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					              ACCURATE OPTICAL FIBER REFRACTIVE INDEX
                 RECONSTRUCTION FROM NEAR FIELD

                                    Α. C. Boucouvalas and C. Α. Thraskias
                          Department of Telecommunications Science and Technology,
                         Faculty of Science and Technology, University of Peloponnese,
                                                 Tripoli, 22100,
                                                     Greece,
                                        e-mail: { acb, tst06059 }@uop.gr


                                                    ABSTRACT
           A new and efficient algorithm is proposed for calculating directly and accurately the
           refractive index profile of cylindrical waveguides from knowledge of the mode near field.
           This inverse problem is solved using transmission line techniques. From Maxwell’s
           equations, we derive an equivalent transmission-line circuit for a cylindrical dielectric
           waveguide. Based on an analytical method, that computes the error in the reconstructed
           refractive index due to inaccuracy in the propagation constant β. The proposed analytical
           method computes the refractive index error directly without the need for curve fitting or
           numerical differentiation. Subsequently we work out the exact value of the propagation
           constant β. The calculation of the propagation constant β is vital for the refractive index
           reconstruction and we show that it implies error minimization in the refractive index
           synthesis from near field. We demonstrate this algorithm with example reconstructions for
           step, triangular and parabolic optical fiber refractive index profiles. Furthermore we present
           example reconstructions for step optical fiber refractive index profiles using higher order
           modes near electric field. This technique is exact, fast and rapidly convergent.


1   INTRODUCTION                                                In this paper we extend the transmission line
                                                           theory and present a novel method for the exact
     The main characteristics of optical waveguides,       reconstruction of the waveguide refractive index
such as bandwidth, spot size, single-mode                  profile using the fundamental or other higher order
propagation conditions, and interwaveguide coupling        modes near electric field data. Although in most cases
coefficients are all related to their refractive index     instrumentation uses the fundamental mode for
profiles. So the characterization of optical fiber         reconstruction, in some cases this may not be easy
refractive index profiles has the fundamental
importance for the determination of waveguide              and especially if a good higher order mode is present
optical properties. Refractive index profile               at a convenient wavelength rather than an operation
measurement, however, is generally very difficult due      wavelength, it may be suitable to use higher order
to small optical fiber dimensions and low refractive       modes. The following section describes the basic
index differences between core and cladding. A             theory our technique is based upon.
number of techniques, [1], have been proposed for
determining the refractive index distribution of optical   2   FORWARD SOLUTION
fibres from the propagation mode near field, and the
most well known rely on the seminal theoretical work            We divide an optical waveguide into a large
by Morishita,[2]. Reference [2] relies on an inverse       number of homogeneous cylindrical layers of
solution of the scalar wave equation for the refractive    thickness δr, permittivity ε, permeability µ and
index profile. In [1], the measurement of the near field   conductivity σ in Fig.1.
intensity is improved using a scanning optical
microscopy technique rather than conventional optics.
Improvements from [2], have been recently reported
in [3], which is a robust method to noise and errors,               θ
and non-iterative, but reported for planar waveguides.                                        ε
     Using transmission line techniques we have
shown that they can be applied in optical fibres and                                                            z
can determine exactly the mode propagation constants
[4], and cutoff wavelengths of waveguide modes [5].
                                                               δr
In general from knowledge of the refractive index,
complete waveguide characterization can be achieved
                                                           Figure 1: Homogeneous cylindrical layer
using this powerful technique [6].
      The E and H components of Maxwell’s equations                                                    After algebraic derivatives, (1) and (2) can be
 for any such layer can be written as:                                                            transformed into:

                                                                                                 ∂Vs        −γ s
                                                                                                                  2
                                                                                                                      
  β rEθ − lEZ = ωµ rH r                                                                                 =            Is 
                                                                                                   ∂r     jωε 0 nF 
                                                                                                                                                   (4)
  lH Z − β rH θ = (ωε − jσ ) rEr                                                           (1)   ∂I s                
  ∂ (ωµ rH r )                                                                                        = − jωε 0 nFVs
                                                                                                                      
               = − jωµ (lH θ + β rH Z )                                                          ∂r                  
       ∂r                               
                                                                                                 ∂Vd        −γ d
                                                                                                                  2

