Patent Publication Number: US-9423236-B2

Title: Apparatus for optical interferometric measurement and method for the same

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     The present application is based on and claims priority to Japanese patent application No. 2014-114884, filed Jun. 3, 2014, the disclosure of which is hereby incorporated by reference herein in its entirety. 
     BACKGROUND 
     1. Technical Field 
     This invention is related to an apparatus for optical interferometric measurement (optical interferometric measurement apparatus) and a method for the same. In particular, this invention is related to the apparatus and the method adapted to measure a movement of an object in shorter than or equal to the nanometer scale by acquiring a shift of a phase of an interference signal. The shift can be measured by using a point-spread-function that is determined by applying Fourier transform to the interference signal. 
     Conventionally, an optical interferometric measurement apparatus has been known. The conventional apparatus is configured to acquire measurement data by periodically sampling a spectrum interference signal, which is detected by an interferometer, and to determine a point-spread-function by applying Fourier transform to the acquired measurement data, thereby measuring a movement of an object based on a shift amount of a peak position in the determined point-spread-function. 
     This conventional optical interferometric measurement apparatus is, however, not able to detect (measure) a movement of the object in shorter than or equal to a micrometer scale since the point-spread-function itself spreads in the micrometer scale (i.e., the resolution of the point-spread-function is micrometer scale). 
     2. Description of Related Art 
     Here, an optical interferometric measurement apparatus that can measure a movement of an object in the sub-nanometer scale is taught by Non Patent Literature 1 (NPL 1): B. Braaf et al., “Phase-stabilized optical frequency domain imaging at 1-μm for the measurement of blood flow in the human choroid,” Optics Express, USA, October 2011, Vol 19, No. 22, pp. 20886-20903. The apparatus of NPL1 is equipped with an optical interferometer for measurement (measuring interferometer) and an optical interferometer for reference (referring interferometer), acquires monitoring data by sampling a spectrum interference signal detected by the referring interferometer and measurement data by sampling a spectrum interference signal detected by the measuring interferometer, calculates correlation between the acquired monitoring data and the acquired measurement data, and removes a noise of the spectrum interference signal detected by the measuring interferometer, thereby achieving a measurement in the sub-nanometer scale. 
     SUMMARY 
     The apparatus of NPL 1 is configured to acquire monitoring data by sampling a spectrum interference signal detected by the referring interferometer, to acquire measurement data by sampling a spectrum interference signal detected by the measuring interferometer, and to calculate correlation between the monitoring data and the measurement data. Although this can achieve a measurement in the sub-nanometer scale, the calculation of the correlation takes a long time. 
     To solve the disadvantage, it is an object of the present invention to provide an apparatus and a method for optical interferometric measurement that can measure a movement in the sub-nanometer scale at a high processing speed. 
     To achieve the above object, an aspect of the present invention provides an optical interferometric measurement apparatus including an interference optical system adapted to output a monitoring interference signal and a measurement interference signal in accordance with light emitted from a wavelength-swept light source, and a controller adapted to measure a movement of an object to be measured. The controller includes a storage that stores monitoring data acquired by sampling the monitoring interference signal in each period of the light source and measurement data acquired by sampling the measurement interference signal in each period of the light source, a calculator that calculates a departure of a sweep start frequency from an average value of the sweep start frequencies in each period based on the monitoring data and converts the calculated departure into a difference in the number of sampling points, and a Fourier transformation unit that applies Fourier transform to the measurement data in each period. The calculator defines a maximum difference in the number of sampling points and aligns a sampling width of the measurement data in each period by removing data as many as the defined maximum difference in the number of sampling points from a sampling start point and retroactively from a sampling end point, The Fourier transformation unit applies the Fourier transform to the measurement data the width of which has been aligned by the calculator. Further, the controller determines a phase of the measurement interference signal based on the Fourier-transformed measurement data and measures the movement of the object based on the determined phase. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a schematic view for showing an example of an optical interferometric measurement apparatus according to an embodiment of the present invention; 
         FIG. 2  is an explanatory view for schematically showing wavelength and a period of a wavelength-swept light source illustrated in  FIG. 1 ; 
         FIG. 3  is an explanatory view for showing a relation between a change of an optical frequency and an optical path difference; 
         FIG. 4  is an explanatory view for showing raw data representing temporal change in distance, which corresponds to temporal fluctuation of a phase, detected when an object to be measured is intentionally moved, and for showing processed data acquired from the raw data by using a low-pass filter; 
         FIG. 5  is an explanatory view for showing several processed data acquired from corresponding raw data detected when the object is intentionally moved by 10 nanometers to 50 nanometers; 
         FIG. 6  is an explanatory view for showing a waveform of a spectrum interference signal without a phase error and a waveform of a spectrum interference signal having a phase error; 
         FIG. 7  is an explanatory view for explaining the phase error caused by a shift of sweep start frequencies; 
         FIG. 8  is an explanatory view for showing a waveform of the spectrum-measurement interference signal having the phase error that is caused by the shift of sweep start frequencies; and 
         FIG. 9  is an explanatory view for showing the sampling data of  FIG. 8  spectrum interference signal after aligning a sampling width of the sampling data. 
     
