Patent Publication Number: US-2023160687-A1

Title: Wafer thickness, topography, and layer thickness metrology system

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     There are no related applications. 
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     There was no federal sponsorship for this research and development. 
     THE NAMES OF THE PARTIES TO A JOINT RESEARCH AGREEMENT 
     There was no joint development agreement. 
     REFERENCE TO A “SEQUENCE LISTING,” A TABLE, OR A COMPUTER PROGRAM LISTING APPENDIX SUBMITTED ON A COMPACT DISC AND AN INCORPORATION-BY-REFERENCE OF THE MATERIAL ON THE COMPACT DISC 
     There is no appendix. There is no disc. 
     STATEMENT REGARDING PRIOR DISCLOSURES BY THE INVENTOR OR A JOINT INVENTOR 
     There was no public prior disclosure of the invention. 
     BACKGROUND OF THE INVENTION 
     Our invention is related to the determining the topography and thickness of slabs of materials. More particularly, the present invention in one of its embodiments relates to measuring the thickness of slabs of materials using a spectroscopic system. The invention can be used for measurement of the topography and thickness of semiconductors and other materials ranging from 0.1 micrometer up to 10 mm. The invention can be used to measure the thickness of slabs of homogenous materials, thickness of layered materials, optical thickness of individual layers, refractive indices of uniform materials, and the roughness of the interfaces between material layers. 
     The invention may be used in applications involving back-end processing of semiconductor chips, grinding and polishing of patterned and blanket wafers, and processing of micro-electromechanical-systems, such as but not limited to pressure monitors, micro-mirrors and similar advanced devices and structures. 
     BRIEF SUMMARY OF THE INVENTION 
     Measurement of the thickness of thin wafers is important in modern chip packaging. At the backend phase of manufacturing, wafers are thinned using mechanical or chemical means. Since various chips are often stacked, it is important to accurately control the thickness and flatness of individual chips. Often the thickness and flatness metrology tools are in proximity of other machinery causing mechanical vibrations, such as grinders or lapping machines. The invention describes a metrology system allowing for the reduction of the errors caused by vibration of the production floor and allowing for measurements of the thickness of wafers in motion. This is accomplished by performing measurements of spectra containing interference signals containing distance information using a plurality of probes positioned on both sides of the measured wafer on the same detector at the same time. The invention reduces number of spectrometers, and detectors used in measurement and provides excellent synchronization of the measured signals. 
    
    
     
       BRIEF DESCRIPTIONS OF THE SEVERAL VIEWS OF THE DRAWINGS 
         FIG.  1    represents how a sample wafer  8  is positioned on the motorized table 4. 
         FIG.  2    represents a metrology system including optical and electrical connections. 
         FIG.  3    represents metrology unit  1000  and the way it is connected to probes  3  and  8 . 
         FIG.  4    represents a metrology unit similar with computer-controlled shutters  1201  and  1301 . 
         FIG.  5    represents probe  2  when connected to the metrology unit shown in  FIG.  3     
         FIG.  6    represents probe  3  when connected to the metrology unit shown in  FIG.  4   . In this case, the optical fiber  201  is replaced by the optical fiber  291 . 
         FIG.  7    represents the probe with adjustable length of the reference arm L reference . 
         FIG.  8    represents the probe with adjustable optical length of the reference arm L reference  using color filter wheel equipped with refractive slabs of material for introducing additional delay. 
         FIG.  9    represents spectrometer for simultaneous measurement of the spectra of radiation emanating from fibers  201 , and  301 . 
         FIG.  10    represents the two-dimensional detector  100001  and recorded spectra. 
         FIG.  11    represents a simulated signal detected by the system shown in  FIG.  4   . 
         FIG.  12    represents the spectrum of the signal when sample  8  is illuminated only by probe  2 . 
         FIG.  13    represents the signal presented in  FIG.  12    fitted with a gaussian function. 
         FIG.  14    represents the signal shown in  FIG.  12    fitted with function with interference fringes. 
         FIG.  15    represents the spectrum of signal when sample  8  is only illuminated by probe  3 . 
         FIG.  16    represents the signal presented in  FIG.  15    fitted with a gaussian function. 
         FIG.  17    represents the signal shown in  FIG.  15    fitted with the function with fringes. 
         FIG.  18    represents the signal presented in  FIG.  11    fitted with a gaussian function. 
         FIG.  19    represents the signal presented in  FIG.  15    fitted with the final fitting function. 
         FIG.  20    shows a simulated signal like that shown in  FIG.  11    where the noise is small. 
         FIG.  21    shows a fitted function to the data shown in  FIG.  20   . 
         FIG.  22    shows a multilayer sample  8  comprising two layers layer  81  and layer  82  . . .  23   
         FIG.  23    describes procedure for finding differential spectrum using the first probe. 
         FIG.  24    describes procedure for finding differential spectrum using the second probe. 
         FIG.  25    describes procedure for finding procedure for finding the interface proximal to the first probe. 
         FIG.  26    describes procedure for finding procedure for finding the interface proximal to the second probe. 
         FIG.  27    describes procedure for finding multiple interfaces using the first probe. 
         FIG.  28    describes procedure for finding multiple interfaces using the second probe. 
         FIG.  29    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a slab of gold coated material, and the first probe is positioned at the distance 0.250 mm from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  30    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  29   . 
         FIG.  31    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  30    using parameter ΔD=0.010 mm. 
         FIG.  32    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  31    using parameter ΔD=0.010 mm using algorithm shown in  FIG.  25   . 
         FIG.  33    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a slab of gold coated material, and the first probe is positioned at the distance 0.150 mm from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  34    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  33   . 
         FIG.  35    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  33    using parameter Δt=0.010 mm. 
         FIG.  36    represents filtered and integrated differential spectrum IDFFTDS1 calculated from the spectrum shown in  FIG.  35    using parameter Δt=0.010 mm. 
         FIG.  37    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a slab of gold coated material, and the first probe is positioned at the distance 0.050 mm from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  38    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  37   . 
         FIG.  39    represents differential spectrum IDFFTDS1 calculated from the spectrum shown in  FIG.  38    using parameter Δt=0.010 mm. 
         FIG.  40    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  39    using parameter Δt=0.010 mm. 
         FIG.  41    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a slab of gold coated material, and the first probe is positioned at the distance −0.050 mm (negative 0.050 mm) from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  42    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  29   . 
         FIG.  43    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  30    using parameter Δt=0.010 mm. 
         FIG.  44    represents filtered and integrated differential spectrum IDFFTDS1 calculated from the spectrum shown in  FIG.  31    using parameter Δt=0.010 mm. 
         FIG.  45    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a slab of gold coated material, and the first probe is positioned at the distance −0.150 mm from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  46    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  45   . 
         FIG.  47    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  46    using parameter Δt=0.010 mm. 
         FIG.  48    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  47    using parameter Δt=0.010 mm. 
         FIG.  49    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a multilayer sample, and the first probe is positioned at the distance 0.250 mm from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  50    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  49   . 
         FIG.  51    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  50    using parameter Δt=0.010 mm. 
         FIG.  52    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  51    using parameter Δt=0.010 mm. 
         FIG.  53    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a multilayer sample, and the first probe is positioned at the distance 0.150 mm from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  54    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  53   . 
         FIG.  55    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  54    using parameter Δt=0.010 mm. 
         FIG.  56    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  55    using parameter Δt=0.010 mm. 
         FIG.  57    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a multilayer sample, and the first probe is positioned at the distance 0.050 mm from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  58    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  57   . 
         FIG.  59    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  58    using parameter Δt=0.010 mm. 
         FIG.  60    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  59    using parameter Δt=0.010 mm. 
         FIG.  61    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a multilayer sample, and the first probe is positioned at the distance −0.050 mm (negative) from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  62    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  61   . 
         FIG.  63    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  62    using parameter Δt=0.010 mm. 
         FIG.  64    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  63    using parameter Δt=0.010 mm. 
         FIG.  65    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a multilayer sample, and the first probe is positioned at the distance −0.150 mm (negative) from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  66    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  65   . 
         FIG.  67    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  66    using parameter Δt=0.010 mm. 
         FIG.  68    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  67    using parameter Δt=0.010 mm. 
         FIG.  69    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a multilayer sample, and the first probe is positioned at the distance −0.250 mm (negative) from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  70    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  69   . 
         FIG.  71    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  70    using parameter Δt=0.010 mm. 
         FIG.  72    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  71    using parameter Δt=0.010 mm. 
         FIG.  73    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a multilayer sample, and the first probe is positioned at the distance −0.350 mm (negative) from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  74    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  73   . 
         FIG.  75    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  74    using parameter Δt=0.010 mm. 
         FIG.  76    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  75    using parameter Δt=0.010 mm. 
         FIG.  77    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a multilayer sample, and the first probe is positioned at the distance −0.450 mm (negative) from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  78    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  77   . 
         FIG.  79    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  78    using parameter Δt=0.010 mm. 
         FIG.  80    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  79    using parameter Δt=0.010 mm. 
         FIG.  81    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a multilayer sample, and the first probe is positioned at the distance −0.550 mm (negative) from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  82    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  81   . 
         FIG.  83    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  82    using parameter Δt=0.010 mm. 
         FIG.  84    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  83    using parameter Δt=0.010 mm. 
         FIG.  85    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the sample is a multilayer sample, and the first probe is positioned at the distance −0.650 mm (negative) from the surface proximal to it. The second probe is positioned at the distance 0.200 mm from the surface proximal to the second probe. 
         FIG.  86    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  85   . 
         FIG.  87    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  86    using parameter Δt=0.010 mm. 
         FIG.  88    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  87    using parameter Δt=0.010 mm. 
     
