Abstract:
In a OCT interferometer it is necessary to balance dispersion within the reference aim with dispersion within the object aim. This is normaly done by replicating within the reference aim the components found in the object aim. This adds to the complexity and cost of the OCT interferometer. A method is provided for determining the design of and designing a simplified OCT interferometer, in which the reference aim contains only a single piece of glass of a single glass type. This reduces the cost and complexity of the OCT interferometer, and reduces power loss and undesired reflections within the reference arm.

Description:
FIELD OF INVENTION 
       [0001]    This invention relates to balancing dispersion in broadband interferometers. 
       BACKGROUND 
       [0002]    In optical coherence tomography (OCT), the longitudinal resolution in an A-scan depends partly on the degree of similarity between dispersion in a reference arm and dispersion in an object arm. Dispersion is the term given when the propagation constant of a wave has a nonlinear dependence on frequency. The zero-th derivative of the phase function of the wave with respect to the frequency indicates the phase delay, the first derivative indicates the group delay, and the second derivative indicates the group delay dispersion. 
         [0003]    Generally the dispersive properties of a component will be known and the problem is to determine what effect this will have on a time varying signal. In interferometers the effect is well known and the problem then is to balance the dispersion in each arm. 
         [0004]    A dispersion imbalance in broadband interferometers causes a reduction of signal-to-noise and a broadening of the coherence profile, which in turn means a reduction in longitudinal resolution in an OCT A-scan. The presence of dispersive elements in the interferometer is not in itself the issue, but the dispersion in both arms of the interferometer must be balanced to achieve optimal signal-to-noise and resolution. Generally this is achieved by duplicating any element located in the object arm within the reference arm. For example, lenses are usually made up of different glass types or various thicknesses, and there are generally more in the object arm than in the reference arm. Each element within each arm may contribute to dispersion of the light within the arm, generally in different ways. A common technique therefore is to include extra pieces of glass in the reference path, matching the glass types and mean thicknesses in each lens in the object path, to compensate for the additional dispersion present in the object path. Each glass type is usually bonded together to form a compound rod which is included in the reference path. While this technique does ensure the mean dispersion in each arm of the interferometer is well matched, the cost and complexity of the interferometer are increased. In addition, there is an increase in multiple reflections due to boundaries between different materials and power loss. 
         [0005]    There is a need to provide a system which equalized the dispersion in the arms of an interferometer without the cost and complexity of reproducing each functional element located in the object arm within the reference arm. 
       SUMMARY 
       [0006]    By using a single glass type with a given thickness within the reference arm in order to mimic the dispersion characteristics of various elements within the object arm, the design of an OCT interferometer can be greatly simplified. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0007]    The features and advantages of embodiments of the invention will become more apparent from the following detailed description of the preferred embodiment(s) with reference to the attached figures, wherein: 
           [0008]      FIG. 1  shows a hypothetical optical pathway; 
           [0009]      FIG. 2  shows a diagram of an example reference arm of an interferometer according to one embodiment of the invention; 
           [0010]      FIG. 3  shows a schematic representation of a reference arm of an interferometer according to one embodiment of the invention; and 
           [0011]      FIG. 4  shows a flowchart of a method by which the reference arm of an OCT interferometer is designed according to one embodiment of the invention. 
       
    
    
       [0012]    It is noted that in the attached figures, like features bear similar labels. 
       DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0013]    OCT interferometers contain various glass components, such as lenses, with varying physical properties, and generally the phase function φ(ω) is highly nonlinear. However for relatively narrow band sources (typical of OCT systems) and for frequencies far from an absorption edge (typically found in the ultraviolet for most glasses) the phase function can be accurately approximated by a Taylor series expansion: 
         [0000]    
       
         
           
             
               φ 
                
               
                 ( 
                 ω 
                 ) 
               
             
             ≈ 
             
               
                 φ 
                 0 
               
               + 
               
                 
                   ( 
                   
                     ω 
                     - 
                     
                       ω 
                       0 
                     
                   
                   ) 
                 
                 · 
                 
                   φ 
                   1 
                 
               
               + 
               
                 
                   
                     ( 
                     
                       ω 
                       - 
                       
                         ω 
                         0 
                       
                     
                     ) 
                   
                   2 
                 
                 · 
                 
                   
                     φ 
                     2 
                   
                   2 
                 
               
               + 
               … 
               + 
               
                 
                   
                     ( 
                     
                       ω 
                       - 
                       
                         ω 
                         0 
                       
                     
                     ) 
                   
                   n 
                 
                 · 
                 
                   
                     φ 
                     n 
                   
                   
                     n 
                     ! 
                   
