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					    Supplementary Information for
    “Backward      self-induced transparency in metamaterials”
Jianhua Zeng, Jianying Zhou*, Gershon Kurizki†, and Tomas Opatrny
         * email: stszjy@mail.sysu.edu.cn
         † email: gershon.kurizki@weizmann.ac.il




Included in this document:
Supplementary Discussion 1
Supplementary Discussion 2




                                            1
Supplementary Discussion 1
Eigenstate solutions of SIT backward soliton
     We consider the propagation of an EM pulse along the z axis, normally to an input
interface of a MM at z  0 , with the constitutive relations deviation from (1a):

     Dx   0 Ex  pe Kx    Px , By  0 H y   pm  0m  M y   .
                    2                                  2     2



We first transform the Maxwell equations in the frequency domain (  t  it ) and
then rewrite them in the time domain, we have
       e          Re  12      2     
              2                 pe     m,
        t             t              t    z
        m                           
               pm  0 m 
                  2       2
                                    e,
        t                      t       z
                                                                                     (S1)
       2
                  
              e       e ,
       t  2
                   t
        2        
              m        0 m    m .
                            2

        t 2
                    t
They are supplemented by the Bloch equations
       W      W  1  2i  *   ,
                          e  12  12 
        t       T1
         12            
               i0 12  12  i eW .                                                 (S2)
          t             T2

Writing  e ,  y ,  ,  , 12 , W as functions of
                                             1 
          tz v,                ,           
                          t  z  z           v 
We can rewrite the Maxwell-Bloch equations as
e          Re  12       2      1 m
       2                 pe             ,                                       (S3a)
                             v 
m                      1 e
        pm  0 m 
           2       2
                                      ,                                             (S3b)
                       v 
 2       
       e       e ,                                                               (S3c)
  2
           
 2        
       m        0 m    m ,
                      2
                                                                                    (S3d)
  2
             
W       W  1  2i  *   ,
                         e  12    12                                            (S3e)
           T1


                                                  2
 Re  12                                     Im  12 
                 0 Im  12  ,                             0 Re  12   eW .                    (S3f)
                                                
     Based on the fact that the expressions for the fields  e and m are similar to those in
a uniform SIT medium, and likewise for the functions  and  , we obtain the following
ansatz:
         e  0 1      sec h   ,
                                                                                                       (S4a)
          m   e v,                                                                                    (S4b)

           0 sec h  ,                                                                            (S4c)
           0 sec h   ,                                                                            (S4d)
         Re  12   C0 sec h ,                                                                     (S4e)

                              0 pe
                               2 2

         W  1                               sec h2   ,                                            (S4f)
                              0         2



where  0 is the amplitude of the solitary pulse,  its width and v its group velocity;

 is a real perturbation parameter that quantifies the deviation from solitonic SIT
solution as a function of  e and  m .
    Upon substituting these expressions into Eqs. (S3), our system is reduced to a set of
algebraic equations for the amplitude modulation and the pulse width.
     2 C0   pe 0   ,
               2
                                                                                (S5a)
     0 C 0  1   2 C 0  0   ,                                                                        (S5b)
         2
           pm          2
                         0m   
                              v  , 0    0 1  v    2
                                                                                                           (S5c)
             2           ,
           e     0            0                  0
                                                     2
                                                               0       0
                                                                           2
                                                                                                         (S5d)
           v  2     v     
           m         0            0                  0
                                                         2
                                                                   0           0
                                                                                   2   2
                                                                                       0m   0    .   (S5e)
      It follows from Eqs. (S5d) and (S5e) that the following relations should be satisfied
for stable SIT propagation:
               
           e  0                                                             (S6a)
                0
                v
          m  0                                                               (S6b)
                 0
Conversely, when  e and  m are of the order of  , the solutions of  and  become
unstable, and the SIT BW soliton does not form.
     The solutions for the above parameters are, under the conditions (S6a), (S6b)



                                                                                   3
              pe
               2

      C0         0                                                          (S7a)
             2
              0  0 C 0   0 
                                                                               (S7b)
                         C0

     0 
                   1  v 
                          2

                                    0 ,                                        (S7c)
             v  pm  0 m 
                  2     2


             0
     0           ,                                                           (S7d)
             2
               0v
     0              ,                                                          (S7e)
               02m
               2


The soliton amplitude is found to satisfy 0  2 , as in the case of usual SIT; which
implies that the area under the  e envelope is 2 .
     Remarkably, we have checked that the SIT properties continue to hold even when
we introduce different damping terms, i.e., e  m .Simulation results show that the SIT
can be preserved when e  m (  m  103 rad fs ) is within 0.5, . The parameter
                                                                 5
  0.2 is more than 100 times of loss terms  e and  m .

