Abstract:
An ellipsometer for determining thickness and ellipsometric parameters (Ψ and Δ) of a thin film material. The apparatus includes a light source emitting light, a transmitting optical system that has a polarizer, modulator and an optical compensator for conveying polarized modulated light for incidence on a film, and a receiving optical system that has an analyzer and conveys the reflected light to a photodetector device. The apparatus is used for full range measurement of ellipsometric parameters by applying two-phase detection method. It also determines thickness of thin films with a high degree of accuracy.

Description:
FIELD 
       [0001]    Exemplary embodiments relate to ellipsometry, and, more particularly, to a system and method for determining ellipsometric parameters and thickness of thin films. 
       BACKGROUND 
       [0002]    Ellipsometry is a related art optical technique that uses polarized light to probe the properties of a sample. Ellipsometry has applications in many different fields which may include but are not limited to semiconductor physics, microelectronics and biology, and further applications that may pertain to but are not limited to basic research and industrial applications. Ellipsometry is a sensitive measurement technique and provides unequalled capabilities for thin film metrology. To be useful, the measurement system must be able to determine the thickness of films with a high degree of accuracy. 
         [0003]    In related art elipsometric methods, incident polarized light is made to fall on the film whose thickness has to be measured and the reflected light is received by the photodetector for phase detection and calculation of ellipsometric parameters and thickness of film. Errors in measurement result from slight imperfections in the optical elements within the measurement system, or may be caused by misalignment errors when constructing the measurement platform. If the individual optical elements of the modulated ellipsometry are not perfectly aligned, the resulting elliptical polarized state of the light cause phase errors. Moreover, the non-perfect sinusoidal retardance variation of the modulator may induce errors in the subsequent demodulation signal processing procedure. 
         [0004]    Further, these related art methods depend on amplitude and/or intensity of the detected signal for determination of thickness of thin films. The amplitude and/or intensity is/are affected by environmental disturbances and optical misalignments. Thus, using these related methods would not be suitable for accurate determination of ellipsometric parameters and thickness of thin film. 
         [0005]    Thus, there is an unmet need for an improved system and method for accurate determination of ellipsometric parameters and thickness of thin films. 
       SUMMARY 
       [0006]    Aspects of the exemplary embodiments relate to measuring thickness of thin film with high accuracy using a modulated ellipsometric apparatus. 
         [0007]    It is an object of the exemplary embodiments to achieve full range-measurement of ellipsometric parameters with high accuracy using a two-phase detection method. 
         [0008]    It is another object of the exemplary embodiments to determine properties including but not limited to stress, strain, thickness, refractive indices, dielectric constants, magneto-optical parameters. 
         [0009]    It is still another object of the exemplary embodiments to measure optical material properties including but not limited to pre-tilt angle, tilt angle, azimuth angle and phase retardation of liquid crystal displays and birefringence materials. 
         [0010]    It is still another object of the exemplary embodiments to provide spectroscopic, in-situ, image measurement for isotropic multilayer material. 
         [0011]    It is still another object of the exemplary embodiments to provide dynamic measurement of properties of sample including but not limited to measuring properties of sample when sample is vibrating. 
         [0012]    It is still another object of the exemplary embodiments to provide a modulated ellipsometer for thin film thickness measurement. The modulated ellipsometer comprising: a light source; a polarizer for polarizing the light received from the light source; a modulator for modulating the polarized light; an optical compensator for altering the polarization state of light received from the modulator and directing light onto the sample; an analyzer for polarizing the reflected light received from the sample; a detector for receiving light from the analyzer for two-phase detection corresponding to two different orientations of the optical compensator. 
         [0013]    It is still another object of the exemplary embodiments to provide a method for thin film thickness measurement. The method comprising: emitting light from a light source; polarizing the light from the light source; modulating the polarized light; altering the polarization state of the light received from the modulator and directing the light onto the sample; polarizing the reflected light received from the sample using an analyzer; receiving the light from the analyzer for two-phase detection corresponding to two different orientations of the optical compensator. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0014]      FIG. 1  shows the ellipsometric apparatus  100  in accordance with an exemplary embodiment; 
           [0015]      FIG. 2  illustrates a first exemplary configuration of the ellipsometric apparatus  100 ; 
           [0016]      FIG. 3  illustrates a second exemplary configuration of the ellipsometric apparatus  100 ; 
           [0017]      FIG. 4  illustrates an exemplary optical model for an ambient/thin film/substrate structure; 
           [0018]      FIG. 5  is a graph showing an exemplary ellipsometric parameter Ψ result of the simulation; 
           [0019]      FIG. 6  is a graph showing an exemplary ellipsometric parameter Δ result of the simulation; 
           [0020]      FIG. 7  is a graph showing an exemplary first phase Φ 1  result of the simulation; 
           [0021]      FIG. 8  is a graph showing an exemplary second phase Φ 2  result of the simulation; 
           [0022]      FIG. 9  is a graph showing an exemplary ellipsometric parameter Ψ result by the inversed calculation; 
           [0023]      FIG. 10  is a graph showing an exemplary ellipsometric parameter Δ result by the inversed calculation; 
           [0024]      FIG. 11  is a graph showing a thickness result by the inversed calculation; 
           [0025]      FIG. 12  is a graph showing a correlation between input Ψ and extracted Ψ when θ i  Error=±0.01°; 
           [0026]      FIG. 13  is a graph showing a correlation between input Δ and extracted Δ when θ i  Error=±0.01°; 
           [0027]      FIG. 14  is a graph showing a correlation between input thickness and extracted thickness in the case when θ i  Error=±0.01°; 
           [0028]      FIG. 15  is a graph showing a correlation between input  4 ′ and extracted IP when Phase Error=±0.01°; 
           [0029]      FIG. 16  is a graph showing a correlation between input A and extracted A when Phase Error=±0.01°; 
           [0030]      FIG. 17  is a graph showing a correlation between input thickness and extracted thickness when Phase Error=±0.01°; 
           [0031]      FIG. 18  is a graph showing a correlation between input Ψ and extracted Ψ when θ i  Error=±0.01 and the Phase Error=±0.01°; 
           [0032]      FIG. 19  is a graph showing a correlation between input Δ and extracted Δ when θ i  Error=±0.01 and the Phase Error=±0.01*; 
           [0033]      FIG. 20  is a graph showing a correlation between input thickness and extracted thickness when θ i  Error=±0.01 and the Phase Error=±0.01°; 
           [0034]      FIG. 21  is a graph showing experimental results of measured Φ 1  (deg); 
           [0035]      FIG. 22  is a graph showing the correlation of simulated Φ 1  and measured Φ 1 ; 
           [0036]      FIG. 23  is a graph showing experimental results of measured Φ 2  (deg); 
           [0037]      FIG. 24  is a graph showing the correlation of simulated Φ 2  and measured Φ 2 ; 
           [0038]      FIG. 25  is a graph showing experimental results of measured Ψ (deg); 
           [0039]      FIG. 26  is a graph showing the correlation of simulated Ψ and measured Ψ; 
           [0040]      FIG. 27  is a graph showing experimental results of measured Δ (deg); 
           [0041]      FIG. 28  is a graph showing the correlation of simulated Δ and measured Δ; 
           [0042]      FIG. 29  is a graph showing experimental results of measured thickness (nm); and 
           [0043]      FIG. 30  is a graph showing the correlation of measured thickness and known thickness. 
       