                 
               
             
           
         
       
       
         
           
             
               where 
                
               
                   
               
                
               
                 φ 
                 n 
               
             
             = 
             
               
                 
                    
                   
                     
                         
                       n 
                     
                      
                     φ 
                   
                 
                 
                    
                   
                     ω 
                     n 
                   
                 
               
                
               
                  
                 
                   ω 
                   = 
                   
                     ω 
                     0 
                   
                 
               
             
           
         
       
     
         [0014]    and ω 0  is the center frequency of the light source. 
         [0015]    Given these conditions only terms up to second order are required, i.e. up to φ 2 . In bulk materials the phase function after propagation through a thickness of glass given by x, is related to the propagation constant of the material β such that: 
         [0000]      φ(ω)=β(ω)· x,  
 
         [0016]    and therefore after expansion: 
         [0000]    
       
         
           
             
               φ 
               n 
             
             = 
             
               
                 
                   
                      
                     
                       
                           
                         n 
                       
                        
                       φ 
                     
                   
                   
                      
                     
                       ω 
                       n 
                     
                   
                 
                  
                 
                    
                   
                     ω 
                     = 
                     
                       ω 
                       0 
                     
                   
                 
               
               = 
               
                 
                   
                     x 
                     · 
                     
                       
                          
                         
                           
                               
                             n 
                           
                            
                           β 
                         
                       
                       
                          
                         
                           ω 
                           n 
                         
                       
                     
                   
                    
                   
                      
                     
                       ω 
                       = 
                       
                         ω 
                         0 
                       
                     
                   
                 
                 = 
                 
                   x 
                   · 
                   
                     
                       β 
                       n 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0017]    For propagation in a vacuum, to which propagation in air is a good approximation, the dependence of β on ω is known explicitly: 
         [0000]    
       
         
           
             
               β 
                
               
                 ( 
                 ω 
                 ) 
               
             
             = 
             
               
                 k 
                  
                 
                   ( 
                   ω 
                   ) 
                 
               
               = 
               
                 ω 
                 c 
               
             
           
         
       
     
         [0018]    The values of β n  in a vacuum are found by differentiation to be: 
         [0000]    
       
         
           
             
               
                 β 
                 0 
               
               = 
               
                 
                   k 
                   0 
                 
                 = 
                 
                   
                     ω 
                     0 
                   
                   c 
                 
               
             
             , 
             
               
                 β 
                 1 
               
               = 
               
                 1 
                 c 
               
             
             , 
             
               
                 β 
                 2 
               
               = 
               0. 
             
           
         
       
     
         [0019]    So far only a distributed form of dispersion, in which the phase shifts caused by propagation are directly proportional to the distance travelled, has been considered. There are also cases where the phase shifts can occur over very small regions of space. The use of dielectric coatings to produce mirrors, beam splitters and anti-reflection coatings are examples. The dispersive properties of these devices are dominated by their structure rather than the materials used to make them. These devices are best described by the total phase shift, φ(ω), that they induce after transmission or reflection of the input signal, in which case the previous derivations for φ(ω) can still be used. 
         [0020]    In a more complicated system made up from several elements, all that needs to be calculated is the total phase shift after passing through the system. Referring to  FIG. 1 , a hypothetical optical pathway is shown. The optical pathway shown is merely a hypothetical example of an optical pathway, and is intended to show how the invention works. The optical pathway shown in  FIG. 1  comprises a first material  12  with a length x 1 , a second material  14  with a length x 2 , and a coating  16 . The total phase shift can be written as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       φ 
                       T 
                     
                      
                     
                       ( 
                       ω 
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         φ 
                         1 
                       
                        
                       
                         ( 
                         ω 
                         ) 
                       
                     
                     + 
                     
                       
                         φ 
                         2 
                       
                        
                       
                         ( 
                         ω 
                         ) 
                       
                     
                     + 
                     
                       
                         φ 
                         3 
                       
                        
                       