                                                                                Id                      =
∂[(ωε − jσ )rEr ]                                              ∂r     jωε 0 nF 
                  = −(σ + jωε )(lEθ + β rEZ )                                                                      (5)
       ∂r
                                                              ∂I d                
                                                                   = − jωε 0 nFVd
∂(lHθ + β rHZ )       γ                                                            
                        2
                                           l                                       
                 =−        ωµrHr + β HZ − Hθ             (2) ∂r
      ∂r             jωµ                   r            
                                                    l 
                                                                                  l 2         2 nk 0 β l
∂(lEθ + β rEZ )         γ
                          2                                             2     2       2 2
                                                              where γ s = β + ( ) − n k 0 ∓                ( - for HE, +
                =−            (ωε − jσ )rEr + β EZ − Eθ                d         r
                                                                                                  2
                                                                                            (β r ) + l
                                                                                                         2
      ∂r            σ + jωε                         r 
                                                              for EH modes).
                                                                                                       Equations (4) and (5) represent two independent
                                   l
 where γ = β + ( ) − ω µε + jωµσ
              2           2                 2       2
                                                                                                  transmission lines with voltages Vs , Vd and currents
                r
                                                                                                  I s , I d . The corresponding characteristic impedances
      We           restrict             our             analysis   to     the       case          are:
  σ = 0, µ = µ 0 , ε = n ε 0 , where n is the refractive
                                        2


 index of the layer at distance r from the axis.                                                             γs 
                                                                                                  Zs =
                                                                                                       jωε 0 nF 
     We define the following variable voltages and
 currents:                                                                                                      
                                                                                                                                                    (6)
                                                                                                         γd 
                                                                                                  Zd =
 Vs =
         VM
                  + VE         n                (sum)
                                                                                                      jωε 0 nF 
                                                                                                                
            n                                                
                                                             
         VM                                                                                          The above equations are recognized as the well
 Vd =             − VE         n                (difference)                                      known transmission line equations with the solution
            n                                                
                                                                                                 represented by the following electric circuit Fig. 2.
                                                                                           (3)
                              IE                             
  Is = IM         n+                            (sum)
                               n                             
                                                             
                          IE
  Id = IM         n−                            (difference) 
                               n                             
                                                             
 where

         lH θ + β rH Z
 VM =                               Z0             (magnetic voltage)
                     jF
         ωµ rH r                                                                                  Figure 2: Equivalent circuit for an optical fibre
  IM =                                             (magnetic current)                             cylindrical thin layer
              jZ 0
         lEθ + β rEZ                                                                                                  δr 
 VE =                              Z0               (electric voltage)                            Z B = Z s tanh(γ s    )
                  F                                                                                      d         d 2 
                                                                                                                          
  I E = ωε 0 n rEr                                                                                                        
                  2
                                                    (electric current)                                     Zs                                        (7)
                                                                            2       2             ZP =       d            
                                                                        (β r) + l                      sinh(γ s δ r )     
  Z 0 = 120π      the free space impedance, F =                                         .                     d           
                                                                            r
where δ r is the length of the transmission line.                             Vs                           Is
     The resonance frequency of the cascade of those             VE =                  , IM =
electric circuits for a fixed wavelength represents the                2 n(r )             2 n( r ) 
mode propagation constants β of the relevant
                                                                                                    
waveguide with certain refractive index profile.                       V n( r )             I n(r ) 
                                                                                                    
     The technique can further be extended in order to           VM = s              , IE = s                           (12)
be used for plotting the electric or magnetic field
                                                                             2                2     
components of the waveguides as follows:                         I E = ωε 0 n ( r ) rEr
                                                                              2
                                                                                                    
     From the above equations, we can derive the                                                    
Electric current I and Electric field Er . We know                                                  
                                                                                                    
that:
                                                                 Hence:

VHE =
            VM
                      + VE      n( r )
                                                                             IE              Z0 IE
                                                                Er =                  =
             n( r )                                                     ωε 0 n r  2
                                                                                           k0 n 2 ( r ) r
                                                          (8)
            VM
VEH =                 − VE      n( r ) 
             n( r )                    
                                       
