures and components have not been described in detail, so as to not obscure the following description. Further, the examples disclosed are for exemplary purposes only and other examples may be employed in lieu of, or in combination with, the examples disclosed. It should also be noted the examples presented herein should not be construed as limiting of the scope of embodiments of the present invention, as other equally effective examples are possible and likely. 
         [0026]    Embodiments in accordance with the present invention disclose a dual-polarized sunglass, which is a new type of sunglass that can overcome both polarization from reflected sunlight (i.e., glare) and is compatible with the linear polarized light emitted by LCD displays. The sunglass may be fitted in a frame, either as one lens covering both eyes or separate lenses for each eye. The frame may be fitted to a wearer&#39;s head, for example as conventional glasses or sunglasses, or as a helmet-mounted visor, or the like. 
         [0027]    Mathematical Background 
         [0028]    In order to describe operation of a circular polarizer, a 4×4 Mueller matrix formulation and Stokes vector parameters are used. The 4×4 Mueller matrix describes a transformation caused by a polarizing element and the Stokes vector describes the polarization state of an optical beam. 
         [0029]    Conventional circular polarizers are constructed using a linear horizontal polarizer with its transmission axis rotated through +45° from the x axis followed by a quarter-waveplate with its fast axis along the horizontal x axis and its slow axis along the y axis. The respective Mueller matrices of the two elements having the form shown in Equations (1) and (2). 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                     QWP 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     
                       L 
                       + 
                       
                         45 
                          
                         
                             
                         
                          
                         P 
                       
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0030]    The incident light is described by a Stokes vector S having the general form shown in Equation (3). 
         [0000]    
       
         
           
             
               
                 
                   S 
                   = 
                   
                     ( 
                     
                       
                         
                           
                             S 
                             0 
                           
                         
                       
                       
                         
                           
                             S 
                             1 
                           
                         
                       
                       
                         
                           
                             S 
                             2 
                           
                         
                       
                       
                         
                           
                             S 
                             3 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0031]    The first parameter S 0  describes the intensity of the optical beam and the three remaining parameters S 1 , S 2 , and S 3  describe the polarization state of the optical beam. The parameters S 1  and S 2  describe the linear polarization state and S 3  describes the circular polarization state. If all four parameters appear in Equation (3), the light is elliptical polarized. 
         [0032]      FIG. 1  illustrates a circular polarizer  100  created by joining together a linear polarizing element  102  with quarter waveplate element  104 , such that the linear polarizing element  102  faces an incident light ray  106 . The incident light ray  106 , which may be either unpolarized light or be horizontally polarized road glare, is thereby transformed into an outgoing circularly polarized light ray  108 , which then may be seen by observer  110  such as a driver, in accordance with an embodiment of the present invention. The Mueller matrix for circular polarizer  100  has the form shown in Equation (4). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           M 
                           CP 
                         
                         = 
                           
                          
                         
                           
                             M 
                             QWP 
                           
                           × 
                           
                             M 
                             
                               L 
                               + 
                               
                                 45 
                                  
                                 
                                     
                                 
                                  
                                 P 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             2 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0033]    Equation (4) describes a circular polarizer as indicated by the subscript “CP”. For an incident beam represented by the Stokes vector S as shown in Equation (3), the Stokes vector S OBS  of the observed emerging beam  108  shown in  FIG. 1  using Equations (3) and (4) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           S 
                           OBS 
                         
                         = 
                           
                          
                         
                           
                             M 
                             CP 
                           
                           × 
                           S 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     1 
                                   
                                   
                                     0 
                                   
                                   
                                     1 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     1 
                                   
                                   
                                     0 
                                   
                                   
                                     1 
                                   
                                   
                                     0 
                                   
                                 
                               
                               ] 
                             
                           
                           × 
                           
                             [ 
                             
                               
                                 
                                   
                                     S 
                                     0 
                                   
                                 
                               
                               
                                 
                                   
                                     S 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     S 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     S 
                                     3 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 S 
                                 0 
                               