    
    
     DETAILED DESCRIPTION 
     Hereinafter, an apparatus and a method for optical interferometric measurement according to an embodiment of the present invention will be explained with reference to the drawings. 
     Embodiment 
       FIG. 1  is a schematic overview showing an optical interferometric measurement apparatus according to an embodiment of the present invention. The optical interferometric measurement apparatus has a wavelength-swept light source  1 .  FIG. 2  is an explanatory view for showing an example of wavelength of the light source  1 . Note a conventional wavelength-swept light source can be applied to the embodiment as the light source  1 . 
     Here, f represents an optical frequency, ΔF represents a frequency sweep width (wavelength sweep width), T represents a period (duration of a cycle) of the wavelength-swept light source  1 , and fs represents a sweep start frequency (start frequency) of the light source  1  for each period T. Further, Δf shown in  FIG. 3  represents a beat frequency generated in response to a difference in length (gap) between a reference optical path and a measurement optical path. 
     The optical interferometric measurement apparatus is equipped with a first interference optical system  2  as a monitoring interference optical system and a second interference optical system  3  as a measurement interference optical system. The first interference optical system  2  includes a first reference optical path  2   a , a first measurement optical path  2   b , and a combined optical path  2   m . The light emitted from the wavelength-swept light source  1  is guided to both the first reference optical path  2   a  and the first measurement optical path  2   b . Note in this specification, the first and second interference optical systems  2 ,  3  are collectively called an “interference optical system.” 
     The first reference optical path  2   a  includes a photocoupler  2   c , an optical fiber  2   d , a photocoupler  2   e , a circulator  2   f , a collimator lens  2   h , and a reference mirror  2   g.    
     The first measurement optical path  2   b  includes the photocoupler  2   c , the optical fiber  2   d , the photocoupler  2   e , a circulator  2   i , a collimator lens  2   j , and a measurement mirror  2   k . The combined optical path  2   m  has a photocoupler  2   n  and a balanced detector  2   p.    
     The light emitted from the wavelength-swept light source  1  is guided to the photocoupler  2   c  and divided into light guided to the first interference optical system  2  and light guided to the second interference optical system  3  by the photocoupler  2   c . The light guided to the optical fiber  2   d  of the first interference optical system  2  is further divided by the photocoupler  2   e  and guided to the circulators  2   f  and  2   i  respectively. 
     The light guided to the circulator  2   f  is collimated into a parallel beam by the collimator lens  2   h  and guided to the reference mirror  2   g . The parallel beam is then reflected by the reference mirror  2   g  and returned to the collimator lens  2   h . The returned light is condensed by the collimator lens  2   h  and guided back to the circulator  2   f.    
     The light guided to the circulator  2   i  is collimated into a parallel beam by the collimator lens  2   j  and guided to the measurement mirror  2   k . The parallel beam is then reflected by the measurement mirror  2   k  and returned to the collimator lens  2   j . The returned light is condensed by the collimator lens  2   j  and guided back to the circulator  2   i.    
     The reference mirror  2   g  and measurement mirror  2   k  are fixed. A gap (difference in length) between the reference mirror  2   g  and measurement mirror  2   k  is represented as Δz and is expressed by: Δz=z 2 −z 1 , where z 1  represents a total optical path length of the light passing through the first reference optical path  2   a , and z 2  represents a total optical path length of the light passing through the first measurement optical path  2   b.    
     The light reflected by the reference mirror  2   g  and guided back to the circulator  2   f  and the light reflected by the measurement mirror  2   k  and guided back to the circulator  2   i  are further guided to the photocoupler  2   n  on the combined optical path  2   m . Both of the lights guided to the photocoupler  2   n  are then combined so as to interfere with each other and guided to the balanced detector  2   p . The balanced detector  2   p  outputs a spectrum-monitoring interference signal (monitoring interference signal) S(t) in accordance with the lights guided thereto. The spectrum-monitoring interference signal S(t) includes the beat frequency Δf that is generated in response to the gap (difference in length) Δz. 