                                                                 and:

                                                                          Z0 IE
                                 
                                IE                               Er =                                                    (13)
I HE = I M       n( r ) +                                                 2
                                                                         n (r )r
                          n( r ) 
                                 
                                                          (9)
                          IE                                                                                    β
I EH = I M       n( r ) −                                        where r = rk 0 , δ r = δ rk 0 , β =                 .
                          n( r ) 
                                                                                                 k0
                                                                      Therefore if we already know the refractive index
   We wish to determine the E/M field of the HE                  as a function of radius, we can use the (13) to plot the
mode in terms of I HE and VHE variables, we can set              Electric fields, E r and subsequent Magnetic fields
I EH = VEH =0 when the HE modes are of interest. This            precisely.
implies that
                                                                 3      INVERSE PROBLEM

                      IE
IM       n( r ) =                                                     The equivalent circuit for a cylindrical thin
            n( r ) 
                                                                dielectric layer, Fig.1, of constant refractive index n
                                                         (10)   and thickness δr at distance r from the core is
  VM
        = VE n(r )                                              represented as an electric circuit in Fig. 3. For
  n(r )            
                   
                                                                 determining the refractive index profile from
                                                                 knowledge of Er , we assume the following boundary
Substituting into (8) and (9), the following equations           condition: At r = ∞, we assume n = n2 (silica
can be derived:                                                  refractive index). At r = ∞,            Z prev = 0 and

                                                                 n ( ∞ ) = n2 are assumed.
VHE = 2VE             n(r ), I HE = 2 I M n( r ) 
                                                   
               VM                          IE           (11)
VHE = 2                     , I HE = 2             
                 n( r )                    n(r )   

        Note that I HE , VHE are also referred to as I s , Vs
respectively. Hence:




                                                                 Figure 3: Equivalent circuit for a cylindrical thin
                                                                 layer at constant refractive index n and thickness δ r
                                                                 at distance r from the core
                                                                                                         −1
    We know that β is the effective refractive index                             1       1 
and for typical waveguides lies between n1 and n2.          where Z n = Z B +           +  , (n=1,2..), is
                                                                              Z +Z          
                                                                                          Zp 
Furthermore, as we know λ0 , r , δ r and n ( ∞ ) = n2 ,                        B   n −1
                                                            the total characteristic impedance of cylindrical layers.
I ( r + δ r ) and V ( r + δ r ) can be calculated for any   Using the derivative (17) we can study the refractive
radius. Hence we work out n( r ) as follows:                index changes versus the radius r for various β. From
                                                            (7) we also know that the equivalent T-circuit
                                                            impedances are functions of propagation constant β.
                2I E ( r )                                 Hence, the derivative (17) is a function of the
   I s (r ) =
                                                           propagation constant can be calculated. We first work
                  n( r )
                                                  (14)          ∂Z n ∂Z n
             I E (r )Z0                                    out        ,      , given by:
   Er ( r ) = 2                                                   ∂r     ∂n
              n (r )r  
                                                                                                 −2 ∂ΖΒ ∂Ζn−1 ∂Ζ p 
and finally
                                                            ∂Zn ∂ΖΒ      1      1                 ∂r + ∂r              
                                                                =   +          +                                  + ∂r  (18)
                                                             ∂r                      
                                                                  ∂r  ZB + Zn−1 Z p               ( ZB + Zn−1 )
                                                                                                                   2    2
                                                                                                                      Zp 
                   I s (r ) Z0                                                                                           
   Er (r ) =                                                                                                             
                      32                            (15)
                  2n       (r )r
                                                                                       −2  ∂ΖΒ ∂Ζn−1   ∂Ζ p 
                                                            ∂Zn   ∂ΖΒ     1      1   ∂n + ∂n              
                 I s ( r ) Z0                                   =    +          +                   + ∂n (19)
   n( r ) = (                   )2 3                (16)     ∂n        ZB + Zn−1 Z p   ( Z + Z )2 Z p 
                                                                   ∂n                                    2

                2Er ( r ) r                                                             B n−1
                                                                                                            
                                                                                                             
                                                                                                                          