                               + 
                               
                                 S 
                                 2 
                               
                             
                             2 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                               
                               
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                               
                               
                                 
                                   1 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0034]    In Equation (5), S 1 =S 2 =0 and S 3 =1. This result shows the light  108  that emerges from circular polarizer  100  and viewed by the observer  110  is right circular polarized. Thus, regardless of the polarization state of the incident beam the emerging beam  108  will always be right circular polarized. 
         [0035]    Glare occurs when an incident unpolarized beam is reflected from a dielectric surface, such as the reflection of unpolarized sunlight from the surface of the road or water. Atmospheric scattering may also produce glare. While driving, glare is often observed in reflections close to or at the Brewster angle of incidence. This is shown by  FIG. 2 , in which an incident unpolarized ray  201  impinges surface  207 . Surface  207  is ordinarily a dielectric surface  207  (e.g., asphalt, water, oil, etc.). Unpolarized ray  201  is shown having both a vertical polarization component (denoted by double-headed arrows on unpolarized ray  201 , indicating vectors parallel to the plane of  FIG. 2 ) and a horizontal polarization component (denoted by dots on unpolarized ray  201 , indicating vectors perpendicular to the plane of  FIG. 2 ). 
         [0036]    Upon impinging dielectric surface  207 , unpolarized ray  201  is split into a polarized reflected ray  203  and/or a slightly polarized refracted ray  205 . Ray  205  is illustrated as having a greater vertical polarization component than horizontal polarization component. At the Brewster angle, only the horizontal polarization component (denoted by dots) remains, so the reflected light ray  203  is linear horizontal polarized (LHP). The reflected light ray  203  may also be referred to herein as glare or a glare beam. 
         [0037]    The Stokes vector for the glare beam is linear horizontally polarized having the form shown in Equation (6). 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                     GLARE 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           1 
                         
                       
                       
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0038]    Used directly, the conventional circular polarizer  100  will not remove the glare in the configuration of  FIG. 1  without appropriate selection of an orientation angle. The glare can be removed by rotating the configuration shown in  FIG. 1  through an angle of +45°. Upon doing this Equation (4) is transformed to the form shown in Equation (7). 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                     CPROT 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             
                               - 
                               1 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             1 
                           
                           
                             
                               - 
                               1 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0039]    The Stokes vector of the reflected light that is now incident on the driver&#39;s eye is now found by matrix multiplying Equation (6) by Equation (7), to produce the form shown in Equation (8). 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                     REFL 
                   
                   = 
                   
                     
                       
                         M 
                         CPROT 
                       
                       × 
                       
                         S 
                         GLARE 
                       
                     
                     = 
                     
                       
                         
                           
                             1 
                             2 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                             
                             ] 
                           
                         
                         × 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                           ] 
                         
                       
                       = 
                         
                        
                       
                         [ 
                         
                           
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0040]    Thus, the Stokes vector of the observed light is zero after passing through the rotated circular polarizer. Since the first element of the Stokes vector is the intensity of the observed light, and since the first element is zero, the rotated circular polarizer has blocked the glare. 
         [0041]    We now consider light emitted by an LCD display. The polarization of the light emitted by most LCD displays is linearly −45° polarized (L−45P). The Stokes vector for L−45P light is of the form shown in Equation (9). 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                     
                       L 
                       - 
                       
                         45 
                          
                         
                             
                         
                          
                         P 
                       
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           
                             - 
                             1 
                           
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0042]      FIG. 3  illustrates an embodiment in accordance with the present invention when viewing light emitted by the LCD display.  FIG. 3  illustrates a linear polarizer  300  created by joining together a quarter waveplate element  302  with a linear polarizing element  304 , such that the circular polarizing element  302  faces an incident light ray  306 . The incident light ray  306 , which may be either unpolarized light or be light emitted by an LCD display, is thereby transformed into an outgoing linearly polarized light ray  308 , which then may be seen by observer  310  such as a driver wearing sunglasses in accordance with an embodiment of the present invention. The Mueller matrix for linear polarizer  300  has the form shown in Equation (10). 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                     LP 
                   
                   = 
                   
                     
                       
                         M 
                         
                           L 
                           - 
                           
                             45 
                              
                             
                                 
                             
                              
                             P 
                           
                         
                       
                       × 
                       
                         M 
                         QWP 
                       
                     
                     = 
                     
                       
                         1 
                         2 
                       
                        
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 - 
                                 1 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 - 
                                 1 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0043]    In general, the configuration shown in  FIG. 3  may be rotated through an angle θ. The angle θ may represent an offset in the coordinate frame of reference between that of the LCD display and that of sunglasses in accordance with an embodiment of the present invention. The rotation may arise if, for example, a mobile device with an LCD display is placed by the user at certain positions or angles, of if the mobile device moves as a result of vehicle motion, or if the user tilts their head in certain axes of rotation while looking at the LCD display, and so forth. The rotation operation transforms the Muller matrix to a matrix having the form shown in Equation (11). 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                     LPROT 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                         
                           