    
    
     DETAILED DESCRIPTION 
       [0044]    Disclosed herein is an improved method and apparatus for ellipsometry that will aid in the measurement and characterization of thin films. Numerous specific details are provided such as examples of components and/or mechanisms to provide a thorough understanding of the various exemplary embodiments. One skilled in the relevant art will recognize however, that an exemplary embodiment can be practiced without one or more of the specific details, or with other apparatus, systems, assemblies, methods, components, materials, parts, and/or the like. In other instances, well-known structures, materials or operations are not specifically shown or described in detail to avoid obscuring aspects of exemplary embodiments and for the sake of clarity. 
         [0045]      FIG. 1  illustrates an apparatus  100  for measuring thickness of a sample  106  in accordance with an exemplary embodiment. The apparatus  100  includes a light source  101 , a filter  102 , a polarizer  103 , an electro-optic (EO) modulator  104 , an optical compensator  105 , the sample  106 , an analyzer  107 , a lens  108 , a photo-detector  109 , a lock-in amplifier  110  and a function generator  111 . The sample  106  can be a static or a vibrating thin film. 
         [0046]    In an exemplary embodiment, the sample  106  comprises single-layer thin film applied on a substrate. However, the sample  106  is not limited thereto, and other samples as understood by those skilled in the art may be substituted therefore without departing from the scope of the inventive concept. 
         [0047]    In another exemplary embodiment, the sample  106  comprises multilayer isotropic thin films. 
         [0048]    In yet another exemplary embodiment, the sample  106  comprises multilayer anisotropic thin films. 
         [0049]    The light source  101  emits light onto the sample  106  for reflection. In an exemplary embodiment, the light source  101  used is He—Ne Laser. However, other light sources can also be used without departing from the scope of the present inventive concept. 
         [0050]    The light emitted from the light source  101  is polarized by the polarizer  103 . The polarized light received from the polarizer  103  enters the EO (electro-optic) modulator  104  which via the function generator  111  produces a modulated light. 
         [0051]    In an exemplary embodiment, the saw tooth signal  112  is applied to the EO modulator  104  at a frequency of 1 kHz. However, other signals at other frequencies can also be used without departing from the scope of the present inventive concept. 
         [0052]    The modulated light enters the optical compensator  105  and then onto the sample  106 . The slow axis of the optical compensator  105  is disposed at a first angle (that is, 0° with respect to x-axis in a first configuration as shown in  FIG. 2 ) and at a second angle (that is, −45° with respect to x-axis in a second configuration as shown in  FIG. 3 ). The light after reflection from the sample  106  passes through the analyzer  107  and is detected by the photo-detector  109 . The signal received by the photo-detector  109  is locked in phase by using the lock-in amplifier  110  which receives a reference signal from the function generator  111 . 
         [0053]    In an exemplary embodiment, the optical compensator  105  is a quarter-wave plate. However, other optical compensators can also be used without departing from the scope of the present inventive concept. 
         [0054]      FIG. 2  and  FIG. 3  illustrate two configurations of the apparatus  100  in accordance with exemplary embodiments.  FIG. 2  shows the first configuration in which the slow axis of the polarizer  103  and the optical compensator  105  (quarter-wave plate) are adjusted to 0° (the first angle) with respect to x-axis, and the slow axis of the analyzer  107  is adjusted to −45° with respect to x-axis.  FIG. 3  shows the second configuration in which the slow axis of the polarizer  103  is adjusted to 0°, the slow axis of the optical compensator  105  is adjusted to −45° (the second angle) with respect to x-axis, and the slow axis of the analyzer  107  is adjusted to −45°, respectively with respect to x-axis. 
         [0055]    In the first configuration of the apparatus  100 , the light vector (E 1 ) of the light emerging from the photo-detector  109  is determined by using following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                           1 
                         
