# Patent

## Title
Polarization imaging apparatus with auto-calibration

## Abstract

A polarization imaging apparatus measures the Stokes image of a sample. The apparatus consists of an optical lens set, a first variable phase retarder (VPR) with its optical axis aligned 22.5°, a second variable phase retarder with its optical axis aligned 45°, a linear polarizer, a imaging sensor for sensing the intensity images of the sample, a controller and a computer. Two variable phase retarders were controlled independently by a computer through a controller unit which generates a sequential of voltages to control the phase retardations of the first and second variable phase retarders. A auto-calibration procedure was incorporated into the polarization imaging apparatus to correct the misalignment of first and second VPRs, as well as the half-wave voltage of the VPRs. A set of four intensity images, I0, I1, I2 and I3 of the sample were captured by imaging sensor when the phase retardations of VPRs were set at (0,0), (π,0), (π,π) and (π/2,π), respectively. Then four Stokes components of a Stokes image, S0, S1, S2 and S3 were calculated using the four intensity images.

## Background
CROSS REFERENCE TO RELATED APPLICATION
This is a continuation-in-part of U.S. patent application Ser. No. 11/701,738, filed on Feb. 2, 2007 now U.S. Pat. No. 7,701,561, which claims the benefit of provisional application Ser. No. 60/772,779 filed on Feb. 13, 2006, the entire teachings of all of which are incorporated herein by this reference.


STATEMENT OF GOVERNMENTAL INTEREST
This invention was made partially with Government supports under grant No. NNJ05JC13C awarded by NASA and No. DE-FG02-08ER85109 awarded by Department of Energy. The government has certain rights in this invention.


BACKGROUND OF THE INVENTION
1. Technical Field
The invention relates to optical imaging apparatus, particularly related to polarization imaging apparatus, more specifically related to a high speed polarization imaging system that can measure Stokes components from a sample at different optical wavelengths.
2. Technical Background
The light scattered by a tissue has interacted with the ultrastructure of the tissue, which imprinted some intrinsic properties of the tissue. Tissue ultrastructure extends from membranes to membrane aggregates to collagen fibers to nuclei to cells. Photons are most strongly scattered by those structures whose size matches the photon wavelength. It has been demonstrated that light scattering can provide structural and functional information about the tissue. One important biomedical application of optical imaging and spectroscopy is non-invasive or minimally invasive detection of pre-cancerous and early cancerous changes in human epithelium, such as dysplasia or carcinoma in situ.
In recent years there has been an increasing interest in the propagation of polarized light in randomly scattering media. The investigation of backscattered light is of particular interest since most medical applications aimed at the in-vivo characterization of biological tissue rely on backscattered light. By recording the spatially dependent response of a medium to a polarized point source, one may obtain information about the scattering particles that are not accessible to mere intensity measurements. A diagnostic imaging modality based on near-infrared (NIR) radiation offers several potential advantages over existing radiological techniques. First, the radiation is non-ionizing, and therefore reasonable doses can be repeatedly employed without harm to the patient. Second, optical methods offer the potential to differentiate between soft tissues, due to their different absorption or scatter at NIR wavelengths that are indistinguishable using other modalities. And third, specific absorption by natural chromophores (such as oxy-haemoglobin) allows functional information to be obtained. NIR imaging research has focused on a variety of possible clinical applications.
The fact that the polarization state of the light contains useful information has been shown in many literatures, for example. Rahmann and Canterakis in “Reconstruction of specular surfaces using polarization imaging,” IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 149-155, 2001, describe how the polarization state of light can be used for the reconstruction of specular surfaces to determine the shape of 3-D objects. They use the fact that light reflected by dielectrics and metals becomes partially linearly polarized and that the direction of polarization depends on the orientation of the reflecting surface. Demos and Alfano in “Deep subsurface imaging in tissues using spectral and polarization filtering,” Optics Express, vol. 7, no. 1, pp. 23-28, 2000, demonstrate a technique based on polarization imaging that allows for optical imaging of the surface as well as structures beneath the surface.
There have been several polarization imaging schemes published. In U.S. Pat. No. 6,437,856, inventor Steven Louis Jacques, which issued Aug. 20, 2002, and U.S. Pat. No. 5,847,394, inventors Robert R. Alfano, et al., which issued Dec. 8, 1998, a set of measurements at different polarization orientations are taken to render a new image that is independent of the light reflected from the surface of a tissue sample and that is dependent of the light scattered from deep tissue layers. Especially, a linearly parallel polarized light is used for illumination and two images are acquired, one image selecting linearly parallel (Par) polarized light (i.e., parallel to the light source-tissue-camera plane) and one image selecting linearly perpendicular (Per) polarized light (i.e., perpendicular to the light source-tissue-camera plane). A new image is obtained by (Par−Per) or (Par−Per)/(Par+Per).
Many polarimetric sensing technologies have been developed to capture the Stokes polarization information. People use rotating retarder, rotating polarizer to obtain Stokes parameters from several successive frames of image. In these mechanically-rotation approaches, however, the exact spatial registration between various frames is difficult due to changes of viewing angle and the polarization properties during the period of between successive frames in most target detection applications.
In U.S. Pat. No. 6,798,514, inventor James Maurice Daniels, which issued Sep. 28, 2004, the light passes through two liquid crystal waveplates and a polarizing filter before falling on a light sensitive element device to measure intensity of light in four different polarizations. However, the same difficulty also applied to the approach using liquid crystal for electro-optic polarization modulation, which usually has response time in the order of 100 ms. Another drawback for the liquid crystal approach is that polarization-rotating liquid crystals are not commercially available in many important IR bands.
SUMMARY OF THE INVENTION
It is therefore the main objects of the invention to provide a polarization imaging apparatus which can consistently provide high speed and broadband Stokes images in real-time.
It is also the object of the invention to provide a polarization imaging apparatus with an auto-calibration procedure.
The key feature of the polarization imaging apparatus is that it provides fast adjustable optical phase control through high speed electro-optical variable phase retarders made from transparent electro-optic ceramics.
A preferred embodiment of the polarization imaging apparatus is shown in FIG. 1, which consists of a polarization imaging sensing unit 100, an electronic driver 101 and a computer 102. The imaging sensing unit 100 consists of an optical assembly 11, a linear polarizer 14 with its optical axis 18, a first variable phase retarder (VPR) 12 with its optical axis 16 aligned 22.5° to axis 18, a second variable phase retarder 13 with its optical axis 17 aligned 45° to axis 18, and a imaging sensor 15 for sensing the intensity images of the sample. The imaging sensor 15 can be a focal plan array (FPA) detector, a CMOS or a CCD sensor. Two VPRs 12 and 13 were controlled independently by a computer 102 through a controller unit 101 which generates a sequential of voltages to control the phase retardations of VPRs 12 and 13. A set of four intensity images, I0, I1, I2 and I3 of the sample were captured by imaging sensor 15 when the phase retardations of VPRs 12 and 13 were set at (0,0), (π,0), (π,π) and (π/2,π), respectively Then four Stokes components of a Stokes image, S0, S1, S2 and S3 can be calculated by computer 102 using the following formula:




  
    
      
        {
        
          
            
              
                
                  S
                  0
                
                =
                
                  0.5
                  ×
                  
                    (
                    
                      
                        I
                        0
                      
                      +
                      
                        I
                        1
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  S
                  1
                
                =
                
                  0.5
                  ×
                  
                    (
                    
                      
                        I
                        0
                      
                      -
                      
                        I
                        1
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  S
                  2
                
                =
                
                  0.5
                  ×
                  
                    (
                    
                      
                        I
                        2
                      
                      -
                      
                        S
                        0
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  S
                  3
                
                =
                
                  0.5
                  ×
                  
                    (
                    
                      
                        I
                        3
                      
                      -
                      
                        S
                        0
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        1
        )
      
    
  




Another key feature of the invention is that a simplified time sequential control and measurement algorithm which provides a fast measurement and converts sequential intensity images to Stokes images.
In one aspect, the present invention has a calibration procedure that needs one known polarization state to accurately calculate all of the parameters, such as the orientation of first and second variable phase retarter α1 and α2, respectively, as well as half-wave phase retardance δ1π of first variable phase retarder, quarter-wave phase retardance δ1π/2 of first variable phase retarder, and half-wave phase retardance δ2π of second variable phase retarder.
In one aspect, the present invention includes the ferro-electric materials, such as PLZT, La modified PMN-PT, and La modified PZN-PT to form the electric voltage controlled ferro-electric variable phase retarder.
Referring to FIG. 5 is another preferred embodiment of the polarization imaging apparatus consists of an illumination unit 50, a polarization imaging sensing unit 500, a controller unit 101 and a computer 102. The said illumination unit and imaging sensing unit are aligned in a reflection configuration. In addition to a light source, the illuminator may consist of parts such as a collimator, a polarizer or polarization modulator, and a fixed wavelength filter or tunable wavelength filter. Tunable optical band-pass filter is used to produce desired wavelength for the imaging, from visible to infrared (IR). The illuminator can produce a linear or arbitrary polarized light beam.
The illumination unit 50, and a polarization imaging sensing unit 500 in FIG. 5 can be further aligned in a transmission configuration, as illustrated in FIG. 6.
Additional features and advantages of the invention will be set forth in the detailed description which follows, and in part will be readily apparent to those skilled in the art from that description or recognized by practicing the invention as described herein, including the detailed description which follows, the claims, as well as the appended drawings.
It is to be understood that both the foregoing general description and the following detailed description present embodiments of the invention, and are intended to provide an overview or framework for understanding the nature and character of the invention as it is claimed. The accompanying drawings are included to provide a further understanding of the invention, and are incorporated into and constitute a part of this specification. The drawings illustrate various embodiments of the invention and together with the description serve to explain the principles and operations of the invention.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is the schematic of a preferred embodiment of polarization imaging apparatus with variable retarders in accordance with the present invention;
FIG. 2 shows the measurement steps for polarization imaging apparatus in accordance with the present invention;
FIG. 3 shows the schematic drawing of a high speed variable retarder using a electro-optic ceramic;
FIG. 4 shows measured responses of a high speed variable retarder;
FIG. 5 is a schematic view of a preferred embodiment of reflective polarization imaging apparatus in accordance with the present invention;
FIG. 6 is a schematic view of a preferred embodiment of transmissive polarization imaging apparatus in accordance with the present invention.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
It is to be understood that the invention is not limited in its application to the details of the particular arrangement shown since the invention is capable of other embodiments. Also, the terminology used herein is for the purpose of description and not of limitation.
FIG. 1 shows a preferred embodiment of the polarization imaging apparatus in accordance with the present invention. It consists of an imaging sensing unit 100, an electronic driver 101 and a computer 102. The imaging sensing unit 100 further consists of an optical assembly 11, a polarizer 14 with its optical axis 18, a first variable phase retarder (VPR) 12 with its optical axis 16 aligned 22.5° to axis 18, a second variable phase retarder 13 with its optical axis 17 aligned 45° to axis 18, a imaging sensor 15 for sensing the intensity images of the sample. The imaging sensor 15 can be a focal plan array (FPA) detector, a CMOS or a CCD sensor. The imaging sensing unit then is assembled inside a metal host (not shown). Two VPRs 12 and 13 were controlled independently by a computer 102 through a controller 101 which generates a sequential of voltages to control the phase retardations of the VPRs 12 and 13. The intensity images were captured through the imaging sensor 15, and processed by computer 102 to convert the sequential intensity images to Stokes images.
Using Mueller's calculus, we can obtain the Stokes parameters of an input light 10 by measuring the output light intensity of the sensor 15. The transformation done by the phase retarders can be described in the following equation,

Sout=(S0,outS1,outS2,outS3,out)T=Mr2*Mr1*Sin.  (2)

where Sin=(S0 S1 S2 S3)T is the unknown input Stokes vector of the target 10, which consists of four Stokes components. The parameter S0 is the intensity of the light (including un-polarized component), S1 is the intensity difference between horizontal linearly polarized and the vertical linearly polarized components, S2 is the intensity difference between 45° linearly polarized and the 135° linearly polarized components, and S3 is the difference between left- and right-circularly polarized intensities.