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 θ 
                               
                                
                               
                                   
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0044]    The L−45P light emitted by the LCD display is now observed by observer  310  and is described by a Stokes vector formed by multiplying the Muller matrix of Equation (11) by the Stokes vector for L−45P light as shown in Equation (9), to produce a Stokes vector having the form shown in Equation (12). 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                     OBS 
                   
                   = 
                   
                     
                       
                         M 
                         LPROT 
                       
                       × 
                       
                         S 
                         
                           L 
                           - 
                           
                             45 
                              
                             
                                 
                             
                              
                             P 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             1 
                             2 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     θ 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     θ 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       cos 
                                     
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     θ 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       - 
                                       2 
                                     
                                      
                                     
                                         
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     θ 
                                   
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                             
                             ] 
                           
                         
                         × 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                             
                             
                               
                                 0 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         [ 
                         
                           
                             
                               1 
                             
                           
                           
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   cos 
                                 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                             
                           
                           
                             
                               0 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0045]    S OBS  of Equation (12) represents the Stokes vector, observed by a driver, of the light emitted by an LCD display and passing through the polarizer of  FIG. 3  rotated by θ degrees, in accordance with an embodiment of the present invention. Equation (12) shows that regardless of the rotation angle of polarizer  300  with respect to the LCD display, the intensity of the light seen by driver  310  (as evidenced by the first element of the S OBS  vector) does not vary with rotation angle. This characteristic is also true regardless of the orientation angle of the linear polarized light emitted by the LCD display. This is readily shown by representing the polarization state emitted by the LCD display as having the form shown in Equation (13). 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                     LCD 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             S 
                             0 
                           
                         
                       
                       
                         
                           
                             S 
                             1 
                           
                         
                       
                       
                         
                           
                             S 
                             2 
                           
                         
                       
                       
                         
                           
                             S 
                             3 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0046]    Multiplying the Muller matrix of Equation (11) by the Stokes vector of Equation (13) yields a Stokes vector having the form shown in Equation (14). 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                     OBS 
                   
                   = 
                   
                     
                       
                         M 
                         LPROT 
                       
                       × 
                       
                         S 
                         LCD 
                       
                     
                     = 
                     
                       
                         
                           
                             1 
                             2 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     θ 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     θ 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       cos 
                                     
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     θ 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       - 
                                       2 
                                     
                                      
                                     
                                         
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     θ 
                                   
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                             
                             ] 
                           
                         
                         × 
                         
                           [ 
                           
                             
                               
                                 
                                   S 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   S 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   S 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   S 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         
                           
                             
                               S 
                               0 
                             
                             + 
                             
                               S 
                               3 
                             
                           
                           2 
                         
                          
                           
                         [ 
                         
                           
                             
                               1 
                             
                           
                           
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   cos 
                                 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                             
                           
                           
                             
                               0 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0047]    Thus, Equation (14) shows that regardless of the orientation of the linear polarized light emitted by the LCD display, the intensity seen by the driver will always be constant, as evidenced by the first element of the S OBS  vector. 
         [0048]    This result is in contrast to the behavior seen by viewing the LCD display through a rotated linear polarizer not having the structure of linear polarizer  300 , i.e., without being joined to quarter waveplate  302 . The Mueller matrix for a rotated linear polarizer is of the form shown in Equation (15). 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                     POLROT 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             
                               
                                 cos 
                                 2 
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θsin 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θsin 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             
                               
                                 sin 
                                 2 
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0049]    Multiplying the Muller matrix of Equation (15) by an arbitrary Stokes vector of light emitted by the LCD display, as given in Equation (13), produces a Stokes vector having the form shown in Equation (16). 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                     OBS 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
                                 S 
                                 0 
                               
                               + 
                               
                                 
                                   S 
                                   1 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               + 
                               
                                 
                                   S 
                                   2 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   S 
                                   0 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               + 
                               
                                 
                                   S 
                                   1 
                                 
                                  
                                 
                                   cos 
                                   2 
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               + 
                               
                                 
                                   S 
                                   2 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θcos 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   S 
                                   0 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               + 
                               
                                 
                                   S 
                                   1 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θsin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               + 
                               
                                 
                                   S 
                                   2 
                                 
                                  
                                 
                                   sin 
                                   2 
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                             
                           
                         
                         
                           
                             
                               0 
                                
                               
                                   
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0050]    Equation (16) shows that the intensity as viewed by an observer, i.e., the first element of the S OBS  vector, is of the form shown in Equation (17). 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                      
                     