     The second interference optical system  3  includes a second reference optical path  3   a , a second measurement optical path  3   b , and a combined optical path  3   m . The light emitted from the wavelength-swept light source  1  is guided to both the second reference optical path  3   a  and the second measurement optical path  3   b.    
     The second reference optical path  3   a  includes the photocoupler  2   c , an optical fiber  3   d , a photocoupler  3   e , a circulator  3   f , a collimator lens  3   h , and a reference mirror  3   g.    
     The second measurement optical path  3   b  includes the photocoupler  2   c , the optical fiber  3   d , the photocoupler  3   e , a circulator  3   i , a collimator lens  3   j , and an object to be measured  3   k . The combined optical path  3   m  has a photocoupler  3   n  and a balanced detector  3   p . Note although  FIG. 3  schematically shows a mirror as the object  3   k , this is only an example. The object  3   k  may be blood flow, for example. 
     The light emitted from the wavelength-swept light source  1  is divided by the photocoupler  2   c . The light divided and guided to the optical fiber  3   d  of the second interference optical system  3  is further divided by the photocoupler  3   e  and then guided to the circulators  3   f  and  3   i  respectively. 
     The light guided to the circulator  3   f  is collimated into a parallel beam by the collimator lens  3   h  and guided to the reference mirror  3   g . The parallel beam is then reflected by the reference mirror  3   g  and returned to the collimator lens  3   h . The returned light is condensed by the collimator lens  3   h  and guided back to the circulator  3   f.    
     The light guided to the circulator  3   i  is collimated into a parallel beam by the collimator lens  3   j  and guided to the object to be measured  3   k . The parallel beam is then reflected by the object  3   k  and returned to the collimator lens  3   j . The returned light is condensed by the collimator lens  3   j  and guided back to the circulator  3   i.    
     The light reflected by the reference mirror  3   g  and guided back to the circulator  3   f  and the light reflected by the object  3   k  and guided back to the circulator  3   i  are further guided to the photocoupler  3   n  on the combined optical path  3   m . Both of the lights guided to the photocoupler  3   n  are then combined so as to interfere with each other and guided to the balanced detector  3   p.    
     The balanced detector  3   p  outputs a spectrum-measurement interference signal (measurement interference signal) S(t) in accordance with the lights guided thereto. The spectrum-measurement interference signal S(t) includes the beat frequency Δf that is generated in response to a gap (difference in length) between the reference mirror  3   g  and the object  3   k . Note in this specification, the spectrum-measurement interference signal and the spectrum-monitoring interference signal are both expressed by the same reference character, S(t). 
     The spectrum-monitoring interference signal S(t) and the spectrum-measurement interference signal S(t) are inputted to a processor (controller)  4 . The processor  4  includes sampling units  4   a ,  4   a ′, storages (memories)  4   b ,  4   b ′, a calculator  4   c , and a Fourier Transformation unit  4   d . The sampling unit  4   a  samples the spectrum-monitoring interference signal S(t) in each period T of the wavelength-swept light source  1 . The sampled data is stored in the storage  4   b  as monitoring data. 
     The sampling unit  4   a ′ samples the spectrum-measurement interference signal S(t) in each period T of the wavelength-swept light source  1 . Note the sampling unit  4   a ′ is synchronized with the sampling unit  4   a  to sample the spectrum-measurement interference signal S(t). The sampled data is stored in the storage  4   b ′ as measurement data. 
     The monitoring data and the measurement data respectively stored in the storages  4   b ,  4   b ′ are sent to the calculator  4   c . The calculator  4   c  eliminates (removes) some data from the measurement data as described later. The remaining measurement data is sent to the Fourier transformation unit  4   d  and applied Fourier transform in each period T. 
     The Fourier transformation unit  4   d  determines a point-spread-function PSF(f) by multiplying the spectrum-measurement interference signal S(t) and a window-function W(t). To be specific, the point-spread-function PSF(f) is determined by the following equation:
 