    
    
     DETAILED DESCRIPTION OF INVENTION 
       FIG.  1    represents how a sample wafer  8  is positioned on the motorized XY positioning table 4. Table 4 is provided with multiple opening  6  allowing optical access to wafer  8  from both above and below of the wafer. Table 4 is motorized using the motors  304  which are attached to frame  1  of the tool. To the same tool are attached linear motion stages  31  and  21  which position the optical probes  2  and  3  respectively. Stages  31  and  21  can be used to adjust the distance between probes  3  and  2  and the measured sample  8  for purpose of optimizing, and analysis of the optical signal. 
       FIG.  2    represents a metrology system including optical and electrical connections. The optical probes  2  and  3  are connected by optical fibers  201  and  301  to an optical source, data acquisition, and analysis module (OSDAM)  1000 . OSDAM  1000  is connected to a controlling computer  3000  by cable  1001 . In one embodiment the OSDAM module  1000  is mounted inside the 19-inch rack  12 . The motorized stages  21 ,  31 , and motors  304  controlling XY table 4 shown in  FIG.  1    are connected to the motion control module (MCM)  2000  by an electrical harness  202 . The MCM is connected to computer  3000  using the electrical cable  2001 . 
       FIG.  3    represents metrology unit  1000  and the way it is connected to probes  3  and  8  and computer  3000 . Light source  1110  is connected by an electrical harness  1001  to computer  3000 . Computer  3000  controls light source  1110 . The light source is producing radiation which is coupled to optical fiber  1109  and directed to 2×2 optical coupler  1108 . Optical coupler  1108  is directing radiation to two distinct optical fibers  301  and  201  connected to two optical probes  3  and  2  respectively. Probes  3  and  2  are directing light to sample  8  and collect reflected radiation from sample  8 . The reflected radiation from probe  3  is transmitted through the optical fiber  301  to 2×2 optical coupler  1108  and the reflected radiation from probe  2  is transmitted through the optical fiber  201  to 2×2 optical coupler  1108 . Radiation reflected from probes  2  and  3  is transmitted from 2×2 optical coupler  1108  through optical fiber  1107  to computer-controlled filter  1106 . Filter  1106  is connected through the electrical harness  1001  to computer  3000 . Filter  1106  is connected through optical fiber  1105  to the second optical filter  1104 . Optical filter  1104  is connected through the electrical harness  1001  to computer  3000 . Optical filter  1104  is connected through the optical fiber  1103  to optical computer-controlled spectrograph  1002 . The spectrograph  1002  is connected through the optical harness  1001  to computer  3000 . 
       FIG.  4    represents a metrology unit similar as shown in  FIG.  3    where the fiber  201  was replaced by a fiberoptic assembly comprising fiber  291  connected to a computer-controlled shutter  1201  connected by optical fiber  1211  to 2×2 optical coupler  1108 , and where the fiber  301  was replaced by a fiberoptic assembly comprising fiber  391  connected to a computer-controlled shutter  1301  connected by optical fiber  1311  to 2×2 optical coupler  1108 . Computer-controlled shutters  1201  and  1301  are connected through the electrical harness  1001  to the computer  3000 . 
       FIG.  5    represents probe  2  when connected to the metrology unit shown in  FIG.  3   . The radiation propagates through the optical fiber  201  and is emitted at the end of fiber  201  and forms a divergent beam  91 . The divergent beam  91  is collimated by lens  901  and forms collimated beam  92 . The collimated beam  92  impinges beam splitter  902 . The beam splitter  902  divides beam  92  into two portions: the first portion  102  propagating towards reflector  100  and the second portion  93  propagating through the beam-forming lens  903  and forming beam  94  impinging sample  8 . Reflector  100  reflects beam  102  and forms reflected beam  101 , while sample  8  reflects beam  94  and forms beam  95 . Beam  95  propagates through lens  903  forms beam  96 . Beam  96  is combined with optical beam  101  by the beam splitter  902  and the combined beam forms beam  96  impinging lens  901 . Beam propagating through lens  901  is forming focused beam  97  which is coupled to the optical fiber  201 . 
       FIG.  6    represents probe  3  when connected to the metrology unit shown in  FIG.  4   . In this case, the optical fiber  201  is replaced by the optical fiber  291 . 
       FIG.  7    represents the probe with adjustable length of the reference arm L reference . The reflector  100  is mounted on linear motion stage  110 . The linear motion  110  is connected by the harness  1001  and controlled by computer  3000 . 
       FIG.  8    represents the probe with adjustable optical length of the reference arm L reference . Polished and anti-reflection coated plates having preset optical thicknesses are mounted on motorized rotary motion stage  130 . The actuator of the rotary motion  110  is connected by the harness  1001  and controlled by computer  3000 . 
       FIG.  9    represents spectrometer for simultaneous measurement of the spectra of radiation emanating from fibers  201 , and  301 . The radiation emanating from the fiber  201  forms the beam  20000  and impinges the surface of the collimating lens  40000 . The radiation emanating from the fiber  301  forms the beam  30000  is impinges the surface of the collimating lens  40000 . The lens  40000  is collimating and combining beams  20000  and  30000  and directs the combined beam  50000  towards grating  60000 . Grating  60000  diffracts the impinging radiation and produces beam  70000 . Beam  70000  is transmitted through the focusing lens  80000 . Lens  80000  separates radiation  70000  into two components: the first component  90002  originating from fiber  201 , and the second component  90003  originating from fiber  301 . Beams  90002  and  90003  form an image in the image plane  100000  and can be measured by the detector or plurality of detectors positioned in plane  100000 . 
       FIG.  10    represents the two-dimensional detector  100001  and recorded spectra. The spectra  100003  and  100002  can be measured by the detector  100001  simultaneously, and since they do not overlap, they can be separately analyzed. 
       FIG.  11    represents a simulated signal detected by the system shown in  FIG.  1    when the optical path of radiation detected by probe  2  differs from than the optical path of the radiation detected by probe  3  by much more than the coherence length of the radiation. The simulated signal is comprising two incoherent contributions from two probes  2 ,  3 , and random noise. It was simulated using Equations (15)-(21) where the following parameters were used: 
     f 0 =2.29·10 14  Hz, w=3.172·10 12  Hz, A 1 =1000 arb. units, A 2 =1500 arb. units, f 1 =3.172·10 11  Hz, f 2 =9.517·10 10  Hz, φ 1 =0, φ 2 =0, B=100, N 0 =200. 
       FIG.  12    represents the spectrum of the signal when sample  8  is illuminated only by probe  2 . It was simulated using prescription given by Equations 15-21 where the following parameters were used: 
     f 0 =2.29·10 14  Hz, w=3.172·10 12  Hz, A 1 =0 arb. units, A 2 =1500 arb. units, f 1 =3.172·10 11  Hz, f 2 =9.517·10 10  Hz, φ 1 =0, φ 2 =0, B=100, N 0 =200, m 1 =0.50, m 2 =0.40. 
     The simulated signal comprises of two incoherent contributions from two probes  2 ,  3 , and random noise, and is simulated by following (1): 
       TotSignal( f )=| S   1 ( f )+ S   2 ( f )+Bckg( f )+ N ( f )|  (1)
 