                       ( 
                       θ 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           S 
                           0 
                         
                         + 
                         
                           
                             S 
                             1 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           θ 
                         
                         + 
                         
                           
                             S 
                             2 
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           θ 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0051]    Thus, as the alignment between an LCD display and a conventional linear polarizer changes, e.g., if the LCD display is moved or if an observer rotates their head with respect to the LCD display, the intensity will vary if conventional linearly polarized sunglasses are used. This accounts for unexpected visual artifacts when a user views an LCD display through conventional linearly polarized sunglasses. 
         [0052]    The visual artifacts may be exacerbated because linear polarizers are often fabricated from polyvinyl alcohol polymers that are doped with iodine. The polyvinyl alcohol polymers are arranged as strands, and the iodine atoms bond to the strands in order to polarize the light passing through. Because iodine is a very large atom, its outermost atomic shell contains seven electrons that are held very loosely. This allows the incident radiation to excite the electrons which then re-emit the incident radiation perpendicular to the strands. There is also a certain amount of dichroism in the polarizers and iodine naturally dyes the polyvinyl so the result is that, e.g., Polaroids and commercial polarizers have a reddish-purple tint. 
         [0053]    Certain less common LCD displays may emit light that is either right or left circular polarized. In this condition, the incident beam from the LCD display is represented by the Stokes vector given by Equation (3) and the Mueller matrix for the rotated right circular polarizer has the form shown in Equation (18). 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                     RCPROT 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             1 
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             0 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0054]    The Stokes vector of the beam incident on the eye is then found by matrix multiplying Equation (18) with Equation (3), to have the form shown in Equation (19). 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                     EYE 
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 
                                   S 
                                   0 
                                 
                                 - 
                                 
                                   
                                     S 
                                     1 
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                                 + 
                                 
                                   
                                     S 
                                     2 
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                               
                             
                           
                           
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                           
                           
                             
                               
                                 
                                   S 
                                   0 
                                 
                                 - 
                                 
                                   
                                     S 
                                     1 
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                                 + 
                                 
                                   
                                     S 
                                     2 
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             S 
                             0 
                           
                           - 
                           
                             
                               S 
                               1 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             θ 
                           
                           + 
                           
                             
                               S 
                               2 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             θ 
                           
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             
                               1 
                             
                           
                           
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                           
                           
                             
                               1 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0055]    Thus, the light incident on the eye is right circularly polarized and the intensity of light on the eye has the form shown in Equation (20). 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     EYE 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           S 
                           0 
                         
                         - 
                         
                           
                             S 
                             1 
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           θ 
                         
                         + 
                         
                           
                             S 
                             2 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           θ 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0056]    For either right or left circular polarized light emitted by the circularly polarized LCD display, S 1 =S 2 =0 so Equation (20) reduces to have the form shown in Equation (21). 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     EYE 
                   
