Patent Publication Number: US-9417608-B2

Title: Apparatus and method for generating interference fringe pattern

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to U.S. provisional application No. 61/794,265 filed Mar. 15, 2013 entitled “APPARATUS AND METHOD FOR GENERATING HOLOGRAMS,” the entire contents of which are hereby incorporated by reference. 
    
    
     BACKGROUND 
     1. Field of the Disclosure 
     Aspects of the disclosure generally are related to an apparatus and a method for generating an interference fringe pattern. More specifically, the disclosure is related to a holographic microscope. 
     2. Description of the Related Art 
     For real-time dual-wavelength imaging, Kühn et al. disclosed a technique to perform two-wavelength digital holographic microscopy measurements with a single hologram acquisition (Jonas Kühn, Tristan Colomb, Frédéric Montfort, Florian Charrière, Yves Emery, Etienne Cuche, Pierre Marquet, and Christian Depeursinge, Optics Express, Vol. 15, No. 12, pp. 7231-7242 (2007)). 
     Since two semiconductor laser diodes at λ1=679.57 nm and λ2=759.91 nm are used for obtaining a two-wavelength hologram, the spatial frequencies of each wavelength in the Fourier spectrum of the hologram is filtered (selected) separately. 
     The reason why Kühn et al. uses two laser sources to obtain the two-wavelength hologram is that they can create a synthetic beat wavelength which is much longer than either of the laser wavelengths. The range of the measurable sample depth is limited by the light wavelength used. Therefore, using the synthetic beat wavelength may increase the range of the sample depth measured. 
     SUMMARY 
     Embodiments of the present disclosure provide an apparatus and a method for generating holograms. 
     According to an aspect of the present embodiment, it is provided that an apparatus for generating holograms comprising: a laser source configured to emit a laser beam with a frequency of v; an acoustic optical modulator configured to generate, from the laser beam, a first beam with a frequency of v 1  and a second beam with a frequency of v 2  different from the frequency of v 1 ; a first beam splitter configured to split the first beam into a first reference beam and a first object beam, the first object beam being led to a sample with a first incident angle; a second beam splitter configured to split the second beam into a second reference beam and a second object beam, the second object beam being led to the sample with a second incident angle; and a detector configured to detect, at an exposure time, an image composed of a first interference fringe pattern, based on the first reference beam and the first object beam, and a second interference fringe pattern, based on the second reference beam and the second object beam, wherein the inverse of Δv (Δv=|v 2 −v 1 |) is shorter than the exposure time so that the detector does not detect interference fringe patterns due to a first pair of the first and second reference beams, a second pair of the first and second object beams, a third pair of the first reference beam and the second object beam, and a fourth pair of the second reference beam and the first object beam. 
     Further features of the present invention will become apparent from the following description of exemplary embodiments with reference to the attached drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates an apparatus for generating holograms. 
         FIGS. 2A and 2B  illustrate an apparatus for generating holograms explained in an embodiment 1. 
         FIGS. 3A, 3B, and 3C  illustrate an apparatus for generating holograms explained in an embodiment 2. 
         FIG. 4A  illustrates an apparatus for generating holograms explained in an embodiment 3. 
         FIG. 4B  illustrates an apparatus for generating holograms explained in an embodiment 4. 
         FIG. 5  is an interference fringe pattern. 
         FIG. 6  is an interference fringe pattern. 
         FIGS. 7A to 7D  are calculated images based on interference between reference beams 1 and 2. 
         FIGS. 8A to 8D  are calculated images based on interference between object beams 1 and 2. 
         FIGS. 9A to 9D  are calculated images based on interference between reference beam 1 and object beam 2. 
         FIGS. 10A to 10D  are calculated images based on interference between reference beam 2 and object beam 1. 
         FIGS. 11A to 11D  are calculated images based on interference between four all beams. 
         FIG. 12  is a chart to explain that no interference fringe is obtained. 
         FIGS. 13A to 13Q  are simulation results depending on time. 
         FIGS. 14A to 14D  are simulation results showing time dependent images. 
         FIG. 15  is a chart to explain that no interference fringe is obtained. 
         FIGS. 16A and 16B  are images with time-dependent terms and a final image with all terms, respectively. 
         FIGS. 17A and 17B  are images with time-dependent terms and a final image with all terms, respectively. 
         FIGS. 18A and 18B  are Fourier-spectrums obtained by using a hologram image when there is the cross-talk and isn&#39;t it, respectively. 
     
    
    