                         = 
                           
                          
                         
                           
                             A 
                              
                             
                               ( 
                               
                                 - 
                                 
                                   45 
                                   ° 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             S 
                              
                             
                               ( 
                               
                                 Ψ 
                                 , 
                                 Δ 
                               
                               ) 
                             
                           
                           · 
                           
                             Q 
                              
                             
                               ( 
                               
                                 0 
                                 ° 
                               
                               ) 
                             
                           
                           · 
                           
                             EO 
                              
                             
                               ( 
                               
                                 - 
                                 
                                   45 
                                   ° 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             P 
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                               ( 
                               
                                 0 
                                 ° 
                               
                               ) 
                             
                           
                           · 
                           
                             E 
                             in 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
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                                 1 
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                                 0 
                               
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                               r 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
         [0056]    where E 0  is the amplitude of the incident electric field, P(0°) represents the Jones matrix of the polarizer  103  aligned with x-axis, Q(0°) represents the Jones matrix of the optical compensator  105 , whose slow axis is aligned with x-axis, and S(Ψ,Δ) represents the Jones matrix of the sample  106 . Furthermore, E 0 (−45°, ωt) represents the Jones matrix of the EO modulator  104  driven by a saw tooth voltage waveform with an angular frequency ω and its slow axis is oriented at −45° related to the x-axis, and A(−45°) represents the Jones matrix of the analyzer  107  whose transmission axis forms an angle −45° with the x-axis. 
         [0057]    As a result, the intensity of the detected signal is given by following equation: 
         [0000]        I   1   =I   dc (1+sin 2Ψ*cos Δ*sin ω t +(−cos 2Ψ)*cos ω t )= I   dc   +R   1  sin(ω t+Φ   1 )  (Equation 2)
 
         [0058]    where I dc =E 0   2 /4 is the dc component of the output intensity, and E 0   2  is the intensity of the input light. R 1  represents the amplitude, and Φ 1  represents the first phase. The first phase Φ 1  corresponding to the first configuration is obtained as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
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                             = 
                               
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                     ⇒ 
                     
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                       1 
                     
                   
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                      
                     
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         [0059]    In the second optical configuration of the ellipsometry apparatus  100  as shown in  FIG. 3 , the polarizer  103  is adjusted to 0°, the optical compensator  105  (quarter-wave plate) is adjusted to −45° (the second angle), and the analyzer  107  is adjusted to −45°, respectively. The light vector emerging from the configuration is determined by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                           2 
                         
                         = 
                           
                          
                         
                           
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                                   ° 
                                 
                               
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                             S 
                              
                             
                               ( 
                               
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                                 , 
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                               ) 
                             
                           
                           · 
                           
                             Q 
                              
                             
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                                   ° 
                                 
                               
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                           · 
                           
                             P 
                              
                             
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                                 0 
                                 ° 
                               
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                             in 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
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                               [ 
                               
                                 
                                   
                                     
                                       E 
                                       0 
                                     
                                   
                                 
                                 
                                   