The quantities Mr1, and Mr2 are the Mueller matrices representing two variable phase retarders 12 and 13, respectively. These are given for the specific orientation by




  
    
      
        
          
            M
            
              r
              ⁢
              
                
              
              ⁢
              1
            
          
          ⁡
          
            (
            δ1
            )
          
        
        =
        
          [
          
            
              
                1
              
              
                0
              
              
                0
              
              
                0
              
            
            
              
                0
              
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      1
                      +
                      
                        cos
                        ⁡
                        
                          (
                          δ1
                          )
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      1
                      -
                      
                        cos
                        ⁡
                        
                          (
                          δ1
                          )
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    -
                    
                      1
                      
                        2
                      
                    
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      δ1
                      )
                    
                  
                
              
            
            
              
                0
              
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      1
                      -
                      
                        cos
                        ⁡
                        
                          (
                          δ1
                          )
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      1
                      +
                      
                        cos
                        ⁡
                        
                          (
                          δ1
                          )
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      δ1
                      )
                    
                  
                
              
            
            
              
                0
              
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      δ1
                      )
                    
                  
                
              
              
                
                  
                    -
                    
                      1
                      
                        2
                      
                    
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      δ1
                      )
                    
                  
                
              
              
                
                  cos
                  ⁡
                  
                    (
                    δ1
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        3
        )
      
    
  
  
    
      
        
          
            M
            
              r
              ⁢
              
                
              
              ⁢
              2
            
          
          ⁡
          
            (
            δ2
            )
          
        
        =
        
          
            1
            2
          
          ⁡
          
            [
            
              
                
                  1
                
                
                  0
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    cos
                    ⁡
                    
                      (
                      δ2
                      )
                    
                  
                
                
                  0
                
                
                  
                    -
                    
                      sin
                      ⁡
                      
                        (
                        δ2
                        )
                      
                    
                  
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    sin
                    ⁡
                    
                      (
                      δ2
                      )
                    
                  
                
                
                  0
                
                
                  
                    cos
                    ⁡
                    
                      (
                      δ2
                      )
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        4
        )
      
    
  




where δ1 and δ2 are the two phase retardance introduced by the two variable phase retarders 12 and 13, respectively.
To be able to measure the four components of the input Stokes vector, the variable phase retarders need to be set to at least four different states labeled as k. In the following, the linkage between input state Stokes parameters and measured light intensity is detailed. The Mueller matrix for the cascaded variable retarders can be generally cast into an alternative form so that the state of polarization incident on the polarizer can be expressed as,




  
    
      
        
          (
          
            
              
                
                  S
                  
                    0
                    ,
                    out
                  
                
              
            
            
              
                
                  S
                  
                    1
                    ,
                    out
                  
                
              
            
            
              
                
                  S
                  
                    2
                    ,
                    out
                  
                
              
            
            
              
                
                  S
                  
                    3
                    ,
                    out
                  
                
              
            
          
          )
        
        =
        
          
            M
            ·
            
              (
              
                
                  
                    
                      S
                      
                        0
                        ,
                        in
                      
                    
                  
                
                
                  
                    
                      S
                      
                        1
                        ,
                        in
                      
                    
                  
                
                
                  
                    
                      S
                      
                        2
                        ,
                        in
                      
                    
                  
                
                
                  
                    
                      S
                      
                        3
                        ,
                        in
                      
                    
                  
                
              
              )
            
          
          =
          
            
              (
              
                
                  
                    
                      m
                      00
                    
                  
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      m
                      11
                    
                  
                  
                    
                      m
                      12
                    
                  
                  
                    
                      m
                      13
                    
                  
                
                
                  
                    0
                  
                  
                    
                      m
                      21
                    
                  
                  
                    
                      m
                      22
                    
                  
                  
                    
                      m
                      23
                    
                  
                
                
                  
                    0
                  
                  
                    
                      m
                      31
                    
                  
                  
                    
                      m
                      32
                    
                  
                  
                    
                      m
                      33
                    
                  
                
              
              )
            
            ·
            
              (
              
                
                  
                    
                      S
                      
                        0
                        ,
                        in
                      
                    
                  
                
                
                  
                    
                      S
                      
                        1
                        ,
                        in
                      
                    
                  
                
                
                  
                    
                      S
                      
                        2
                        ,
                        in
                      
                    
                  
                
                
                  
                    
                      S
                      
                        3
                        ,
                        in
                      
                    
                  
                
              
              )
            
          
        
      
    
    
      
        (
        5
        )
      
    
  




The actually detected signal through the polarizer is essentially




  I
  =
  
    
      1
      2
    
    ⁢
    
      
        (
        
          
            S
            
              0
              ,
              out
            
          
          +
          
            S
            
              1
              ,
              out
            
          
        
        )
      
      ·
    
  




Without worrying about a global factor, the factor 1⁄2 was thrown away. Thus the intensity can be expressed in terms of three of the Muller matrix elements,

I=S0,out+S1,out=m11S1,in+m12S2,in+m13S3,in  (6)

It is clear that the output intensity can be linked to the input Stokes parameters through generating four known states of polarization at the input so that we have,




  
    
      
        
          (
          
            
              
                
                  I
                  1
                
              
            
            
              
                
                  I
                  2
                
              
            
            
              
                
                  I
                  3
                
              
            
            
              
                
                  I
                  4
                
              
            
          
          )
        
        =
        
          
            A
            ·
            
              (
              
                
                  
                    
                      S
                      
                        0
                        ,
                        in
                      
                    
                  
                
                
                  
                    
                      S
                      
                        1
                        ,
                        in
                      
                    
                  
                
                
                  
                    
                      S
                      
                        2
                        ,
                        in
                      
                    
                  
                
                
                  
                    
                      S
                      
                        3
                        ,
                        in
                      
                    
                  
                
              
              )
            
          
          =
          
            
              (
              
                
                  
                    
                      m
                      00
                      1
                    
                  
                  
                    
                      m
                      11
                      1
                    
                  
                  
                    
                      m
                      12
                      1
                    
                  
                  
                    
                      m
                      13
                      1
                    
                  
                
                
                  
                    
                      m
                      00
                      2
                    
                  
                  
                    
                      m
                      11
                      2
                    
                  
                  
                    
                      m
                      12
                      2
                    
                  
                  
                    
                      m
                      13
                      2
                    
                  
                
                
                  
                    
                      m
                      00
                      3
                    
                  
                  
                    
                      m
                      11
                      3
                    
                  
                  
                    
                      m
                      12
                      3
                    
                  
                  