                         ( 
                         ω 
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           β 
                           1 
                         
                          
                         
                           ( 
                           ω 
                           ) 
                         
                       
                       · 
                       
                         x 
                         1 
                       
                     
                     + 
                     
                       
                         
                           β 
                           2 
                         
                          
                         
                           ( 
                           ω 
                           ) 
                         
                       
                       · 
                       
                         x 
                         2 
                       
                     
                     + 
                     
                       
                         φ 
                         3 
                       
                        
                       
                         ( 
                         ω 
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
         [0021]    where here the subscript T refers to the total phase shift and the subscripts  1 ,  2 , and  3  refer to the first material  12 , the second material  14 , and the coating  16  respectively. Expanding each term as a Taylor series leads to expressions for the phase delay, group delay and GDD: 
         [0022]    φ T0 =β 10 ·x 1 +β 20 ·x 2 +φ 30 =Total Phase Delay 
         [0023]    φ T1 =β 11 ·x 1 +β 21 ·x 2 +φ 31 =Total Group Delay 
         [0024]    φ T2 =β 12 ·x 1 +β 22 ·x 2 +φ 32 =Total GDD 
         [0025]    where the notation AF ij  represents the j th  derivative with respect to frequency for the i th  material. 
         [0026]    To balance dispersion within the arms of an interferometer the phase function φ(ω) in each arm must be equal. However some simplifications can be made. In OCT the signal is always produced by generating a carrier, either by sweeping the path length or in spectral OCT by sweeping across the frequency spectrum of the source. In either case this means the constant phase term φ 0  in each arm can be ignored, and only the second and third terms φ 1  and φ 2  must be balanced for the reference and object arms. 
         [0027]    Referring to  FIG. 2 , a diagram of an example reference arm used in an interferometer is shown according to one embodiment of the invention. The reference arm  20  includes a first air portion  22 , a glass portion  24  with a length x glass  and composed of glass of a single glass type, a second air portion  26 , a bulk component  28 , a third air portion  30 , and a discrete component  32 . 
         [0028]    The bulk component is a component necessary to produce an output signal from the reference arm, such as a lens which collimates the light within the reference arm. The discrete component is a component necessary to produce an output signal from the reference arm, such as a mirror which reflects light reaching the end of the reference arm. More generally, there are zero or more bulk components and one or more discrete components. These components are the optical components within the reference arm necessary for the reference arm to produce an output signal. They are not components added merely to compensate for dispersion within the object arm. 
         [0029]    The dispersions in the segments of the reference arm are additive. Therefore the bulk component(s) and the discrete components can be considered as a single known component for the purposes of dispersion. Similarly, the various air portions can be considered as a single air portion for the purposes of dispersion. Referring to  FIG. 3 , a schematic representation of a reference arm used in OCT is shown according to one embodiment of the invention. The schematic representation  40  of the reference arm includes an air portion  42  with a length x air , a glass portion  44  with a length x glass  of a single glass type, and a known components portion  46 . The known components portion  46  represents at least one necessary and known optical component present in the reference arm. The at least one necessary and known optical component is necessary in order for the reference arm to produce an output signal, and is not present merely to compensate for dispersion in the object arm. 
         [0030]    The schematic representation of a reference arm shown in  FIG. 3  can be used to represent the example reference arm shown in  FIG. 2 . Because dispersion is additive, the various air portions  22 ,  26 , and  30  of  FIG. 2  can be represented as a single air portion  42  in  FIG. 3  whose length x air  is equal to the sum of the lengths of the individual air portions in  FIG. 2 . The glass portion  24  of  FIG. 2  is represented simply by an identical glass portion  44 . The various known components  28  and  32  can be represented as a single known components portion  46 . The dispersion Φ known  of the known components portion  46  is equal to the sum of the dispersions of the known components, i.e. 
         [0031]    Φ known =ΣΦ component    
         [0032]    where Φ component  is the dispersion of a particular optical component. 
         [0033]    The total group delay for light passing through the reference arm is the sum of the group delay through each portion. The total group delay in the reference arm is therefore 
         [0000]      Φ ref,1   =x   air ·β air,1   +x   glass ·β glass,1 +Φ known, 1 ,
 
         [0034]    and since the dispersion of light passing through air is effectively the same as the dispersion of light passing through a vacuum, this simplifies to 
         [0000]      Φ ref,1   =x   air   /c+x   glass ·β glass,1 +Φ known, 1 ,
 
         [0035]    Similarly, the total GDD in the reference arm becomes 
         [0000]      Φ ref,2   =x   glass ·β glass,2 +Φ known, 2 .
 