                                     0 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 0 
                               
                                
                               t 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
         [0060]    As a result, the intensity of the detected signal is given by following equation: 
         [0000]        I   2   =I   dc [1+(cos 2Ψ)sin ω t +(−sin 2Ψ sin Δ)cos ω t]=I   dc   +R   2  sin(ω t+Φ   2 )  (Equation 5)
 
         [0061]    where I dc =E 0   2 /4 is the dc component of the output intensity, and E 0   2  is the intensity of the input light. R 2  represents the amplitude, and Φ 2  represents the second phase. The second phase Φ 2  is obtained as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               I 
                               2 
                             
                             = 
                               
                              
                             
                               
                                 I 
                                 dc 
                               
                                
                               
                                 [ 
                                 
                                   1 
                                   + 
                                   
                                     
                                       ( 
                                       
                                         cos 
                                          
                                         
                                             
                                         
                                          
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         Ψ 
                                       
                                       ) 
                                     
                                      
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     ω 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           - 
                                           sin 
                                         
                                          
                                         
                                             
                                         
                                          
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         Ψsin 
                                          
                                         
                                             
                                         
                                          
                                         Δ 
                                       
                                       ) 
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     ω 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                              
                             
                               
                                 I 
                                 dc 
                               
                               + 
                               
                                 
                                   R 
                                   2 
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         ω 
                                          
                                         
                                             
                                         
                                          
                                         t 
                                       
                                       + 
                                       
                                         Φ 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                              
                             
                               
                                 I 
                                 dc 
                               
                               + 
                               
                                 
                                   I 
                                   dc 
                                 
                                  
                                 
                                   
                                     
                                       
                                         cos 
                                         2 
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       Ψ 
                                     
                                     + 
                                     
                                       
                                         sin 
                                         2 
                                       
                                        
                                       2 
                                        
                                       
                                         Ψsin 
                                         2 
                                       
                                        
                                       Δ 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                              
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     ω 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                   + 
                                   
                                     
                                       tan 
                                       
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             - 
                                             sin 
                                           
                                            
                                           
                                               
                                           
                                            
                                           2 
                                            
                                           Ψsin 
                                            
                                           
                                               
                                           
                                            
                                           Δ 
                                         
                                         
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           Ψ 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                              
                             
                               
                                 I 
                                 dc 
                               
                               + 
                               
                                 
                                   I 
                                   dc 
                                 
                                  
                                 
                                   
                                     
                                       
                                         cos 
                                         2 
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       Ψ 
                                     
                                     + 
                                     
                                       
                                         sin 
                                         2 
                                       
                                        
                                       2 
                                        
                                       
                                         Ψsin 
                                         2 
                                       
                                        
                                       Δ 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                              
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     ω 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                   + 
                                   
                                     
                                       tan 
                                       
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           - 
                                           
                                             ( 
                                             
                                               tan 
                                                
                                               
                                                   
                                               
                                                
                                               2 
                                                
                                               
                                                   
                                               
                                                
                                               Ψ 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             sin 
                                              
                                             
                                                 
                                             
                                              
                                             Δ 
                                           
                                           ) 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                      
                     
                       
 
                     
                     ⇒ 
                     
                       Φ 
                       2 
                     
                   
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           - 
                           
                             ( 
                             
                               tan 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               Ψ 
                             
                             ) 
                           
                         
                          
                         
                           ( 
                           
                             sin 
                              
                             
                                 
                             
                              
                             Δ 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
         [0062]    As shown in Equation 3 and Equation 6, first phase Φ 1  and second phase Φ 2  are derived from the detected signal. Also, the I dc  term which is affected by the environmental noise or intensity fluctuation is eliminated in first phase Φ 1  and second phase Φ 2 . Therefore, the two-phase detection and its calculation is not dependent on amplitude and intensity. 
         [0063]    The ellipsometric parameters (Ψ, Δ) are determined by using the above calculated first phase Φ 1  and second phase Φ 2  as: 
         [0000]    
       
         
           
             
               
                 
                   Δ 
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         tan 
                          
                         
                             
                         
                          
                         
                           Φ 
                           1 
                         
                         × 
                         tan 
                          
                         
                             
                         
                          
                         
                           Φ 
                           2 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     7 
                   
                   ) 
                 
               
             
             
               
                 
                   Ψ 
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           - 
                           
                             
                               tan 
                                
                               
                                   
                               
                                
                               
                                 Φ 
                                 2 
                               
                             
                             
                               sin 
                                
                               
                                   
                               
                                
                               Δ 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     8 
                   
                   ) 
                 
               
             
           
         
       
     