     DESCRIPTION OF THE EMBODIMENTS 
     Embodiments according to the present invention will be described below with reference to the attached drawings. 
     As for holographic microscopy, reducing acquisition time for holograms is desired, especially for the three-dimensional imaging. Therefore, the present embodiment explains a new method and apparatus with multi-angle illumination on the object. 
       FIG. 1  illustrates an apparatus for generating composite multiplex holograms using two independent lasers ( 1000  and  1050 ). The laser beam with the light frequency v 1  is expanded and spatially filtered by two lenses L 1  ( 1017 ), L 2  ( 1018 ) and a pinhole ( 1112 ). 
     The laser beam from the laser  1050  is split into a reference beam (RBJ  1915  and an object beam (OBJ  1950  by a beam splitter/coupler  1019 . The reference beam, RB 1 , is focused by a lens (L 3 )  1007  and diverging to the camera  1009  via beam splitters/couplers ( 1004 ,  1008 ), mirrors  1005 ,  1006 . The object beam, OB 1 , is angle-controlled by the 2D galvo mirror  1020  and going through a microscope, which consists of a condenser and objective lens ( 1015  and  1016 ), and the sample via beam splitters/couplers ( 1019 ,  1023 ), the mirrors ( 1112 ,  1020 ,  1014 ), and lenses ( 1021 ,  1022 ,  1013 ). The object beam is led to the camera via a mirror  1017  and a tube lens  1117 . The reference beam, RB 1  ( 1915 ), and the object beam, OB 1  ( 1950 ), are combined on the camera, having a detector, with an angle between them. The wavefront curvatures of RB 1  and OB 1  are matched so that one set of parallel fringes due to two beams interference is formed on the camera  1009  when there is no sample in the object beam, OB 1 , path. Once a sample is placed in the object beam OB 1 , that sample changes the light phase such that the wavefront of the object beam, OB 1 , is modulated. This causes the parallel fringes to be modulated. The modulation of the fringes is used to extract the refractive index property of the sample integrated along the direction of the object beam, OB 1 . 
     Another laser beam with the light frequency v 2  is expanded and spatially filtered by two lenses ( 1001 ,  1003 ) and a pinhole  1002 . The laser beam is split into a reference beam (RB 2 ,  1910 ) and an object beam (OB 2 ,  1955 ). The two reference beams are overlapped through the beam splitter  1004 . The reference beam, RB 2 , is focused by a lens  1007  (L 3 ) and diverging to the camera  1009 . The object beam, OB 2 ,  1955 , is angle controlled by the 2D galvo mirror  1010  and going through lenses  1011  and  1012 , a microscope, which consists of a condenser and objective lens ( 1015 ,  1016 ), and the sample. The reference beam, RB 2 , and the object beam, OB 2 , are combined on the camera  1009  with an angle between them. The wavefront curvatures of RB 2  and OB 2  are matched so that one set of parallel fringes due to two beams interference is formed on the camera when there is no sample in the object beam, OB 2 , path. Once a sample is placed in the object beam, OB 2 , that sample changes the light phase such that the wavefront of the object beam, OB 2 , is modulated. This causes the parallel fringes to be modulated. The modulation of the fringes is used to extract the refractive index property of the sample integrated along the direction of the object beam, OB 2 . 
     The orientation of the first set of fringes, generated by the interferences between RB 1  and OB 1  should be different from that of the second set of fringes, generated by the interferences between RB 2  and OB 2  to create the separation between two corresponding images in the Fourier space. 
     Furthermore, two lasers ( 1000 ,  1050 ) are independent and the beams coming from two lasers do not interfere and give other sets of interference fringes. It is important to have two beams not interfere with each other. Using two independent lasers is one way of realizing this condition. 
     Instead of using two laser sources to obtain a two-wavelength hologram, an apparatus and a method using one laser source will be explained below. 
     It is desirable for our disclosure to use single laser source for simplicity instead of two sources. Nevertheless, one drawback is that we will have a “cross-talk” spectrum in the Fourier space due to the interference between two object beams. However, it is discovered that there is substantially no cross-talk when two light beams have difference light frequencies. This embodiment describes the method to generate two light beams using a single light source (laser) to avoid the cross-talk. Even though these two beams will interfere, the time average of the interference becomes spatially uniform. Thus, no “cross-talk” spectrum will show up in the Fourier space. 
     Embodiment 1 
     An apparatus for generating composite multiplex holograms is illustrated in  FIG. 2A . This configuration uses one laser  2000  (wavelength λ 1 ˜543 nm). The laser beam is first split into two laser beams, LB 1  and LB 2  with an acoustic optical modulator (AOM)  2100  with a modulation frequency of Δv. Two laser beams, LB 1  and LB 2 , will have different light frequencies, v 1  and v 2 , respectively. The difference between v 1  and v 2  is Δv. For example, Δv used is 200 MHz, but the exact number is not important as long as the period of oscillation (5 ns) is much shorter than the camera exposure time (i.e., a period for a detector included in the camera to be exposed to capture an image). v 1  is (c/λ 1 ), and v 2  is (c/λ 1 )−Δv (which is 200 MHz in the current configuration). Δv used may be set at between 5 MHz and 5 GHz, for example. Thus, the AOM generates a least two beams v 1  and v 1 . In  FIG. 2A , reference numbers of  FIG. 1  are used for avoiding redundant explanations. 
     In one embodiment, the number of AOMs needed is the number of laser beams minus one. For two laser beams, only one AOM is needed. 
     The first laser beam, LB 1 , with the light frequency v 1  is expanded and spatially filtered by two lenses (L 1   1017  and L 2   1018 ) and a pinhole  1112 . The laser beam is split into a reference beam (RB 1 ,  1915 ) and an object beam (OB 1 ,  1950 ). The reference beam, RB 1 , is focused by a lens (L 3   1007 ) and diverging to the camera  1009  via the beam splitters ( 1019 ,  1004 ), mirrors ( 1005 ,  1006 ). 
     The object beam, OB 1 , is angle-controlled by the 2D galvo mirror  2020  and going through a microscope, which consists of a condenser  1015  and objective lens  1016 , and the sample  1118 . The reference beam, RB 1 , and the object beam, OB 1    1915 , are combined on the camera  1009  with an angle between them, via the beam splitters ( 1004 ,  1018 ), mirrors ( 1005 ,  1006 ), and the lens  1007 . The wavefront curvatures of RB 1  and OB 1  are matched so that one set of parallel fringes due to two beams interference is formed on the camera when there is no sample in the object beam, OB 1 , path. Once a sample is placed in the object beam OB 1 , that sample changes the light phase such that the wavefront of the object beam, OB 1 , is modulated. This causes the parallel fringes to be modulated. The modulation of the fringes is used to extract the refractive index property of the sample integrated along the direction of the object beam, OB 1 . 
     Another laser beam, LB 2 , from the same laser is split with the light frequency v 2  is expanded and spatially filtered by two lenses ( 1001 ,  1003 ) and a pinhole  1002 . The laser beam is split into a reference beam (RB 2 ,  1910 ) and an object beam (OB 2 ,  1955 ). The reference beam, RB 2 , is focused by a lens (L 3 ) and diverging to the camera via the mirrors ( 1005 ,  1006 ), the lens  1007 , and the beam splitter  1018 . The object beam, OB 2 , is angle controlled by the 2D galvo mirror  1010  and going through a microscope, which consists of the condenser and objective lens ( 1015 ,  1017 ), and the sample  1118 . The reference beam, RB 2 , and the object beam, OB 2 , are combined on the camera with an angle between them. The wavefront curvatures of RB 2  and OB 2  are matched so that one set of parallel fringes due to two beams interference is formed on the camera when there is no sample in the object beam, OB 2 , path. Once a sample is placed in the object beam, OB 2 , that sample changes the light phase such that the wavefront of the object beam, OB 2 , is modulated. This causes the parallel fringes to be modulated. The modulation of the fringes is used to extract the refractive index property of the sample integrated along the direction of the object beam, OB 2 . 
     The first reference beam  1915  may be multiplexed with the second object beam  1955  through the beam splitter  1004 . To avoid such multiplex, the beam splitters  1019 ,  1004 , and  2990  can be arranged as described in  FIG. 2B . The first reference beam  1915  is overlapped with the second reference beam  1910  through the beam splitter  2990 . 
     The orientation of the first set of fringes, generated by the interferences between RB 1  and OB 1  should be different from that of the second set of fringes, generated by the interferences between RB 2  and OB 2  to create the separation between two corresponding images in the Fourier space. 
     Furthermore, two laser beams, LB 1  and LB 2 , do interfere and give other sets of interference fringes. However, these sets of interference fringes are changing in space with a frequency of Δv. Therefore, effectively, the fringes are averaged out in time as long as the camera integration time is much longer than the inverse of Δv. This condition can be met because the camera integration time can be set on the order of milli-second (or mini-second), whereas the inverse of Δv can be set on the order of sub micro-second. 
     2D galvo mirrors ( 1010 ,  2020 ) are positioned on the conjugate planes of the sample plane so the angles of the beams coming from the 2D galvo mirrors correspond to the illumination angles on the sample  1118 . The amount of the angle movement depends on (1) the amount of the illumination angle range and the magnification of the optical relay system between the galvo mirror and the sample. Instead of the 2D galvo mirror  2020  to change an incident angle of the object beam OB 1  to the sample, a mirror which is not supposed to move can be used. 
     In general, Electric field of each beam is explained by the following equations, when phase difference between reference beam 1 and reference beam 2 is assumed to be zero at t=0):
 