                   = 
                   
                     
                       S 
                       0 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0057]    Equation (21) shows the intensity is constant and is invariant of rotation. Accordingly, dual-polarized sunglass for LCD displays emitting right or left circular polarized light will include a vertically oriented polarizer  100  to block the glare, together with a circular polarizer mounted on the bottom. It is not necessary for the circular polarizer to be flipped and the emphasis must be made that this configuration is only applicable to the emission of right or left circularly polarized light. Analysis shows that both linear and circular polarized displays can be seen through the circular polarized portion of the sunglasses. 
         [0058]    Thus, the foregoing mathematical analysis shows that an embodiment in accordance with the present invention, which combines the structure of  FIG. 1  and  FIG. 3 , may be used to eliminate the glare and view a LCD display simultaneously. 
         [0059]    A driver wearing sunglasses in accordance with an embodiment of the present invention may view the road through the structure of  FIG. 1  in order to eliminate glare, while viewing an LCD display at substantially any angle through the structure of  FIG. 3  in order to eliminate artifacts observed when the linearly polarized light emitted by an LCD display passes through a conventional linear polarizer. 
         [0060]    Application of the Mathematical Background 
         [0061]    Lenses for dual-polarized sunglasses in accordance with an embodiment of the present invention may be prepared from a single sheet of a circular polarizer, appropriately cut in accordance with the diagrams of  FIG. 1  and  FIG. 3 . The linear polarizing side of polarizer  100  is oriented toward a source of glare, and may be used in the upper part of a dual-polarized sunglass lens in order to block the glare. The circular polarizing side of the polarizer  300  is oriented toward an LCD display, and may be used in the lower part of a dual-polarized sunglass lens in accordance with an embodiment of the present invention. Using the same sheet the circular polarizer tends to lessen intensity variations between polarizer  100  and polarizer  300 . The linear and circular elements are fused seamlessly to form embodiments in accordance with the present invention. 
         [0062]      FIG. 4  illustrates a schematic view of sunglasses  400  in accordance with an embodiment of the present invention. Sunglasses  400  are illustrated from a point of view as seen by light propagating toward sunglasses  400 , i.e., as if by seen by an outside observer on the face of a person wearing sunglasses  400 . Sunglasses  400  are illustrated as including a first lens  402  covering a first eye and a substantially similar second lens  404  covering a second eye, which may be connected by a bridge piece  406 , either directly or by coupling to a frame supporting the first lens  402  and/or second lens  404 . It should be understood that other physical embodiments are possible, such as a single lens that covers both eyes and presents substantially the same optical characteristics to both eyes (such as a visor or helmet), or as clip-ons that clip individually or together to a conventional pair of eyeglasses, and so forth. Bridge piece  406  may be omitted if first lens  402  and second lens  404  are independently secured, such as by individual clip-ons to a conventional pair of eyeglasses. In another, more durable embodiment, first lens  402  and second lens  404  may be coupled to one or more relatively thin plates or layers of shatter-resistant light-transmissive material (e.g., glass, resin, plastic, etc.) so as to make dual-polarized sunglasses. The plates or layers may be provided between first lens  402  and second lens  404 , or outside the combination of the first lens  402  and second lens  404 . 
         [0063]    Another embodiment may be to make optical quality eyeglasses (e.g., prescription sunglasses) and impregnating the linear and circular materials, e.g., liquid crystals into a transmissive substrate material such as glass, plastic, resin, or the like. Another embodiment may be to provide first lens  402  and second lens  404  as contact lenses. 
         [0064]    Lenses  402  and  404  each include a first region  408  and a second region  410 . First region  408  includes a polarized material in accordance with  FIG. 1 . Second region  410  includes a polarized material in accordance with  FIG. 3 . The relative sizes and positions of first region  408  and second region  410  may be chosen such that a roadway in front of a driver ordinarily will be seen while looking through first region  408 , and a dashboard and other interior portions of a vehicle below the dashboard ordinarily will be seen while looking through second region  410 , when embodiments of the present invention are worn in a typical manner by a driver or front-seat passenger of a vehicle. 
         [0065]    Although sunglasses  400  are illustrated as having particular shapes for first region  408  and second region  410 , regions  408  and  410  may have substantially any shape such that an upper portion of lenses  402  and  404  include first region  408  and a lower portion of lenses  402  and  404  include second region  410 . 
         [0066]    Another embodiments in accordance with the present invention may be to fabricate lens  402  or  404  as a contact lens, but weighted such that second region  410  is relatively heavier, so that the contact lens will tend to seek the correct orientation when placed on a user&#39;s cornea. 
         [0067]    Additional, non-polarization specific additives may be added to any of the layers or regions discussed above, or added as a separate layer, in order to provide additional light blockage by the sunglasses, or a desired color (e.g., gray, green, yellow, etc.), effect (e.g., mirrored, gradient, scratch resistance, lettering on the lenses, etc.), and so forth. 
         [0068]      FIG. 5  illustrates a schematic view of an embodiment of a contact lens  500  in accordance with the present invention. Contact lens  500  may be fabricated as substantially concentric regions, such that an inner region  508  is fabricated in accordance with  FIG. 1  and an outer region is fabricated in accordance with  FIG. 3 . The relative sizes and positions of first region  508  and second region  510  may be chosen such that a roadway ordinarily will be seen by the fovea of a driver while looking through first region  508 , and a dashboard and other interior portions of a vehicle below the dashboard ordinarily will be seen while looking through second region  510 , when embodiments of the present invention are worn in a typical manner by a driver or front-seat passenger of a vehicle. Contact lens  500  does not need to have a weighted side since the design is rotationally invariant. 