         [0036]    Broadly, in producing a reference arm within an interferometer, the group delay and the group dispersion delay (GDD) of an object arm of the interferometer are determined. The GDD of the object arm is matched with the GDD of a hypothetical reference arm comprising an air portion, a glass portion comprising glass of a single glass type, and a known components portion. The glass type and the length of the glass portion is determined from the matching of the GDDs. In addition, the group delay of the object arm is matched with the group delay of the hypothetical reference arm. The length of the air portion is determined from the matching of the group delays, the glass type, and the length of the glass portion. A real reference arm can then be produced having an air portion and a glass portion each having the same properties as the air portion and the glass portion of the hypothetical reference arm. 
         [0037]    Referring to  FIG. 4 , a flowchart of a method by which a reference arm is produced according to one embodiment of the invention is shown. At step  60  the total group delay of the object arm, Φ obj,1  is determined. As explained above, the total group delay is the sum of the group delays of the components within the object arm. The components within the object arm are known once the design of the object arm is complete, including their respective glass types and thicknesses. The total dispersion of the object arm can then be determined by referring to dispersion data for each glass type, although other methods of determining the total dispersion of the object arm can be used such as by using a modelling package. In one embodiment the additional dispersion from a patient&#39;s eye is included in the total dispersion of the object arm, for example by using the dispersion properties of water. 
         [0038]    At step  62  the total GDD of the object arm, Φ obj,2 , is determined. The GDD is determined in the same way as is the total group delay, i.e. by adding the dispersive contribution of each component within the object arm, including possibly the eye. Of course the precise order of these two determinations is not important and they could in fact be determined simultaneously. 
         [0039]    At step  64  the glass type and the length X g lass of the glass portion  44  are determined by requiring that the dispersion in the reference arm is the same as the dispersion in the object arm, as desired in an OCT interferometer. Setting the GDD in each arm to be the same, 
         [0000]        x   glass ·β glass,2 +Φ known,2 =Φ obj,2 .
 
         [0040]    Since the GDD of the object arm was determined at step  62  and the GDD Φ known,2  of known components within the reference arm is known, a glass type can be chosen such that the above equality is satisfied while still giving a length x glass  of the glass portion  44  that is practical for an OCT interferometer. Once the glass type and length x glass  of the glass portion  44  are determined, then at step  66  the length xair of the air portion  42  is determined by setting the group delay in each arm to be the same: 
         [0000]        x   air   /c+x   glass ·β glass,1 +Φ known,1 =Φ obj,1 .
 
         [0041]    Since the glass type of the glass portion  44 , and hence β glass,1 , the length x glass , the group delay Φ obj,1  within the object arm, and the group delay Φ known,1  of known and necessary components within the reference arm are known, the length x air  of the air portion  42  can be determined. Of course if the total length of x air  and x glass  would not result in a practical length for the reference arm of an OCT interferometer, then a different glass type of the glass portion  44  can be used. 
         [0042]    Once the glass type, the length of the glass portion, and the total length of the air portion or air portions are determined, a reference arm for use in an OCT interferometer can be built having these properties. The length of any one or more of the air portion or air portions can be set so that the total length of the air portion or air portions is equal to that determined total air portion length x air . Similarly an OCT interferometer having such a reference arm can be built. 
         [0043]    The invention has been described with reference to an OCT interferometer. More generally, the invention may be used to provide the reference arm of any interferometer. 
         [0044]    The invention has been described as placing a single glass portion of determined length and type in the reference arm of an OCT interferometer in order to compensate for differences in dispersion between the object arm and the reference arm of the OCT interferometer. Alternatively, a single glass can be placed in the object arm in order to compensate for such differences if the dispersion in the reference arm would otherwise be higher than that in the object arm, the length and type of glass being determined as described above. 
         [0045]    The embodiments presented are exemplary only and persons skilled in the art would appreciate that variations to the embodiments described above may be made without departing from the spirit of the invention. The scope of the invention is solely defined by the appended claims.