         [0064]    The two phase-modulated ellipsometry described above is a full-range measurement, because the range of Δ is defined before being understood whether the value of 2Ψ is smaller than 90° or not. The range of 2Ψ is defined from the Equation 5, and the term I dc  cos 2Ψ is determined whether it is positive or not. If the value of 2Ψ&lt;90°, measured second phase Φ 2 &lt;0, and measured first phase Φ 1 &lt;0, then Δ is located at I-quadrant. If the value of 2Ψ&lt;90°, measured second phase Φ 2 &lt;0, and measured first phase Φ 1 &gt;0, then Δ is located at II-quadrant. If the value of 2Ψ&lt;90°, measured second phase Φ 2 &gt;0, and measured first phase Φ 1 &gt;0, then Δ is located at the III-quadrant. If the value of 2Ψ&lt;90°, measured second phase Φ 2 &gt;0, and measured first phase b&lt;0, then Δ is located at the IV-quadrant. If the value of 2Ψ&gt;90°, measured second phase Φ 2 &gt;0, and measured first phase Φ 1 &gt;0, then Δ is located at the I-quadrant. If the value of 2Ψ&gt;90°, measured second phase Φ 2 &gt;0, and measured first phase Φ 1 &lt;0, then Δ is located at II-quadrant. If the value of 2Ψ&gt;90°, measured second phase Φ 2 &lt;0, and measured first phase (1), &lt;0, then Δ is located at the III-quadrant. If the value of 2Ψ&gt;90°, measured second phase Φ 2 &lt;0, and measured first phase Φ 1 &gt;0, then Δ is located at IV-quadrant. Thus, a full scale (i.e. 0˜360°) measurement of ellipsometric parameters (Ψ and Δ) is obtained, and hence a full-range (i.e. 0°˜180°) measurement of the optical properties is achieved. 
         [0065]    The ellipsometric parameters calculated from Equation 7 and Equation 8 above are used to combine the Fresnel equations (for s- and p-polarized waves). Thus the equation is obtained: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               r 
                               
                                 12 
                                 , 
                                 p 
                               
                             
                              
                             
                               r 
                               
                                 01 
                                 , 
                                 s 
                               
                             
                              
                             
                               r 
                               
                                 12 
                                 , 
                                 s 
                               
                             
                           
                           - 
                           
                             
                               r 
                               
                                 01 
                                 , 
                                 p 
                               
                             
                              
                             
                               r 
                               
                                 12 
                                 , 
                                 p 
                               
                             
                              
                             
                               
                                 r 
                                 
                                   12 
                                   , 
                                   s 
                                 
                               
                                
                               
                                 ( 
                                 
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     Ψ 
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       Δ 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         X 
                         2 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             r 
                             
                               12 
                               , 
                               p 
                             
                           
                           + 
                           
                             
                               r 
                               
                                 01 
                                 , 
                                 p 
                               
                             
                              
                             
                               r 
                               
                                 01 
                                 , 
                                 s 
                               
                             
                              
                             
                               r 
                               
                                 12 
                                 , 
                                 s 
                               
                             
                           
                           - 
                           
                             
                               r 
                               
                                 12 
                                 , 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   Ψ 
                                   Δ 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               r 
                               
                                 01 
                                 , 
                                 p 
                               
                             
                              
                             
                               r 
                               
                                 12 
                                 , 
                                 p 
                               
                             
                              
                             
                               
                                 r 
                                 
                                   12 
                                   , 
                                   s 
                                 
                               
                                
                               
                                 ( 
                                 
                                   tan 
                                    
                                   
                                       
                                   
                                    
                                   
                                     Ψ 
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       Δ 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                        
                       X 
                     
                     + 
                     
                       ( 
                       
                         
                           r 
                           
                             01 
                             , 
                             p 
                           
                         
                         - 
                         
                           
                             r 
                             
                               01 
                               , 
                               s 
                             
                           
                            
                           
                             ( 
                             
                               tan 
                                
                               
                                   
                               
                                
                               
                                 Ψ 
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   Δ 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
         [0066]    where r jk  (t jk ) is the amplitude reflection (transmission) coefficient at each interface as illustrated in  FIG. 4 . 
         [0067]    θ 0 , θ 1  are the angles which the incident rays and the refracted rays make to the normal of the interface respectively, θ 2  is the angle which the ray entering the medium (with refractive index n 2 ) makes to the normal of the interface as shown in  FIG. 4 . 
         [0000]    
       
         
           
             
               
                 
                   
                     r 
                     
                       jk 
                       , 
                       p 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               N 
                               k 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               j 
                             
                           
                           - 
                           
                             
                               N 
                               j 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               k 
                             
                           
                         
                         
                           
                             
                               N 
                               k 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               j 
                             
                           
                           + 
                           
                             
                               N 
                               j 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               k 
                             
                           
                         
                       
                       · 
                       
                         r 
                         
                           jk 
                           , 
                           2 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             N 
                             j 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             j 
                           
                         
                         - 
                         
                           
                             N 
                             k 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             k 
                           
                         
                       
                       
                         
                           
                             N 
                             j 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             j 
                           
