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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 the normalised propagation constant b using the HE21 [1] L. Dhar, H. J. Lee, E. J. Laskowski, S. K. Buratto, mode. Table 4 shows the calculated error. H. M. Presby, C. Narayanan, C.C. Bahr, P.J. Anthony, M. J. Cardillo, Refractive index profiling of optical waveguides using near-field scanning optical microscopy, OFC 1996, paper ∆ = 0.0047 FB-5, pp.303-304. ∆ = 0.0013 ∆ = 0.000662 [2] K. Morishita, Refractive-Index-Profile Determination of Single Mode Optical Fibres by a Propagation-Mode Near Field Scanning Technique, IEEE J. Lightwave technology Vol LT-1, No 3, Sept.1983, pp.445-449. [3] Jin-Hong Lin and Cha’o-Kuang Chen, An inverse Algorithm to Calculate the refractive Index Profiles of Periodically Segmented Waveguides from the Measured Near-Field Intensities, IEEE J. Lightwave Technology, Vol.20, No 1, Jan 2002, pp.58-64. [4] C D Papageorgiou and A C Boucouvalas, Propagation constants of cylindrical dielectric waveguides with arbitrary refractive index profile, using the Resonance technique, Electronics Letters, Vol 18, No18, 2 September 1982, pp. 768-788. [5] A C Boucouvalas and C D Papageorgiou, Cutoff frequencies in optical fibres of arbitrary refractive Figure 10: The ripple of the derivative dn/dr versus the index profile using the resonance technique, normalised propagation constant b for three different IEEE Journal of Quantum Electronics, Vol QE- step index optical fibres with the same V = 5 and 18, No 12, Dec. 1982, pp. 2027-2031. different ∆ using the HE21 mode. [6] A C Boucouvalas and S C Robertson, Optical waveguide transverse transmission line equations and their use in determining mode properties, Table 4 JOERS Advanced Fibre Measurement Symposium, National Physical Laboratories, London, 10 September 1985. The accuracy of our method in the calculation of the [7] A C Boucouvalas and X. Qian, Optical Fiber propagation constant using the HE21 mode. Refractive Index Profile Synthesis from Near Field IEEE GLOBECOM 2003 - Optical Networking and Systems, San Francisco, CA, bexact − b min November 2003, pp. 2669-2673. Three different bexact bmin % bexact 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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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.

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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.

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