     where S 1 (f) is the term corresponding to probe  2  and is modeled in the simulation by the following formula: 
     
       
         
           
             
               
                 
                   
                     
                       
                         S 
                         1 
                       
                       ( 
                       f 
                       ) 
                     
                     = 
                     
                       
                         
                           R 
                           1 
                         
                         ( 
                         f 
                         ) 
                       
                       [ 
                       
                         1 
                         + 
                         
                           
                             m 
                             1 
                           
                           · 
                           
                             cos 
                             ⁡ 
                             ( 
                             
                               
                                 f 
                                 
                                   f 
                                   1 
                                 
                               
                               - 
                               
                                 φ 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                   ) 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where m 1  is modulation depth which depends on reflectivity of sample  8 , f 1  is the frequency of fringes corresponding to the distance between probe  2  and sample  8 , φ 1  is the phase of these fringes and where R 1 (f) is the spectrum of the radiation reflected by probe  2  and depends slowly on the frequency f. For simulation, we modeled it by gaussian: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       1 
                     
                     ( 
                     f 
                     ) 
                   
                   = 
                   
                     
                       A 
                       1 
                     
                     ⁢ 
                     
                       e 
                       
                         - 
                         
                           
                             ( 
                             
                               
                                 f 
                                 - 
                                 
                                   f 
                                   0 
                                 
                               
                               w 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     In Equation 1y, A 1  is the amplitude of the signal reflected from probe  2 , w is the spectral width of the light emitted by the light. 
     Similarly, S 2 (f) is the term corresponding to probe  3  and is modeled in the simulation by the following formula 
     
       
         
           
             
               
                 
                   
                     
                       S 
                       2 
                     
                     ( 
                     f 
                     ) 
                   
                   = 
                   
                     
                       
                         R 
                         2 
                       
                       ( 
                       f 
                       ) 
                     
                     [ 
                     
                       1 
                       + 
                       
                         
                           
                             m 
                             2 
                           
                           · 
                           cos 
                         
                         ⁢ 
                            
                         
                           ( 
                           
                             
                               f 
                               
                                 f 
                                 2 
                               
                             
                             - 
                             
                               φ 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     where m 2  is modulation depth which depends on reflectivity of sample  8 , f 2  is the frequency of fringes corresponding to the distance between probe  3  and sample  8 , φ 2  is the phase of these fringes and where R 2  (f) is the spectrum of the radiation reflected by probe  3  and depends slowly on the frequency f. For simulation, it is modeled by a gaussian with finite offset B: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       2 
                     
                     ( 
                     f 
                     ) 
                   
                   = 
                   
                     
                       
                         A 
                         2 
                       
                       ⁢ 
                       
                         e 
                         
                           - 
                           
                             
                               ( 
                               
                                 
                                   f 
                                   - 
                                   
                                     f 
                                     0 
                                   
                                 
                                 w 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     + 
                     B 
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     where A 2  is the amplitude of the signal reflected from probe  3 , w is spectral width of the light emitted by the light source f 2  is the frequency of fringes corresponding to the distance between probe  3  and sample  8 , φ 2  is the phase of these fringes. 
     The background of the detector is modeled by a constant and in this simulation does not depend on frequency: 
       Bckg( f )= B   (6)
 
     where B=const. 
     The random noise of the detector, spectrometer, and optical system is 
         N ( f )=rand( N   0 )  (7)
 
     where rand(σ) is a random variable having an expected value of 0 and standard deviation N 0 . 
     The absolute value operator |.| in (1) assures that the signal does not become negative in the case of very unlikely high noise values produced by the random variable generated in Equation 5y. 
       FIG.  13    represents the signal presented in  FIG.  12    fitted with a gaussian function with the background given by Equation 8y: 
     
       
         
           
             
               
                 
                   = 
                   
                     
                       
                         B 
                         1 
                       
                       ~ 
                     
                     + 
                     
                       
                         
                           
                             A 
                             1 
                           
                           ⁢ 
                           e 
                         
                         ~ 
                       
                       
                         - 
                         
                           
                             ( 
                             
                               
                                 f 
                                 - 
                               
                               
                                 
                                   w 
                                   1 
                                 
                                 ~ 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     where {tilde over (B)} 1 ,  ,  ,   are fitting parameters and the fitted values of these parameters are: 
       =143.961 arb. units,  =1.49·10 3  arb. units,  =2.290·10 14  Hz,  =3.077·10 12  Hz. 
       FIG.  14    represents the signal shown in  FIG.  12    fitted with function 
         S   1 ( f )= + [1+ ·cos( f · − )]  (9)
 
     where  ,   are defined with Equations 31-35 and where values of  ,  , {tilde over (f)} 0 , {tilde over (w)} are kept constant, and the only fitting parameters are  ,  ,  . The fitted values of these parameters are: 
       =−0.524,  =1.052·10 −11  s,  =−106.09 rad 
       FIG.  15    represents the spectrum of signal when sample  8  is only illuminated by probe  3 . It was simulated using prescription given by Equations 15-21 where the following parameters were used: 
     f 0 =2.29·10 14  Hz, w=3.172·10 12  Hz, A 1 =1000 arb. units, A 2 =0 arb. units, f 1 =3.172·10 11  Hz, f 2 =9.517·10 10  Hz, φ 1 =0 rad, φ 2 =0 rad, B=100, N 0 =200. 
       FIG.  16    represents the signal presented in  FIG.  15    fitted with a gaussian function with the background given by Equation 8y: 
     