 E   r1 ( x,y,t )= E   1 exp└ i ( k   r1,x   ·x+k   r1,y   ·y− 2 πv   1   t )┘,
 
 E   o1 ( x,y,t )= E   1 exp[ i ( k   o1,x   ·x+k   o1,y   ·y− 2 πv   1   t−φ   1 )],
 
 E   r2 ( x,y,t )= E   2 exp[ i ( k   r2,x   ·x+k   r2,y   ·y− 2 πv   2   t )],
 
 E   o2 ( x,y,t )= E   2 exp[ i ( k   o2,x   ·x+k   o2,y   ·y− 2 πv   2   t−φ   2 )].
 
     E r1 , E 1 , E o1 , E r2 , E 2 , and E o2  are related to electric fields. k r1,x , k r1,y , k o1,x , k o1,y , k r2,x , k r2,y , k o2,x , and k o2,y  are related to the beam angles. Φ 1  and Φ 2  are related to phase. 
     For experiment, the beam angles and intensities can be arranged as follows:
 
 k   r1,x   =k   r1,y =0 ,k   r2,x   =k   r2,y =0,
 
 k   o1,x   =k   1   ,k   o1,y =0 ,k   o2,x =0 ,k   o2,y   =k   2 ,then,
 
 E   r1 ( x,y,t )=√{square root over ( I   r1 )}exp[ i (−2 πv   1   t )],
 
 E   o1 ( x,y,t )=√{square root over ( I   o1 )}exp[ i ( k   1   ·x− 2 πv   1   t−φ   1 )],
 
 E   r2 ( x,y,t )=√{square root over ( I   r2 )}exp[ i (−2 πv   2   t )],
 
 E   o2 ( x,y,t )=√{square root over ( I   o2 )}exp[ i ( k   2   ·y− 2 πv   2   t−φ   2 )].
 
