Paper UBICC 344

8/7/2019 Paper UBICC 344

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ACCURATE OPTICAL FIBER REFRACTIVE INDEX

RECONSTRUCTION FROM NEAR FIELD

. C. Boucouvalas and C. . 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 Maxwells

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 . The proposed analyticalmethod computes the refractive index error directly without the need for curve fitting ornumerical differentiation. Subsequently we work out the exact value of the propagation

constant . The calculation of the propagation constant is vital for the refractive indexreconstruction and we show that it implies error minimization in the refractive indexsynthesis 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 INTRODUCTIONThe main characteristics of optical waveguides,such as bandwidth, spot size, single-mode

propagation conditions, and interwaveguide couplingcoefficients are all related to their refractive index

profiles. So the characterization of optical fiberrefractive index profiles has the fundamentalimportance for the determination of waveguideoptical properties. Refractive index profilemeasurement, however, is generally very difficult dueto small optical fiber dimensions and low refractiveindex differences between core and cladding. Anumber of techniques, [1], have been proposed fordetermining the refractive index distribution of opticalfibres from the propagation mode near field, and the

most well known rely on the seminal theoretical work by Morishita,[2]. Reference [2] relies on an inversesolution of the scalar wave equation for the refractiveindex profile. In [1], the measurement of the near fieldintensity is improved using a scanning opticalmicroscopy technique rather than conventional optics.Improvements from [2], have been recently reportedin [3], which is a robust method to noise and errors,and non-iterative, but reported for planar waveguides.

Using transmission line techniques we haveshown that they can be applied in optical fibres andcan determine exactly the mode propagation constants[4], and cutoff wavelengths of waveguide modes [5].In general from knowledge of the refractive index,

complete waveguide characterization can be achievedusing this powerful technique [6].

In this paper we extend the transmission line

theory and present a novel method for the exact

reconstruction of the waveguide refractive index profile using the fundamental or other higher ordermodes near electric field data. Although in most cases

instrumentation uses the fundamental mode for

reconstruction, in some cases this may not be easy

and especially if a good higher order mode is present

at a convenient wavelength rather than an operation

wavelength, it may be suitable to use higher ordermodes. The following section describes the basic

theory our technique is based upon.

2 FORWARD SOLUTIONWe divide an optical waveguide into a large

number of homogeneous cylindrical layers ofthickness r, permittivity , permeability andconductivity in Fig.1.

Figure 1: Homogeneous cylindrical layer

r z


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TheEandHcomponents of Maxwells equationsfor any such layer can be written as:

( )

( )( )

Z

Z

Z

r

r

r

rE lE rH

lH rH j rE

rH j lH rH

r

=

=

= +

(1)

2

2

Z

[( ) ]( )( )

( )

( )( ) E

Z

Z

Z

Z

r

r

r

j rE j lE rE

r

lH rH lrH H H

r j r

lE rE l j rE E

r j r

= + +

+= +

+= +

+

(2)

where2 2 2 2

( )l

jr

= + +

We restrict our analysis to the case2

0 00, , n = = = , where n is the refractive

index of the layer at distance rfrom the axis.We define the following variable voltages and

currents:

(sum)

(difference)

(sum)

(difference)

M

E

M

E

E

M

E

d M

d

s

s

VV V n

n

VV V n

n

I I I n

n

I I I n

n

= +

=

= +

=

(3)

where

0M

ZlH rH

V ZjF

+

= (magnetic voltage)

0

M

rrHIjZ

= (magnetic current)

0

E

ZlE rE

V ZF

+

= (electric voltage)

2

0

E r I n rE = (electric current)

0 120Z = the free space impedance,

2 2( )r l

F r

+

= .

After algebraic derivatives, (1) and (2) can be

transformed into:

0

0

2

s ss

ss

VI

r j nF

I j nFV

r

=

=

(4)

2

0

0

d d

d

d

d

VI

r j nF

I j nFV

r

=

=

(5)

where2 2 2 2 2 0

0 2 2

2( )

( )sd

nk lln k

r r l

= +

+ ( - forHE, +

forEHmodes).