    Since we know Er ( r ) and I s ( r ) , hence we can                                                       ∂Ζ p       ∂Ζ Β
                                                            Notice that for equations (18) and (19)                  ,          ,
calculate n ( r ) for every r recursively .                                                                    ∂r        ∂r
     We know that the calculation of the exact              ∂Ζ p           ∂Ζ Β
propagation constant β is necessary for solving the                 and              are also required and are given by:
inverse problem of refractive index reconstruction. In       ∂n             ∂n
[7], we have shown empirically that the error in the
reconstructed refractive index increases with use of                         l 2     
                                                                       Ζ0    − β 2 
incorrect propagation constant β but it is minimum at
                                                            ∂Z p                       
the exact value. Here we present an analysis which
                                                                 =          r                                         (20)
confirms this result and we demonstrate
reconstruction of the index profiles using this
                                                             ∂r             2  l 2  
                                                                                             2

                                                                   nδ rk0  r  β +    
method. In the following we provide exact analytical                                    
                                                                                     r  
formulas for the calculation of the radial refractive                      
index gradient with respect to radius and as a function
of the required propagation constant β . For a certain
β can simply start to calculate the radial refractive       ∂Z B δ r 2  ∂γ 2         ∂Ζ p 
index change with respect to r. Starting with                   =            Zp +γ 2                                    (21)
                                                             ∂r   2  ∂r               ∂r 
 β = k0 n2 we change β within k 0 n2 ≤ β ≤ k0 n1 and
recalculate the refractive index profile until the error
in the radial refractive index, (17) at certain β is        ∂Z p                         Ζ0
minimum. When the error is minimum we have the                     =−                                                    (22)
                                                             ∂n                     2  l 2 
exact β, and the reconstructed refractive index is also                   n rδ rk0  β +   
                                                                           2
exact. The analytical computation of the refractive                                     r 
index error can be achieved with use of derivative                                 
(17). From (7) we know equivalent T-circuit
impedances are functions of refractive index and
                                                            ∂Z B δ r 2  ∂γ 2         ∂Ζ p 
radius r, so the following derivative can be extended           =            Zp +γ 2                                    (23)
as follows:                                                  ∂n   2  ∂n               ∂n 

        ∂Z n
   ∂n
             |n=n0                                          Where
      =  ∂r                                                                      2
                                                                                    2nk0 β l
                                                   (17)             l
   ∂r ∂Z n |                                                γ = β +   − n 2 k0 −
                                                             2      2          2

                                                                                   (β r) + l2
                                                                                        2
         ∂n
              r =r0                                                 r
∂γ 2   2l 2         4nk0 β 3 rl
     =− 3 +
 ∂r
                  (( β r )              )
                                            2
       r                     2
                                 + l2

∂γ 2                2k 0 β l
     = −2 k 0 n −
            2

 ∂n               (β r ) + l2
                        2




4     NUMERICAL RESULTS AND DISCUSSION


     Fig. 4 shows the reconstructed refractive index
profile of a step index optical fibre of refractive index
n1= 1.51508 and n2=1.508 and V=2.3 using the
fundamental mode near electric field data. In the fig.
4 we assume we have full knowledge of the exact
propagation constant β. Fig. 5 shows the                    Figure 5: The reconstructed refractive index using (16)
reconstructed refractive index profile of the same step     with β ≠ βexact. The normalised propagation constant
index optical fibre for an incorrect propagation            used is b = 0.5112.
constant β ≠ βexact. We observe that the
reconstructed refractive index is sensitive to the
propagation constant β. Hence, if we want to
minimize the error in the refractive index
reconstruction we require to work out the exact value
of the propagation constant.




