# Patent

## Title
Reconfigurable spectroscopic ellipsometer

## Abstract

A Mueller ellipsometer of the type having a first rotating element on an incident beam side of a sample and a second rotating element on a reflected beam side of the sample and a detector having an integration time, having a controller for selectively and separately adjusting (1) a first angular frequency of the first rotating element and (2) a second angular frequency of the second rotating element.

## Background
FIELD
This invention relates to the field of spectroscopy. More particularly, this invention relates to measuring critical dimensions of integrated circuits using an adaptation of a Mueller ellipsometer.
INTRODUCTION
Spectroscopic ellipsometry is used extensively in integrated circuit metrology and process control, such as for measuring the thicknesses of film stacks and the critical dimensions of integrated circuit structures during etch, lithography, and other fabrication processes. As the term is used herein, “integrated circuit” includes devices such as those formed on monolithic semiconducting substrates, such as those formed of group IV materials like silicon or germanium, or group III-V compounds like gallium arsenide, or mixtures of such materials. The term includes all types of devices formed, such as memory and logic, and all designs of such devices, such as MOS and bipolar. The term also comprehends applications such as flat panel displays, solar cells, and charge coupled devices.
One of the most critical challenges is the low sensitivity caused by the ever-reducing dimensions of the structures being measured. For example, the lateral dimensions of gate structures have been continuously reduced, typically down to a few tens of nanometers, while at the same time there remains a critical need to maintain extremely high measurement precision and tool-to-tool matching.
Mueller ellipsometry is often used, where the sample is described by a 4-by-4 matrix, where each of the elements in the matrix is a set of spectra.
Commonly-used ellipsometers include a light source that reflects a light beam off of a sample and into a detector. Between the light source and the sample there is a polarizer. Between the sample and the detector, there is an analyzer. Either or both of polarizer and the analyzer can rotate.
A rotating quarter-wave plate (or alternately a photoelastic modulator, acousto-optic modulator, liquid-crystal modulator, or other polarization-sensitive phase modulation means) is optionally placed between the sample and one or both of the polarizer and the analyzer. These elements are commonly called compensators. Typically, only one element on either side of the sample is rotated during a measurement.
Different ones of these configurations produce different numbers of harmonic spectra, where some produce a sufficient number of harmonic spectra to completely populate the Mueller matrix. However, additional improvements to a Mueller spectroscopic ellipsometer are desirable.
SUMMARY OF THE CLAIMS
The above and other needs are met by a Mueller ellipsometer of the type having a first rotating element on an incident beam side of a sample and a second rotating element on a reflected beam side of the sample and a detector having an integration time, having a controller for selectively and separately adjusting (1) a first angular frequency of the first rotating element and (2) a second angular frequency of the second rotating element.
In various embodiments, the first rotating element is a polarizer In other embodiments the first rotating element is a compensator disposed between a polarizer and the sample along the incident beam In some embodiments the second rotating element is an analyzer In other embodiments the second rotating element is a compensator disposed between the sample and the analyzer along the reflected beam In some embodiments the controller additionally selectively and separately adjusts the integration time In some embodiments the ellipsometer is a spectroscopic ellipsometer. In other embodiments the ellipsometer is an angle resolved ellipsometer.



BRIEF DESCRIPTION OF THE DRAWING
Further advantages of the invention are apparent by reference to the detailed description when considered in conjunction with the FIGURE, which is not to scale so as to more clearly show the details, wherein like reference numbers indicate like elements throughout the view, and which depicts a basic exemplary embodiment of a spectroscopic ellipsometer according to an embodiment of the present invention.