PSF( f )= C∫S ( t ) W ( t )exp( i 2πft) dt   (1)
 
     where C is a coefficient, and i represent an imaginary unit. Note the window-function W(t) is used to smooth the point-spread-function PSF(f). 
     As illustrated in  FIG. 3 , in the second interference optical system  3 , time required for the light (reference light) to return from the reference mirror  3   g  and time required for the light (measurement light) to return from the object  3   k  differ from each other depending on the difference in the lengths of the optical paths. This time difference is expressed as a delay time Δt. Note a corresponding delay time Δt in the first interference optical system  2  is a time difference between the time required for the light to return from the reference mirror  2   g  and time required for the light to return from the measurement mirror  2   k.    
     Accordingly, the frequency of the measurement light increases by Δf compared to the frequency of the reference light fs. Here, the frequency difference Δf represents the beat frequency of the spectrum interference signal S(t). 
     The beat frequency Δf corresponds to the variable f of the point-spread-function PSF(f). The variable f is expressed by:
 
 f=ΔF·Δt/T   (2)
 
     where ΔF represents the frequency sweep width. From the equation (2), the following equation is introduced:
 
 c·f·T/ΔF=cΔt   (3)
 
     where c represents the speed of light. 
     Here, the gap Δz (difference in length of the optical paths) is described as cΔt. Therefore, by determining the frequency sweep width ΔF and the variable f, it is possible to calculate the gap Δz. However, the accuracy of the gap Δz calculated as explained above is limited to a scale of a half value width of the point-spread-function PSF(f). In other words, it is not possible to calculate the gap Δz in less than or equal to the scale of the half value width of the point-spread-function PSF(f), particularly, it is limited to the micrometer scale. 
     To achieve a measurement in shorter than or equal to the nanometer scale, the embodiment uses phase data (phase) included in the point-spread-function PSF(f). However, the phase fluctuates. Hence, a low-pass filter having a cutoff frequency of 25 Hz is used to remove the temporal fluctuation of the phase. 
     The optical interferometric measurement apparatus according to the embodiment determines the point-spread-function PSF(f) by applying Fourier transform to the spectrum-measurement interference signal S(t) acquired by using the reference mirror  3   g  and the object  3   k  in the second interference optical system  3 . The apparatus then determines the phase φ from the point-spread-function PSF(f) and converts the temporal fluctuation of the determined phase φ into distances.  FIG. 4  shows raw data of the distances. In this example, a movable minor is used as the object  3   k.    
     In  FIG. 4 , the horizontal axis shows time, and the vertical axis shows movements (change of the distances) of the object  3   k , which correspond to the temporal fluctuation of the phase φ. The line Q 1  represents the raw data of the change of the distances (i.e., the temporal fluctuation of the phase φ), and the line Q 2  represents processed data. The processed data is acquired by removing the temporal fluctuation of the phase φ with the low-pass filter and then by converting the data into distances. 
     Theoretically, the temporal fluctuation of the phase φ (phase error φe) does not appear unless the object  3   k  moves. However, the phase φ (raw data) determined based on the point-spread-function PSF(f) generally has temporal fluctuation as shown in  FIG. 4 . Using the low-pass filter removes the temporal fluctuation and gives a stable result of the phase φ. 
     If the object  3   k  is intentionally moved (shifted), the phase φ shifts (fluctuates). Accordingly, it is possible to detect a movement (shift) of the object  3   k  based on the fluctuation of the phase φ. The processed data Q 2  of  FIG. 4  shows the fluctuation of the phase φ when the object  3   k  is moved by 50 nm with respect to the reference minor  3   g.    