       
         
           
             = 
             
               + 
               
                 
                   e 
                   
                     - 
                     
                       
                         ( 
                         
                           
                             f 
                             - 
                           
                           
                             
                               w 
                               2 
                             
                             ~ 
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
             
           
         
       
     
     where  ,  ,  ,   are fitting parameters and the fitted values of these parameters are: 
       =117.198 arb. units,  =937.631 arb. units,  =2.289·10 14  Hz,  =3.193·10 12  Hz. 
       FIG.  17    represents the signal shown in  FIG.  15    fitted with the following function 
         S   2 ( f )= + [1+ ·cos( f · − )]
 
     where  +  are defined with Equations 31-35, and where values of  ,  , {tilde over (f)} 0 , {tilde over (w)} are kept constant, and only fitting parameters are  ,  ,  . The fitted values of these parameters are: 
       =−0.389,  =3.180·10 −12 ,  =9.438. 
       FIG.  18    represents the signal presented in  FIG.  15    fitted with a gaussian function with the background given by Equation 10y: 
     
       
         
           
             
               
                 
                   = 
                   
                     + 
                     
                       
                         - 
                         
                           
                             ( 
                             
                               
                                 f 
                                 - 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     where  ,  ,  ,   are fitting parameters and the fitted values of these parameters are: 
       =106.861 arb. units,  =2.449·10 3  arb. units,  =2.29·10 14  Hz,  =3.181·10 12  Hz. 
       FIG.  19    represents the signal presented in  FIG.  11    fitted with function given by Equation 11y: 
     
       
         
           
             
               
                 
                   = 
                   
                     + 
                     
                       
                         
                           - 
                           
                             
                               ( 
                               
                                 
                                   f 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       [ 
                       
                         1 
                         + 
                         
                           
                             · 
                             cos 
                           
                           ⁢ 
                              
                           
                             ( 
                             
                               
                                 f 
                                 · 
                               
                               - 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             · 
                             cos 
                           
                           ⁢ 
                              
                           
                             ( 
                             
                               
                                 f 
                                 · 
                               
                               - 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     where  ,  ,  ,  ,  ,   are fitting parameters. The starting fitting values of  ,  ,  ,  ,  ,   were  ,  ,  ,  ,  ,  . The final fitted parameters  ,  ,  ,  ,  ,   are: 
     The values of the fitted parameters are: 
       =−0.15,  =3.185·10 −12  Hz −1 ,  =−10.487 rad,  =−0.293,  =1.050·10 −12  Hz −1 ,  =−88.474 rad. 
     It is important to notice that the value 1/ =3.140·10 11  Hz is a good approximation of f 1 =3.172·10 11  Hz, and similarly the value of 1/ =9.524·10 10  Hz is a good approximation of f 2 =9.517·10 10  Hz. Therefore, the method produces results consistent with simulated parameters. 
       FIG.  20    shows a simulated signal similar to that shown in  FIG.  11    where the noise is described by the parameter N 0 =20 and whose amplitude is 10 times smaller than in the case of the signal shown in  FIG.  7   . This case more realistically simulates the performance of the actual system. Other than N 0  all parameters are the same. 
       FIG.  21    shows a fitted function in form  31  to the data shown in  FIG.  16   . In this case, we get  =3.153·10 −12  Hz −1 , or  =3.1716·10 11  Hz which is a very good approximation of f 1 =3.172·10 11  Hz, and  =1.051·10 −12  Hz −1 , or 1/ =9.522·10 10  Hz which is a very good approximation of f 2 =9.517·10 10  Hz. 
       FIG.  22    shows a multilayer sample  8  comprising two layers layer  81  and layer  82 . There are three interfaces in the sample  8 : the interface between layer  81  and air denoted by  810 , interface between layer  81  and  82  denoted by  812 , and the interface between layer  82  and air denoted by  820 . 
       FIG.  23    describes procedure to find differential Fourier transform spectrum using and about the displacement of the first probe comprising following steps:
         STEP 1: Physically move the first probe by ΔD1/2, where ΔD1 is of the order of 10% of coherence length of the light propagating through the first probe.   STEP 2: Measure and record spectrum S0.   STEP 3: Move the first probe by ΔD1, where ΔD1 is of the order of 10% of coherence length of the light propagating through the first probe.   STEP 4: Measure and record spectrum S1 STEP 5: Calculate normalized difference between spectra ΔS1=(S0−S1)/ΔD1, when the first probe is displaced by ΔD1.   STEP 6: Calculate discrete Fourier spectrum of the difference of earlier taken spectra DFFTDS1=DFFT(ΔS1). The discrete Fourier transform can be taken directly from difference spectrum ΔS1, or after appropriate apodization and zero padding.   STEP 7: Filter DFFTDS1, as a filter a threshold or other similar filter can be used       

     
       
         
           
             
               DFFTDS 
               ⁢ 
               
                 1 
                 i 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         DFFTDS 
                         ⁢ 
                         
                           1 
                           i 
                         
                       
                       , 
                     
                   
                   
                     
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           DFFTDS 
                           ⁢ 
                           
                             1 
                             i 
                           
                         
                         
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                       &lt; 
                       T 
                     
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         
                           ❘ 
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                           DFFTDS 
                           ⁢ 
                           
                             1 
                             i 
                           
                         
                         
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                       ≥ 
                       
                         T 
                         ⁢ 
                             
                         where 
                         ⁢ 
                             
                         T 
                         ⁢ 
                             
                         is 
                         ⁢ 
                             
                         threshold 
                       
                     
                   
                 
               
             
           
         
       
         
         
           
             Integrate filtered spectrum using formula: 
             STEP 8: Calculate integrated spectrum IDFFTDS1 i =Σ j=1   i DFFTDS1 i . 
             STEP 9: Find peaks in integrated IDFFTDS2. Peaks can be found using standard peak finding algorithms rejecting peaks which have width smaller than preset value and larger than preset value. 
           
         
       
    
       FIG.  24    describes procedure to find differential Fourier transform spectrum using and about the displacement of the second probe comprising similar steps as procedure described in  FIG.  25   . 
       FIG.  25    describes algorithm for identification of observed interference from the proximal surface in the reflection spectra comprising following steps:
         STEP 1: Position first probe in place where L Sample, Proximal interface &gt;L reference      STEP 2: Decrease L Sample , by Δt where Δt is less than the optical thickness of the thinnest layer by moving probe&#39;s motion stage. Record new position of probe&#39;s motion stage.   STEP 3: Find peaks in IDFFTDS1 according to  FIG.  23   .   STEP 4: Check if there is a visible a negative peak in IDFFTDS1. If negative peaks becomes present that means that the proximal interface is at distance shorter than L reference , or L Sample, Proximal interface &lt;L reference . Or that proximal interface is now at distance closer then L reference , and at the last step it crossed value L reference . If there is not such positive peak keep searching and go to STEP 2.   STEP 5: using last spectrum calculate position of the proximal interface using position of the positive peak and current position of the motion stage.       