DETAILED DESCRIPTION
According to various embodiments of the present invention, there is provided an ellipsometer having two rotating polarizing elements (one on each side of the sample), where the angular frequencies of the two rotating polarizing elements are separately adjustable. These two parameters, coupled with the acquisition time of the detector that is integrating the signals, determines the magnitudes of the harmonics that are produced, and the degree to which the Mueller matrix is populated.
With reference now to the FIGURE, there is depicted a basic exemplary embodiment of a spectroscopic ellipsometer 100 for taking measurements at a position 110 on a sample 108. A broadband light source 114 produces an incident beam 112 that passes through a polarizer 120 and an optional compensator means 124, before acquiring the sample 108. The light reflected from the sample 108 comprises a reflected beam 102 that passes through optional compensator means 126 and analyzer 118 before attaining the detector 104. A controller 128 is operable to selectively adjust the angular frequencies of the appropriate one or more of the elements 120, 124, 126, and 118, and the integration time of the detector 104.
As an example, consider an ellipsometer where the polarizer and the analyzer rotate at frequencies fp and fa (in hertz) with angular frequencies of [ωp, ωa], respectively. For a sample described by a Mueller matrix Mij, i, j=0, 1, 2, 3, propagating the stokes vector from the light source to the detector yields the detector signal:




  
    
      
        
          
            S
            D
          
          =
          
            
              S
              F
            
            ⁡
            
              [
              
                
                  
                    
                      
                        f
                        0
                      
                      +
                      
                        
                          f
                          1
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            
                              ω
                              a
                            
                            -
                            
                              ω
                              p
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          2
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            
                              ω
                              a
                            
                            -
                            
                              ω
                              p
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          3
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          ω
                          p
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          4
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          ω
                          p
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          5
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          ω
                          a
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          6
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          ω
                          a
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          7
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            
                              ω
                              a
                            
                            +
                            
                              ω
                              p
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          8
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            
                              ω
                              a
                            
                            +
                            
                              ω
                              p
                            
                          
                          )
                        
                        ⁢
                        t
                      
                    
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        1
        )
      
    
  




where the harmonic coefficients fn, n=0, 1, . . . , 8 are directly related to the sample Mueller elements:




  
    
      
        
          
            S
            F
          
          =
          
            
              I
              i
            
            4
          
        
        ⁢
        
          

        
        ⁢
        
          
            f
            0
          
          =
          
            M
            00
          
        
        ⁢
        
          

        
        ⁢
        
          
            f
            1
          
          =
          
            
              
                M
                11
              
              +
              
                M
                22
              
            
            2
          
        
        ⁢
        
          

        
        ⁢
        
          
            f
            2
          
          =
          
            
              
                M
                12
              
              -
              
                M
                21
              
            
            2
          
        
        ⁢
        
          

        
        ⁢
        
          
            f
            3
          
          =
          
            M
            01
          
        
        ⁢
        
          

        
        ⁢
        
          
            f
            4
          
          =
          
            -
            
              M
              02
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            f
            5
          
          =
          
            M
            10
          
        
        ⁢
        
          

        
        ⁢
        
          
            f
            6
          
          =
          
            -
            
              M
              20
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            f
            7
          
          =
          
            
              
                M
                11
              
              -
              
                M
                22
              
            
            2
          
        
        ⁢
        
          

        
        ⁢
        
          
            f
            8
          
          =
          
            -
            
              
                
                  
                    M
                    12
                  
                  +
                  
                    M
                    21
                  
                
                2
              
              .
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




It is generally preferable to integrate the detector signal at a certain time period, accumulating all of the photons collected during this period of time, and recording the corresponding photoelectron signals as the so-called “sums.” Various embodiments of the present invention use “32-sum,” where for a data acquisition time of T, detector signals in each interval of T/32 are accumulated and recorded, as given by:




  
    
      
        
          
            S
            m
          
          =
          
            
              ∫
              
                mT
                /
                32
              
              
                
                  (
                  
                    m
                    +
                    1
                  
                  )
                
                ⁢
                
                  T
                  /
                  32
                
              
            
            ⁢
            
              
                
                  S
                  D
                
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              
                d
                t
              
            
          
        
        ,
      
    
    
      
        (
        3
        )
      
    
  




where m=0, 1, . . . , 31. Defining the functions as:




  
    
      
        
          cos
          ⁢
          
            
          
          ⁢
          
            c
            ⁡
            
              (
              
                x
                ,
                m
              
              )
            
          
        
        =
        
          {
          
            
              
                
                  
                    
                      
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            mTx
                          
                          -
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  m
                                  +
                                  1
                                
                                )
                              
                            
                            ⁢
                            Tx
                          
                        
                        Tx
                      
                      ,
                    
                  
                  
                    
                      x
                      ≠
                      0
                    
                  
                
                
                  
                    
                      1
                      ,
                    
                  
                  
