Abstract:
An apparatus is provided for determining a target wavelength λ of a target photon beam. The apparatus includes a photon emitter, a pre-selection polarizer, a prism composed of a Faraday medium, a post-selection polarizer, a detector and an analyzer. The photon emitter projects a monochromatic light beam at the target wavelength λ substantially parallel to a magnetic field having strength B. The target wavelength is offset from established wavelength λ′ as λ=λ′+Δλ by wavelength difference of Δλ&lt;&lt;λ. The Faraday prism has Verdet value V. After passing through the pre-selection polarizer, the light beam passes through the prism and is incident to an interface surface at incidence angle θ 0  to the normal of the surface and exits into a secondary medium as first and second circularly polarized light beams separated by target separation angle δ and having average refraction angle θ. The secondary medium has an index of refraction of n 0 . After passing the post-selection polarizer, the detector measures target pointer rotation angle A w  based on the target separation angle δ. The analyzer determines the target wavelength λ by calculating offset pointer rotation angle ΔA w =A w −A′ w  from calibrated pointer rotation angle A′ w  based on established separation angle δ′ that corresponds to the established wavelength λ′, and by estimating the wavelength difference based on 
     
       
         
           
             
               
                 Δ 
                  
                 
                     
                 
                  
                 λ 
               
               ≈ 
               
                 - 
                 
                   
                     2 
                      
                     ɛπ 
                      
                     
                         
                     
                      
                     
                       n 
                       0 
                     
                      
                     Δ 
                      
                     
                         
                     
                      
                     
                       A 
                       w 
                     
                      
                     cos 
                      
                     
                         
                     
                      
                     
                       θ 
                       ′ 
                     
                   
                   
                     VB 
                      
                     
                         
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       θ 
                       0 
                     
                   
                 
               
             
             , 
           
         
       
     
     in which ε is an amplification factor. A method is provided incorporating operations described for the apparatus.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
       [0001]    The invention is a Continuation-in-Part, claims priority to and incorporates by reference in its entirety U.S. patent application Ser. No. 13/134,486 filed Jun. 6, 2011 titled “Magnetic Field Detection Using Faraday Effect” and assigned Navy Case 99670. 
     
    
     STATEMENT OF GOVERNMENT INTEREST 
       [0002]    The invention described was made in the performance of official duties by one or more employees of the Department of the Navy, and thus, the invention herein may be manufactured, used or licensed by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor. 
     
    
     BACKGROUND 
       [0003]    The invention relates generally to detection of small wavelength differences. In particular, this invention relates to a quantum enhanced method for determining small differences in coherent photon wavelengths using weak value amplification of a Faraday Effect. 
         [0004]    Detecting the presence of two nearly coincident wavelengths can become necessary for various operations, such as for detection of small Doppler shifts. However, conventional devices and processes exhibit deficiencies in cost and/or portability. 
       SUMMARY 
       [0005]    Conventional wavelength discrimination devices yield disadvantages addressed by various exemplary embodiments of the present invention. Various exemplary embodiments provide an apparatus for determining a target wavelength λ of a target photon beam wavelength λ using a prism composed of a Faraday medium having Verdet value V. 
         [0006]    The apparatus includes a photon emitter, a pre-selection polarizer, a prism composed of a Faraday medium, a post-selection polarizer, a detector and an analyzer. The photon emitter projects a monochromatic light beam at the target wavelength λ substantially parallel to a magnetic field having strength B. The target wavelength is offset from established wavelength λ′ as λ=λ′+Δλ by wavelength difference of Δλ&lt;&lt;λ. 
         [0007]    The light beam passes through the pre-selection polarizer and the prism. The beam is incident to an interface surface at incidence angle θ 0  to the normal of the surface and is refracted into a secondary medium as first and second circularly polarized light beams separated by separation angle δ and having average refraction angle θ. The secondary medium has an index of refraction of n 0 . 
         [0008]    The two circularly polarized light beams pass through the post-selection polarizer and reach the detector, which measures target pointer rotation angle A w  based on the separation angle δ. The analyzer determines the wavelength difference Δλ, first by calculating offset pointer rotation angle ΔA w =A w −A′ w  from calibrated pointer rotation angle A′ w  based on the separation angle θ′ that corresponds to the established wavelength λ′, and second by estimating the wavelength difference based on 
         [0000]    
       
         
           
             
               
                 Δ 
                  
                 
                     
                 
                  
                 λ 
               
               ≈ 
               
                 - 
                 
                   
                     2 
                      
                     ɛπ 
                      
                     
                         
                     
                      
                     
                       n 
                       0 
                     
                      
                     Δ 
                      
                     
                         
                     
                      
                     
                       A 
                       w 
                     
                      
                     cos 
                      
                     
                         
                     
                      
                     
                       θ 
                       ′ 
                     
                   
                   
                     VB 
                      
                     
                         
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       θ 
                       0 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    in which ε is an amplification factor. Various exemplary embodiments also provide a method that incorporates operations described for the apparatus. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0009]    These and various other features and aspects of various exemplary embodiments will be readily understood with reference to the following detailed description taken in conjunction with the accompanying drawings, in which like or similar numbers are used throughout, and in which: 
           [0010]      FIG. 1  is a diagram view of an optical diagram; and 
           [0011]      FIG. 2  is a schematic view of a wavelength discriminator apparatus. 
       