                                                                                          (a)

                                                                                                   ∆ = 0.000662
                                                                                                   ∆ = 0.0013
                                                                                                   ∆ = 0.0026




    Figure 4: The reconstructed refractive index using
    (16) with β = βexact. The normalized value of the
    exact propagation constant used is b = 0.5032




                                                                                         (b)

                                                            Figure 6 (a): The derivative dn/dr versus the
                                                            normalised radius r using the fundamental mode near
                                                            field data with β≠βexact. The normalised propagation
constant used is b = 0.2889, (b): The ripple of the                       derivative dn/dr versus β for the triangular refractive
derivative dn/dr versus the normalised propagation
                                                                          index optical fiber. Fig.8(a), shows an example
constant b for three different step index optical fibres
                                                                          refractive index reconstruction for a parabolic
with the same V = 2.3 and different ∆ using the                           refractive index optical fiber. Fig.8(b), shows the
fundamental mode near field data.
                                                                          oscillation of the derivative dn/dr versus β for the
     Fig.6(a) shows the derivative dn/dr versus the                       parabolic refractive index optical fiber. Finally, table
normalised radius r with β ≠ βexact. Since in this                        2, shows the accuracy of our method for the triangular
example we study a step index optical fiber, the                          and parabolic refractive index optical fibres.
derivative dn/dr should be zero in the core of the
fiber. The oscillations of the derivative in the core
prove the existence of error in the refractive index
reconstruction which we expect to be near zero when
the exact β is used.
     The error in the cladding is forced to zero using
the knowledge of the cladding silica refractive index.
     Fig.6(b) shows that there is a minimum ripple
(error) in the reconstructed index at a specific β value.
We observe that the ripple oscillations increase with
incorrect β. At the minimum ripple point we have the
exact β, and Table 1 shows the calculated error for b,
the derived normalised β, (0<b<1).

                          Table 1

  The accuracy of our method in the calculation of the
                     propagation constant.                                                              (a)


                                                   bexact − b min
Three     different bexact         bmin                             (%)
                                                      bexact
optical fibres
∆= 0.000662 ,V=2.3    0.5032       0.5033              0.01987

∆= 0.0013 , V=2.3     0.5032       0.5039               0.14

∆= 0.0026 , V=2.3     0.5032       0.5041               0.18



    In order to derive the exact value of the
propagation constant β we start to calculate the
derivative dn/dr versus the radius r with β = n2 and
                                                                                                          (b)
repeat the process with a                 β        change within
n2 ≤ β ≤ n1 until the ripple is minimum, Fig.5(b). At                     Figure 7 (a): A triangular refractive index
                                                                          reconstruction from the fundamental mode near field.
this minimum ripple point we have the exact β , and                        (b): The ripple of the derivative dn/dr versus the
the reconstructed refractive index is also exact. The
measurement of the oscillation can be achieved with                       normalised propagation constant β for the triangular
the standard deviation. In the Fig.6(b) the three curves                  refractive index optical fiber.
correspond to three different step index optical fibers
with the same V = k 0 a n1 − n2(   2
                                            )
                                          2 1/ 2
                                                    and different

 ∆ = ( n1 − n2 ) / n1 . So the three optical fibers have
the same normalised propagation constant and, hence,
the three curves have the same minimum.
     Fig.7(a), shows an example refractive index
reconstruction for a triangular refractive index optical
fiber. Fig.7(b), shows the index oscillation, the
                                                               Fig. 9(a) shows the reconstructed refractive index
                                                               profile of a step index optical fibre of refractive index
                                                               n1= 1.51508 and n2=1.508 and V=5 using the HE12
                                                               mode near field data. We demonstrate that the method
                                                               reconstructs the refractive index profile successfully
                                                               for this and other higher mode fields. Fig.9(b) shows
                                                               the ripple of the derivative dn/dr versus the
                                                               normalised propagation constant b using the HE12
                                                               mode.




                              (a)




                                                                                             (a)


                                                                                                       ∆ = 0.0047
                                                                                                       ∆ = 0.0013
                                                                                                       ∆ = 0.000662


                              (b)

Figure 8 (a): A parabolic refractive index
reconstruction from the fundamental mode near field.
 (b): The ripple of the derivative dn/dr versus the
normalised propagation constant β for the parabolic
refractive index optical fiber.