Equations (4) and (5) represent two independent

transmission lines with voltages ,s dV V and currents

,s dI I . The corresponding characteristic impedances

are:

0

0

ss

dd

Z j nF

Z

j nF

=

=

(6)

The above equations are recognized as the wellknown transmission line equations with the solutionrepresented by the following electric circuit Fig. 2.

Figure 2: Equivalent circuit for an optical fibrecylindrical thin layer

tanh( )2

sinh( )

B

P

s sd d

sd

s

d

Z Z

Z

Z

r

r

=

=

(7)


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where r is the length of the transmission line.The resonance frequency of the cascade of those

electric circuits for a fixed wavelength represents themode propagation constants of the relevantwaveguide with certain refractive index profile.

The technique can further be extended in order to be used for plotting the electric or magnetic fieldcomponents of the waveguides as follows:

From the above equations, we can derive the

Electric current I and Electric field rE . We know

that:

( )( )

( )( )

M

M

HE E

EH E

VV V n r

n r

VV V n r

n r

= +

=

(8)

( )( )

( )( )

E

HE M

E

EH M

I I I n r

n r

I I I n r

n r

= +

=

(9)

We wish to determine the E/M field of the HE

mode in terms ofHE

I andHE

V variables, we can set

EHI =

EHV =0 when theHEmodes are of interest. This

implies that

( )( )

( )( )

M

M

E

E

II

VV

n rn r

n rn r

=

=

(10)

Substituting into (8) and (9), the following equations

can be derived:

2 ( ) , 2 ( )

2 , 2( ) ( )

HE E HE M

M E HE HE

V V I I

V IV I

n r n r

n r n r

= =

= =

(11)

Note thatHE

I ,HE

V are also referred to as sI , sV

respectively. Hence:

20

( )

,2 ( ) 2 ( )

( ) ( ),

2 2

s sE M

s sM E

E r

V IV I

V IV I

I E

n r n r

n r n r

n r r

= =

= =

=

(12)

Hence:

0

2 2

0 0( )

EEr

Z IIE

n r k n r r = =

and:

0

2( )

Er

Z IEn r r

= (13)

where0 0

0

, ,r rk r rk k

= = = .

Therefore if we already know the refractive index

as a function of radius, we can use the (13) to plot the

Electric fields, rE and subsequent Magnetic fields

precisely.

3 INVERSE PROBLEMThe equivalent circuit for a cylindrical thin

dielectric layer, Fig.1, of constant refractive index n

and thickness r at distance r from the core isrepresented as an electric circuit in Fig. 3. Fordetermining the refractive index profile from

knowledge of rE , we assume the following boundary

condition: At r = , we assume n = n2 (silica

refractive index). At r = , 0Zprev = and

2( )n n = are assumed.

Figure 3: Equivalent circuit for a cylindrical thin

layer at constant refractive index n and thickness r

at distance r from the core


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We know that is the effective refractive index

and for typical waveguides lies between n1 and n2.

Furthermore, as we know0

, r, r and2

( )n n = ,

( ) I r r + and ( )V r r+ can be calculated for any

radius. Hence we work out ( )n r as follows:

0

2

2 ( )( )

( )

( )( )

( )

Es

Er

II

I ZE

rr

n r

rr

n r r

=

=

(14)

and finally

0

3 2

( )( )

2 ( )

sr

I r Z E r

n r r

= (15)

2 30( )

( ) ( )2 ( )

s

r

I Z

E

rn r

r r= (16)

Since we know ( )rE r and ( )sI r , hence we can

calculate ( )n r for every rrecursively .We know that the calculation of the exact

propagation constant is necessary for solving theinverse problem of refractive index reconstruction. In[7], we have shown empirically that the error in the

reconstructed refractive index increases with use ofincorrect propagation constant but it is minimum atthe exact value. Here we present an analysis whichconfirms this result and we demonstratereconstruction of the index profiles using thismethod. In the following we provide exact analyticalformulas for the calculation of the radial refractiveindex gradient with respect to radius and as a functionof the required propagation constant . For a certain can simply start to calculate the radial refractiveindex change with respect to r. Starting with