    
    
     DETAILED DESCRIPTION 
       [0012]    In the following detailed description of exemplary embodiments of the invention, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration specific exemplary embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention. Other embodiments may be utilized, and logical, mechanical, and other changes may be made without departing from the spirit or scope of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined only by the appended claims. 
         [0013]    In accordance with a presently preferred embodiment of the present invention, the components, process steps, and/or data structures may be implemented using various types of operating systems, computing platforms, computer programs, and/or general purpose machines. In addition, those of ordinary skill in the art will readily recognize that devices of a less general purpose nature, such as hardwired devices, or the like, may also be used without departing from the scope and spirit of the inventive concepts disclosed herewith. General purpose machines include devices that execute instruction code. A hardwired device may constitute an application specific integrated circuit (ASIC) or a floating point gate array (FPGA) or other related component. 
         [0014]    This disclosure provides an overview of how a combined application of a recently discovered Faraday effect and weak value amplification can be used to measure Δλ=λ−λ′ (thereby detecting λ) when λ′ is known. The methodology has potential spectroscopic utility in such areas as measuring small Doppler shifts and detecting the presence of otherwise indistinguishable chemical or biological spectroscopic markers. Such measurements can be made using either classically intense laser light or single photon streams. For example, let λ and λ′ be two wavelengths such that λ=λ′+Δλ with Δλ being small. 
         [0015]    A longitudinal magnetic field induces a circular differential refraction of a linearly polarized photon beam at the boundary between a Faraday medium and a medium with negligible Verdet constant as reported by A. Ghosh et al., “Observation of the Faraday effect . . . ” in  Phys. Rev. A  76, 055402 (2007). See either http://www.rowland.harvard.edukjf/fischedimages/PRA — 76 — 055402.pdf or else http://anciv.org/PS_cache/physics/pdf/0702/0702063v1.pdf for details. 
         [0016]    This differential refraction is independent of the photon&#39;s pathlength through the Faraday medium and occurs within a few wavelengths of the boundary. The Verdet constant V represents an optical parameter that describes the strength of Faraday rotation from interaction between light and a magnetic field for a particular material, named for French physicist Émile Verdet. 
         [0017]      FIG. 1  depicts an optical diagram  100  with accompanying coordinate reference frame  110  for a monochromatic linearly polarized coherent laser beam  120  of a wavelength λ. The beam  120  forms either a classically intense continuum or a stream of single photons that is incident from a Faraday medium  130  to a secondary medium  140  of negligible Verdet constant separated by an interface boundary  150 . The Faraday and secondary media  130  and  140  have respective indices of refraction n ±  and n 0 . The subscripts plus (+) and minus (−) respectively correspond to right and left circular polarized radiation. The diagram  100  illustrates the paths taken by the light beam  120  at the interface between the Faraday medium  130  and the secondary medium  140  with negligible Verdet constant. 
         [0018]    The beam  120  has an angle of incidence of θ 0  from the normal to the interface  150 . In the presence of a longitudinal magnetic field {right arrow over (B)} (having strength B), the beam  120  refracts at angles θ ±  from normal at the interface  150  into two circularly polarized beams having an angular divergence δ approximated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     δ 
                     ≈ 
                     
                       
                         - 
                         
                           
                             λ 
                              
                             
                                 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                           
                           
                             π 
                              
                             
                                 
                             
                              
                             
                               n 
                               0 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             θ 
                           
                         
                       
                        
                       VB 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where angle 
         [0000]    
       
         
           
             θ 
             = 
             
               
                 1 
                 2 
               
                
               
                 ( 
                 
                   
                     θ 
                     + 
                   
                   + 
                   
                     θ 
                     - 
                   
                 
                 ) 
               
             
           
         
       
     
         [0000]    is the average of θ +  and θ − , which are respectively the right- and left-circularized refraction angles, and V is the Verdet constant for the Faraday medium  130 . 
         [0019]    As shown, the incident beam&#39;s direction of propagation determines the y-axis of the reference frame  110 . The x-axis is in the plane containing the beam  120  and the normal to the interface at the point of incidence. The origin of the reference frame  110  is defined by the perpendicular intersection of the x-axis with the y-axis at the interface  150 . The usual z-axis (into the plane) with positive direction {circumflex over (z)}={circumflex over (x)}×ŷ completes the reference frame  110 . 
         [0020]    The longitudinal magnetic field B is assumed to be present and parallel to the positive y-axis. If the photon distribution of the incident beam  120  has a Gaussian distribution symmetric about the positive y-axis with mean value at x=0, then the refracted beams exhibit Gaussian distributions that are symmetric about their refracted paths which are along the vectors {right arrow over (u)} ±  in the x−y plane of the reference frame  110 . More specifically, the refracted beams exhibit photon distribution mean values which are rotationally displaced around the z-axis through distinct angles θ ± −θ 0  from the positive y-axis in the direction of vectors {right arrow over (u)} ±  in the x-y plane, respectively. 
         [0021]    This refraction process can be described from a quantum mechanical measurement perspective using the mean value of the intensity distribution profile produced by a detector as a measurement pointer. This description maintains validity for both a single photon stream and a classically intense beam. 
         [0022]    In particular, an Hermitean operator Â can be constructed and used to form a Hamiltonian operator Ĥ that describes a photon-interface interaction which produces the required geometry of the refraction process. Let |+          and |−          be the right and left circular polarization eigenstates, respectively, of the photon circular polarization operator {circumflex over (σ)} which obey the eigenvalue equation: 
         [0000]      {circumflex over (σ)}|±         =±|±           (2)
 
         [0000]    and have the orthogonality properties: 
         [0000]                ±|±         =1
 
         [0000]      and 
         [0000]                ±|∓         =0.  (3)
 
         [0023]    One can define the “which path” operator Â as: 
         [0000]        Â≡(θ   + −θ 0 )|+                   +|+(θ − −θ 0 )|−                   −|  (4)
 
         [0000]    and the associated interaction Hamiltonian H can be expressed as: 
         [0000]        Ĥ=ÂĴ   z δ( t−t   0 ).  (5)
 