                    
                      m
                      13
                      3
                    
                  
                
                
                  
                    
                      m
                      00
                      4
                    
                  
                  
                    
                      m
                      11
                      4
                    
                  
                  
                    
                      m
                      12
                      4
                    
                  
                  
                    
                      m
                      13
                      4
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    
                      S
                      
                        0
                        ,
                        in
                      
                    
                  
                
                
                  
                    
                      S
                      
                        1
                        ,
                        in
                      
                    
                  
                
                
                  
                    
                      S
                      
                        2
                        ,
                        in
                      
                    
                  
                
                
                  
                    
                      S
                      
                        3
                        ,
                        in
                      
                    
                  
                
              
              )
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




where mijk is the Mueller matrix element at the k-th state of variable retarders, and Ii is the intensity measured with the i-th state of variable retarder. From Equation (7), it can be noticed that once the matrix A is known, the input Stokes parameters can be found out the inverse the matrix so that,




  
    
      
        
          (
          
            
              
                
                  S
                  
                    0
                    ,
                    in
                  
                
              
            
            
              
                
                  S
                  
                    1
                    ,
                    in
                  
                
              
            
            
              
                
                  S
                  
                    2
                    ,
                    in
                  
                
              
            
            
              
                
                  S
                  
                    3
                    ,
                    in
                  
                
              
            
          
          )
        
        =
        
          
            A
            
              -
              1
            
          
          ·
          
            (
            
              
                
                  
                    I
                    1
                  
                
              
              
                
                  
                    I
                    2
                  
                
              
              
                
                  
                    I
                    3
                  
                
              
              
                
                  
                    I
                    4
                  
                
              
            
            )
          
        
      
    
    
      
        (
        8
        )
      
    
  




The very question now is to find out the matrix A. It can be calculated from each Muller matrix in the system or, it can be done by generating four known states of polarization so that,




  
    
      
        A
        =
        
          
            (
            
              
                
                  
                    S
                    
                      0
                      ,
                      in
                    
                    1
                  
                
                
                  
                    S
                    
                      0
                      ,
                      in
                    
                    2
                  
                
                
                  
                    S
                    
                      0
                      ,
                      in
                    
                    3
                  
                
                
                  
                    S
                    
                      0
                      ,
                      in
                    
                    4
                  
                
              
              
                
                  
                    S
                    
                      1
                      ,
                      in
                    
                    1
                  
                
                
                  
                    S
                    
                      1
                      ,
                      in
                    
                    2
                  
                
                
                  
                    S
                    
                      1
                      ,
                      in
                    
                    3
                  
                
                
                  
                    S
                    
                      1
                      ,
                      in
                    
                    4
                  
                
              
              
                
                  
                    S
                    
                      2
                      ,
                      in
                    
                    1
                  
                
                
                  
                    S
                    
                      2
                      ,
                      in
                    
                    2
                  
                
                
                  
                    S
                    
                      2
                      ,
                      in
                    
                    3
                  
                
                
                  
                    S
                    
                      2
                      ,
                      in
                    
                    4
                  
                
              
              
                
                  
                    S
                    
                      3
                      ,
                      in
                    
                    1
                  
                
                
                  
                    S
                    
                      3
                      ,
                      in
                    
                    2
                  
                
                
                  
                    S
                    
                      3
                      ,
                      in
                    
                    3
                  
                
                
                  
                    S
                    
                      3
                      ,
                      in
                    
                    4
                  
                
              
            
            )
          
          ·
          
            (
            
              
                
                  
                    I
                    1
                    1
                  
                
                
                  
                    I
                    1
                    2
                  
                
                
                  
                    I
                    1
                    3
                  
                
                
                  
                    I
                    1
                    4
                  
                
              
              
                
                  
                    I
                    2
                    1
                  
                
                
                  
                    I
                    2
                    2
                  
                
                
                  
                    I
                    2
                    3
                  
                
                
                  
                    I
                    2
                    4
                  
                
              
              
                
                  
                    I
                    3
                    1
                  
                
                
                  
                    I
                    3
                    2
                  
                
                
                  
                    I
                    3
                    3
                  
                
                
                  
                    I
                    3
                    4
                  
                
              
              
                
                  
                    I
                    4
                    1
                  
                
                
                  
                    I
                    4
                    2
                  
                
                
                  
                    I
                    4
                    3
                  
                
                
                  
                    I
                    4
                    4
                  
                
              
            
            )
          
        
      
    
    
      
        (
        9
        )
      
    
  




where Si,inj is the i-th Stokes parameter generate by the j-th retardance. Thus we are able to link the input state polarization to the intensity measured at four retardance states. The Mueller matrix elements needed for finding the input state of polarization can be figured out through the use of four known state of polarization.

The measurement procedure is based on specifying above general principle for practical applications. As the configuration shown in FIG. 1, we can find out the useful Muller matrix elements (according to equation (6)) as follows:

m00=1

m11=cos2(δ1/2)·cos(δ2)−sin(δ1)·sin(δ2)√2

m12=sin2(δ1/2)

m13=−cos2(δ1/2)·sin(δ2)−sin(δ1)·cos(δ2)/√2  (10)

where δ1 and δ2 are the retardance of two VPRs 12 and 13, respectively. Although their values could be many different combinations for finding out the Stokes vector, the following 4-state procedure is considered to be the simplest yet most effective approach.
Referring to FIG. 2 shows the procedure diagram to accomplish one measurement of Stokes parameters.
Four intensity images (I0-I3) are needed and taken sequentially, with two variable phase retarders 12 and 13 controlled by a sequence of voltages which were generated by controller 101. Phase retardation changes 21 and 22 of VPR 12 and VPR 13, respectively. I0 is taken with 0 volt applied to both 12 and 13. I1 is taken with the half-wave voltage Vπ applied to 12 and 0 volt applied to 13. I2 is taken with V, is applied to both retarder 12 and 13. I3 is taken with a quarter-wave voltage Vπ/2 applied to 12 and Vπ applied to 13. The four Stokes vector images S0-S3 can be calculated from the intensity as follows:




  
    
      
        {
        
          
            
              
                
                  S
                  0
                
                =
                
                  0.5
                  ×
                  
                    (
                    
                      
                        I
                        0
                      
                      +
                      
                        I
                        1
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  S
                  1
                
                =
                
                  0.5
                  ×
                  
                    (
                    
                      
                        I
                        0
                      
                      -
                      
                        I
                        1
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  S
                  2
                
                =
                
                  0.5
                  ×
                  
                    (
                    
                      
                        I
                        2
                      
                      -
                      
                        S
                        0
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  S
                  3
                
                =
                
                  0.5
                  ×
                  
                    (
                    
                      
                        I
                        3
                      
                      -
                      
                        S
                        0
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




Table 1 listed the state of retardance of first and second variable phase retarders and the corresponding four intensities measured in each equal length time slot. The intensities also can be expressed as a function of Stokes parameters S0-S3 of the input Stokes vector according to equation (2). Table 1 also listed the formula for extracting the Stokes parameters S0-S3 from these intensities.