       FIG.  26    The similar algorithm as described in  FIG.  25    applied to the second probe. 
       FIG.  27   : describes procedure for finding a multitude of optical interfaces using first probe comprising following steps:
         STEP 1: Position first probe in position where L Sample, Proximal interface &gt;L reference  Set number of interfaces found M found =0. At this all peaks in IDFFTDS1 are positive.   STEP 2: Decrease L Sample , by Δt where Δt is less than the optical thickness of the thinnest layer by moving the first probe&#39;s motion stage. Record new position of the probe&#39;s motion stage.   STEP 3: Find peaks in DFFTDS1 according to  FIG.  23       STEP 3: Check if there is a new visible (M found +1)-th positive peak found in DFFTDS1, which is not caused by the multiple reflection inside the same layer. If there is not such a new peak, continue searching for it and go to STEP 2.   STEP 4: Calculate optical position of number of interface M_found using position of the positive peak having smallest position in of FFTDS1 closest to and the current position of the motion stage Increment number of interfaces found M_found:=M_found+1   STEP 5: Calculate optical position of interface number M found  using position of the positive peak having smallest position in IDFFTDS1 closest to and the current position of the motion stage Increment number of interfaces found M found :=M found +1.   STEP 6: Check if you found preset number of interfaces. If you did not return to STEP 2.   STEP 7: Report found optical positions of each of M found  interfaces.       
       FIG.  28    describes similar procedure as  FIG.  27    for the second probe. 
       FIG.  29    simulated spectrum containing signal from the first and the second probe. Both probes have golden flat mirrors in their reference arms having complex refractive index equal to 0.44+8.2i, and where the length D=0.250 mm for the first probe, and D=0.200 mm for the second probe, where D is the difference between sample arm length and reference arm length for each of probes. The spectrum of the light fed into the probe illuminating sample has from 2000*exp(−(freq(i)−fcenter){circumflex over ( )}2/fsigwidth{circumflex over ( )}2), where I is a pixel index in our example i=1 . . . 512, and df=3E8*(1/1210E−9−1/1410E−9)/512, and fcenter=3E8/1310E−9; and freq(i)=fcenter+(i−256)*df. Noise and detector dark detector signal were simulated by adding (100+50*randn(1,1)) to the light source spectrum where randn(1,1) is a random number having expectation equal to 1 and standard deviation equal to 1. To avoid occasional negative value of the intensity of the noisy source an absolute value of the source intensity was used in a place of source intensity. 
       FIG.  30   : represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  29   , no apodization and no zero padding was used. 
       FIG.  31    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  30    using parameter ΔD=0.010 mm. Threshold was 0 (no filtering was used). 
       FIG.  32    represents nominally filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  31    using parameter ΔD=0.010 mm. Threshold was 0 (effectively no filtering was used). 
       FIG.  33   ,  FIG.  37   ,  FIG.  41   , and  FIG.  45    show the same signal as in  FIG.  29    for different position of the first probe D=0.150 mm, D=0.050 mm, D=−0.050 mm, and D=−0.150 mm respectively. All other parameters remain the same. 
       FIG.  34   ,  FIG.  38   ,  FIG.  42   , and  FIG.  46    show the same signal as in  FIG.  30    for different position of the first probe D=0.150 mm, D=0.050 mm, D=−0.050 mm, and D=−0.150 mm respectively. All other parameters remain the same. 
       FIG.  35   ,  FIG.  39   ,  FIG.  43   , and  FIG.  47    show the same signal as in  FIG.  31    for different position of the first probe D=0.150 mm, D=0.050 mm, D=−0.050 mm, and D=−0.150 mm respectively. All other parameters remain the same. 
       FIG.  36   ,  FIG.  40   ,  FIG.  44   , and  FIG.  48    show the same signal as in  FIG.  32    for different position of the first probe D=0.150 mm, D=0.050 mm, D=−0.050 mm, and D=−0.150 mm respectively. All other parameters remain the same. 
       FIG.  49    represents signal recorded by 1D detector being incoherent sum of the reflection signals from the first and the second probe, when the transparent sample is a multilayer sample  8  shown in  FIG.  22   . The thickness the complex refractive index of layers  81 , and  82  were equal to 0.100 mm and n=1.5, and 0.075 and n=3.5 respectively. The first probe is positioned at the distance 0.250 mm from the surface  810  proximal to it. The second probe is positioned at the distance 0.200 mm from the surface  820  proximal to the second probe. Noise and detector dark detector signal were simulated by adding (100+50*randn(1,1)) to the light source spectrum where randn(1,1) is a random number having expectation equal to 1 and standard deviation equal to 1. To avoid occasional negative value of the intensity of the noisy source an absolute value of the source intensity was used in a place of source intensity. 
       FIG.  50    represents a magnitude of the discrete Fourier transform of the signal shown in  FIG.  49   , no apodization and no zero padding was used. 
       FIG.  51    represents differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  50    using parameter ΔD=0.010 mm. 
       FIG.  52    represents filtered and integrated differential spectrum DFFTDS1 calculated from the spectrum shown in  FIG.  51    using parameter ΔD=0.010 mm. Threshold of the filter was set to 0.3. 
       FIG.  53   ,  FIG.  57   ,  FIG.  61   ,  FIG.  65   ,  FIG.  69   ,  FIG.  73   ,  FIG.  77   ,  FIG.  81   , and  FIG.  85    show the same signal as in  FIG.  49    for different position of the first probe D=0.150 mm, 0.050 mm, −0.050 mm, −0.150 mm, −0.250 mm, −0.350 mm, −0.450 mm, −0.550, and −0.650 mm respectively. All other parameters remain the same. 
       FIG.  54   ,  FIG.  58   ,  FIG.  62   ,  FIG.  66   ,  FIG.  70   ,  FIG.  74   ,  FIG.  78   ,  FIG.  82   , and  FIG.  86    show the same signal as in  FIG.  50    for different position of the first probe D=0.150 mm, 0.050 mm, −0.050 mm, −0.150 mm, −0.250 mm, −0.350 mm, −0.450 mm, −0.550, and −0.650 mm respectively. All other parameters remain the same. 
       FIG.  55   ,  FIG.  59   ,  FIG.  63   ,  FIG.  67   ,  FIG.  71   ,  FIG.  75   ,  FIG.  79   ,  FIG.  83   , and  FIG.  87    show the same signal as in  FIG.  51    for different positions of the first probe D=0.150 mm, 0.050 mm, −0.050 mm, −0.150 mm, −0.250 mm, −0.350 mm, −0.450 mm, −0.550, and −0.650 mm respectively. All other parameters remain the same. 
       FIG.  56   ,  FIG.  60   ,  FIG.  64   ,  FIG.  68   ,  FIG.  72   ,  FIG.  76   ,  FIG.  80   ,  FIG.  84   , and  FIG.  88    show the same signal as in  FIG.  52    for different positions of the first probe D=0.150 mm, 0.050 mm, −0.050 mm, −0.150 mm, −0.250 mm, −0.350 mm, −0.450 mm, −0.550, and −0.650 mm respectively. All other parameters remain the same. 
     MODE OF OPERATION OF INVENTION 
     The measurement of thickness of wafer  8  denoted as T can be performed using system presented in  FIG.  1    by measuring the distance h 1  between probe  2  and the upper surface of the wafer  8  and the distance h 2  between probe  3  and the bottom surface of the wafer  8 . If the distance between probes D is known, then the thickness of the wafer  8  T is given by: 
         T=D−h   1   −h   2   (12)
 
     The distance D between probes can be found by placing the calibration standard block of known thickness T calib  between the probes  2  and  3  and measuring the distance between the upper surface of the calibration standard block h 1,calib , and the distance between the lower surface of the calibration standard block h 2,calib . Directly from (12) we get the distance between the probes. 
         D=T   calib   +h   1,calib   +h   2,calib   (13)
 