         [0024]    Here the Dirac delta function δ(t−t 0 ) encodes the fact that the refraction occurs within a few wavelengths of the interface  150  by modeling the refraction effectively as an impulsive interaction between a photon of the beam  120  and the interface  150  at time t 0 . The “which path” operator Â accounts for the refractive angular displacements of the initial photon beam  120  at the interface  150 . The operator Ĵ z  constitutes the measurement pointer&#39;s z-component of angular momentum, and couples the refractive angular displacements to the measurement pointer. One can note that: 
         [0000]      └ Â,Ĵ   z ┘=0,  (6)
 
         [0000]    and that |±          are eigenstates of Â with respective eigenvalues (θ ± −θ 0 ). 
         [0025]      FIG. 2  shows an elevation schematic  200  of an apparatus that employs this angular divergence. A laser  210  emits a photon beam  220  as analogous to the beam  120 . The beam  220  passes through a pre-selection polarizer or polarizer  230  to reach a Faraday medium  240  (in the form of a prism), analogous to the medium  130 . The refracted beam passes a post-selection polarizer or polarizer  250  to reach a detector  260  that measures the intensity distribution of the refracted polarization post-selected light beams. 
         [0026]    For an initial photon polarization state |ψ i           , i.e., the pre-selected state, and an initial (Gaussian) pointer state |φ         , the initial state of the combined pre-selected system and measurement pointer prior to the interaction at the interface  150  at time t 0  constitutes the tensor product state |ψ i           |φ         . Note that the beam  220  has passed through the pre-selection filter  230  prior to its entry into the Faraday medium  240 . 
         [0027]    Immediately following the measurement&#39;s impulsive interaction, the combined system is in the state: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                              
                             Ψ 
                             〉 
                           
                           = 
                             
                            
                           
                             
                                
                               
                                 
                                   - 
                                   
                                     i 
                                     ℏ 
                                   
                                 
                                  
                                 
                                   ∫ 
                                   
                                     
                                       H 
                                       ^ 
                                     
                                      
                                     
                                        
                                       t 
                                     
                                   
                                 
                               
                             
                              
                             
                                
                               
                                 ψ 
                                 i 
                               
                               〉 
                             
                              
                             
                                
                               ϕ 
                               〉 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                                
                               
                                 
                                   - 
                                   
                                     i 
                                     ℏ 
                                   
                                 
                                  
                                 
                                   A 
                                   ^ 
                                 
                                  
                                 
                                   
                                     J 
                                     ^ 
                                   
                                   z 
                                 
                               
                             
                              
                             
                                
                               
                                 ψ 
                                 i 
                               
                               〉 
                             
                              
                             
                                
                               ϕ 
                               〉 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where use has been made of the fact that the integral of the delta function is: 
         [0000]      ∫δ( t−t   0 ) dt= 1.  (8)
 
         [0000]    Now let the initial polarization state be expressed as: 
         [0000]      |ψ i             =a|+             +b| −         ,  (9)
 
         [0000]    in which a and b are complex numbers that satisfy the condition          ψ i |ψ i           =1, and rewrite eqn. (7) as: 
         [0000]    
       
         
           
             
               
                 
                   
                      
                     Ψ 
                     〉 
                   
                   = 
                   
                     
                       
                          
                         
                           
                             - 
                             
                               i 
                               ℏ 
                             
                           
                            
                           
                             A 
                             ^ 
                           
                            
                           
                             
                               J 
                               ^ 
                             
                             z 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             a 
                              
                             
                                
                               + 
                               〉 
                             
                           
                           + 
                           
                             b 
                              
                             
                                
                               - 
                               〉 
                             
                           
                         
                         ) 
                       
                     
                      
                     
                       
                          
                         ϕ 
                         〉 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0028]    Because of the orthogonality proportion of eqn. (9), the n th  power of Â assumes the form: 
         [0000]        Â   n =(θ + −θ 0 ) n |+                   +|+(θ − −θ 0 ) n |−                   −|, n= 0,1,2, . . .   (11)
 
         [0000]    Then the exponential term of the system state of eqn. (7) can be written as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                            
                           
                             
                               - 
                               
                                 i 
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                                 J 
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                         = 
                           
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                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             ∞ 
                           
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                                 [ 
                                 
                                   
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                                 ] 
                               
                               n 
                             
                             
                               n 
                               ! 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
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                               ∑ 
                               
                                 n 
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                               ∞ 
                             
                              
                             
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         - 
                                         
                                           i 
                                           ℏ 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             + 
                                           
                                           - 
                                           
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                                             - 
                                           
                                         
                                         ) 
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           J 
                                           ^ 
                                         
                                         z 
                                       
                                     
                                     ] 
                                   
                                   n 
                                 
                                 
                                   n 
                                   ! 
                                 
                               
                                
                               
                                  
                                 + 
                                 〉 
                               
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                                 〈 
                                 + 
                                  
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             ∞ 
                           
                            
                           
                             
                               
                                 
                                   [ 
                                   
                                     
                                       - 
                                       
                                         i 
                                         ℏ 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           - 
                                         
                                         - 
                                         
                                           θ 
                                           0 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         J 
                                         ^ 
                                       
                                       z 
                                     
                                   
                                   ] 
                                 
                                 n 
                               
                               
                                 n 
                                 ! 
                               