0 2k n = we change within 0 2 0 1k n k n and

recalculate the refractive index profile until the error

in the radial refractive index, (17) at certain isminimum. When the error is minimum we have theexact, and the reconstructed refractive index is alsoexact. The analytical computation of the refractiveindex error can be achieved with use of derivative(17). From (7) we know equivalent T-circuitimpedances are functions of refractive index andradius r, so the following derivative can be extendedas follows:

0

0

n

n

n n

r r

Zn r

Zr

n

=

=

=

|

|(17)

where

1

1

1 1n B

B n p

Z Z Z Z Z

= + + +

, (n=1,2..), is

the total characteristic impedance of cylindrical layers.Using the derivative (17) we can study the refractiveindex changes versus the radius r for various . From

(7) we also know that the equivalent T-circuitimpedances are functions of propagation constant .Hence, the derivative (17) is a function of the

propagation constant can be calculated. We first work

outnZ

r

,

nZ

n

, given by:

( )

1

2 2

1

2

1

1 1p

pB

n

n

npB n

Z r r r

r r ZZ Z Z Z Z

+ = + +

+

++

(18)

( )

1

2 2

1

2

1

1 1p

pB

n

n

npB n

Z n n n

n n ZZ Z Z Z Z

+ = + +

+

++

(19)

Notice that for equations (18) and (19)p

r

,

r

,

p

n

and

n

are also required and are given by:

2

2

0

22

2

0

pZ

lr

rl

n rk r r

= +

(20)

2 22

2

pBp

ZZ

r

r r r

=

+

(21)

02

2 2

0

p

Z

n ln r rk

r

= +

(22)

2 22

2

pBp

ZZ

r

n n n

=

+

(23)

Where

( )

2

2 2 2 2 00 2 2

2nk lln k

r r l

+

+

=


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( )( )

32 2

0

23 2 2

42 nk rl l

r r r l

+

= +

( )

22 0

0 2 2

2

2

k l

k nn r l

=

+

4 NUMERICAL RESULTS AND DISCUSSIONFig. 4 shows the reconstructed refractive index

profile of a step index optical fibre of refractive indexn1= 1.51508 and n2=1.508 and V=2.3 using thefundamental mode near electric field data. In the fig.4 we assume we have full knowledge of the exact

propagation constant. Fig. 5 shows thereconstructed refractive index profile of the same step

index optical fibre for an incorrect propagationconstant exact. We observe that thereconstructed refractive index is sensitive to the

propagation constant. Hence, if we want tominimize the error in the refractive indexreconstruction we require to work out the exact valueof the propagation constant.

Figure 4: The reconstructed refractive index using

(16) with = exact. The normalized value of the

exact propagation constant used is b = 0.5032

Figure 5: The reconstructed refractive index using (16)

with exact. The normalised propagation constantused is b = 0.5112.

(a)

(b)

Figure 6 (a): The derivative dn/dr versus the

normalised radius r using the fundamental mode nearfield data with exact. The normalised propagation

= 0.000662

= 0.0013

= 0.0026


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constant used is b = 0.2889, (b): The ripple of the

derivative dn/dr versus the normalised propagation

constant b for three different step index optical fibreswith the same V = 2.3 and different using the

fundamental mode near field data.

Fig.6(a) shows the derivative dn/dr versus thenormalised radius r with exact. Since in thisexample we study a step index optical fiber, thederivative dn/dr should be zero in the core of thefiber. The oscillations of the derivative in the core

prove the existence of error in the refractive indexreconstruction which we expect to be near zero whenthe exactis used.

The error in the cladding is forced to zero usingthe knowledge of the cladding silica refractive index.

Fig.6(b) shows that there is a minimum ripple(error) in the reconstructed index at a specificvalue.We observe that the ripple oscillations increase withincorrect. At the minimum ripple point we have the

exact, andTable 1 shows the calculated error for b,the derived normalised, (0


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(a)

(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 for the parabolic

refractive index optical fiber.