     The values of h 1  and h 2  are found by analysis of the spectral fringes observed in the spectra of the radiation reflected from sample  2  or  3  respectively, as described in the section below. 
     It is important that measurements of h 2  are performed while h 1  does not change and similarly that a measurement of h 1  is performed when h 2  does not change. For person skilled in the art, it is obvious that the same applies to h 1,calib  and h 1,calib . This may be a non-trivial task when a single detector is used and when h 1  and h 2  are time-dependent due to mechanical noise or mechanical motion of measured sample  8 . 
     The fringes in the spectra of light reflected from probes  2 , and  3  result in the interference of the radiation propagating in the reference arm of the probe (radiation  102  reflected by the reflector  100  and forming radiation  101 ) and the radiation propagating through the sample arm of the probe (Radiation  96  transmitted through the lens  93  and forming radiation  94  impinging sample  8 , reflected portion of the  94  forms beam  95  which is again transmitted through the lens  903  and forms beam  96 ) as shown in  FIG.  5   . An example of the distance probe used in our system is shown in  FIG.  5   . The radiation entering the probe has electric field 
         E   in ( t )= E   in (ω) e   −iωt   (14)
 
     As described in the description of  FIG.  5   , this beam is split into two portions: a portion impinging sample surface, and a reference arm portion impinging the reference reflective element. 
     The reflected portion of the sample beam portion of the radiation is of the form 
     
       
         
           
             
               
                 
                   
                     
                       E 
                       
                         sample 
                         , 
                            
                         
                           back 
                           ⁢ 
                              
                           reflected 
                         
                       
                     
                     ( 
                     t 
                     ) 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       r 
                       sample 
                     
                     ⁢ 
                     
                       t 
                       
                         sample 
                         ⁢ 
                            
                         path 
                       
                     
                     ⁢ 
                     
                       
                         E 
                         in 
                       
                       ( 
                       ω 
                       ) 
                     
                     ⁢ 
                     
                       e 
                       
                         
                           
                             - 
                             i 
                           
                           ⁢ 
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           ft 
                         
                           
                         + 
                         
                           ik 
                           ⁢ 
                           2 
                           ⁢ 
                           
                             L 
                             sample 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     where the length of the optical path in the sample beam is denoted as 2L sample , the amplitude complex reflection coefficient of the sample is denoted r sample , and k is the wavevector in vacuum. The change of the phase and transmission for the beam traveling to and reflected by the samples of optical elements residing in the sample arm including portion of beam-splitter, and lenses. By the length of the optical path, we understand the physical length of the light corrected by a factor of 1/n for the portion of the path when optical beam travels through the medium having optical refractive index n. 
     Similarly, the electric field of optical radiation back reflected from the reference beam is given by: 
     
       
         
           
             
               
                 
                   
                     
                       E 
                       
                         reference 
                         , 
                            
                         
                           back 
                           ⁢ 
                              
                           reflected 
                         
                       
                     
                     ( 
                     t 
                     ) 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       r 
                       reference 
                     
                     ⁢ 
                     
                       t 
                       
                         reference 
                         ⁢ 
                            
                         path 
                       
                     
                     ⁢ 
                     
                       
                         E 
                         0 
                       
                       ( 
                       ω 
                       ) 
                     
                     ⁢ 
                     
                       e 
                       
                         
                           
                             - 
                             i 
                           
                           ⁢ 
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           ft 
                         
                           
                         + 
                         
                           ik 
                           ⁢ 
                           2 
                           ⁢ 
                           
                             L 
                             reference 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     where the length of the optical path in reference beam is denoted as 2L reference , the amplitude complex reflection coefficient of the reference reflection element is denoted r reference , the complex transmission coefficient t referencepath  describes changes of phase and amplitude of the reference beam due to presence of optical elements in the reference arm. 
     Both the reference and sample beams are recombined by beam-splitter  902  shown in  FIG.  5   . The resulting beam intensity is given by: 
         I   total,reflected   =     |E   sample,backreflected ( t )+ E   reference,backreflected ( t )| 2     (17)
 
     By combining Equations 14y through  17   y , we get 
     
       
         
           
             
               
                 
                   
                     I 
                     
                       total 
                       , 
                          
                       reflected 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           1 
                           4 
                         
                         ⁢ 
                         
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 r 
                                 sample 
                               
                               ⁢ 
                               
                                 t 
                                 
                                   s 
                                   ⁢ 
                                   a 
                                   ⁢ 
                                   mple 
                                   ⁢ 
                                      
                                   path 
                                 
                               
                             
                             
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                           2 
                         
                       
                       + 
                       
                         
                           1 
                           4 
                         
                         ⁢ 
                         
                           
                             
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                                 r 
                                 reference 
                               
                               ⁢ 
                               
                                 t 
                                 
                                   reference 
                                   ⁢ 
                                      
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                           2 
                         
                       
                       + 
                       
                         
                           1 
                           4 
                         
                         ⁢ 
                         
                           b 
                           ˜ 
                         
                         ⁢ 
                            
                         cos 
                         ⁢ 
                            
                         
                           ( 
                           
                             k 
                             ⁢ 
                             2 
                             ⁢ 
                             
                               ( 
                               
                                 
                                   L 
                                   reference 
                                 
                                 - 
                                 
                                   L 
                                   sample 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           I 
                           in 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     where I in  is the intensity of light entering probe, and {tilde over (b)}=r sample t samplepath r reference t referencepath +(r sample t samplepath r reference t referencepath )* where the second summand is the complex conjugate of the first. 
     Since factors r sample , t samplepath , r reference , t referencepath  are slowly varying functions of the frequency of radiation used in our system, and since k=2πf/c, we can express the frequency dependent reflection function of the probe as: 
         R   DP ( f )= I   total,reflected   /I   in   (19)
 
     or using (17) and (18) we get 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       
                         D 
                         ⁢ 
                         P 
                       
                     
                     ( 
                     f 
                     ) 
                   
                   = 
                   
                     
                       A 
                       ⁡ 
                       ( 
                       f 
                       ) 
                     
                     + 
                     
                       
                         B 
                         ⁡ 
                         ( 
                         f 
                         ) 
                       
                       ⁢ 
                          
                       cos 
                       ⁢ 
                          
                       
                         ( 
                         
                           
                             
                               
                                 2 
                                 · 
                                 2 
                                 · 
                                 π 
                                   
                                 · 
                                 f 
                               
                               c 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   L 
                                   reference 
                                 
                                 - 
                                 
                                   L 
                                   sample 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             φ 
                             ⁡ 
                             ( 
                             f 
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     where A(f), B(f), and φ(f) are slowly varying functions of ω, where “slowly varying” means that their values vary less than 10% the spectral width of the bandwidth of low coherence light source  1102  shown in  FIG.  3   . Since the probe is employing low coherence light sources having central wavelength in range 800-2000 nm, and bandwidth is typically 5% and always smaller than 20% of the central wavelength, we can replace slowly varying functions A(f), B(f), and φ(f) by some constants equal to their values at the central wavelength of light source: A, B, and φ respectively. If in addition, we introduce a new parameter characterizing frequency of the fringes in the reflectance spectrum of the distance probe: 
     
       
         
           
             
               
                 