     Further,  FIG. 5  shows several processed data Q 2  representing the fluctuations of the phase φ when the object  3   k  is moved by 10 nm, 20 nm, 30 nm, 40 nm, and 50 nm. 
     As clearly illustrated in  FIGS. 4 and 5 , it becomes possible to detect the movement of the object  3   k  in the nanometer scale by removing the temporal fluctuation of the phase φ (phase error φe). In the above explanation, the temporal fluctuations are removed by using the low-pass filter. However, the movement of the object  3   k  is unpredictable. For example, if the movement of the object  3   k  has the same period as the period of the temporal fluctuation of the phase, the low-pass filter may cutoff not only the fluctuation of the phase but also the fluctuation caused by the movement of the object  3   k . As a result, removing the temporal fluctuation of the phase φ (phase error φe) by using the low-pass filter is not preferable and is unreliable. 
     To overcome such a deficiency, the inventors of the present invention considered about a relation between the phase error φe and a difference in the number of sampling points (sampling shift). 
     When the gap (difference in length) Δz between the reference minor  2   g  and the measurement mirror  2   k  of the first interference optical system  2  is set to be, for example, Δz=5 mm, the wavelength sweep width of the optical frequency of the wavelength-swept light source  1  is set to be, for example, 42 nm, the center wavelength of the wavelength-swept light source  1  is set to be, for example, λ0=1550 nm (i.e., the optical frequency width is 5.2 terahertz (THz)), and the sweep time is set to be, for example, 20 μs (i.e., the sweep frequency is 50 kHz); the beat frequency Δf of the spectrum-monitoring interference signal S(t) is determined to be 9.0 MHz. Accordingly, when the signal S(t) does not have a phase error φe, the spectrum interference signal S(t) is drawn as the line (A) of  FIG. 6 . 
     On the other hand, when the spectrum interference signal S(t) has a phase error φe in each period T, the signal S(t) is drawn as the line (B) of  FIG. 6 . 
     The phase error φe is expressed by:
 
φ e= 2π× sh×Δf/Sf    (4)
 
     where sh represents a departure from the sampling start point (i.e., the sampling shift, to be specific, difference in the number of sampling points), Δf represents the beat frequency of the spectrum interference signal, and Sf represents the sampling frequency (i.e., the number of sampling points per unit time S). 
     As a result, when the sampling frequency Sf is set to be, for example, 500 MHz, the beat frequency Δf is set to be, as explained above, 9 MHz, and the sampling shift is set to be  1 ; the phase error φe is determined to be φe=0.113 radian. By converting this phase error φe into a distance using the center wavelength (λ0=1550 nm), the gap Δz is determined to be Δz=14 nm. Since the sampling shift (difference in the number of sampling points) is normally 10 or more, the shift likely causes an inacceptable problem in the measurement in the nanometer scale. Note that the apparatus and the method of the embodiment uses a conventional equation to convert the phase error φe into a distance (length). This conventional equation is explained later. 
     As is known, intensities of optical interference fringes of the spectrum-monitoring interference signal S(t) can be expressed based on motion equations of optical vibrations on a complex plate. To be specific, the vibrations of the light passing through the first reference optical path  2   a  are expressed by: exp(i2πfz 1 /c), and the vibrations of the light passing through the first measurement optical path  2   b  are expressed by: exp(i2π(f+Δf)z 2 /c). Accordingly, the intensities of the optical interference fringes of the spectrum-monitoring interference signal S(t) are expressed by the square of the absolute value of: {exp(i2πfz 1 /c)+exp(i2π(f+Δf)z 2 /c)}. 
     By replacing (2πfz 1 /c) with α (i.e., α=2πfz 1 /c) and {2π(f+Δf)z 2 /c} with β (i.e., β=2π(f+Δf)z 2 /c), the intensities of the optical interference fringes of the spectrum-monitoring interference signal S(t) are determined as follows: 
     