                    
                      x
                      =
                      0
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              sin
              ⁢
              
                
              
              ⁢
              
                c
                ⁡
                
                  (
                  
                    x
                    ,
                    m
                  
                  )
                
              
            
            =
            
              {
              
                
                  
                    
                      
                        
                          -
                          
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                mTx
                              
                              -
                              
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      m
                                      +
                                      1
                                    
                                    )
                                  
                                
                                ⁢
                                Tx
                              
                            
                            Tx
                          
                        
                        ,
                      
                    
                    
                      
                        x
                        ≠
                        0
                      
                    
                  
                  
                    
                      
                        1
                        ,
                      
                    
                    
                      
                        x
                        =
                        0
                      
                    
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        4
        )
      
    
  




and applying equations (1) and (2) to equation (3) yields the sums of a rotating polarizer and analyzer configuration related to the sample and system characteristics, as:




  
    
      
        
          
            G
            mn
          
          =
          
            
              
                T
                32
              
              ⁡
              
                [
                
                  
                    
                      
                        1
                        ,
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      a
                                    
                                    -
                                    
                                      ω
                                      p
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                        ,
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      a
                                    
                                    -
                                    
                                      ω
                                      p
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                        ,
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    ω
                                    p
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                        ,
                      
                    
                  
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    ω
                                    p
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                        ,
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    ω
                                    a
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                        ,
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    ω
                                    a
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                        ,
                      
                    
                  
                  
                    
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      a
                                    
                                    +
                                    
                                      ω
                                      p
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                        ,
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      a
                                    
                                    +
                                    
                                      ω
                                      p
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                      
                    
                  
                
                ]
              
            
            T
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              f
              n
            
            =
            
              
                [
                
                  
                    f
                    0
                  
                  ,
                  
                    f
                    1
                  
                  ,
                  
                    f
                    2
                  
                  ,
                  
                    f
                    3
                  
                  ,
                  
                    f
                    4
                  
                  ,
                  
                    f
                    5
                  
                  ,
                  
                    f
                    6
                  
                  ,
                  
                    f
                    7
                  
                  ,
                  
                    
                  
                  ⁢
                  
                    f
                    8
                  
                
                ]
              
              T
            
          
          ,
          
            

          
          ⁢
          
            m
            =
            0
          
          ,
          2
          ,
          ⋯
          ⁢
          
            
          
          ,
          
            31
            ;
            
              
            
            ⁢
            
              n
              =
              0
            
          
          ,
          1
          ,
          ⋯
          ⁢
          
            
          
          ,
          
            
              8
              ⁢
              
                
              
              ⁢
              S
            
            =
            
              Gf
              .
            
          
        
      
    
    
      
        (
        5
        )
      
    
  




The linear equations Gf=S provides nine harmonic coefficients from 32 sums signals. This is an over-determined problem and one method to solve it is the so-called singular value decomposition method. Solutions of these linear equations indicate that the magnitudes of these harmonics are related to certain system parameters, namely, the angular frequencies and the acquisition time.
For example, a dual-rotating element ellipsometer having an acquisition time of one second is seen to exhibit different measurement capabilities (as measured by the standard deviation of repeated measurements), when using rotational frequencies of [2, 3] hertz in one case and [4, 5] hertz in another case for the polarizer and analyzer, respectively. Which frequency pair, and the exact values of the desired frequencies, will differ according to the specific configuration of the ellipsometer (which elements are present and rotating, etc.) and the material and structure of the sample being measured.
The above method can be applied to other Mueller spectroscopic ellipsometers with dual rotating polarizing elements. For instance, for a dual rotating compensator ellipsometer, one can express the detector signal as 25 harmonics related to the rotating frequencies of the compensators in the illumination side (ωg) and in the analyzer side (ωq), as shown below:




  
    
      
        
          S
          D
        
        =
        
          
            
              S
              F
            
            ⁡
            
              [
              
                
                  
                    
                      
                        f
                        0
                      
                      +
                      
                        
                          f
                          1
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            
                              ω
                              g
                            
                            -
                            
                              ω
                              q
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          2
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            
                              ω
                              g
                            
                            -
                            
                              ω
                              q
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          3
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        4
                        ⁢
                        