                   
                     F 
                     
                       D 
                       ⁢ 
                       P 
                     
                   
                   = 
                   
                     c 
                     
                       2 
                       · 
                       
                         ( 
                         
                           
                             L 
                             reference 
                           
                           - 
                           
                             L 
                             
                               s 
                               ⁢ 
                               a 
                               ⁢ 
                               m 
                               ⁢ 
                               p 
                               ⁢ 
                               l 
                               ⁢ 
                               e 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Using (21) we can rewrite (20) in the form 
         R   DP ( f )= A+B  cos(2π f/F   DP +φ)  (22)
 
     The reflectance spectrum of the probe R(f) exhibits fringes in spectral domain spaced by |F DP |. 
     Fringes in the reflectance spectrum of the probe are discussed also in  FIG.  10    and  FIG.  10   . 
     The spacing between fringes 
       Λ=| F   DP |  (23)
 
     can be found using many well-known numerical methods including but not limited to best fit, zero crossing and Fourier techniques. The spacing between fringes can be used to find distance between the probe and the sample using Equation 21. By (23) we can assure that the sign of F DP  is positive by gradually increasing L sample  by moving it by the means of the motion stage  21  in case of the probe  2  and  31  in case of the probe  3  from the initial position L sample =L reference  until the interference between radiation traveling in the sample and in the reference arm is observed. Alternatively In Equation 10 we can assure that the sign of F DP  is negative by gradually decreasing L sample  by moving it by the means of the motion stage  21  in case of the probe  2  and  31  in case of the probe  3  from the position L Sample &gt;&gt;L reference , where sign “&gt;&gt;” means much more than the coherence length of the radiation, until the interference between radiation travelling in the sample and in the reference arm is observed. 
     If sign and absolute value of quantity F DP  is known, then the value of F DP  is known. Then we can find L sample  from Equation 10: 
     
       
         
           
             
               
                 
                   
                     
                       F 
                       
                         D 
                         ⁢ 
                         P 
                       
                     
                     = 
                     
                       c 
                       
                         
                           2 
                           · 
                           2 
                         
                         ⁢ 
                         
                           π 
                           ⁡ 
                           ( 
                           
                             
                               L 
                               reference 
                             
                             - 
                             
                               L 
                               
                                 s 
                                 ⁢ 
                                 a 
                                 ⁢ 
                                 m 
                                 ⁢ 
                                 p 
                                 ⁢ 
                                 l 
                                 ⁢ 
                                 e 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                     
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     L 
                     sample 
                   
                   = 
                   
                     
                       L 
                       reference 
                     
                     - 
                     
                       c 
                       
                         
                           2 
                           · 
                           2 
                         
                         ⁢ 
                         
                           π 
                           · 
                           
                             F 
                             
                               D 
                               ⁢ 
                               P 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     Usually, L reference  does not change during the measurement, and it forms a constant offset to the measured value of L sample . People skilled in the art will notice that this constant offset L reference  defines a reference plane in the topography measurements and does not affect the values of measured bow and warp, waviness, or other commonly used parameters. 
     Thickness measurement using fringes generated by a two or more probes is described below. When measuring thickness of the wafer L sample  corresponds to (has value equal to) h 1  for probe  2  and h 2  for probe  3  as used in the Equation 1. Equation 2 shows how the absolute thickness of the wafer can be calculated when h 1 , h 2 , h 1calib , and h 2calib  are known. 
     The effect of the vibration on measured thickness can be reduced by simultaneous measurement of the spectra of the radiation reflected by probes  2 ,  3 . This is the subject of this invention. 
     In the first embodiment of this invention, one can simply use as two one channel spectrographs which are synchronized and detect radiation reflected by probes  2  and  3  simultaneously. 
     In the second embodiment of this invention, one can use one two-channel spectrographs detecting radiation reflected by probes  2  and  3  simultaneously. An example of such a two-channel spectrograph is shown in  FIG.  9   x   . The two-channel spectrograph is equipped with two separate and synchronized linear detectors positioned in the focal plane  100000  and detecting spectra projected by beams  90003  and  90002 . 
     In the third embodiment of this invention, one can use one two-channel spectrograph detecting radiation reflected by probes  2  and  3  simultaneously. An example of such a two-channel spectrograph is shown in  FIG.  9   x   . The two-channel spectrograph is equipped with a single two-dimensional (array) detector  100001  simultaneously detecting spectra  100003  and  100002  shown in  FIG.  10    projected by beams  90003  and  90002  respectively. 
     The fourth embodiment of this invention employs an OSDAM shown in  FIG.  3    and  FIG.  4   . In this design, the absolute value of the difference of the optical paths of the of the radiation propagating through probe  2  and  3  is much larger or the coherence length of radiation. This assures that the radiation reflected from probes  2  and  3  is added by 2×2 coupler as incoherent sum. The radiation reflected from the probe  2  and the radiation reflected by probe  3  propagating through fiber  1107  do not interfere one with another. The filtered spectrum of this radiation is detected by a single channel spectrograph  1102  equipped with a single linear detector. Since the signals originating from probes  2  and  3  are separated by much more than the coherence length of the light source when they are combined by 2×2 coupler  1108  their intensity spectra add as spectra from two incoherent sources without interference (without producing additional interference fringes). The detected spectrum by the detector of the spectrograph  1102  is a sum of spectra produced by the radiation reflected from sample  2  and  3 .  FIGS.  11 - 19    explain how the signal detected by this single channel spectrograph can be analyzed and how the spacing between the interference fringes in the spectra of the light reflected by the probe  2  and  3  which are Λ 2 , and Λ 3 . or alternatively the frequency of the fringes in frequency space be measured simultaneously. When Λ 2  and Λ 3  are known then h 1 , h 2  can be found as shown above and after performing calibration with calibration standard block thickness of the measured sample can be found using (1). 
       FIG.  7    represents the signal generated by radiation reflected from probes  2  and  3 . It is possible to analyze this signal using Fourier methods and well as using fitting the measured spectra to simulated data. The latter approach offers the advantage of providing more accurate result when very few oscillations are visible in the observed spectrum. First we will describe the fitting method. 
     The spectrum presented in  FIG.  11    is usually modeled using many parameters whose number is typically exceeding eight. The direct application of the standard procedure minimizing chi-square such as Simplex, or Levenberg-Marquardt methods having many fitting parameters often leads to poor convergence or convergence to local minima not corresponding to actual physical parameters including fringe spacings Λ 2 , and Λ 3 . We describe a robust method of finding fringe spacings Λ 2 , and Λ 3  by means of the procedure consisting of the following steps: 
     Step A: Finding approximate value of Λ 3  using the signal reflected from the second probe by measurement and analysis of the first signal alone by system employing an OSDAM shown in the  FIG.  12    where the signal from the first probe can be selected by optical switches  1201  and  1301 :
         I. initially fitting an envelope function to the spectrum reflected from the second probe as shown in the  FIG.  9       II. using the fitted envelope parameters as starting parameters to fit the entire spectrum from the second probe including oscillations produced by interference between the sample and the reference arm. This step produces fitted spectrum of the radiation reflected from the first probe as shown in  FIG.  14   .       

     Step B: finding the approximate value of Λ 2  using signal reflected from the first probe by measurement and analysis of the first signal alone by system employing an OSDAM shown in the  FIG.  15    where the signal from the first probe can be selected by optical switches  1201  and  1301 :
         i. initially fitting an envelope function to the spectrum reflected from the first probe as shown in the  FIG.  12       ii. using the fitted envelope parameters as starting parameters to fit the entire spectrum from the first probe including oscillations produced by interference between the sample and the reference arm. This step produces fitted spectrum of the radiation reflected from the first probe as shown in  FIG.  13   .       