     I r1 , I o1 , I r2 , and I o2  mean intensity of the reference beam 1, object beam 1, reference beam 2, and object beam 2, respectively. 
     E total  and I total  mean the total electric field and the total intensity, respectively. 
     
       
         
           
             
               
                 
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                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             1 
                                                             
                                                             * 
                                                           
                                                           ⁡ 
                                                           
                                                             ( 
                                                             
                                                             x 
                                                             , 
                                                             y 
                                                             , 
                                                             t 
                                                             
                                                             ) 
                                                           
                                                         
                                                       
                                                       ] 
                                                     
                                                     + 
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             
                                                             E 
                                                             
                                                             r 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             1 
                                                             
                                                             
                                                             ⁡ 
                                                             
                                                             ( 
                                                             
                                                             x 
                                                             , 
                                                             y 
                                                             , 
                                                             t 
                                                             
                                                             ) 
                                                             
                                                           
                                                           · 
                                                           
                                                             
                                                             E 
                                                             
                                                             o 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             2 
                                                             
                                                             * 
                                                             
                                                             ⁡ 
                                                             
                                                             ( 
                                                             
                                                             x 
                                                             , 
                                                             y 
                                                             , 
                                                             t 
                                                             
                                                             ) 
                                                             
                                                           
                                                         
                                                         + 
                                                         
                                                           
                                                             
                                                             E 
                                                             
                                                             o 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             2 
                                                             
                                                             
                                                             ⁡ 
                                                             
                                                             ( 
                                                             
                                                             x 
                                                             , 
                                                             y 
                                                             , 
                                                             t 
                                                             
                                                             ) 
                                                             
                                                           
                                                           · 
                                                           
                                                             
                                                             E 
                                                             
                                                             r 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             1 
                                                             
                                                             * 
                                                             
                                                             ⁡ 
                                                             
                                                             ( 
                                                             
                                                             x 
                                                             , 
                                                             y 
                                                             , 
                                                             t 
                                                             
                                                             ) 
                                                             
                                                           
                                                         
                                                       
                                                       ] 
                                                     
                                                     + 
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             
                                                             E 
                                                             
                                                             r 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             2 
                                                             
                                                             
                                                             ⁡ 
                                                             
                                                             ( 
                                                             
                                                             x 
                                                             , 
                                                             y 
                                                             , 
                                                             t 
                                                             
                                                             ) 
                                                             
                                                           
                                                           · 
                                                           
                                                             
                                                             E 
                                                             
                                                             o 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             1 
                                                             
                                                             * 
                                                             
                                                             ⁡ 
                                                             
                                                             ( 
                                                             
                                                             x 
                                                             , 
                                                             y 
                                                             , 
                                                             t 
                                                             
                                                             ) 
                                                             
                                                           
                                                         
                                                         + 
                                                         
                                                           
                                                             
                                                             E 
                                                             
                                                             o 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             1 
                                                             
                                                             
                                                             ⁡ 
                                                             
                                                             ( 
                                                             
                                                             x 
                                                             , 
                                                             y 
                                                             , 
                                                             t 
                                                             
                                                             ) 
                                                             
                                                           
                                                           · 
                                                           
                                                             
                                                             E 
                                                             
                                                             r 
                                                             ⁢ 
                                                             
                                                               
                                                             
                                                             ⁢ 
                                                             2 
                                                             
                                                             * 
                                                             
                                                             ⁡ 
                                                             
                                                             ( 
                                                             
                                                             x 
                                                             , 
                                                             y 
                                                             , 
                                                             t 
                                                             
                                                             ) 
                                                             
                                                           
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     The total intensity I total  has 16 products (4 fields times 4 fields). Those 16 products can be grouped into 7 terms. The first term is the sum of the intensities all four beams. The second term is the interference term due to the v 1  reference beam and v 1  object beam. The third term is the interference term due to the v 2  reference beam and v 2  object beam. The fourth term is the interference term due to the v 1  reference beam and v 2  reference beam. The fifth term is the interference term due to the v 1  object beam and v 2  object beam. The sixth term is the interference term due to the v 1  reference beam and v 2  object beam. The seventh term is the interference term due to the v 1  object beam and v 2  reference beam. 
     Since E r1 , E o1 , E r2 , and E o2  can be rewritten as follows.
 
 E   r1 ( x,y,t )= E   1 exp└ i ( k   r1,x   ·x+k   r1,y   ·y− 2 πv   1   t )┘,
 
 E   o1 ( x,y,t )= E   1 exp[ i ( k   o1,x   ·x+k   o1,y   ·y− 2 πv   1   t−φ   1 )],
 
 E   r2 ( x,y,t )= E   2 exp[ i ( k   r2,x   ·x+k   r2,y   ·y− 2 πv   2   t )],
 
 E   o2 ( x,y,t )= E   2 exp[ i ( k   o2,x   ·x+k   o2,y   ·y− 2 πv   2   t−φ   2 )].
 
     Therefore, I total  can be rewritten as follows. 
     