TABLE 1





Time slot
1
2
3
4







State of VPR1 (δ1)
0
π
π
π/2


State of VPR2 (δ2)
0
0
π
π


Measured intensity
I0
I1
I2
I3


Relation to Stokes
S0,in + S1,in
S0,in − S1,in
S0,in + S2,in
S0,in + S3,in


 








Derived parameters




  {
  
    
      
        
          
            
              S
              0
            
            =
            
              0.5
              ×
              
                (
                
                  
                    I
                    0
                  
                  +
                  
                    I
                    1
                  
                
                )
              
            
          
        
      
      
        
          
            
              S
              1
            
            =
            
              0.5
              ×
              
                (
                
                  
                    I
                    0
                  
                  -
                  
                    I
                    1
                  
                
                )
              
            
          
        
      
      
        
          
            
              S
              2
            
            =
            
              0.5
              ×
              
                (
                
                  
                    I
                    2
                  
                  -
                  
                    S
                    0
                  
                
                )
              
            
          
        
      
      
        
          
            
              S
              3
            
            =
            
              0.5
              ×
              
                (
                
                  
                    I
                    3
                  
                  -
                  
                    S
                    0
                  
                
                )
              
            
          
        
      
    
     
  













With this simple converting formula, Stokes images can be obtained using very simple calculation from the intensity images captured by an imaging sensor.
Considering the misalignment of both retarders VPR 12 and VPR 13, and the inaccuracy of their actual half-wave voltages in a polarization system, an auto-calibration technique is necessary to maintain its accuracy. The auto-calibration procedures are for correcting the errors caused by misalignment the optical axis 16 of VPR 12 and optical axis 17 of VPR 13, and the inaccuracy of the voltages applied to each of VPRs.
Assuming:

    
    
        α1 the actual orientation of the optical axis 16 of variable phase retarder VPR 12. The ideal value is 22.5°;
        α2 the actual orientation of the optical axis 17 of variable phase retarder VPR 13. The ideal value is 45°;
        δ1π the actual phase retardance when a half-wave voltage V, is applied to VPR 12.
        The ideal phase retardance is π;
        δ1π/2 the actual phase retardance when a quarter-wave voltage Vπ/2 is applied to VPR 12. The ideal phase retardance is π/2; and
        δ2π the actual phase retardance when a half-wave voltage V, is applied to VPR 13. The ideal phase retardance is π.
    
    


where initial quarter-wave voltage and half-ware voltages of VPR 12 and VPR 13 are measured before assembly.
The components of Mueller matrix in Equation (5) will be:

m11=(cos2+2α1+sin22α1 cos δ1)(cos22α2+sin22α2 cos δ2)+sin 2α1 cos 2α1(1−cos δ1)sin 2α2 cos 2α2(1−cos δ2)−sin 2α1 sin δ1 sin 2α2 sin δ2  (12)

m12=(cos22α1+sin22α1 cos δ1)sin 2α2 cos 2α2(1−cos δ2)+sin 2α1 cos 2α1(1−cos δ1)(sin22α2+cos22α2 cos δ2)+sin 2α1 sin δ1 cos 2α2 sin δ2  (13)

m13=−(cos22α1+sin22α1 cos δ1)sin 2α2 sin δ2+sin 2α1 cos 2α1(1−cos δ1)cos 2α2 sin δ2−sin 2α1 sin δ1 cos δ2  (14)

The parameter α1, α2, δ1π/2, δ1π and δ2π can be solved using a following sets of measurement procedure with a known polarization incident light, such as a horizontal polarized light:




  
    
      
        
          [
          
            
              
                
                  S
                  
                    out
                    ⁡
                    
                      (
                      
                        
                          π
                          /
                          2
                        
                        ,
                        0
                      
                      )
                    
                  
                
              
            
            
              
                
                  S
                  
                    out
                    ⁡
                    
                      (
                      
                        π
                        ,
                        0
                      
                      )
                    
                  
                
              
            
            
              
                
                  S
                  
                    out
                    ,
                    
                      (
                      
                        0
                        ,
                        π
                      
                      )
                    
                  
                
              
            
            
              
                
                  S
                  
                    out
                    ⁡
                    
                      (
                      
                        π
                        ,
                        π
                      
                      )
                    
                  
                
              
            
            
              
                
                  S
                  
                    out
                    ⁡
                    
                      (
                      
                        
                          π
                          /
                          2
                        
                        ,
                        π
                      
                      )
                    
                  
                
              
            
          
          ]
        
        =
        
          
            [
            
              
                
                  1
                
                
                  
                    m
                    11
                    
                      
                        π
                        /
                        2
                      
                      ,
                      0
                    
                  
                
                
                  
                    m
                    12
                    
                      
                        π
                        /
                        2
                      
                      ,
                      0
                    
                  
                
                
                  
                    m
                    13
                    
                      
                        π
                        /
                        2
                      
                      ,
                      0
                    
                  
                
              
              
                
                  1
                
                
                  
                    m
                    11
                    
                      π
                      ,
                      0
                    
                  
                
                
                  
                    m
                    12
                    
                      π
                      ,
                      0
                    
                  
                
                
                  
                    m
                    13
                    
                      π
                      ,
                      0
                    
                  
                
              
              
                
                  1
                
                
                  
                    m
                    11
                    
                      0
                      ,
                      π
                    
                  
                
                
                  
                    m
                    12
                    
                      0
                      ,
                      π
                    
                  
                