       
         
           
             
               
                 
                   
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                               ⅈ 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 + 
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   β 
                                 
                               
                               ) 
                             
                           
                           2 
                         
                       
                       = 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 + 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   β 
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 + 
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   β 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         = 
                         
                           
                             
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               α 
                             
                             + 
                             
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               β 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 α 
                                 · 
                                 cos 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               β 
                             
                             + 
                             
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               α 
                             
                             + 
                             
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               β 
                             
                             + 
                             
                               2 
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 α 
                                 · 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               β 
                             
                           
                           = 
                           
                             2 
                             + 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 α 
                                 · 
                                 cos 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               β 
                             
                             + 
                             
                               2 
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 α 
                                 · 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               β 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Here, based on the formulas of trigonometrical function: cos(α+β)=cos α·cos β−sin α·sin β, cos(α−β)=cos α·cos β+sin α·sin β, the following equations are introduced:
 
cos(α+β)+cos(α−β)=2 cos α·cos β
 
cos(α−β)−cos(α+β)=2 sin α·sin β
 
     Accordingly, the above equation (6) is expressed by:
 
2+2 cos(α−β).   (7)
 
     In this equation, α=2πfz 1 /c, β=2π(f+Δf)z 2 /c, and Δz=z 2 −z 1 . Thus, 
     
       
         
           
             
               α 
               - 
               β 
             
             = 
             
               
                 
                   ( 
                   
                     2 
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         fz 
                         1 
                       
                       / 
                       c 
                     
                   
                   ) 
                 
                 - 
                 
                   { 
                   
                     2 
                     ⁢ 
                     
                       π 
                       ⁡ 
                       
                         ( 
                         
                           f 
                           + 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             f 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         z 
                         2 
                       
                       / 
                       c 
                     
                   
                   } 
                 
               
               = 
               
                 
                   
                     { 
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           f 
                           ⁡ 
                           
                             ( 
                             
                               
                                 z 
                                 2 
                               
                               - 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 z 
                               
                             
                             ) 
                           
                         
                         / 
                         c 
                       
                     
                     } 
                   
                   - 
                   
                     { 
                     
                       2 
                       ⁢ 
                       
                         π 
                         ⁡ 
                         
                           ( 
                           
                             f 
                             + 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               f 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           z 
                           2 
                         
                         / 
                         c 
                       
                     
                     } 
                   
                 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               fz 
                               2 
                             
                             / 
                             c 
                           
                         
                         - 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             z 
                             / 
                             c 
                           
                         
                       
                       ) 
                     
                     - 
                     
                       ( 
                       
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               fz 
                               2 
                             
                             / 
                             c 
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               fz 
                               2 
                             
                             / 
                             c 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     - 
                     
                       ( 
                       
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             z 
                             / 
                             c 
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               fz 
                               2 
                             
                             / 
                             c 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     Consequently, the intensities of the optical interference fringes of the spectrum-monitoring interference signal S(t) are determined by: 
     
       
         
           
             
               
                 
                   
                     2 
                     + 
                     
                       2 
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           α 
                           - 
                           β 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       2 
                       + 
                       
                         2 
                         ⁢ 
                         cos 
                         ⁢ 
                         
                           { 
                           
                             - 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   f 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     z 
                                     / 
                                     c 
                                   