       
         
           
             
               
                 I 
                 total 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   t 
                 
                 ) 
               
             
             = 
             
               
                 
                   
                     E 
                     t 
                   
                   ⁡ 
                   
                     ( 
                     
                       x 
                       , 
                       y 
                       , 
                       t 
                     
                     ) 
                   
                 
                 · 
                 
                   
                     E 
                     t 
                     * 
                   
                   ⁡ 
                   
                     ( 
                     
                       x 
                       , 
                       y 
                       , 
                       t 
                     
                     ) 
                   
                 
               
               = 
               
                 
                   [ 
                   
                       
                   
                   ⁢ 
                   
                     
                       I 
                       
                         r 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     + 
                     
                       I 
                       
                         o 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     + 
                     
                       I 
                       
                         r 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     + 
                     
                       I 
                       
                         o 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                   
                   ] 
                 
                 + 
                 
                   2 
                   · 
                   
                     
                       
                         I 
                         
                           r 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                       · 
                       
                         I 
                         
                           o 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                     
                   
                   · 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             k 
                             1 
                           
                           · 
                           x 
                         
                         - 
                         
                           ϕ 
                           1 
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   2 
                   · 
                   
                     
                       
                         I 
                         
                           r 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       · 
                       
                         I 
                         
                           o 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                     
                   
                   · 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             k 
                             2 
                           
                           · 
                           y 
                         
                         - 
                         
                           ϕ 
                           2 
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   [ 
                   
                     
                       
                         
                           I 
                           
                             r 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         · 
                         
                           I 
                           
                             r 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                       
                     
                     · 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             π 
                             · 
                             Δ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             v 
                             · 
                             t 
                           
                         
                         ) 
                       
                     
                   
                   ] 
                 
                 + 
                 
                     
                   
                     
                       [ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             
                               I 
                               
                                 o 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             · 
                             
                               I 
                               
                                 o 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                           
                         
                         · 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       k 
                                       1 
                                     
                                     · 
                                     x 
                                   
                                   - 
                                   
                                     
                                       k 
                                       2 
                                     
                                     · 
                                     y 
                                   
                                 
                                 ) 
                               
                               - 
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   π 
                                   · 
                                   Δ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   v 
                                   · 
                                   t 
                                 
                               
                               - 
                               
                                 ( 
                                 
                                   
                                     ϕ 
                                     1 
                                   
                                   - 
                                   
                                     ϕ 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                     + 
                     
                       [ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             
                               I 
                               
                                 r 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             · 
                             
                               I 
                               
                                 o 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                           
                         
                         · 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       k 
                                       2 
                                     
                                     · 
                                     y 
                                   
                                   - 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       π 
                                       · 
                                       Δ 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       v 
                                       · 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               - 
                               
                                 ϕ 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                     + 
                     
                       [ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             
                               I 
                               
                                 r 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             · 
                             
                               I 
                               
                                 o 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                           
                         
                         · 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       k 
                                       1 
                                     
                                     · 
                                     x 
                                   
                                   - 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       π 
                                       · 
                                       Δ 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       v 
                                       · 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               - 
                               
                                 ϕ 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     I total  can be also rewritten as follows. 
     
       
         
           
             
               
                 I 
                 total 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 t 
                 
                     
                 
               
               ⁢ 
               
                 { 
                 
                   
                     [ 
                     
                       
                         I 
                         
                           r 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                       + 
                       
                         2 
                         · 
                         
                           
                             
                               I 
                               
                                 r 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             · 
                             
                               I 
                               
                                 o 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                           
                         
                         · 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   k 
                                   1 
                                 
                                 · 
                                 x 
                               
                               - 
                               
                                 ϕ 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         I 
                         
                           o 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                     
                     ] 
                   
                   + 
                   
                     [ 
                     
                       
                         I 
                         
                           r 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       + 
                       
                         2 
                         · 
                         
                           
                             
                               I 
                               
                                 r 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             · 
                             
                               I 
                               
                                 o 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                           
                         
                         · 
                         
                             
                           
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       k 
                                       2 
                                     
                                     · 
                                     y 
                                   
                                   - 
                                   
                                     ϕ 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               I 
                               
                                 o 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         [ 
                         
                           
                             
                               
                                 I 
                                 
                                   r 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                               · 
                               
                                 I 
                                 
                                   r 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                             
                           
                           · 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   π 
                                   · 
                                   Δ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   v 
                                   · 
                                   t 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       + 
                       
                           
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     I 
                                     
                                       o 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   · 
                                   
                                     I 
                                     
                                       o 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                   
                                 
                               
                               · 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             k 
                                             1 
                                           
                                           · 
                                           x 
                                         
                                         - 
                                         
                                           
                                             k 
                                             2 
                                           
                                           · 
                                           y 
                                         
                                       
                                       ) 
                                     
                                     - 
                                     
                                       2 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         π 
                                         · 
                                         Δ 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         v 
                                         · 
                                         t 
                                       
                                     
                                     - 
                                     
                                       ( 
                                       
                                         
                                           ϕ 
                                           1 
                                         
                                         - 
                                         
                                           ϕ 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                           + 
                           
                               
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         I 
                                         
                                           r 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       · 
                                       
                                         I 
                                         
                                           o 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                         
                                       
                                     
                                   
                                   · 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 k 
                                                 2 
                                               
                                               · 
                                               y 
                                             
                                             - 
                                             
                                               2 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 π 
                                                 · 
                                                 Δ 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 v 
                                                 · 
                                                 t 
                                               
                                             
                                           
                                           ) 
                                         
                                         - 
                                         
                                           ϕ 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                               + 
                               
                                   
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           I 
                                           
                                             r 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             2 
                                           
                                         
                                         · 
                                         
                                           I 
                                           
                                             o 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             1 
                                           
                                         
                                       