                          (
                          
                            
                              ω
                              g
                            
                            -
                            
                              ω
                              q
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          4
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        4
                        ⁢
                        
                          (
                          
                            
                              ω
                              g
                            
                            -
                            
                              ω
                              q
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          5
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          ω
                          g
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          6
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          ω
                          g
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          7
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          ω
                          q
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          8
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          ω
                          q
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          9
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  ω
                                  g
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  ω
                                  q
                                
                              
                            
                            )
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          10
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  ω
                                  g
                                
                              
                              -
                              
                                2
                                ⁢
                                
                                  ω
                                  q
                                
                              
                            
                            )
                          
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          11
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ω
                                  g
                                
                              
                              -
                              
                                4
                                ⁢
                                
                                  ω
                                  q
                                
                              
                            
                            )
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          12
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ω
                                  g
                                
                              
                              -
                              
                                4
                                ⁢
                                
                                  ω
                                  q
                                
                              
                            
                            )
                          
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          13
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        4
                        ⁢
                        
                          ω
                          g
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          14
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        4
                        ⁢
                        
                          ω
                          g
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          15
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        4
                        ⁢
                        
                          ω
                          q
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          16
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        4
                        ⁢
                        
                          ω
                          q
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          17
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            
                              ω
                              g
                            
                            +
                            
                              ω
                              q
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          18
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            
                              ω
                              g
                            
                            +
                            
                              ω
                              q
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          19
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  ω
                                  g
                                
                              
                              +
                              
                                2
                                ⁢
                                
                                  ω
                                  q
                                
                              
                            
                            )
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          20
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ω
                                  g
                                
                              
                              +
                              
                                2
                                ⁢
                                
                                  ω
                                  q
                                
                              
                            
                            )
                          
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          21
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ω
                                  g
                                
                              
                              +
                              
                                4
                                ⁢
                                
                                  ω
                                  q
                                
                              
                            
                            )
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          22
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ω
                                  g
                                
                              
                              +
                              
                                4
                                ⁢
                                
                                  ω
                                  q
                                
                              
                            
                            )
                          
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          23
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        4
                        ⁢
                        
                          (
                          
                            
                              ω
                              g
                            
                            +
                            
                              ω
                              q
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          24
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        4
                        ⁢
                        
                          (
                          
                            
                              ω
                              g
                            
                            +
                            
                              ω
                              q
                            
                          
                          )
                        
                        ⁢
                        t
                      
                    
                  
                
              
              ]
            
          
          .
        
      
    
    
      
        (
        6
        )
      
    
  




Similarly, one can relate these harmonic coefficients to 32-sum:




  
    
      
        
          
            
              G
              mn
            
            =
            
              
                
                  T
                  32
                
                ⁡
                
                  [
                  
                    
                      
                        
                          1
                          ,
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              c
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        ω
                                        g
                                      
                                      -
                                      
                                        ω
                                        q
                                      
                                    
                                    16
                                  
                                  ,
                                  m
                                
                                )
                              
                            
                          
                          ,
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              c
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        ω
                                        g
                                      
                                      -
                                      
                                        ω
                                        q
                                      
                                    
                                    16
                                  
                                  ,
                                  m
                                
                                )
                              
                            
                          
                          ,
                        
                      
                    
                    
                      
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      g
                                    
                                    -
                                    
                                      ω
                                      q
                                    
                                  
                                  8
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              c
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        ω
                                        g
                                      
                                      -
                                      
                                        ω
                                        q
                                      
                                    
                                    8
                                  
                                  ,
                                  m
                                
                                )
                              
                            
                          
                          ,
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    ω
                                    g
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                        
                      
                    
                    
                      
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    ω
                                    g
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    ω
                                    q
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    ω
                                    q
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                        
                      
                    
                    
                      
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    
                                      2
                                      ⁢
                                      
                                        ω
                                        g
                                      
                                    
                                    -
                                    
                                      ω
                                      q
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              c
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        2
                                        ⁢
                                        
                                          ω
                                          g
                                        
                                      
                                      -
                                      
                                        ω
                                        q
                                      
                                    
                                    16
                                  
                                  ,
                                  m
                                
                                )
                              
                            
                          
                          ,
                        
                      
                    
                    
                      