Table 2

Fig. 9(a) shows the reconstructed refractive index

profile of a step index optical fibre of refractive indexn1= 1.51508 and n2=1.508 and V=5 using the HE12mode 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 thenormalised propagation constant b using the HE12mode.

(a)

(b)

Figure 9 (a): The reconstructed refractive index

using the HE12 mode. (b): The ripple of thederivative dn/dr versus the normalised propagation

constant b for three different step index opticalfibres with the same V = 5 and different using the

HE12mode.

The accuracy of our method in the calculation of the

propagation constant.

exact min ( )%

minexact

exact

Triangular

refractive

index

1.510877 1.510872 0.00033093

Parabolic

refractive

index

1.512256 1.512251 0.00033063

= 0.0047

= 0.0013

= 0.000662


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Table 3

Fig.10 shows the ripple of the derivative dn/dr versus

the normalised propagation constant b using theHE21mode. Table 4 shows the calculated error.

Figure 10: The ripple of the derivative dn/dr versus the

normalised propagation constant b for three differentstep index optical fibres with the same V = 5 and

different using theHE21mode.

Table 4

5 CONCLUSIONSIn this paper, a new and accurate refractive index

profile synthesis technique has been developed. Thismethod is based on the Transmission-Line technique.

An exact analytic method for calculating the mode ,is presented during the refractive index reconstructionfrom the near field. The accurate calculation of the

propagation constant is very important for therefractive index synthesis as it results in the errorminimization in the reconstructed refractive index.The method requires knowledge of the near field ofthe optical fibre and the reconstruction is theoreticallyexact. Simulation results demonstrate the potential ofthis new method.

6 REFERENCES[1] L. Dhar, H. J. Lee, E. J. Laskowski, S. K. Buratto,

H. M. Presby, C. Narayanan, C.C. Bahr, P.J.Anthony, M. J. Cardillo, Refractive index profiling of optical waveguides using near-fieldscanning optical microscopy, OFC 1996, paperFB-5, pp.303-304.

[2] K. Morishita, Refractive-Index-ProfileDetermination of Single Mode Optical Fibres bya Propagation-Mode Near Field ScanningTechnique, IEEE J. Lightwave technology VolLT-1, No 3, Sept.1983, pp.445-449.

[3] Jin-Hong Lin and Chao-Kuang Chen, An inverseAlgorithm to Calculate the refractive IndexProfiles of Periodically Segmented Waveguidesfrom the Measured Near-Field Intensities, IEEEJ. Lightwave Technology, Vol.20, No 1, Jan2002, pp.58-64.

[4] C D Papageorgiou and A C Boucouvalas,Propagation constants of cylindrical dielectricwaveguides with arbitrary refractive index

profile, using the Resonance technique,Electronics Letters, Vol 18, No18, 2 September1982, pp. 768-788.

[5] A C Boucouvalas and C D Papageorgiou, Cutofffrequencies in optical fibres of arbitrary refractiveindex profile using the resonance technique,IEEE Journal of Quantum Electronics, Vol QE-18, No 12, Dec. 1982, pp. 2027-2031.

[6] A C Boucouvalas and S C Robertson, Opticalwaveguide transverse transmission line equationsand their use in determining mode properties,JOERS Advanced Fibre MeasurementSymposium, National Physical Laboratories,

London, 10 September 1985.[7] A C Boucouvalas and X. Qian, Optical Fiber

Refractive Index Profile Synthesis from NearField IEEE GLOBECOM 2003 - Optical

Networking and Systems, San Francisco, CA,November 2003, pp. 2669-2673.


propagation constant using theHE12 mode.

Three different

optical fibres

bexact bmin %minbexact b

bexact

= 0.0047 , V=5 0.2149 0.2151 0.093

= 0.0013 , V=5 0.2149 0.2152 0.14

= 0.000662, V=5 0.2149 0.2148 0.046


propagation constant using theHE21mode.

Three different

optical fibres

bexact bmin %minbexact b

bexact

= 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

= 0.0047

= 0.0013

= 0.000662


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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 joinedBournemouth University. In 1996 he became aProfessor 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 headof 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 industrystandard 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). In2002 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 numerousJournals 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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