                             
                              
                             
                                
                               - 
                               〉 
                             
                              
                             
                               〈 
                               - 
                                
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                  
                                 
                                   
                                     - 
                                     
                                       i 
                                       ℏ 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         + 
                                       
                                       - 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     
                                       J 
                                       ^ 
                                     
                                     z 
                                   
                                 
                               
                                
                               
                                  
                                 + 
                                 〉 
                               
                                
                               
                                 〈 
                                 + 
                                  
                               
                             
                             + 
                             
                               
                                  
                                 
                                   
                                     - 
                                     
                                       i 
                                       ℏ 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         - 
                                       
                                       - 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     
                                       J 
                                       ^ 
                                     
                                     z 
                                   
                                 
                               
                                
                               
                                  
                                 - 
                                 〉 
                               
                                
                               
                                 〈 
                                 - 
                                  
                               
                             
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where i=√{square root over (−1)} is the imaginary unit and 
         [0000]    
       
         
           
             ℏ 
             = 
             
               h 
               
                 2 
                  
                 π 
               
             
           
         
       
     
         [0000]    represents the reduced Planck constant. This result correlates refraction angle rotations with polarization. 
         [0029]    The exponential operators constitute the rotation operators {circumflex over (R)} z : 
         [0000]    
       
         
           
             
               
                 
                   
                      
                     
                       
                         - 
                         
                           i 
                           ℏ 
                         
                       
                        
                       
                         ( 
                         
                           
                             θ 
                             ± 
                           
                           - 
                           
                             θ 
                             0 
                           
                         
                         ) 
                       
                        
                       
                         
                           J 
                           ^ 
                         
                         z 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           R 
                           ^ 
                         
                         z 
                       
                        
                       
                         ( 
                         
                           
                             θ 
                             ± 
                           
                           - 
                           
                             θ 
                             0 
                           
                         
                         ) 
                       
                     
                     ≡ 
                     
                       
                         
                           R 
                           ^ 
                         
                         z 
                         ± 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    These operators rotate the x- and y-axes through angles (θ ± −θ 0 ) around the z-axis of the reference frame  110 . The rotation notation is consistent with the convention used by A. Messiah,  Quantum Mechanics , v. 2, p. 1068 (1961). 
         [0030]    This enables the system state in the {|{right arrow over (r)}         } representation to be rewritten as: 
         [0000]                  {right arrow over (r)}|Ψ             =a|+                       {right arrow over (r)}|{circumflex over (R)}   z   +   |φ             +b|−                       {right arrow over (r)}|{circumflex over (R)}   z   − |φ         .  (14)
 
         [0000]    The associated pointer state distribution in the {|{right arrow over (r)}         }-representation is then: 
         [0000]      |           {right arrow over (r)}Ψ             |   2   =|a|   2   |             {right arrow over (r)}|{circumflex over (R)}   z   +             φ|   2   +|b|   2   |             {right arrow over (r)}|{circumflex over (R)}   z   − |         φ| 2 ,  (15)
 
         [0000]    and clearly corresponds to a sum of two Gaussian distributions |         {right arrow over (r)}|{circumflex over (R)} z   ± |φ         | 2  which are each symmetrically distributed about the vectors {right arrow over (u)} ± , respectively. 
         [0031]    A final photon polarization state |ψ f            that is post-selected can be expressed as: 
         [0000]      |ψ f             =c|+             +d|−             (16)
 
         [0000]    in which c and d represent complex numbers that satisfy the condition          ψ f |ψ f           =1. Note that the post-selection polarizer  250  receives the beam after refraction by the Faraday medium  240 . From this, the resulting pointer state becomes: 
         [0000]      |Φ         ≡         ψ f |Ψ         = ac*{circumflex over (R)}   z   + |φ         + bd*{circumflex over (R)}   z   − |φ         ,  (17)
 
         [0000]    in which the asterisk denotes the complex conjugate, and its {|{right arrow over (r)}         }-representation distribution is: 
         [0000]      |           {right arrow over (r)}|Φ             |   2   =|ac*|   2   |             {right arrow over (r)}|{circumflex over (R)}   z   + |φ         | 2   +|bd*|   2 |           {right arrow over (r)}|{circumflex over (R)}   z   − φ         | 2 +2 Reac*bd*             {right arrow over (r)}|{circumflex over (R)}   z   + |φ                     {right arrow over (r)}|{circumflex over (R)}   2   − |φ         *.  (18)
 
         [0032]    One may observe that although eqn. (18) constitutes a sum of two Gaussian distributions that are symmetrically distributed around vector {right arrow over (u)} ± , unlike eqn. (15), this distribution also contains an interference term. Careful manipulation of this interference term can be described herein that produces the desired amplification effect. 
         [0033]    In contrast to a strong measurement, a weak measurement of the “which path” operator Â occurs when the uncertainty Δθ in the pointer&#39;s rotation angle is much greater than the separation between Â&#39;s eigenvalues, and when the interaction between a photon and the pointer is sufficiently weak so that the system remains essentially undisturbed by that interaction. In this case, the post-selected pointer state is represented as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                              
                             Φ 
                             〉 
                           
                           = 
                             
                            
                           
                             
                               〈 
                               
                                 
                                   ψ 
                                   f 
                                 
                                  
                                 
                                    
                                   
                                      
                                     
                                       
                                         - 
                                         
                                           i 
                                           ℏ 
                                         
                                       
                                        
                                       
                                         A 
                                         ^ 
                                       
                                        
                                       
                                         
                                           J 
                                           ^ 
                                         
                                         z 
                                       
                                     
                                   
                                    
                                 
                                  
                                 
                                   ψ 
                                   i 
                                 
                               
                               〉 
                             
                              
                             
                                
                               ϕ 
                               〉 
                             
                           
                         
                       
                     
                     
                       
                         
                           ≈ 
                             
                            
                           
                             
                               〈 
                               
                                 
                                   ψ 
                                   f 
                                 
                                  
                                 
                                    
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         
                                           i 
                                           ℏ 
                                         
                                          
                                         
                                           A 
                                           ^ 
                                         
                                          
                                         
                                           
                                             J 
                                             ^ 
                                           
                                           z 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                 
                                  
                                 
                                   ψ 
                                   i 
                                 
                               
                               〉 
                             
                              
                             