                         
                         + 
                         
                           
                             N 
                             k 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             k 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     10 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     i 
                     
                       jk 
                       , 
                       p 
                     
                   
                   = 
                   
                     
                       
                         
                           2 
                            
                           
                             N 
                             j 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             j 
                           
                         
                         
                           
                             
                               N 
                               k 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               j 
                             
                           
                           + 
                           
                             
                               N 
                               j 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               k 
                             
                           
                         
                       
                       · 
                       
                         t 
                         
                           jk 
                           , 
                           2 
                         
                       
                     
                     = 
                     
                       
                         2 
                          
                         
                           N 
                           j 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         
                           θ 
                           j 
                         
                       
                       
                         
                           
                             N 
                             j 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             j 
                           
                         
                         + 
                         
                           
                             N 
                             k 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             k 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     11 
                   
                   ) 
                 
               
             
           
         
       
     
         [0068]    where N j  and N k  are refractive indices of media. 
         [0000]    
       
         
           
             
               
                 
                   X 
                   = 
                   
                     ( 
                     
                       
                          
                         
                           - 
                           
                             
                               4 
                                
                               π 
                                
                               
                                   
                               
                                
                                
                             
                             λ 
                           
                         
                       
                        
                       
                         n 
                         1 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         1 
                       
                        
                       D 
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     12 
                   
                   ) 
                 
               
             
           
         
       
     
         [0069]    where n 1  is refractive index of medium. 
         [0070]    The thickness of the thin film can be calculated by solving the Equation 9, and the thickness (D) is determined by: 
         [0000]    
       
         
           
             
               
                 
                   D 
                   = 
                   
                     
                       
                         m 
                          
                         
                             
                         
                          
                         π 
                       
                       + 
                       
                          
                          
                         
                           ( 
                           
                             
                               ln 
                                
                               
                                  
                                 X 
                                  
                               
                             
                             + 
                             
                                
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 x 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           4 
                            
                           
                               
                           
                            
                           π 
                         
                         λ 
                       
                        
                       
                         n 
                         1 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     13 
                   
                   ) 
                 
               
             
           
         
       
     