                
                  
                    m
                    13
                    
                      0
                      ,
                      π
                    
                  
                
              
              
                
                  1
                
                
                  
                    m
                    11
                    
                      π
                      ,
                      π
                    
                  
                
                
                  
                    m
                    12
                    
                      π
                      ,
                      π
                    
                  
                
                
                  
                    m
                    13
                    
                      π
                      ,
                      π
                    
                  
                
              
              
                
                  1
                
                
                  
                    m
                    11
                    
                      
                        π
                        /
                        2
                      
                      ,
                      π
                    
                  
                
                
                  
                    m
                    12
                    
                      
                        π
                        /
                        2
                      
                      ,
                      π
                    
                  
                
                
                  
                    m
                    13
                    
                      
                        π
                        /
                        2
                      
                      ,
                      π
                    
                  
                
              
            
            ]
          
          ⁡
          
            [
            
              
                
                  
                    S
                    
                      0
                      ,
                      in
                    
                  
                
              
              
                
                  
                    S
                    
                      1
                      ,
                      in
                    
                  
                
              
              
                
                  
                    S
                    
                      2
                      ,
                      in
                    
                  
                
              
              
                
                  
                    S
                    
                      3
                      ,
                      in
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        15
        )
      
    
  




where




   
  
    [
    
      
        
          
            S
            
              0
              ,
              in
            
          
        
      
      
        
          
            S
            
              1
              ,
              in
            
          
        
      
      
        
          
            S
            
              2
              ,
              in
            
          
        
      
      
        
          
            S
            
              3
              ,
              in
            
          
        
      
    
    ]
  




is the Stokes matrix of the input light and





   
  
    [
    
      
        
          
            S
            
              out
              ⁡
              
                (
                
                  
                    π
                    /
                    2
                  
                  ,
                  0
                
                )
              
            
          
        
      
      
        
          
            S
            
              out
              ⁡
              
                (
                
                  π
                  ,
                  0
                
                )
              
            
          
        
      
      
        
          
            S
            
              out
              ,
              
                (
                
                  0
                  ,
                  π
                
                )
              
            
          
        
      
      
        
          
            S
            
              out
              ⁡
              
                (
                
                  π
                  ,
                  π
                
                )
              
            
          
        
      
      
        
          
            S
            
              out
              ⁡
              
                (
                
                  
                    π
                    /
                    2
                  
                  ,
                  π
                
                )
              
            
          
        
      
    
    ]
  




are measured light output. The Mueller matrix components are, according to equations (12) to (13):

(1) m11π/2,0, m12π/2,0, m12π/2,0 represent the m values while a quarter-wave voltage and a zero are applied to VPR 12 and VPR 13, respectively. The detected output is Sout(π/2,0).

(2) m11π,0, m12π,0, m13π,0 represent the m values while a half-wave voltage and a zero voltage are applied to VPR 12 and VPR 13, respectively. The detected output is Sout(π,0).

(3) m110,π, m120,π, m130,π represent the m values while a zero and a half-wave voltage are applied to VPR 12 and VPR 13, respectively. The detected output is Sout(0,π). 

(4), m11π,π, m12π,π, m13π,π represent the m values while half-wave voltages are applied to both VPR 12 and VPR 13. The detected output is Sout(π,π). 

(5) m11π/2,π, m12π/2,π, m13π/2,π represent the m values while a quarter-wave voltage and a half-wave voltage are applied to VPR 12 and VPR 13, respectively. The detected output is Sout(π,π/2).

The Stokes vector of an unknown incident image (S0, S1, S2, S3)T can be obtained using the equation (16):




  
    
      
        
          [
          
            
              
                
                  S
                  0
                
              
            
            
              
                
                  S
                  1
                
              
            
            
              
                
                  S
                  2
                
              
            
            
              
                
                  S
                  3
                
              
            
          
          ]
        
        =
        
          
            
              [
              
                
                  
                    1
                  
                  
                    
                      m
                      11
                      
                        0
                        ,
                        0
                      
                    
                  
                  
                    
                      m
                      12
                      
                        0
                        ,
                        0
                      
                    
                  
                  
                    
                      m
                      13
                      
                        0
                        ,
                        0
                      
                    
                  
                
                
                  
                    1
                  
                  
                    
                      m
                      11
                      
                        π
                        ,
                        0
                      
                    
                  
                  
                    
                      m
                      12
                      
                        π
                        ,
                        0
                      
                    
                  
                  
                    
                      m
                      13
                      
                        π
                        ,
                        0
                      
                    
                  
                
                
                  
                    1
                  
                  
                    
                      m
                      11
                      
                        π
                        ,
                        π
                      
                    
                  
                  
                    
                      m
                      12
                      
                        π
                        ,
                        π
                      
                    
                  
                  
                    
                      m
                      13
                      
                        π
                        ,
                        π
                      
                    
                  
                
                
                  
                    1
                  
                  
                    
                      m
                      11
                      
                        
                          π
                          /
                          2
                        
                        ,
                        π
                      
                    
                  
                  
                    
                      m
                      12
                      
                        
                          π
                          /
                          2
                        
                        ,
                        π
                      
                    
                  
                  
                    
                      m
                      13
                      
                        
                          π
                          /
                          2
                        
                        ,
                        π
                      
                    
                  
                
              
              ]
            
            
              -
              1
            
          
          ⁡
          
            [
            
              
                
                  
                    S
                    
                      out
                      ⁡
                      
                        (
                        
                          0
                          ,
                          0
                        
                        )
                      
                    
                  
                
              
              
                
                  
                    S
                    
                      out
                      ⁡
                      
                        (
                        
                          π
                          ,
                          0
                        
                        )
                      
                    
                  
                
              
              
                
                  
                    S
                    
                      out
                      ,
                      
                        (
                        
                          π
                          ,
                          π
                        
                        )
                      
                    
                  
                
              
              
                
                  
                    S
                    
                      out
                      ⁡
                      
                        (
                        
                          
                            π
                            /
                            2
                          
                          ,
                          π
                        
                        )
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        16
        )
      