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      g
                                    
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        ω
                                        q
                                      
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              c
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        ω
                                        g
                                      
                                      -
                                      
                                        2
                                        ⁢
                                        
                                          ω
                                          q
                                        
                                      
                                    
                                    16
                                  
                                  ,
                                  m
                                
                                )
                              
                            
                          
                          ,
                        
                      
                    
                    
                      
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    ω
                                    g
                                  
                                  8
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    ω
                                    g
                                  
                                  8
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                        
                      
                    
                    
                      
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    ω
                                    q
                                  
                                  8
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    ω
                                    q
                                  
                                  8
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                        
                      
                    
                    
                      
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      g
                                    
                                    +
                                    
                                      ω
                                      q
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              c
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        ω
                                        g
                                      
                                      +
                                      
                                        ω
                                        q
                                      
                                    
                                    16
                                  
                                  ,
                                  m
                                
                                )
                              
                            
                          
                          ,
                        
                      
                    
                    
                      
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    
                                      2
                                      ⁢
                                      
                                        ω
                                        g
                                      
                                    
                                    +
                                    
                                      ω
                                      q
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              c
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        2
                                        ⁢
                                        
                                          ω
                                          g
                                        
                                      
                                      +
                                      
                                        ω
                                        q
                                      
                                    
                                    16
                                  
                                  ,
                                  m
                                
                                )
                              
                            
                          
                          ,
                        
                      
                    
                    
                      
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      g
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        ω
                                        q
                                      
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              c
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        ω
                                        g
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          ω
                                          q
                                        
                                      
                                    
                                    16
                                  
                                  ,
                                  m
                                
                                )
                              
                            
                          
                          ,
                        
                      
                    
                    
                      
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            c
                            ⁢
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      g
                                    
                                    +
                                    
                                      ω
                                      q
                                    
                                  
                                  8
                                
                                ,
                                m
                              
                              )
                            
                          
                          ,
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              c
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        ω
                                        g
                                      
                                      +
                                      
                                        ω
                                        q
                                      
                                    
                                    8
                                  
                                  ,
                                  m
                                
                                )
                              
                            
                          
                        
                      
                    
                  
                  ]
                
              
              T
            
          
          ⁢
          
            

          
          ⁢
          
            f
            n
          
          =
          
            
              [
              
                
                  f
                  0
                
                ,
                
                  f
                  1
                
                ,
                
                  f
                  2
                
                ,
                
                  f
                  3
                
                ,
                
                  f
                  4
                
                ,
                
                  f
                  5
                
                ,
                
                  f
                  6
                
                ,
                
                  f
                  7
                
                ,
                
                  f
                  8
                
                ,
                
                  f
                  9
                
                ,
                
                  f
                  10
                
                ,
                
                  f
                  11
                
                ,
                
                  f
                  12
                
                ,
                
                  

                
                ⁢
                
                  f
                  13
                
                ,
                
                  f
                  14
                
                ,
                
                  f
                  15
                
                ,
                
                  f
                  16
                
                ,
                
                  f
                  17
                
                ,
                
                  f
                  18
                
                ,
                
                  f
                  19
                
                ,
                
                  f
                  20
                
                ,
                
                  f
                  21
                
                ,
                
                  f
                  22
                
                ,
                
                  f
                  23
                
                ,
                
                  f
                  24
                
              
              ]
            
            T
          
        
        ,
        
          

        
        ⁢
        
          m
          =
          0
        
        ,
        2
        ,
        ⋯
        ⁢
        
          
        
        ,
        
          31
          ;
          
            
          
          ⁢
          
            n
            =
            0
          
        
        ,
        1
        ,
        ⋯
        ⁢
        
          
        
        ,
        
          
            24
            ⁢
            
              
            
            ⁢
            S
          
          =
          Gf
        
      
    
    
      
        (
        7
        )
      
    
  




The linear equations Gf=S determine 25 harmonic coefficients from 32 sums signals. This is an over-determined problem and again, one method to solve it is the singular value decomposition method. Solutions of these linear equations indicate that the magnitudes of these harmonics are related to certain system parameters, namely, the angular frequencies and the acquisition time.
Similarly, for the various other dual-rotating element embodiments, one can simply apply the same method to relate the harmonic coefficients to the 32-sum, and find that the harmonic coefficients can be advantageously reconfigured by changing the rotation frequencies and the integration time. As further example, a rotating compensator—rotating analyzer system produces fifteen harmonics, as given by:




  
    
      
        
          S
          D
        
        =
        
          
            
              S
              F
            
            ⁡
            
              [
              
                
                  
                    
                      
                        f
                        0
                      
                      +
                      
                        
                          f
                          1
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            
                              ω
                              a
                            
                            -
                            
                              ω
                              p
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          2
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            
                              ω
                              a
                            
                            -
                            
                              ω
                              p
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          3
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ω
                                  a
                                
                              
                              -
                              
                                4
                                ⁢
                                
                                  ω
                                  p
                                
                              
                            
                            )
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          4
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ω
                                  a
                                
                              
                              -
                              
                                ω
                                p
                              
                            
                            )
                          
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          5
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          ω
                          p
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          6
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          ω
                          p
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          7
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          ω
                          a
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          8
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          ω
                          a
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          9
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            
                              ω
                              a
                            
                            +
                            
                              ω
                              p
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          10
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          (
                          
                            
                              ω
                              a
                            
                            +
                            
                              ω
                              p
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          11
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        4
                        ⁢
                        
                          ω
                          p
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          12
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        4
                        ⁢
                        
                          ω
                          p
                        
                        ⁢
                        t
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          f
                          13
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ω
                                  a
                                
                              
                              +
                              
                                4
                                ⁢
                                
                                  ω
                                  p
                                
                              
                            
                            )
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          f
                          14
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  ω
                                  a
                                
                              
                              +
                              
                                4
                                ⁢
                                
                                  ω
                                  p
                                
                              
                            
                            )
                          
                        
                        ⁢
                        t
                      
                    
                  
                
              
              ]
            
          
          .
        
      
    
    
      
        (
        8
        )
      
    
  




The relation between the harmonic coefficients and the 32-sum can be determined in a similar way as:




  
    
      
        
          
            G
            mn
          
          =
          
            
              
                T
                32
              
              ⁡
              
                [
                
                  
                    
                      
                        1
                        ,
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      a
                                    
                                    -
                                    
                                      ω
                                      g
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                        ,
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      a
                                    
                                    -
                                    
                                      ω
                                      g
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                        ,
                      
                    
                  
                  
                    
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          c
                          ⁢
                          
                            (
                            
                              
                                
                                  
                                    ω
                                    a
                                  
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      ω
                                      g
                                    
                                  
                                
                                16
                              
                              ,
                              m
                            
                            )
                          
                        
                        ,
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      a
                                    
                                    -
                                    
                                      ω
                                      g
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                        ,
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          c
                          ⁢
                          
                            (
                            
                              
                                
                                  ω
                                  p
                                
                                16
                              
                              ,
                              m
                            
                            )
                          
                        
                        ,
                      
                    
                  
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          c
                          ⁢
                          
                            (
                            
                              
                                
                                  ω
                                  p
                                
                                16
                              
                              ,
                              m
                            
                            )
                          
                        
                        ,
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          c
                          ⁢
                          
                            (
                            
                              
                                
                                  ω
                                  a
                                
                                16
                              
                              ,
                              m
                            
                            )
                          
                        
                        ,
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          c
                          ⁢
                          
                            (
                            
                              
                                
                                  ω
                                  a
                                
                                16
                              
                              ,
                              m
                            
                            )
                          
                        
                        ,
                      
                    
                  
                  
                    
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          c
                          ⁢
                          
                            (
                            
                              
                                
                                  
                                    ω
                                    a
                                  
                                  +
                                  
                                    ω
                                    g
                                  
                                
                                16
                              
                              ,
                              m
                            
                            )
                          
                        
                        ,
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      a
                                    
                                    +
                                    
                                      ω
                                      g
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                        ,
                      
                    
                  
                  
                    
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          c
                          ⁢
                          
                            (
                            
                              
                                
                                  ω
                                  p
                                
                                8
                              
                              ,
                              m
                            
                            )
                          
                        
                        ,
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    ω
                                    p
                                  
                                  8
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                        ,
                      
                    
                  
                  
                    