Steps in the derivation of Eqs. (S7)

From Eq. (S3a), we have
                                   pe 0
                                    2

          2 C0    0  C0  
                              2

                                   2
                              pe


From Eq. (S3b),
                    1 e m 1  v  e
                                                 2

           v     v 
        2
        pm
                  2
                  0m


    e                                               e    e 
           0 sec h             20 1      
                                                             2
                                                       e  e         
                                                                         
                          
     0 sec h              0 1       sec h   tan h  
                                                  
                        
    
           0 sec h   tan h  
    
                        1  v 2      1  v 2  
      pm  0m   0  v 0
        2     2
                                           0
                                               v  pm  0 m 
                                                     2      2      0


From Eq. (S3c):




                                                  4
              e   e  
                                   

    2 0
          e   e  
                                2

                             
                                             e   e  
                                                                                              2
2
                        1                  2                   
      2 0 2
                       
                                    4 0
 2
                e  e                           
                                              e  e     
                                                                
                                                                  3
                                                             
                                                               e   e  
                                                                                                                                   2
                 e   e 
                                                        4 0                     1    
                                                                      1                               2
2e  0                      2 0      
                                                       2
                                                                                      0        
                                                                                              2

           e  e
                 
                          
                            2
                                      e  e                       
                                                                e  e    
                                                                                 
                                                                                   3
                                                                                                  e e
                                                                            
                                                                                                              
2 e  0 e   e    e   e     4 0 2 e   e     2 0 1        2 0 2 e   e                                  
                                                                                                      2                                                       2
                                                                                              
                                                                                                          

                                                                                            
2e  0 e 2   e 2    4 0 2 e 2   2  e 2    20 1       2 0 2 e 2   2  e 2  
                                                                                                                                             
e 2  coefficients:
                                              
          2e  0  4 0 2  2 0 1         0 2  2  0   0 2   20   
                                                                                                 
e 2  coefficients:
                                              
              2e  0  4 0 2  2 0 1         0 2  2  0   0 2   20   
                                                                                                 
Constant coefficients: 8 0 2  4 0 1         0 2
                                                                                                               
2 e  0  4 0   e  2 .
                  2


                                               0                                                                                  0    
Under this condition:  0                                     . 2e 0  20     e                                                  should of the
                                                      2
                                                                                                                                     0
same order, a very small parameter.
Conversely, when 2 e  0  4 0 2   e  2 , the solution of  is unstable.

From Eq. (S3d):
                e   e  
                             

     2 0
            e      e     
                                        2

                                     
                                                          e     e     
                                                                                                  2
2
                     1                                 2                        
      2 0 2                        4 0
 2
               e e                                      e   e  
                                                                                   3

                                                                                
                              e   e  
                                                                  2
                                                               
                                                  2  e  e
                                                                        
                                                                                                                1      
2 0      4 0 2                                                    20 m 0      20v   
          2 1                                                                        2           1
       e e                                                                                e e                 e e
                                                      0  m
                              e   e            e   e  
                                                  3                            2

                                                                          

                                                                 
2 0 2  e  e    0 m  e2  e2   0v 1         0 0 m   2   e  e 
                            2                                                  2                                                                       2
                                                                    
e  coefficients:        2     v          v   
  2                                                                         2                           2           2
                                       m       0                      0               0           0                   0m       0        0

e   coefficients:         2     v          v   
   2                                                                       2                           2           2
                                           m       0                  0               0           0                   0m       0        0

Constant coefficients: 4   2   v          v   
                                               0
                                                           2
                                                                          0       0
                                                                                          2       2
                                                                                                  0m          0            0



                                                                                  5
 m  0  2 0 2   m  2 . Under this condition:
                                                                                                  0v
2 0 2   0 v   0 2  0 m  0    0 2   0 v  0 m  0   0 
                             2                             2
                                                                                                         .
                                                                                                  02m
                                                                                                 2


                                            0v   
m 0  0v      m                                 should of the same order, a very small
                                               0
parameters.
Conversely, when 2 m  0  4 0 2   m  2 , the solution of  is unstable.

From Eqs. (S3f):
                    1  Re  12    
                                          C0  sec h    0 pe  sec h  
                                                                      2

Im  12                
                  0                     0              20  2     
Substitute into Eq. (S3e)
W                                                       2  sec h  
      2ie  12  12   4e Imag  12   4e 0 pe 2
                 *

                                                    20       
                                            0 pe  sec h  
                                                2

 40 1       sec h  
                                         20  2     
     0 pe  sec h2  
       2 2


     0  2      
                 0 pe
                  2 2

W  1                       sec h2  
             20        2




From Eqs. (S3f):
                                                             1  Re  12   
                     1  Re  12      Im  12   
                                                                   2

Im  12                         ,                                      
                   0                                    0        2


                                                                   e     e     
                                                                                           2
 2 Re  12    
                                           1                  2                        
                          2C0  2
                                             
                                                          4C0
            2
                                    e  e                        e     e     
                                                                                            3
                                                                                       
  C0 sec h    C0 sec h   tan h  
                                2                        2                      2


                                                                                                    0 pe
                                                                                                      2 2
                                                                                                                       
0 Re  12   eW  0C0 sec h    0 1       sec h    1 
                                                                     
                                                                                                         sec h 2   
                                                                                                                       
                                                                                                 20       2
                                                                                                                       
                                                      0 0C0  0 

    1
    0
          C      C
             0
                  2
                              0   0    0    
                                                               C0




                                                               6
Supplementary Discussion 2
Electric permittivity solution (  TLS ) for two-level systems:
 Bloch equations (2) can be divided into real and imaginary parts:
 Re  12                         Re  12 
                0 Im  12  
    t                                  T2
                                                                                     (S8)
 Im  12                        Im  12 
                0 Re  12                   eW
    t                                T2

By using the relation Px  2 Nd Re 12 , we can rewrite Eqs. (S8) as

  2 Px      2 Px                             Nd 2
         0        0 Px  2 Nd0eW  20
                      2
                                                    WEx                             (S9)
  t 2       T2 t

Using the linear polarization expression Px   0 TLS Ex , it is not hard to get

                       2TLS W
TLS     0                                                                        (S10)
                      i 2TLS T2
                    2     2
                          TLS


here   Nd 2  0 is the cooperative dopant-field coupling constant.




                                                        7

				
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