                                
                               ϕ 
                               〉 
                             
                           
                         
                       
                     
                     
                       
                         
                           ≈ 
                             
                            
                           
                             
                               〈 
                               
                                 
                                   ψ 
                                   f 
                                 
                                  
                                 
                                   ψ 
                                   i 
                                 
                               
                               〉 
                             
                              
                             
                                
                               
                                 
                                   - 
                                   
                                     i 
                                     ℏ 
                                   
                                 
                                  
                                 
                                   A 
                                   w 
                                 
                                  
                                 
                                   
                                     J 
                                     ^ 
                                   
                                   z 
                                 
                               
                             
                              
                             
                                
                               ϕ 
                               〉 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    or else as: 
         [0000]                  {right arrow over (r)} |Φ         ≈         ψ f |ψ i                       {right arrow over (r)}|{circumflex over (R)}   z ( ReA   w )|φ         ,  (20)
 
         [0000]    where the quantity A w  is expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       A 
                       w 
                     
                     = 
                     
                       
                         〈 
                         
                           
                             ψ 
                             f 
                           
                            
                           
                              
                             
                               A 
                               ^ 
                             
                              
                           
                            
                           
                             ψ 
                             i 
                           
                         
                         〉 
                       
                       
                         
                           
                             〈 
                             
                               ψ 
                               f 
                             
                              
                           
                            
                           
                             ψ 
                             i 
                           
                         
                         〉 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and constitutes the weak value of operator Â. Note that rotation angle A w  is generally a complex value that can be directly calculated from the associated theory. One may also note that in response to |ψ i            and |ψ f            being nearly orthogonal, the real value ReA w  can lie far outside the spectrum of eigenvalues for Â. 
         [0034]    The pointer state distribution for eqn. (20) is: 
         [0000]      |           {right arrow over (r)}|Φ             |   2 ≈         ψ f |ψ i           | 2 |           {right arrow over (r)}|{circumflex over (R)}   z ( ReÂ   w )|φ         | 2 ,  (22)
 
         [0000]    and corresponds to a broad distribution that is symmetric around a vector in the x-y plane. That vector can be determined by a rotation of the x- and y-axes through an angle ReA w  about the z-axis. In order that eqn. (20) be valid, both of the two following general weakness conditions for the uncertainty in the pointer rotation angle must be satisfied: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         a 
                         ) 
                       
                        
                       
                           
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       θ 
                     
                      
                     
                       
                          
                         
                           A 
                           w 
                         
                          
                       
                        
                       
                           
                       
                        
                       
                         and 
                          
                         
                           
 
                         
                         ( 
                         b 
                         ) 
                       
                        
                       
                           
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       θ 
                     
                      
                     
                       
                         { 
                         
                           
                             min 
                             
                               ( 
                               
                                 
                                   n 
                                   = 
                                   2 
                                 
                                 , 
                                 3 
                                 , 
                                 … 
                               
                                
                               
                                   
                               
                               ) 
                             
                           
                            
                           
                             
                                
                               
                                 
                                   A 
                                   w 
                                 
                                 
                                   ( 
                                   
                                     A 
                                     n 
                                   
                                   ) 
                                 
                               
                                
                             
                             
                               1 
                               
                                 n 
                                 - 
                                 1 
                               
                             
                           
                         
                         } 
                       
                       
                         - 
                         1 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    as reported by I. M. Duck et al., “The sense in which ‘weak measurement’ of a spin-½ particle&#39;s spin component yields a value  100 ” in  Phys. Rev. D  40, 2112-17 (1989). See http://prd.aps.org/pdf/PRD/v40/i6/p2112 — 1 for details. 
         [0035]    Use of the above pre- and post-selected states |ψ i            and |ψ f           —along with eqn. (11)—provides the following scalar expression for the weak value of the n th  moment of “which path” operator Â: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           A 
                           n 
                         
                         ) 
                       
                       w 
                     
                     = 
                     
                       
                         
                           a 
                            
                           
                               
                           
                            
                           
                             
                               
                                 c 
                                 * 
                               
                                
                               
                                 ( 
                                 
                                   
                                     θ 
                                     + 
                                   
                                   - 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                             n 
                           
                         
                         + 
                         
                           b 
                            
                           
                               
                           
                            
                           
                             
                               
                                 d 
                                 * 
                               
                                
                               
                                 ( 
                                 
                                   
                                     θ 
                                     - 
                                   
                                   - 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                             n 
                           
                         
                       
                       
                         
                           a 
                            
                           
                               
                           
                            
                           
                             c 
                             * 
                           
                         
                         + 
                         
                           b 
                            
                           
                               
                           
                            
                           
                             d 
                             * 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where c* and d* represent complex conjugates of c and d. 
         [0036]    When n=1, then the first moment corresponds to the pointer&#39;s peak intensity. The first moment is: 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     w 
                   
                   = 
                   
                     
                       
                         
                           a 
                            
                           
                               
                           
                            
                           
                             
                               c 
                               * 
                             
                              
                             
                               ( 
                               
                                 
                                   θ 
                                   + 
                                 
                                 - 
                                 
                                   θ 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           b 
                            
                           
                               
                           
                            
                           
                             
                               d 
                               * 
                             
                              
                             
                               ( 
                               
                                 
                                   θ 
                                   - 
                                 
                                 - 
                                 
                                   θ 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           a 
                            
                           
                               
                           
                            
                           
                             c 
                             * 
                           
                         
                         + 
                         
                           b 
                            
                           
                               
                           
                            
                           
                             d 
                             * 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0037]    When the transmission axis of the pre-selection polarizer  230  is set so that: 
         [0000]        a =sin φ,
 
         [0000]      and 
         [0000]        b =cos φ,  (26)
 