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          c
                          ⁢
                          
                            (
                            
                              
                                
                                  
                                    ω
                                    a
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      ω
                                      p
                                    
                                  
                                
                                16
                              
                              ,
                              m
                            
                            )
                          
                        
                        ,
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      ω
                                      a
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        ω
                                        p
                                      
                                    
                                  
                                  16
                                
                                ,
                                m
                              
                              )
                            
                          
                        
                      
                    
                  
                
                ]
              
            
            T
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              f
              n
            
            =
            
              
                [
                
                  
                    f
                    0
                  
                  ,
                  
                    f
                    1
                  
                  ,
                  
                    f
                    2
                  
                  ,
                  
                    f
                    3
                  
                  ,
                  
                    f
                    4
                  
                  ,
                  
                    f
                    5
                  
                  ,
                  
                    f
                    6
                  
                  ,
                  
                    f
                    7
                  
                  ,
                  
                    f
                    8
                  
                  ,
                  
                    f
                    9
                  
                  ,
                  
                    f
                    10
                  
                  ,
                  
                    f
                    11
                  
                  ,
                  
                    f
                    12
                  
                  ,
                  
                    f
                    13
                  
                  ,
                  
                    f
                    14
                  
                
                ]
              
              T
            
          
          ,
          
            

          
          ⁢
          
            m
            =
            0
          
          ,
          2
          ,
          ⋯
          ⁢
          
            
          
          ,
          
            31
            ;
            
              
            
            ⁢
            
              n
              =
              0
            
          
          ,
          1
          ,
          ⋯
          ⁢
          
            
          
          ,
          
            
              14
              ⁢
              
                
              
              ⁢
              S
            
            =
            Gf
          
        
      
    
    
      
        (
        9
        )
      
    
  




For a rotating polarizer—rotating compensator system, the above harmonic analysis still applies, one just needs to exchange frequencies ωa and ωp in equations (6) and (7). Finally, for a system with rotating compensators on both sides of the sample, one will obtain 25 harmonics, and can apply equations (6) and (7) to relate 32-sum to harmonic coefficients, as already described.
It is noted that in the above three embodiments, even though one can get fifteen to twenty-five harmonics, one can still only measure nine Mueller elements. All the methods described and contemplated herein can be applied to both spectroscopic and angle-resolved systems. In a basic embodiment, one method identifies the harmonic or harmonics with the highest sensitivity for the system and sample combination. This harmonic or these harmonics are identified as the optimum harmonic or harmonics, and the angular frequencies of the rotating elements are adjusted accordingly, to enhance the optimum harmonic or harmonics. The angular frequency and the integration time can be adjusted in the system 100 either by user input as determined such as by empirically or modeled, or automatically such as by system 100 configuration.
The foregoing description of embodiments for this invention has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Obvious modifications or variations are possible in light of the above teachings. The embodiments are chosen and described in an effort to provide illustrations of the principles of the invention and its practical application, and to thereby enable one of ordinary skill in the art to utilize the invention in various embodiments and with various modifications as are suited to the particular use contemplated. All such modifications and variations are within the scope of the invention as determined by the appended claims when interpreted in accordance with the breadth to which they are fairly, legally, and equitably entitled.

## Claims

1. In a Mueller ellipsometer of the type having a first rotating element on an incident beam side of a sample and a second rotating element on a reflected beam side of the sample and a detector having an integration time, the improvement comprising a controller for selectively and separately adjusting (1) a first angular frequency of the first rotating element and (2) a second angular frequency of the second rotating element.

2. The ellipsometer of claim 1, wherein the first rotating element is a polarizer.

3. The ellipsometer of claim 1, wherein the first rotating element is a compensator disposed between a polarizer and the sample along the incident beam.

4. The ellipsometer of claim 1, wherein the second rotating element is an analyzer.

5. The ellipsometer of claim 1, wherein the second rotating element is a compensator disposed between the sample and the analyzer along the reflected beam.

6. The ellipsometer of claim 1, wherein the controller additionally selectively and separately adjusts the integration time.

7. The ellipsometer of claim 1, wherein the ellipsometer is a spectroscopic ellipsometer.

8. The ellipsometer of claim 1, wherein the ellipsometer is an angle resolved ellipsometer.