    
  




where Sout(0,0), Sout(π,0), Sout(π,π) and Sout(π/2,π) outputs when the first VPR 12 and second VPR 13 have the phase retardation of (0,0), (π,0), (π,π), and (π/2,π), respectively; and m values can be calculated according to equations (12) to (13) using calibrated VPR parameters α1, α2, δ1π, δ1π/2 and δ2π.
It should be pointed out that the switch of the relative positions of VPR 12 and VPR 13 will not affect the outcome of the Stokes images according to equation (2).
It is to be appreciated that the orientation configuration between the two variable phase plates 12 and 13 might be achieved with an intervening fixed waveplate, so that physical orientation of the two plates could be less constrained. Hence, the 22.5 degree orientation of the second variable phase retarder 13 with respect to the first variable phase retarder 12 is achieved by any combination of relative physical rotation of the second variable phase retarder 13 with respect to the first retarder 12 and rotation of the polarization reference frame between the two variable phase retarders 12 and 13 by a fixed phase retarding plate (not shown but included as part of the present invention).
Any other orientation configuration of first variable phase retarder 12 and second variable phase retarder 13 will also work, as long as the two variable retarders oriented at different angles, however may involve more difficult calculations.
In addition to the simplified measurement algorithm and auto-calibration, a key component in this polarization imaging apparatus is the variable phase retarder. The other means to change the phase retardation, rotating waveplates, Faraday rotators and Pockel's cell have been used in laboratories worldwide. These devices, however, are bulky, high power consumption, slow or expensive.
A preferred variable phase retarder is made from an electro-optic (EO) material. The general requirement for an EO variable phase retarder is that, when a voltage is applied, a predetermined phase shift is produced for the light beam. Also the material is isotropic with no voltage applied, there is no residual birefringence to cause the initial phase shift. Preferably, the material has a high electro-optic coefficient in order to reduce operating voltages to less than 500 volts. Also the mechanical characteristics allow formation of a bar or plate for use as the electrode. Of course, the material must be transparent at the wavelength of interest, e.g., between 450 nm and 7 μm for most of the polarization imaging applications.
These requirements are satisfied by a class of ferroelectric complex oxides which 1) are optically isotropic; 2) have a Curie temperature less than about 490° C., so that electro-optic coefficients are high near room temperature; 3) have a diffusive phase transition, so that the temperature dependence of the electro-optic coefficients is lessened; and 4) which are not permanently poled by moderate electric fields since materials with a low Curie temperature that become permanently poled are less stable. Example material systems include electro-optic lanthanum modified PZT (PLZT), lead manganese niobate (PMN), a solid solution of lead zinc niobate and lead tantalate (PZN-PT), and a solid solution of lead manganese niobate and lead tantalate (PMN-PT). Besides PLZT, without being an exhaustive list the following materials and their solid solution with or without lanthanum modification, may be used: Pb(Zr,Ti)O3, Pb(Mg,Nb)O3, PbTiO3, Pb(Zn,Nb)O3 and PbTaO3. These materials are available from Corning Inc. (Corning, N.Y.) and Boston Applied Technologies, Inc. (Woburn, Mass.), referenced as OptoCeramic materials. These materials are electrical controllable, solid-state, high speed, low insertion loss, and low fabrication cost.
A preferred OptoCeramic material, PMN-PT, for use in the present invention may be described by the general formula

Pb1-zLaz[(MgtNb1-t)xTi1−x]1-z/4O3+x/2−3xt/2−xz/8+3xtz/8 

wherein x is between about 0.6 and about 0.95, z is between about 0 and about 0.08, and t is between about 0.30 and about 0.36. In especially preferred electro-optic ceramic materials of the present invention, z is between about 0.01 and about 0.06. In preferred electro-optic materials of the present invention, t may be between about 0.32 and about 0.34, and x may be between about 0.65 and about 0.90.
According to the present invention, the preferred OptoCeramic PMN-PT is lanthanum modified solid solution of lead manganese niobate and lead tantalate (La:PMN-PT, or PLMNT) with a nominal 3.5/75/25 La/PMN/PT composition. For reference purposes nominal values for this PLMNT are n=2.48 and R=11.5×10−16 m2/V2 at 1.06 μm and n=2.458 and R=6.86×10−16 m2/V2 at 1.55 μm, both at 23° C. PLMNT has a polycrystalline structure with crystal sizes ranging from about 5 to 20 microns. In the form of hot-pressed ceramics, it is optically isotropic and exhibits little birefringence with zero applied voltage.
An electrode geometry of phase retarder which takes advantage of these materials is illustrated by the transverse field configuration illustrated in FIG. 3. A plate 30, for use as the variable phase retarder 12 and 13 of FIG. 1, is shown which has thin metallized electrodes 31 and 32 on the both sides, respectively, of a block section 33 of EO material. The electrodes 31 and 32 are shown connected to a driving voltage 34 provide by controller 102 shown in FIG. 1. The electric field (designated E) is 90° to the direction of light propagation. Since the effect is based on the electric field in the material, it is desirable to arrange the electrodes as close together as possible to minimize the control voltage. The optical axis of the EO element is perpendicular to the electric field.
As an example, an EO phase retarder is made from a PLMNT plate with metallized electrodes, which has dimensions where the width (w) between the electrodes is about 5 mm, a thickness (t) of about 4 mm, and a length (l) of about 5 mm. Such a dimensioned PLMNT plate has an insertion loss of about 0.1 dB and half-wave voltage of 320V for a 700 nm wavelength laser at 23° C.
It will be understood that the particular geometry described is an example and that other device geometries can be used. Different dimensions of w and t can be selected to provide the same π phase shift with different control voltage. The other preferred PLMNT plate has dimensions of, but not limited to a width (w) of about 0.5-5 mm, a thickness (t) of about 1.30-5 mm, and a length (l) of about 1.0-5.0 mm.
Another preferred OptoCeramic material is PLZT with a nominal 9.5/65/35 La/Pb/Zr composition. This composition is known to be transparent in a range from 450 nm to 7 μm.
Yet another preferred OptoCeramic material, PZN-PT, for use in the present invention may be described by the general formula