         [0071]    where m is the order of the thickness. 
         [0072]    In an exemplary embodiment a method for thin film thickness measurement is provided. The method comprises: emitting light from a light source; polarizing the light from the light source; modulating the polarized light; altering the polarization state of the light received from the modulator and directing the light onto the sample; polarizing the reflected light received from the sample using an analyzer; receiving the light from the analyzer for two-phase detection corresponding to two different orientations of the optical compensator. 
         [0073]    The following simulation results show the feasibility of the proposed method in measuring ellipsometric parameters and thickness of the film. Also, if the errors of the incident angle and lock-in amplifier  110  are not too big, the error of Ψ, Δ, and thickness of the sample  106  will not be enlarged. 
       Simulation Results 
     Simulation Results of the Material: 
       [0074]    By using the properties of the sample  106 , a 4×4 matrix analytical model simulates the terms including first phase Φ 1 , second phase Φ 2 , and the ellipsometric parameters (Ψ,Δ) corresponding to the incident angle from 10° to 80°. The Ellipsometric parameter Ψ result of the simulation is illustrated by the curve shown in  FIG. 5 . Similarly, The Ellipsometric parameter Δ result of the simulation is illustrated by the curve shown in  FIG. 6 . Then first phase Φ 1  and second phase Φ 2  results of the simulation are shown in  FIG. 7  and  FIG. 8  respectively and are used in the Equation 7, Equation 8 and Equation 13 to calculate the ellipsometric parameters (Ψ and Δ) and the thickness of the thin film corresponding to the incident angle from 10°˜80°, respectively. The results of the simulation for Ψ, Δ and thickness by the inversed calculation are shown in  FIG. 9 ,  FIG. 10  and  FIG. 11  respectively. These figures confirm the feasibility of the signal processing system according to the invention, and it is applied to the ellipsometric parameters (Ψ, Δ) in the experiments following this section. These simulation results also show the capability of the proposed invention in measuring ellipsometric parameters and the thickness of the film. 
       Simulation Results of Ψ and Δ Error Analysis 
       [0075]    The 4×4 matrix method is used to derive theoretical output ellipsometric parameters (Ψ, Δ) and let the theoretical input of the incident angle and lock-in amplifier  110  have ±0.01° error in variations. Inserting the ±0.01° error of the parameters with a simulated error into the algorithm deduced by 4×4 matrix method, the error of algorithm is understood. The characteristics of the algorithm by using three different cases are mentioned: the incident angle with ±0.01° error, lock-in amplifier  110  with ±0.01° error, and both incident angle and lock-in amplifier  110  exist the ±0.01° error. The simulation shows the results of 0%˜1% error-analysis by using the material illustrated in the subsection above with regard to the 25° incident angle. 
       Simulation Results of Ψ and Δ Error Analysis in θ i  Error=±0.01° 
       [0076]    The 4×4 matrix method is used to derive theoretical output ellipsometric parameters (Ψ, Δ) and let the theoretical input of the incident angle have ±0.01° error in variations. The values of the ellipsometric parameters Ψ=43.2574°, Δ=187.5686°, thickness of the film D=147.1 nm are chosen in order to extract Ψ, Δ, D by using Equation 7, Equation 8 and Equation 13 with ±0.01° error in the incident angle.  FIGS. 12 ,  13  and  14  show the simulated results for Ψ, Δ and D respectively. A correlation between input Ψ and extracted Ψ when θ i  Error=±0.01 (incident angle has ±0.01° error in variations) is shown in  FIG. 12 . Similarly, correlation between input A and extracted Δ is shown in  FIG. 13  and correlation between input thickness and extracted thickness is shown in  FIG. 14 . The three error bars in the three parameters, Ψ, Δ, and D, are ±(4.7276×10 −4 )°, ±(0.0039)°, and ±(1.1912×10 −13 ) nm respectively. 
       Simulation Results of Ψ and Δ Error Analysis in the Phase Error=±0.01° 
       [0077]    The 4×4 matrix method is used to derive theoretical output ellipsometric parameters (Ψ, Δ) and let the theoretical input of the lock-in amplifier have ±0.01° error in variations. The values of the ellipsometric parameters Ψ=43.2574°, Δ=187.5686°, and thickness of the film D=147.1 nm are chosen in order to extracte Ψ, Δ, D by using Equation 7, Equation 8 and Equation 13 with ±0.01° error in the incident angle.  FIGS. 15 ,  16  and  17  show the simulated results for Ψ, Δ and D respectively. A correlation between input Ψ and extracted Ψ when Phase Error=±0.01° is shown in  FIG. 15 . Similarly, correlation between input Δ and extracted Δ is shown in  FIG. 16  and correlation between input thickness and extracted thickness is shown in  FIG. 17 . The three error bars in the three parameters, Ψ, Δ, and D, are ±(4.4128×10 −5 )°, ±(2.3872×10 −4 )°, and ±(3.4564×10 −4 ) nm respectively. 
       Simulation Results of Ψ and Δ Error Analysis in the θ i  Error=±0.01 and the Phase Error=±0.01° 
       [0078]    The 4×4 matrix method is used to derive theoretical output ellipsometric parameters (Ψ, Δ) and let the theoretical input of the incident angle and lock-in amplifier  110  have ±0.01° error in variations. The values of the ellipsometric parameters Ψ=43.2574°, Δ=187.5686°, and thickness of the film D=147.1 nm are chosen in order to extract Ψ, Δ, D by using Equation 7, Equation 8 and Equation 13 with ±0.01° error in the incident angle.  FIGS. 18 ,  19  and  20  show the simulated results for Ψ, Δ and D respectively. A correlation between input Ψ and extracted Ψ when θ i  Error=±0.01 and the Phase Error=±0.01° is shown in  FIG. 18 . Similarly, correlation between input Δ and extracted Δ is shown in  FIG. 19  and correlation between input thickness and extracted thickness is shown in  FIG. 20 . The three error bars in the three parameters, Ψ, Δ, and D, are ±(4.9963×10 −4 )°, ±(0.0071)°, and ±(3.4679×10 −4 ) nm respectively. 
         [0079]    These simulation results have shown that inserting the theoretical input of the incident angle and lock-in amplifier  110  with ±0.01° error in variations into the algorithm causes the maximum error of Ψ, Δ, and thickness. It should be noticed that if the errors of the incident angle and lock-in amplifier  110  are not too large in magnitude, the error of Ψ, Δ, and thickness of the sample  106  will not be enlarged. 
       Experimental Setup and Experimental Results 
     Experimental Setup 
       [0080]    The schematic illustration of the experimental setup used to measure the ellipsometric parameters of the sample  106  is shown in  FIG. 1 . In  FIG. 2 , the polarizer  103  is adjusted to 0°, the quarter-wave plate (optical compensator  105 ) is adjusted to 0° and the analyzer  107  is adjusted to −45°. In  FIG. 3 , the polarizer  103  is adjusted to 0°, the quarter-wave plate (optical compensator  105 ) is adjusted to −45° and the analyzer  107  is adjusted to −45°. Silicon substrate coated with the SiO 2  thin film (147.1 nm) is taken as the sample  106 . The sample stage is rotated at the angle which is equivalent to the incident angle. In the system, the He—Ne laser (SL 02/2, SIOS Co.) is used as a light source  101 . The frequency of the saw tooth signal from a function generator  111  applied to the EO modulator  104  is 1 kHz. The experimental setup includes two sheet polarizers (Sigma Koki, Model: SPF-30C-32) with extinction ratios of 5×10 6 , one quarter wave plate (Sigma Koki, Model: WPQ-6328-4M). The laser beam passes sequentially through a polarizer  103 , electro-optic (EO) modulator  104 , the quarter-wave plate (optical compensator  105 ), reflected by the sample  106 , and an analyzer  107  before being incident on the photo-detector  109 . The signal received by the photo-detector  109  could be locked in phase by using the lock-in amplifier  110  (SRS, Model: SR-830). It should be noticed that the incident light is perpendicular to the sample stage before measuring the sample  106  to calibrate the optical path and the inclination of the sample stage. Furthermore, one should assure the light reflected the sample stage matches the incident laser spot absolutely. This is an important calibration step for the measurement. 
       Experimental Results 
       [0081]    Parameters of the sample  106  are shown in Table 1. First phase Φ 1  (Equation 3) and second phase Φ 2  (Equation 6) are measured using the apparatus  100  as disclosed in the present invention. The experimental results of measured first phase Φ 1  and measured second phase Φ 2  are shown in  FIG. 21  and  FIG. 23 . Phases corresponding to the incident angles Λ i =25° from the lock-in amplifier  110  are obtained and are inserted into the Equation 7, Equation 8 and Equation 13 to calculate the ellipsometric parameters (Ψ and Δ) and the thickness of the thin film. The measured phase (Φ 1  and Φ 2 ), the ellipsometric parameters (Ψ and Δ), and the measured thickness of the sample are shown in Table 2. 
         [0082]    The experimental results of measured IP and measured Δ are shown in  FIG. 25  and  FIG. 27  respectively. Correlation between the simulated and measured values of the two phases (Φ 1  and Φ 2 ), ellipsometric parameters (Ψ and Δ) and thickness is also illustrated in the drawings. The correlation of simulated and measured values of Φ 1  and Φ 2  is shown in  FIG. 22  and  FIG. 24  respectively. The correlation of simulated and measured values of ellipsometric parameters, Ψ and Δ is shown in  FIG. 26  and  FIG. 28 . The experimental results of measured thickness are shown in  FIG. 29  and the correlation of measured thickness and known thickness is shown in  FIG. 30 . 
         [0083]    The experimental results have shown that the standard deviation of Ψ is 0.1313°, the experimental deviation of Δ is 0.6829°, and the experimental deviation of thickness is 0.9355 (nm). 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 PARAMETERS OF SAMPLE 
               