     Step C: Fitting the entire signal shown in  FIG.  11    using starting parameters found in the steps A(ii) and B(ii) above. This step results in the signal shown in  FIG.  15   . 
     The method outlined in steps A-C is much more robust than fitting a large number (more than eight) of parameters directly to spectrum shown in  FIG.  7   . 
     A similar fitting procedure can be used when the system employs an OSDAM shown in  FIG.  3   . In this case one can eliminate fringes corresponding to probe  2  by moving it physically by means of the motion stage  21  away from the sample  8  until interference fringes in spectrum of the radiation reflected by the probe  2  become unresolved by the spectrograph  1102 . Similarly, one can eliminate fringes corresponding to probe  3  by moving it physically by means of the motion stage  31  away from the sample  8  until interference fringes in spectrum of the radiation reflected by the probe  3  become unresolved by the spectrograph  1102 . The spectra with eliminated fringes require less parameters to fit and the fit is less likely to result in the wrong local minimum. 
     When values of Λ 2 , and Λ 3  are close one can avoid danger of misassigning value of Λ 2 , and Λ 3  by slightly changing the distance between probe  2  and sample  8  by means of stage  21 . This change will affect only value of Λ 2  while value of Λ 3  will remain unchanged. This way, one can uniquely identify which of the measured values of Λ 2 , and Λ 3  belongs to probe  2  or  3 . 
     If a sample comprises of several layers such as in the sample shown in  FIG.  22    the radiation spectrum corresponding to probe  2  and probe  3  may exhibit a plurality of oscillations corresponding to interference of signal and reference radiation corresponding to reflections from the external and internal boundaries as well as Fabry-Perot fringes due to multiple reflections between layer boundaries in the sample. 
     One possible method of eliminating effects on signal generated by probe  2  on reflection from internal layers (such as layer  812 , and layer  820  respectively shown in  FIG.  18   ), similarly and effects on signal generated by probe  3  on reflection from internal layers (such as layer  812 , and layer  810  respectively shown in  FIG.  18   ), is to use radiation of such wavelength for which layers  81  and  82  are not transparent. 
     Another method which allows identification of the fringes in the reflected spectrum by probe  2  originating from the interference of the radiation reflected from the reference arm of the probe and front surface  810  comprises of following steps: 
     STEP 1: Increase the initial distance between the distance between sample  8  and the probe  2  by means of motion stage  21  to such position when the fringes originating from the interference between sample and reference arms are no longer resolved by spectrograph  1102 . 
     STEP 2: Gradually decrease the distance between probe  2  and  8  by a known amount until the oscillations from the interference between sample and reference arm become visible. These oscillations are related to interference between radiation reflected from the interface  810  and reflected by the reference arm. These oscillations are uniquely identified since the interface  810  is the closest interface to probe  2 . 
     STEP 3a: Calculate of the new position of the surface  810  using measured L sample  and information about the amount of translation of the stage in STEP 1 and STEP 2 one can measure the absolute position of the sample. 
     STEP 3b: Replace the measured sample by a reference surface such as a reference flat and comparing the position of the surface  810  to the position of the well-known reference surface. 
     Person skilled in art will notice that the similar procedure can be applied to probe  3 . In this case measurement of the position of the interface  820  is consists of the following steps: 
     STEP 1: increase the initial distance between the distance between sample  8  and the probe  3  by means of motion stage  31  to such position when the fringes originating from the interference between sample and reference arms are no longer resolved by spectrograph  1102 . 
     STEP 2: Gradually decrease the distance between probe  2  and  8  by a known amount until the oscillations from the interference between sample and reference arm become visible. These oscillations are related to the interference between radiation reflected from the interface  820  and reflected by the reference arm. These oscillations are uniquely identified since the interface  820  is the closest interface to probe  3 . 
     STEP 3a: Calculate of the new position of the surface  820  using measured L sample  and information about the amount of translation of the stage in STEP 1 and STEP 2, one can measure the absolute position of the sample. 
     STEP 3b: Replace the measured sample by the reference surface such as a reference flat. And comparing position of the surface  820  to the position of the well-known reference surface. 
     A similar method can be used to detect position of the interface  812 . If one would like to detect and identify position of the interface  812  using probe  2 , then after detecting position of the interface  810  by probe  2  in Step 3a above one can continue decreasing the distance between the probe and sample until in addition to oscillations corresponding to the interference of the radiation reflected by reference arm and surface  810  having spacing F DP,810 , a new set of oscillations corresponding to interference of the radiation from the reference arm and radiation reflected from interface  812  having spacing F DP,812  can be observed. People skilled in art will notice that the difference between frequency of these fringes can be used to calculate the optical thickness of the slab  81 . Directly from Equation 14 we get the following expression for the optical thickness T optical  of the layer  81 : 
     
       
         
           
             
               
                 
                   
                     T 
                     optical 
                   
                   = 
                   
                     
                       c 
                       
                         
                           2 
                           · 
                           2 
                         
                         ⁢ 
                         
                           π 
                           · 
                           
                             F 
                             
                               
                                 D 
                                 ⁢ 
                                 P 
                               
                               , 
                               
                                 8 
                                 ⁢ 
                                 1 
                                 ⁢ 
                                 2 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       c 
                       
                         
                           2 
                           · 
                           2 
                         
                         ⁢ 
                         
                           π 
                           · 
                           
                             F 
                             
                               
                                 D 
                                 ⁢ 
                                 P 
                               
                               , 
                               
                                 8 
                                 ⁢ 
                                 1 
                                 ⁢ 
                                 0 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     Where the optical thickness is defined as the product of the thickness of the medium and the group velocity of the radiation propagating in this medium. 
     Other method of measurement of position of the surfaces proximal to probes are described in  FIG.  29 - 48   . Initially as shown in  FIG.  29 - 33    probes are positioned distance such that the reference arms of each probe are shorter than distance to sample. The Fourier spectrum of the radiation detected on 1D detector employed by the spectrometer reveals two peaks corresponding to the signals generated by the first and the second probe. One can identify which peak corresponds to the first probe by translating probe as shown in the  FIG.  30   , and  FIG.  31   . One can also precisely find distance when the interface resides at the distance equal to the length of the reference arm by observing change of the sign the sign of the integrated peak in the IDFFTDS1 signal following procedure described in  FIG.  23   - FIG.  28   . Equation (21)-(23) imply that spacing between fringes decreases as a function of D for D&gt;0 and decrease for D&lt;0. Therefore, one can determine sign of D by calculating difference between two spectra, and by calculating Fourier transform of DFFTDS1. If differentiated peak DFFTDS1 first has negative values for smaller D and positive values for larger D this means that D&gt;0 as shown in  FIG.  31   . One can also integrate this feature and obtain spectrum IDFFTDS1 as shown in the  FIG.  32   . Person skilled in the art will notice that we will get a negative peak (minimum) for D&gt;0 and maximum for D&lt;0. We can use this fact to identify distance between probe and proximal surface to the first probe for nontransparent samples. 
     Person skilled in the art can see that the similar method can be used for the second probe in order to find position of the surface proximal to the second sample. 
     Person skilled in art will notice that this method can be also used for the transparent and multilayer transparent samples. The position of the interface between layers provided the interface is optically accessible can be determined by observing change of the sign of the observed peaks in the integrated differential spectra IDFFTS1 and IDFFTS2 as described in  FIG.  45   - FIG.  88   .