         [0000]    and that of the post-selection polarizer  250  is set so that: 
         [0000]        c =cos χ,
 
         [0000]      and 
         [0000]        d =−sin χ,  (27)
 
         [0000]    then the quantity A w  becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     w 
                   
                   = 
                   
                     
                       Re 
                        
                       
                           
                       
                        
                       
                         A 
                         w 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   θ 
                                   + 
                                 
                                 - 
                                 
                                   θ 
                                   0 
                                 
                               
                               ) 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             φ 
                              
                             
                                 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             χ 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   θ 
                                   - 
                                 
                                 - 
                                 
                                   θ 
                                   0 
                                 
                               
                               ) 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             φ 
                              
                             
                                 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             χ 
                           
                         
                         
                           sin 
                            
                           
                             ( 
                             
                               φ 
                               - 
                               χ 
                             
                             ) 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
         [0038]    One can observe from this that the absolute value of the “which path” scalar |A w | can be made arbitrarily large by choosing φ≈χ, i.e., separated by a small difference term ε. In particular, let χ=φ−ε and φ=π/4 (in which case the pre-selected state is linearly polarized in the x-direction). Consequently. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         a 
                         ) 
                       
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       φ 
                     
                     = 
                     
                       
                         cos 
                          
                         
                             
                         
                          
                         φ 
                       
                       = 
                       
                         
                           2 
                         
                         / 
                         2 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       
                         ( 
                         b 
                         ) 
                       
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       χ 
                     
                     = 
                     
                       
                         
                           2 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             ɛ 
                           
                           + 
                           
                             sin 
                              
                             
                                 
                             
                              
                             ɛ 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       
                         ( 
                         c 
                         ) 
                       
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       χ 
                     
                     = 
                     
                       
                         
                           2 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             ɛ 
                           
                           - 
                           
                             sin 
                              
                             
                                 
                             
                              
                             ɛ 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       
                         and 
                          
                         
                           
 
                         
                         ( 
                         d 
                         ) 
                       
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             φ 
                             - 
                             χ 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         ɛ 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0039]    The previous relation from eqn. (28) for the amplified pointer rotation angle associated with the post-selected circularly polarized beams then becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       A 
                       w 
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               
                                 θ 
                                 + 
                               
                               - 
                               
                                 θ 
                                 0 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 ɛ 
                               
                               + 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 ɛ 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 θ 
                                 - 
                               
                               - 
                               
                                 θ 
                                 0 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 ɛ 
                               
                               - 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 ɛ 
                               
                             
                             ) 
                           
                         
                       
                       
                         2 
                          
                         sin 
                          
                         
                             
                         
                          
                         ɛ 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which can be rewritten (by simplifying grouped terms) alternatively as: 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     w 
                   
                   = 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 θ 
                                 + 
                               
                               - 
                               
                                 θ 
                                 - 
                               
                             
                             ) 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           ɛ 
                         
                         + 
                         
                           
                             [ 
                             
                               
                                 ( 
                                 
                                   
                                     θ 
                                     + 
                                   
                                   + 
                                   
                                     θ 
                                     - 
                                   
                                 
                                 ) 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   θ 
                                   0 
                                 
                               
                             
                             ] 
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           ɛ 
                         
                       
                       
                         2 
                          
                         sin 
                          
                         
                             
                         
                          
                         ɛ 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This quantity is the pointer rotation angle, which can be conveniently related to the angular divergence δ and the difference term ε: 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     w 
                   
                   = 
                   
                     
                       δ 
                       
                         2 
                          
                         tan 
                          
                         
                             
                         
                          
                         ɛ 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           θ 
                           - 
                           
                             θ 
                             0 
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Note that for small difference such that: 
         [0000]      0&lt;ε&lt;&lt;1,  (33)
 
         [0000]    then the rotation angle becomes arbitrarily large in magnitude and can be approximated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       A 
                       w 
                     
                     ≈ 
                     
                       δ 
                       
                         2 
                          
                         
                             
                         
                          
                         ɛ 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and because of this, ε can be called the amplification factor 
         [0040]    The weakness condition constraint follows when eqns. (24) and (25) can be used to obtain the associated weakness condition when incorporating 0&lt;ε&lt;&lt;1 from eqn. (33) into inequalities eqn. (23), along with selections for a, b, c, d, φ, χ. These steps yield: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       a 
                       ) 
                     
                      
                     
                         
                     
                      
                     Δθ 
                   
                    
                   
                     
                       
                          
                         δ 
                          
                       
                       
                         2 
                          
                         
                             
                         
                          
                         ɛ 
                       
                     
                      
                     
                         
                     
                      
                     and 
                      
                     
                         
                     
                      
                     
                       ( 
                       b 
                       ) 
                     
                      
                     
                         
                     
                      
                     Δθ 
                   
                    
                   
                     2 
                      
                     
                       
                          
                         
                           θ 
                           - 
                           
                             θ 
                             0 
                           
                         
                          
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Here use is made of the fact that for 0&lt;ε&lt;&lt;1 being sufficiently small, then: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               min 
                               
                                 ( 
                                 
                                   
                                     n 
                                     = 
                                     2 
                                   
                                   , 
                                   3 
                                   , 
                                   … 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                  
                                 
                                   
                                     A 
                                     w 
                                   
                                   
                                     ( 
                                     
                                       A 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                               
                               
                                 1 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                             
                           
                           ≈ 
                             
                            
                           
                             
                               min 
                               
                                 ( 
                                 
                                   
                                     n 
                                     = 
                                     2 
                                   
                                   , 
                                   3 
                                   , 
                                   … 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                  
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           + 
                                         
                                         - 
                                         
                                           θ 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     - 
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           - 
                                         
                                         - 
                                         
                                           θ 
                                           0 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             + 
                                           