                                     
                                     · 
                                     
                                       cos 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             ( 
                                             
                                               
                                                 
                                                   k 
                                                   1 
                                                 
                                                 · 
                                                 x 
                                               
                                               - 
                                               
                                                 2 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   π 
                                                   · 
                                                   Δ 
                                                 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   v 
                                                   · 
                                                   t 
                                                 
                                               
                                             
                                             ) 
                                           
                                           - 
                                           
                                             ϕ 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                                 } 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     The total light intensity (denoted as I total ) can be grouped into three groups. The first group is the interference pattern (vertical fringes,  FIG. 5 ) from the first beam only. The second group is the interference pattern (horizontal fringes) from the second beam only. The third group (the rest of the terms) is the interference from all other beam combinations. This third group behaves very differently for the case of two beams having the same frequency or different frequencies. When two beams have the same frequency, the third group is “time-independent” and a diagonal interference pattern appears and gives rise to the “cross talk.” When two beams have different frequencies, the third group is “time-dependent” and the time average goes to zero. This is the fundamental reason that the “cross talk” fringes goes away. 
     
       
         
           
             
               
                 I 
                 total 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   t 
                 
                 ) 
               
             
             = 
             
               
                 [ 
                 
                   
                     I 
                     
                       r 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   + 
                   
                     I 
                     
                       o 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   + 
                   
                     I 
                     
                       r 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                     
                   
                   + 
                   
                     I 
                     
                       o 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                     
                   
                 
                 ] 
               
               + 
               
                 2 
                 · 
                 
                   
                     
                       I 
                       
                         r 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     · 
                     
                       I 
                       
                         o 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                   
                 
                 · 
                 
                   cos 
                   ⁡ 
                   
                     ( 
                     
                       
                         
                           k 
                           1 
                         
                         · 
                         x 
                       
                       - 
                       
                         ϕ 
                         1 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 2 
                 · 
                 
                   
                     
                       I 
                       
                         r 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     · 
                     
                       I 
                       
                         o 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                   
                 
                 · 
                 
                   cos 
                   ⁡ 
                   
                     ( 
                     
                       
                         
                           k 
                           2 
                         
                         · 
                         y 
                       
                       - 
                       
                         ϕ 
                         2 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 [ 
                 
                   
                     
                       
                         I 
                         
                           r 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                       · 
                       
                         I 
                         
                           r 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                     
                   
                   · 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           π 
                           · 
                           Δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           v 
                           · 
                           t 
                         
                       
                       ) 
                     
                   
                 
                 ] 
               
               + 
               
                   
                 
                   
                     [ 
                     
                       
                         
                           
                             I 
                             
                               o 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                           · 
                           
                             I 
                             
                               o 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                       
                       · 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               ( 
                               
                                 
                                   
                                     k 
                                     1 
                                   
                                   · 
                                   x 
                                 
                                 - 
                                 
                                   
                                     k 
                                     2 
                                   
                                   · 
                                   y 
                                 
                               
                               ) 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 π 
                                 · 
                                 Δ 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 v 
                                 · 
                                 t 
                               
                             
                             - 
                             
                               ( 
                               
                                 
                                   ϕ 
                                   1 
                                 
                                 - 
                                 
                                   ϕ 
                                   2 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                   + 
                   
                       
                     
                       
                         [ 
                         
                           
                             
                               
                                 I 
                                 
                                   r 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                               · 
                               
                                 I 
                                 
                                   o 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                             
                           
                           · 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         k 
                                         2 
                                       
                                       · 
                                       y 
                                     
                                     - 
                                     
                                       2 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         π 
                                         · 
                                         Δ 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         v 
                                         · 
                                         t 
                                       
                                     
                                   
                                   ) 
                                 
                                 - 
                                 
                                   ϕ 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       + 
                       
                           
                         
                           [ 
                           
                             
                               
                                 
                                   I 
                                   
                                     r 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                                 · 
                                 
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                                     ⁢ 
                                     
                                         
                                     
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                               ⁡ 
                               
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                                         ⁢ 
                                         
                                             
                                         
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                                           · 
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     In order to get the best contrast ratio (100%), the intensities of all four beams should be the same. The following simulated results are calculated with the same beam intensities for simplicity. Since the intensities are not the same in general, this will affect the contrast ratios but not the conclusions we present. 
     When only v 1  (or v 2 ) is present, the result is a well-known interference fringe pattern which is time independent. 
     The interference fringe pattern due to the reference beam r 1  and the object beam o 1  is shown in  FIG. 5 . The pattern is time-independent.
 
 I   total ( x,y,t ) I 1 ( x,y )=[ I   r1   +I   o1 ]+2·√{square root over ( I   r1   ·I   o1 )}·cos( k   1   ·x−φ   1 )
 
     The interference fringe pattern due to the reference beam r 2  and the object beam o 2  is shown in  FIG. 6 . The pattern is time-independent.
 