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       fz 
                                       2 
                                     
                                     / 
                                     c 
                                   
                                 
                               
                               ) 
                             
                           
                           } 
                         
                       
                     
                     = 
                     
                       2 
                       + 
                       
                         2 
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 f 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   z 
                                   / 
                                   c 
                                 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     fz 
                                     2 
                                   
                                   / 
                                   c 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     That is to say, the spectrum-monitoring interference signal S(t) having the intensities of the optical interference fringes of: 4 cos(2πfΔz/c+2πΔfz 2 /c), is outputted from the balanced detector  2   p . Note the intensities of the optical interference fringes are doubled due to output characteristics of the balanced detector  2   p . Further, the constant value “2” does not contribute to the measurement according to the embodiment of this present invention, and thus is neglected. 
     A difference in the sweep start frequency in each period T of the wavelength-swept light source  1  is considered to be the most influencing factor of the phase error φe. In the following, a relation between the difference in the sweep start frequency and the phase error φe is explained. 
     The sweep frequency f is expressed by:
 
 f=fs +(Δ F/T ) t   (9)
 
     where fs represents the sweep start frequency, ΔF represents the frequency sweep width, T represents the period of the light source  1 , and t represents time. 
     As a result, the spectrum interference signal S(t) having the intensities of the optical interference fringes of: 4 cos(2πfΔz/c+2πΔfz 2 /c) and being outputted from the balanced detector  2   p , is expressed by:
 
4 cos {2π( fs +(Δ F/T ) t )Δ z/c+ 2 πΔfz   2   /c}   (10)
 
     When it is assumed that a shift (movement) of the sweep start frequency fs alone causes the phase error φe and that the sweep start frequency fs shifts by δfs in each period T, the term: 2π(fs+(ΔF/T)t)Δz/c in the equation (10), is expressed by:
 
2π( fs +(Δ F/T ) t )Δ z/c+ 2πδ fsΔz/c  
 
     Here, the term: 2πδfsΔz/c, represents the phase error φe caused by the shift of the sweep start frequency fs. Therefore, the phase error φe is expressed by:
 
φ e= 2πδ fsΔz/c   (11)
 
     Consequently, the phase error φe can be converted into the sampling shift sh based on the relation of the phase error φe, which is caused by the shift of the sweep start frequency fs, and the sampling shift (i.e., difference in the number of the sampling points) sh. 
       FIG. 7  is an explanatory view for explaining the relation of the shift δfs of the sweep start frequency fs and the sampling shift sh (i.e., difference in the number of the sampling points). 
     When the sweep start frequency fs shifts from fs to fs+δfs, the relation of the sampling shift sh and the shift δfs of the sweep start frequency fs is expressed in accordance with the formula of the tangent as follows:
 
δ fs /{(1/ S )· sh}=ΔF/T  
 
     where S represents the number of the sampling points per unit time (1 second), and sh represents the difference in the number of sampling points (i.e., sampling shift) with respect to the original sweep start frequency (the average of the sweep start frequencies) fs. 
     As a result, the shift Δfs of the sweep start frequency fs is expressed by:
 
δ fs =(Δ F/T )·{(1/ S )· sh}   (12)
 
     By substituting the equation (12) into the equation (11), the phase error φe is expressed by:
 
φ e= 2π(Δ F/T )·{(1/ S )· sh }(Δ z/c )   (13)
 
     In the equation (3), the variable f corresponds to the beat frequency Δf. Hence, the equation (3) is expressed as: cΔf·T/ΔF=cΔt. Further, the gap Δz is expressed by Δz=cΔt. Accordingly, the equation (3) is re-expressed by: Δf=Δz·ΔF/(c·T) (i.e., ΔF=Δf·c·T/Δz). By substituting this equation into the equation (13), the phase error φe together with the sampling shift sh is expressed by:
 
φ e= 2π(Δ f/S )· sh   (14)
 