Pb1−y−zBayLaz[(ZntNb1-t)xTi1−x]1-z/4O3+x/2−3xt/2−xz/8+3xtz/8 

wherein x is between about 0.5 and about 0.9, y is between about 0.05 and about 0.5, z is between about 0 and about 0.05, and t is between about 0.30 and about 0.36. In especially preferred electro-optic ceramic materials of the present invention, x is between about 0.65 and about 0.85, y is between about 0.1 and about 0.2, z is between about 0.02 and 0.04, and t is between about 0.32 and about 0.34.
Referring to FIG. 4 is the measured variable phase retarder response. The optical response 41 to the applied control voltage 42 is less than 1 microsecond. This means that the speed of setting and resetting of the VPR is as fast as 1.0 microsecond. According to the present invention, the imaging apparatus only needs 2 and 3 settings for the first and the second variable phase retarders 12 and 13 to complete a measurement cycle, respectively. The polarization sampling time could be less than 3 microseconds. This equals the system can generate a set of 4 Stokes component images at a speed of up to 300K frames per second (fps). With a fast enough imaging sensor, a real-time Stokes polarization imaging system can be realized.
In the present invention, another preferred EO material for variable phase retarders 12 and 13 are made from liquid crystals for a polarization imaging system working between the optical wavelengths 350 nm to 2000 nm. Using the invented control algorithm illustrated in FIG. 2, and relationship between sequential intensity images and Stokes images listed in Table 1, a fast polarization imaging system using LC variable retarders can also be configured.
A polarization imaging apparatus shown in FIG. 1 is basically a passive imaging system which using ambient light source to illuminate the sample. Referring to FIG. 5 is a schematic view of a preferred embodiment of an active polarization imaging apparatus in accordance with the present invention. It consists of an illumination unit 50 and an imaging sensing unit 500. The said illumination unit and imaging sensing unit are aligned in a reflection configuration. Both illumination unit 50 and imaging sensing unit 100 can be set to an angle related to sample 56 surface normal 57. In addition to a light source 51, the illuminator may consist of parts such as a collimator 52, a polarizer or polarization modulator 53, and a fixed wavelength filter or tunable wavelength filter 54. Tunable optical band-pass filter was used to produce desired wavelength for the imaging, from visible to infrared (IR). The illuminator could produce a linear or arbitrary polarized light beam.
A preferred light source 51 is a broadband light source, such as a tungsten-halogen lamp which provides a strong intensity over a broad spectrum from UV to NIR (300 nm to 2 μm). Another preferred light source is a light emitting diode (LED) source or laser source.
The scattering light 58 from sample 56 is detected by the imaging sensing unit 500. The optical assembly 11 in 100 shown in FIG. 1 can be an optical objective lens 59 in FIG. 5 with an infinite focal length, which makes the polarization imaging system a Stokes microscope.
The first VPR 12 and second VPR 13 were controlled independently by a computer 102 through a controller unit 101 which generates a sequential of voltages to control the phase retardations of VPRs 12 and 13. A set of four intensity images, I0, I1, I2 and I3 of the sample were captured by imaging sensor 15 when the phase retardations of VPRs 12 and 13 were set at (0,0), (π,0), (π,π) and (π/2,π), respectively Then four Stokes components of a Stokes image, S0, S1, S2 and S3 can be calculated by computer 102 according to equation (1).
Referring to FIG. 6 is a schematic view of a polarization imaging apparatus shown in FIG. 5 configured in a transmission configuration.
It will be apparent to those skilled in the art that various modifications and variations can be made to the present invention without departing from the spirit and scope of the invention. Thus it is intended that the present invention cover the modifications and variations of this invention provided they come within the scope of the appended claims and their equivalents.

## Claims

1. A polarization imaging apparatus for measuring polarization images of a sample comprising:
an optical assembly;
a first variable phase retarder having a first controllable phase retardation;
a second variable phase retarder optically coupled to the first variable phase retarder, the second variable phase retarder having a second controllable phase retardation;
a linear polarizer optically coupled to the second variable phase retarder;
an imaging sensor optically coupled to the polarizer for sensing the intensity images of the sample through the linear polarizer;
a controller for varying the first controllable phase retardation of the first variable phase retarder and for varying the second controllable phase retardation of the second variable phase retarders; and
a computer for controlling the controller, recording the sequential intensity images, calibrating the alignment errors of the first and second variable phase retarders, and converting four sequential intensity images to four components of a Stokes image,
wherein the computer is configured to calibrate the alignment errors of first and second phase retarders using one known input polarization status and a set of sequential intensity images when setting the first and second variable phase retarders at different pairs of phase retardations.

2. The known input polarization status of claim 1 is a linear polarization.

3. The known input polarization status of claim 1 is a circular polarization.

4. The known input polarization status of claim 1 is an elliptical polarization.

5. A set of sequential intensity images of claim 1 wherein the images are measured while the first and second variable phase retarders having 5 different sets of phase retardations.

6. A set of sequential intensity images of claim 1 wherein the images are measured while the first and second variable phase retarders having phase retardations of (π/2,0), (π,0), (0,π), (π,π) and (π/2,π), respectively.

7. A method to measure Stokes images of a sample, the method comprising the steps of:
orienting a first variable phase retarder approximately at a 22.5-degree angle with respect to the linear polarizer;
orienting a second variable phase retarder approximately at a 45-degree angle with respect to the linear polarizer;
illuminating the sample with light having a known polarization status;
sensing a sequential of five intensity images with five sets of phase retardations while varying the first controllable phase retardation of the first variable phase retarder and varying the second controllable phase retardation of the second variable phase retarders;
calculating the optical alignment errors of the first variable phase retarder and the second variable phase retarders;
sensing a sequential of four intensity images with four sets of phase retardations while varying the first controllable phase retardation of the first variable phase retarder and varying the second controllable phase retardation of the second variable phase retarders; and
converting a sequential of four intensity images to four components of a Stokes image.

8. The method in claim 7 wherein the sample is illuminated by a linear polarized light.

9. The method in claim 7 wherein the sample is illuminated by a circular polarized light.

10. The method in claim 7 wherein the sample is illuminated by an elliptical polarized light.

11. The method in claim 7, wherein alignment errors of first and second phase retarders are calibrated using the five sets of phase retardations of first and second variable phase retarders having the phase retardations of (π/2,0), (π,0), (0,π), (π,π) and (π/2,π), respectively.

12. The method in claim 7 further comprising the steps of:
illuminating the sample with a light source;
calibrating the alignment error and phase retardation error of the phase retarders;
adjusting the light source polarization state;
changing the light source wavelength spectrum;
acquiring the intensity images; and
converting to Stokes images.