 I   total ( x,y,t ) I 2 ( x,y )=[ I   r2   +I   o2 ]+2·√{square root over ( I   r2   ·I   o2 )}·cos( k   2   ·x−φ   2 )
 
     The time dependent parts of I total  can be examined. In order to get more insight, we will do it gradually. First, we examine the interference between reference beam 1 and reference beam 2:
 
 I   total ( x,y,t ) [ I   r1   +I   r2 ]+└√{square root over ( I   r1   ·I   r2 )}·cos(2 π·Δv·t )┘Δ v= 200 MHz
 
     As shown in  FIGS. 7A-7D , No interference fringe pattern is observed when the time t=0, 1 ns, and 2.5 ns. As noted in the equation above, there is no spatial component to the interference pattern from the two reference beams. However, there is a temporal component to the intensity which is averaged away as shown in  FIG. 7D . 
     Next, we examine the interference between object beam 1 and object beam 2:
 
 I   total ( x,y,t ) [I o1   +I   o2 ]+└√{square root over ( I   o1   ·I   o2 )}·cos( k   1   ·x−k   2   ·y )−2 π·Δv·t (φ 1 −φ 2 ))┘Δ v= 200 MHz
 
     As shown in  FIGS. 8A-8D , fringe patterns are observed when the time t=0, 1 ns, and 2.5 ns. When the time is averaged, we can see no interference fringes pattern in  FIG. 8D . When two beams (object beams 1 and 2) have the same frequency, it becomes “time-independent” and a diagonal interference pattern appears and gives rise to the “cross talk.” However, when two beams have different frequencies, it becomes “time-dependent” and the time average goes to zero. 
     As shown in  FIGS. 7 and 8 , both time averaged images do not have interference pattern because the interference terms are time dependent and get averaged out when the exposure time of the image sense by a camera is much greater than the period of the oscillation of Δv. 
     Next, the interference between reference beam 1 and object beam 2 is examined according to the following element and the result is shown in  FIGS. 9A-9D .
 
 I   total ( x,y,t ) [I r1   +I   o2 ]+└√{square root over ( I   r1   ·I   o2 )}·cos( k   2   ·y− 2 π·Δv·t )−φ 2 ))┘Δ v= 200 MHz
 
     When two beams (reference beam 1 and object beam 2) have the same frequency, it becomes “time-independent” and a diagonal interference pattern appears and gives rise to the “cross talk.” However, when two beams have different frequencies, it becomes “time-dependent” and the time average goes to zero. 
     Next, the interference between reference beam 2 and object beam 1 is examined according to the following element and the result is shown in  FIGS. 10A-10D .
 
 I   total ( x,y,t ) [I o1   +I   r2 ]+[√{square root over (I r2   ·I   o1 )}·cos( k   1   ·x− 2 π·Δv·t )−φ 2 ))]Δ v= 200 MHz
 
     When two beams (reference beam 2 and object beam 1) have the same frequency, it becomes “time-independent” and a diagonal interference pattern appears and gives rise to the “cross talk.” However, when two beams have different frequencies, it becomes “time-dependent” and the time average goes to zero. 
     As shown in  FIGS. 9D and 10D , both time averaged images do not have interference pattern because the interference terms are time dependent and get averaged out when the exposure time of the image sense is much greater than the period of the oscillation of Δv. 
     Finally, the interference pattern is examined when all beams are on the following equation, and the result is shown in  FIGS. 11A-11D . 
     
       
         
           
             
               
                 I 
                 total 
               
               ⁡ 
               
                 ( 
                 
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                   , 
                   y 
                   , 
                   t 
                 
                 ) 
               
             
             = 
             
               
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                       ⁢ 
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                                 ⁡ 
                                 
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                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               v 
                             
                             = 
                             
                               200 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               MHz 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Very interestingly, the time averaged image shows clear interference pattern as shown in  FIG. 11D . Mathematically, it is because there are still two interference terms (in the equation shown above) that are stationary. Once we drop the time dependent interference terms due to time averaging, the equation becomes as follows.
 
 I   total ( x,y,t ) I 1 ( x,y )+ I   2 ( x,y )=└ I   r1 +2·√{square root over ( I   r1   ·I   o1 )}·cos( k   1   ·x−φ   1 )+ I   o1   ┘+└I   r2 +2·√{square root over ( I   r2   ·I   o2 )}·cos( k   2   ·y−φ   2 )+ I   o2 ┘
 
     More importantly, the total intensity is exactly the sum of two sets of interference fringes associated with v 1  and v 2  without any further interference between v 1  and v 2 . This is exactly what we want. As shown in  FIG. 12 , when interference fringe patterns of  FIG. 5  and  FIG. 6  ( 3201  and  3202 ) are subtracted from the interference fringe of  FIG. 11D  ( 3200 ), no interference fringe of  FIG. 12  ( 3203 ) is obtained. 
     Simulation results according to the following equation are shown in  FIGS. 13A-13Q . 
     
       
         
           
             
               
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                               ⁢ 
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                             = 
                             
                               200 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               MHz 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     As to the time-dependent terms, described as follows, in the total of intensity, an image with time-dependent terms is shown in  FIG. 16A  and a final image with all terms including time-dependent and time-independent terms is shown in  FIG. 16B  when Δv=0. It is understood that cross-talk can be observed due to the pair of beams (reference beams 1 and 2, object beams 1 and 2, reference beam 1 and object beam 2, reference beam 2 and object beam 1). 
     