                         Table 2

                                                                                               (b)
  The accuracy of our method in the calculation of the
                propagation constant.                            Figure 9 (a): The reconstructed refractive index
                                                                 using the HE12 mode. (b): The ripple of the
                                      β exact − β min
               β exact     β min
                                                        ( %)     derivative dn/dr versus the normalised propagation
                                          β exact                constant b for three different step index optical
                                                                 fibres with the same V = 5 and different ∆ using the
Triangular    1.510877     1.510872   0.00033093                 HE12 mode.
refractive
index
Parabolic     1.512256     1.512251   0.00033063
refractive
index
                       Table 3                               5   CONCLUSIONS


 The accuracy of our method in the calculation of the             In this paper, a new and accurate refractive index
                                                             profile synthesis technique has been developed. This
     propagation constant using the HE12 mode.
                                                             method is based on the Transmission-Line technique.
                                                             An exact analytic method for calculating the mode β,
                                     bexact − b min          is presented during the refractive index reconstruction
Three    different bexact bmin                           %
                                         bexact              from the near field. The accurate calculation of the
optical fibres                                               propagation constant β is very important for the
                                                             refractive index synthesis as it results in the error
∆= 0.0047 , V=5    0.2149   0.2151         0.093             minimization in the reconstructed refractive index.
                                                             The method requires knowledge of the near field of
∆= 0.0013 , V=5    0.2149   0.2152         0.14              the optical fibre and the reconstruction is theoretically
                                                             exact. Simulation results demonstrate the potential of
∆= 0.000662, V=5   0.2149   0.2148         0.046             this new method.

Fig.10 shows the ripple of the derivative dn/dr versus       6   REFERENCES
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optical fibres
∆= 0.0047 , V=5    0.6018   0.6021         0.0498

∆= 0.0013 , V=5    0.6018   0.6019         0.0166

∆= 0.000662 , V=5 0.6018    0.6017         0.0166
     Anthony C. Boucouvalas has worked at GEC
Hirst Research Centre, and became Group Leader and
Divisional Chief Scientist until 1987, when he joined
Hewlett Packard (HP) Laboratories as Project
Manager. At HP Labs, he worked in the areas of
optical communication systems, optical networks, and
instrumentation, until 1994, when he joined
Bournemouth University. In 1996 he became a
Professor in Multimedia Communications, and in
1999 became Director of the Microelectronics and
Multimedia Research Centre.
     In 2006 he joined the Department of
Telecommunication Sciences at the University of
Peloponnese in Tripolis, Greece where he is now head
of Department. His current research interests span the
fields of wireless communications, optical fibre
communications and components, multimedia
communications, and human-computer interfaces,
where he has published over 200 papers. He has
contributed to the formation of IrDA as an industry
standard and he is now a Member of the IrDA
Architectures Council.
     He is a Fellow of Fellow of the Royal Society for
the encouragement of Arts, Manufacturers and
Commerce, (FRSA) and a Fellow of IEE, (FIEE). In
2002 he became a Fellow of the Institute of Electrical
and Electronic Engineers (FIEEE), for contributions
to optical fibre components and optical wireless
communications. He is a Member of the New York
Academy of Sciences, and Association for Computing
Machinery (ACM). He is an Editor of numerous
Journals and in the Organising committee of many
conferences.

     Chris A. Thraskias was born in Mesolakkia,
Serres, Greece, on September 23, 1983. He graduated
from the Greek Air Force Academy, Athens, Greece,
in 2005 and he is currently working towards the Ph.D.
degree at University of Peloponnese, Tripoli, Greece.
His doctoral work focuses on design of optical fibers
using the inverse transmission-line technique.
     Mr Thraskias is now Officer in the Hellenic Air
Force and he works as avionics engineer in the
military airport of Kalamata, Greece. He is head of
the Computer Based Training System in the airport of
Kalamata.

				
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Description: UBICC, the Ubiquitous Computing and Communication Journal [ISSN 1992-8424], is an international scientific and educational organization dedicated to advancing the arts, sciences, and applications of information technology. With a world-wide membership, UBICC is a leading resource for computing professionals and students working in the various fields of Information Technology, and for interpreting the impact of information technology on society.
UbiCC Journal UbiCC Journal Ubiquitous Computing and Communication Journal www.ubicc.org
About UBICC, the Ubiquitous Computing and Communication Journal [ISSN 1992-8424], is an international scientific and educational organization dedicated to advancing the arts, sciences, and applications of information technology. With a world-wide membership, UBICC is a leading resource for computing professionals and students working in the various fields of Information Technology, and for interpreting the impact of information technology on society.