                                           - 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       n 
                                     
                                     - 
                                     
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             - 
                                           
                                           - 
                                           
                                             θ 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       n 
                                     
                                   
                                 
                                  
                               
                               
                                 1 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                              
                             
                               
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       + 
                                     
                                     - 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 - 
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       - 
                                     
                                     - 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         + 
                                       
                                       - 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         - 
                                       
                                       - 
                                       
                                         θ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                              
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                                
                               
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       + 
                                     
                                     - 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 - 
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       - 
                                     
                                     - 
                                     
                                       θ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                                
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               ( 
                               
                                 2 
                                  
                                 
                                    
                                   
                                     θ 
                                     - 
                                     
                                       θ 
                                       0 
                                     
                                   
                                    
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   . 
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
         [0041]    Satisfaction of both conditions (a) and (b) of eqn. (35) requires that when 0&lt;ε&lt;&lt;1 from eqn. (33), then the uncertainty Δθ greatly exceeds the absolute ratio value: 
         [0000]    
       
         
           
             
               
                 
                   Δθ 
                    
                   
                     
                        
                       
                         δ 
                         
                           2 
                            
                           ɛ 
                         
                       
                        
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This condition can be satisfied by making the initial Gaussian pointer distribution width sufficiently large. 
         [0042]    Thus, as per eqns. (22) and (34), the rotation of the initial photon distribution axis of symmetry provides an amplified measurement of the angular divergence δ via the weak value of the “which path” operator Â. For a known amplification ε and a measured mean value of the intensity distribution profile produced by the detector  260  corresponding to A w , then angular divergence δ can be estimated from eqn. (34) as: 
         [0000]      δ≈2 εA   w .  (38)
 
         [0043]    The real component of the complex operator A w =ReA w  corresponds to the angle between the direction of the resultant photon distribution peak and the positive y-axis is measured when there is sufficient knowledge of the value of the other parameters (e.g., θ, n 0 , V, etc.) appearing on the right hand side of this expression. 
         [0044]    Consider the case where a target wavelength λ, can be expressed as the sum of a known wavelength λ′ and a difference wavelength Δλ: 
         [0000]      λ=λ′+Δλ.  (39)
 
         [0000]    and similarly average refraction angle θ can be expressed as the sum of a known angle θ′ and a corresponding difference Δθ: 
         [0000]      θ=θ′+Δθ,  (40)
 
         [0000]    so that approximation eqn. (1) can be written as: 
         [0000]    
       
         
           
             
               
                 
                   δ 
                   ≈ 
                   
                     
                       - 
                       
                         
                           
                             ( 
                             
                               
                                 λ 
                                 ′ 
                               
                               + 
                               Δλ 
                             
                             ) 
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             0 
                           
                         
                         
                           π 
                            
                           
                               
                           
                            
                           
                             n 
                             0 
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   θ 
                                   ′ 
                                 
                                 + 
                                 Δθ 
                               
                               ) 
                             
                           
                         
                       
                     
                      
                     VB 
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
         [0000]    or rewritten as the approximation: 
         [0000]    
       
         
           
             
               
                 
                   
                     δ 
                     ≈ 
                     
                       
                         
                           - 
                           
                             
                               
                                 λ 
                                 ′ 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 0 
                               
                             
                             
                               π 
                                
                               
                                   
                               
                                
                               
                                 n 
                                 0 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 ′ 
                               
                             
                           
                         
                          
                         VB 
                       
                       - 
                       
                         
                           
                             Δλ 
                              
                             
                                 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                           
                           
                             π 
                              
                             
                                 
                             
                              
                             
                               n 
                               0 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               ′ 
                             
                           
                         
                          
                         VB 
                       
                     
                     ≡ 
                     
                       
                         δ 
                         ′ 
                       
                       + 
                       Δδ 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where, because Δθ is small, use has been made of the approximation 
         [0000]      cos(θ′+Δθ)≈ cos θ′.  (43)
 
         [0045]    Using eqn. (42) in eqn. (34) renders: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       A 
                       w 
                     
                     ≈ 
                     
                       
                         
                           δ 
                           ′ 
                         
                         
                           2 
                            
                           ɛ 
                         
                       
                       + 
                       
                         Δδ 
                         
                           2 
                            
                           ɛ 
                         
                       
                     
                     ≡ 
                     
                       
                         A 
                         w 
                         ′ 
                       
                       + 
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           A 
                           w 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where A′ w  represents rotation angle corresponding to the mean value of the photon distribution profile associated with the measurement of the known wavelength λ′. Note that the rotation angle A w =A′ w  when there is no wavelength difference or Δλ=0=ΔA w . 
         [0046]    The apparatus represented by the diagram  200  in  FIG. 2  can be used to detect λ=λ′+Δλ from eqn. (39) and estimate Δλ when λ′ is known. In order to accomplish this, the apparatus must first be calibrated so that its pointer value is A′ w  when the source is monochromatic with a known wavelength λ′. In particular, the photon distribution peak A′ w  is determined by enabling monochromatic light of wavelength λ′ to traverse the apparatus that comprises a Faraday medium with Verdet constant V. 
         [0047]    In this example, the parameters θ 0 , n 0 , B are fixed and the polarizers are set per above values to provide an amplification factor ε. When light of wavelength λ (bichromatic or monochromatic) traverses the calibrated apparatus, then the pointer deviates from the calibrated pointer value A′ w  by the amount 
         [0000]    
       
         
           
             
               Δδ 
               
                 2 
                  
                 ɛ 
               
             
             . 
           