             
          
           
               
                   
                 Parameters 
                 Values 
               
               
                   
                   
               
               
                   
                 SiO2 Thickness 
                 147.1 nm 
               
               
                   
                 SiO2 Refractive Index 
                 1.464 
               
               
                   
                 Si Refractive Index 
                 3.88114 
               
               
                   
                 Simulated Ψ (deg) 
                 43.2574 
               
               
                   
                 Simulated Δ(deg) 
                 187.5686 
               
               
                   
                 Simulated Φ 1  (deg) 
                 3.5157 
               
               
                   
                 Simulated Φ 2  (deg) 
                 65.1847 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 Data Number 
                 Φ 1  (deg) 
                 Φ 2  (deg) 
                 Ψ (deg) 
                 Δ (deg) 
                 D (nm) 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 1 
                 3.73 
                 66.75 
                 43.156 
                 188.6283 
                 146.71 
               
               
                 2 
                 3.53 
                 65.72 
                 43.2512 
                 187.787 
                 146.96 
               
               
                 3 
                 3.67 
                 66.21 
                 43.1841 
                 188.2783 
                 147.02 
               
               
                 4 
                 3.45 
                 67.31 
                 43.2926 
                 188.205 
                 146.03 
               
               
                 5 
                 3.63 
                 68.60 
                 43.2083 
                 189.1953 
                 145.36 
               
               
                 6 
                 3.47 
                 62.93 
                 43.2771 
                 186.7664 
                 146.87 
               
               
                 7 
                 4.32 
                 63.91 
                 42.8652 
                 188.7697 
                 148.95 
               
               
                 8 
                 3.97 
                 64.17 
                 43.0350 
                 188.1590 
                 148.75 
               
               
                 9 
                 3.70 
                 62.43 
                 43.1640 
                 187.0604 
                 148.87 
               
               
                 10 
                 3.78 
                 65.54 
                 43.1296 
                 188.2641 
                 147.59 
               
               
                 Standard 
                 0.2600 
                 1.9755 
                 0.1273 
                 0.7421 
                 1.2237 
               
               
                 deviation 
               
               
                   
               
             
          
         
       
     
         [0084]    While the exemplary embodiments have been disclosed for illustrative purposes, those skilled in the art will appreciate that various modifications and other embodiments are possible, without departing from the scope and spirit of the present inventive concept as defined by the appended claims.