     Monitoring data acquired by sampling the spectrum-monitoring interference signal S(t) is stored in the storage  4   b  and then inputted to the calculator  4   c . The calculator  4   c  calculates the sampling shift (i.e., difference in the number of the sampling points) sh with respect to the average value of the sweep start frequencies fs in each period T. Specifically, the calculator  4   c  calculates the shift δfs of the sweep start frequency fs based on the monitoring data with respect to the average value of the sweep start frequency fs in each period T and calculates the difference in the number of sampling points in accordance with the shift δfs. 
     The calculator  4   c  further calculates the maximum sampling shifts (maximum difference in the number of sampling points) shm with respect to the average value of the sweep start frequencies fs in each period T. 
     The calculator  4   c  then aligns the sampling width H of the measurement data in each period T. To be specific, the calculator  4   c  respectively removes measurement data as many as the maximum sampling shift (maximum difference in the number of sampling points) shm from the sampling start point and retroactively from the sampling end point in each period T. 
       FIG. 8  schematically shows the measurement data of the spectrum-measurement interference signal S(t) in each period T. In  FIG. 8 , the measurement data illustrated in the center does not have any phase error, the measurement data illustrated on the left side of the paper has the phase error +φe, and the measurement data illustrated on the right side of the paper has the phase error −φe. 
     As shown in  FIG. 9 , the calculator  4   c  aligns the sampling width H of the measurement data using the maximum sampling shift shm that is determined based on the monitoring data. 
     The measurement data is inputted to the Fourier transformation unit  4   d  in each period T after the sampling width H of the data is aligned. The Fourier transformation unit  4   d  applies the Fourier transform to the aligned measurement data in each period T and determines the point-spread-function PSF(f). As a result, the phase φ without the phase error φe is determined based on the point-spread-function PSF(f). 
     When the position of the object to be measured  3   k  is moved, the phase φ shifts from φ to φ′. The shift of the phase φ is expressed by: φ′−φ=2π·2Δz/λc, where λc represents the center wavelength of the sweep wavelength λ. Accordingly, the gap Δz can be determined in the nanometer scale or in the sub-nanometer scale by: Δz={(φ′−φ)/4π}×λc. Note that the equation to convert the phase difference (φ′−φ) into a distance (gap) is publicly known. 
     As explained above, the processor (controller)  4  of the embodiment stores the monitoring data, which is acquired by sampling the interference signal of the first interference optical system  2  in each period T, and the measurement data, which is acquired by sampling the interference signal of the second interference optical system  3  in each period T, into the storages  4   b  and  4   b ′ (First Step). 
     The processor  4  then calculates the shift δfs of the sweep start frequency fs with respect to the average value of the sweep start frequency fs based on the monitoring data in each period T and calculates the difference in the number of sampling points (sampling shift sh) in accordance with the calculated shift δfs (Second Step). 
     The processor  4  further calculates the maximum sampling shifts (maximum difference in the number of sampling points) shm with respect to the average value of the sweep start frequencies fs in each period T (Third Step). 
     The processor  4  then aligns the sampling width H of the measurement data in each period T (Fourth Step). To be specific, the processor  4  respectively removes measurement data as many as the maximum sampling shifts (maximum difference in the number of sampling points) shm from the sampling start point and retroactively from the sampling end point in each period T. 
     The processor  4  further determines the distance (movement) based on the phase φ determined by applying Fourier transform in each period T to the measurement data acquired in Third Step. Consequently, the apparatus and the method according to the embodiment can detect a movement of the object to be measured  3   k  in the sub-nanometer scale based on the shift of the phase φ at a high processing speed (Fifth Step). 
     Further, when the optical interferometric measurement apparatus according to the embodiment is applied to an apparatus for blood flow speed measurement, it becomes possible to measure the blood flow rate by determining movements of, for example an erythrocyte (red blood-cell), as the object  3   k  per unit time. 
     Although the present invention has been described in terms of an exemplary embodiment, it is not limited thereto. It should be appreciated that variations or modifications may be made in the embodiment described by persons skilled in the art without departing from the scope of the present invention as defined by the following claims.