         
       
     
         [0048]    As numerical examples, consider an additional two instances in which the medium  140  with negligible Verdet constant is air. For the first example, let the Faraday medium  130  be terbium gallium garnet (formula Tb 3 Ga 5 O 12 ) which has a Verdet constant V=−134 rad·Tesla −1 ·m −1  at known wavelength λ′=632.8 nm (red light) so that 
         [0000]    
       
         
           
             
               Δ 
                
               
                   
               
                
               
                 A 
                 w 
               
             
             ≈ 
             
               21.33 
                
               
                 ( 
                 
                   
                     Δλ 
                     · 
                     B 
                   
                   ɛ 
                 
                 ) 
               
                
               
                 rad 
                 · 
                 
                   Tesla 
                   
                     - 
                     1 
                   
                 
                 · 
                 
                   
                     m 
                     
                       - 
                       1 
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0049]    If for this first example, the amplification is ε=10 −4 , the wavelength difference is Δλ=1 pm and magnetic field strength is B=1 Tesla, then rotation angle difference is ΔA w ≈21.33 μrad, provided that the associated weakness condition of minimum rotation angle difference 
         [0000]    
       
         
           
             
               Δθ 
                
               
                 
                   1.35 
                   ɛ 
                 
                 × 
                 
                   10 
                   
                     - 
                     5 
                   
                 
                  
                 rad 
               
             
             = 
             
               0.135 
                
               
                   
               
                
               rad 
             
           
         
       
     
         [0000]    is satisfied. If the detector  260  is 1 m (one meter) from the Faraday medium  240 , then the pointer is translated by −21 μm in the detector plane. Thus, the 1 pm spectrum separation has been amplified by a factor of ˜10 6  at the detector  260 . 
         [0050]    For the second example, let the Faraday medium be MR3-2 Faraday rotator glass which has a Verdet constant V=−31.4 rad·Tesla −1 ·m −1  at known wavelength λ′=1064 nm (and 20° C. temperature) so that the rotation angle difference is 
         [0000]    
       
         
           
             
               Δ 
                
               
                   
               
                
               
                 A 
                 w 
               
             
             ≈ 
             
               4.997 
                
               
                 ( 
                 
                   
                     Δλ 
                     · 
                     B 
                   
                   ɛ 
                 
                 ) 
               
                
               
                 rad 
                 · 
                 
                   Tesla 
                   
                     - 
                     1 
                   
                 
                 · 
                 
                   
                     m 
                     
                       - 
                       1 
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0000]    If ε=10 −3 , Δλ=1 nm and B=1 Tesla, then ΔA w ≈4.997 μrad provided that the associated weakness condition 
         [0000]    
       
         
           
             
               Δθ 
                
               
                 
                   5.317 
                   ɛ 
                 
                 × 
                 
                   10 
                   
                     - 
                     6 
                   
                 
                  
                 rad 
               
             
             = 
             
               5.317 
                
               
                   
               
                
               m 
                
               
                   
               
                
               rad 
             
           
         
       
     
         [0000]    is satisfied. 
         [0051]    If the detector  260  is 1 m (one meter) distant from the Faraday medium, then the pointer translates by ˜5.0 μm in the detector plane. Thus, the 1 nm spectral separation has been amplified by a factor of ˜10 3  at the detector  260  for the medium  140  with negligible Verdet constant (and unitary refraction index) being air. 
         [0052]    As described above, known beam wavelength λ′, magnetic field strength B and angle-of-incidence θ 0  are established á priori. The refraction angle θ′ represents the average of the refraction angles for the circularly polarized beams determined for the known wavelength λ′ based on the indices of refraction n ±  of the Faraday medium  130 . 
         [0053]    For small differences such that Δλ≡λ−λ′&lt;&lt;λ between unknown and known wavelengths, the average angular refraction angle difference between the unknown (i.e., target) and known refractions Δθ≡θ−θ′ is small. This enables the average refraction angle to be reasonably approximated by the known value as θ≈θ′. The approximation cos θ≈ cos θ′ can be made, because of the relation: 
         [0000]      cos(θ′+Δθ)=cos θ′ cos Δθ−sin θ′ sin Δθ≈ cos θ′−Δθ sin θ′≈ cos θ′.  (45)
 
         [0054]    The distribution peak rotation angle A′ w  corresponding to the known wavelength λ′ can be established from a calibration measurement. For a small amplification factor E such that 0&lt;ε&lt;&lt;1, the rotation angle A w  corresponding to the unknown wavelength λ provides an estimate of the divergence δ from eqn. (38) as δ=2εA w . Similarly, the known rotator angle A′ w  the estimate for the corresponding divergence δ′. 
         [0055]    In response to a small change in wavelength from the known λ′ to an unknown λvalue, the measured rotator angle becomes A w =A′ w +ΔA w  with the difference ΔA w  corresponding to the change in measured rotation angle due to the offset wavelength that provides a measure of change in dispersion angle from eqn. (42) as Δδ. Subtracting the calibrated values and rearranging terms enables the wavelength difference to be determined as: 
         [0000]    
       
         
           
             
               
                 
                   Δλ 
                   = 
                   
                     
                       - 
                       
                         
                           π 
                            
                           
                               
                           
                            
                           
                             n 
                             0 
                           
                            
                           Δδcos 
                            
                           
                               
                           
                            
                           
                             θ 
                             ′ 
                           
                         
                         
                           VB 
                            
                           
                               
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             0 
                           
                         
                       
                     
                     ≈ 
                     
                       - 
                       
                         
                           
                             2 
                              
                             ɛπ 
                              
                             
                                 
                             
                              
                             
                               n 
                               0 
                             
                              
                             Δ 
                              
                             
                                 
                             
                              
                             
                               A 
                               w 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               ′ 
                             
                           
                           
                             VB 
                              
                             
                                 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
           
         
       
     
         [0056]    While certain features of the embodiments of the invention have been illustrated as described herein, many modifications, substitutions, changes and equivalents will now occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the embodiments.