       
         
           
             
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                                   ⁢ 
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                             ⁡ 
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       
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                                       · 
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                                       ⁢ 
                                       
                                           
                                       
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                                         · 
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                                       ⁢ 
                                       
                                           
                                       
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                                         · 
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                         ] 
                       
                     
                   
                 
               
             
           
         
       
     
     However, there is no cross-talk of them when Δv is not equal to zero. An image with time-dependent terms is shown in  FIG. 17A  and a final image with all terms including time-dependent and time-independent terms is shown in  FIG. 17B  when Δv is not equal to zero. 
     A Fourier-spectrum obtained by using a hologram image when there is the cross-talk is shown in  FIG. 18A . References  4900  and  4902  drawn in round dot are information from illumination angle 2 and angle 1, respectively. The cross-talk area  4905  drawn in square dot is illustrated in  FIG. 18A . Due to the cross-talk area, it may not be able to add further illumination angle. On the other hand, there is no cross-talk area in  FIG. 18B . References  4903  and  4904  drawn in round dot are information from illumination angle 2 and angle 1, respectively. Therefore, it may be increase the number of illumination angles to obtain holograms. 
     With a phase object (a rectangle, which introduces a phase delay of π/2, in the center of the image plane), the simulation results showing time dependent images are shown in  FIGS. 14A-14D . 
     Again, the total intensity is exactly the sum of two sets of interference fringes associated with v 1  and v 2  without any further interference between v 1  and v 2 . This is exactly what we want. As shown in  FIG. 15 , when interference fringe patterns  5201  and  5202 , which are obtained by using only one angle illumination per hologram, are subtracted from the interference fringe  5200 , there is no interference fringe of  FIG. 15  ( 5203 ). 
     Embodiment 2 
     Another apparatus for generating composite multiplex holograms is illustrated in  FIG. 3 . This configuration uses one laser  2000 . The laser beam is first split into two laser beams, LB 1  and LB 2  with an acoustic optical modulator (AOM)  2100  with a modulation frequency of Δv. Two laser beams, LB 1  and LB 2 , will have different light frequencies, v 1  and v 2 , respectively. The difference between v 1  and v 2  is Δv. 
     This configuration has all the benefits afforded by the configuration presented in  FIG. 2A . Same reference number is used as in  FIG. 2A  for avoiding redundant explanations. In the previous configuration ( FIG. 2A ), the directions of two object beams, OB 1  and OB 2 , are controlled independently using two galvo mirrors. Optical fibers  3085 ,  3095  and a rotating plate  3070  are used. The optical fiber may be a single mode fiber. 
     The embodiment 2 is useful for increasing the number of the object beams. In the configuration presented in  FIG. 3A , the directions of two object beams, LB 1  and LB 2  coupled to the fibers  3095  and  3085  via the lens  3090  and  3080  respectively, are controlled by the single rotating plate  3070 . The rotating plate is used for changing the angle of the beam propagating. Rotating the plate where the ends of the fibers mount can perform the circular scan for multiple angle illumination satisfies our purpose. 
     The rotating plate  3070  is positioned such that the tips of the fibers are on the Fourier plane of the sample plane. Therefore, the positions of the fiber tips determine the angles of the corresponding illuminations. 
     By using the rotating plate, the incident angle of the beam on the sample can be changed. Instead of connecting both fibers  3085  and  3095  to the rotating plate, only one of the optical fiber ( 3085  or  3095 ) can be connected to the rotating plate, as illustrated in  FIG. 3B  which is an enlargement of  3099  in  FIG. 3A . As illustrated in  FIG. 3C , each optical fiber can be connected to each rotating plate ( 3070 A,  3070 C). A reference numbers  3060 A  3060 B are lenses. Polarizers  3000  and  3050  are used as necessary. Furthermore, there is no need to add rotating plate if we want to use more and more object beams with different incident angles. Instead of using two polarizer  3000  and  3050 , one polarizer can be placed between the beam coupler  1018  and the detector (camera  1009 ). 
     Embodiment 3 
     In the embodiment 3, the reference beams  1910  and  1915  has different paths each other as illustrated in  FIG. 4A . The reference beam  1915  is transmitted through the beam splitter  4019  and the mirror  1005 , and the reference beam  1910  is transmitted through a beam splitter  4020  and a coupler  4030 . 
     Embodiment 4 
     In the embodiment 4, the reference beams  1910  and  1915  has different paths each other as illustrated in  FIG. 4B . The reference beam  1915  is transmitted through the beam splitter  1019  and a coupler  1491 , and the reference beam  1910  is transmitted through a beam splitter  1004  and the mirror  1005 . 
     Other Embodiment 
     According to the above explained apparatuses, composite multiplex holograms which are created with multi-incident angles of the object beam can be obtained, and the holograms can be used for reconstructing a three-dimensional image of the sample. The Reconstructed image of the sample can be calculated, for example, as following steps by using a calculation unit, such as a computer. (i) After an interference fringe pattern is obtained by the camera  1009  with a detector as illustrated in  FIG. 18B , a spatial frequency spectrum is obtained by a calculation based on a numerical 2D FFT (Fast Fourier Transform). (ii) The 1st order spectrum will be cropped (selectively collected) by using a computational aperture according to the object beam angle. (iii) Then, finally calculate a reconstructed image by numerical 3D IFFT (Inverse Fast Fourier Transform) spectrum of the object by using the cropped spectrum. 
     While the embodiments according to the present invention have been described with reference to exemplary embodiments, it is to be understood that the present invention is not limited to the above described embodiments. The scope of the following claims is to be accorded the broadest interpretation so as to encompass all such modifications and equivalent structures and functions.