         [0069]      FIG. 6A  illustrates a cross-sectional view of an embodiment of a lens  600  in accordance with the present invention. Lens  600  includes a first polarizer  100  as shown in  FIG. 1 , and a second polarizer  300  as shown in  FIG. 3 . Glare  610  and light  620  from an LCD display impinge lens  600  from the left as shown. Glare  610  and light  620  generally arrive to the user from different viewing angles or lines of sight. As illustrated, first polarizer  100  and second polarizer  300  have substantially the same width. A transition region  608  may be provided, such as by fusing, in order to lessen physical or optical discontinuities arising from joining first polarizer  100  and second polarizer  300 . 
         [0070]      FIG. 6B  illustrates a cross-sectional view of an embodiment of a lens  650  in accordance with the present invention. Lens  650  includes a layer  652  functioning as a substrate for both first polarizer  660  and second polarizer  670 . First polarizer  660  generally corresponds to first polarizer  100  of  FIG. 1 , and second polarizer  670  generally corresponds to first polarizer  300  of  FIG. 3 . Layer  652  is constructed from a material having a first polarization property, and ordinarily functions as a circular polarizer. Layer  654  is constructed from a material having a second polarization property, and ordinarily functions as a linear polarizer such as L+45P or L−45P. Layer  656  is constructed from a material having a polarization property different than the first polarization property, and ordinarily functions as a linear polarizer such as L−45P or L+45P. A transition region  658  may be provided to lessen physical or optical discontinuities arising from joining first polarizer  660  and second polarizer  670 . Lens  650  may be fabricated by applying layer  654  and/or layer  656  as a film or coating over the respective side of substrate layer  652 . 
         [0071]    It is aesthetically preferable that first and second polarized regions (e.g., polarizers  100  and  300 , polarizers  660  and  670 , etc.) have similar darkness (i.e., optical transmission attenuation), tint, and thickness, and minimal physical discontinuity, so that the difference between polarized regions is not readily apparent except in the presence of polarized light. However, this does not preclude intentional application of effects such as a tinting-based darkness gradient. 
         [0072]    Additional layers or coatings may be applied to lens  600  and/or lens  650  for other purposes, such as to provide strength, shatter resistance, scratch resistance, UV protection, and so forth. Although lens  600  and/or lens  650  are illustrated as planar, in a plane perpendicular to the plane of  FIGS. 6A and 6B , it will be understood that lens  600  and/or lens  650  may be curved in order to provide a shape that is more contoured to the human face and/or head, or to provide certain embodiments such as a helmet-mounted visor. Normally, front and back major surfaces of lens  600  and  650 , if curved, will be curved away from the expected location of the light source. 
         [0073]      FIG. 7  illustrates a perspective view of a pair of sunglasses  700  with two lenses, with each lens having an upper portion  702  corresponding to region  408  of  FIG. 4 , and a lower portion  704  corresponding to region  410  of  FIG. 4 , in accordance with an embodiment of the present invention. A contrast in darkness or transmission intensity between upper portion  702  and lower portion  704  has been intentionally overemphasized in order to better illustrate the relative positions of portions of  702  and  704 . Embodiments in accordance with the present invention preferably may have substantially uniform darkness or transmission intensity throughout substantially the entirety of each lens of sunglasses  700 . 
         [0074]    In another embodiment, a lower portion of the lens may be more light transmissive, so that it is easier for the wearer to read an LCD display through the lower portion while the wearer is in a car that is darker inside the car compared to outside the car. 
         [0075]    While the foregoing is directed to embodiments of the present invention, other and further embodiments of the present invention may be devised without departing from the basic scope thereof. It is understood that various embodiments described herein may be utilized in combination with any other embodiment described, without departing from the scope contained herein. Further, the foregoing description is not intended to be exhaustive or to limit the invention to the precise form disclosed. Modifications and variations are possible in light of the above teachings or may be acquired from practice of the invention. Certain exemplary embodiments may be identified by use of an open-ended list that includes wording to indicate that the list items are representative of the embodiments and that the list is not intended to represent a closed list exclusive of further embodiments. Such wording may include “e.g.,” “etc.,” “such as,” “for example,” “and so forth,” “and the like,” etc., and other wording as will be apparent from the surrounding context. 
         [0076]    No element, act, or instruction used in the description of the present application should be construed as critical or essential to the invention unless explicitly described as such. Also, as used herein, the article “a” is intended to include one or more items. Where only one item is intended, the term “one” or similar language is used. Further, the terms “any of” followed by a listing of a plurality of items and/or a plurality of categories of items, as used herein, are intended to include “any of,” “any combination of,” “any multiple of,” and/or “any combination of multiples of” the items and/or the categories of items, individually or in conjunction with other items and/or other categories of items. 
         [0077]    Moreover, the claims should not be read as limited to the described order or elements unless stated to that effect. In addition, use of the term “means” in any claim is intended to invoke 35 U.S.C. §112, ¶  6 , and any claim without the word “means” is not so intended.