Reconstruction of Dielectric Constants of Core and Cladding of...

Research ArticleReconstruction of Dielectric Constants of Core and Cladding ofOptical Fibers Using Propagation Constants Measurements

E M Karchevskii1 A O Spiridonov1 A I Repina1 and L Beilina2

1 Kazan Federal University Kremlevskaya 18 Kazan 420008 Russia2 Chalmers University of Technology and University of Gothenburg 42196 Gothenburg Sweden

Correspondence should be addressed to E M Karchevskii ekarchevyandexru

Received 17 June 2014 Revised 6 September 2014 Accepted 7 September 2014 Published 18 September 2014

Academic Editor Ernest M Henley

Copyright copy 2014 E M Karchevskii et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We present new numerical methods for the solution of inverse spectral problem to determine the dielectric constants of core andcladding in optical fibers These methods use measurements of propagation constants Our algorithms are based on approximatesolution of a nonlinear nonselfadjoint eigenvalue problem for a system of weakly singular integral equationsWe study three inverseproblems and prove that they are well posed Our numerical results indicate good accuracy of new algorithms

1 Introduction

In this paper we investigate the problem of determinationof dielectric constants of core and cladding in optical fibersusing measurements of propagation constant Such kinds ofproblems arise in determination of electromagnetic parame-ters in nanocomposite materials in nanoengineering Theseparameters cannot be measured experimentally because ofnanocomposite type of materials [1]

There are many nondestructive material characterizationtechniques for obtaining permittivities of dielectric materials(see a short review of recent results in [2]) For the firsttime the problem of complex permittivity determination inclosed rectangular waveguides using propagation constantmeasurements was investigated in [3] For open dielectricwaveguides of arbitrary cross-section such problems can beset up as inverse eigenvalue problems [4] of the theory ofoptical waveguides [5] Problems described in [4] are basedon integral formulations of the underlying equations

Inverse problems for determining of dielectric permittiv-ity were studied in many works see for example [6ndash9] andreferences therein In [10] authors present reconstructionof complex permittivity using finite difference time-domain(FDTD) method In [11 12] an analytical method for recon-struction of permittivity of a lossy arbitrary shaped bodyinside a three-dimensional waveguide using transmitted and

reflected data was developed Results of [11 12] are based onthe volume singular integral equation (VSIE) method [8 9]for the system of Maxwellrsquos equations

The new approximately globally convergent method forreconstruction of permittivity function was developed in[13] This method was further verified on computationallysimulated and on experimental data in [14ndash22] An adaptiveversion of the globally convergent method is developed andcomputationally verified for the first time in [23] In recentworks [16 17 21] reconstruction of the spatially distributeddielectric functions and shapes of objects placed in the airas well as of buried objects in the dry sand from blindbackscattered experimental data using two-stage numericalprocedure of [13 15 24] is presented In the first stage in[16 21 22] the approximately globally convergent methodof [13] is applied to get a good first approximation for theexact permittivity function Then the local adaptive finiteelement method of [25 26] is used in the second stage torefine the solution obtained in the first stage Results of thisstage are presented in [15 17] An approximately globallyconvergent algorithm in a frequency domain for the caseof reconstruction of dielectric function in an optical fiber isrecently presented in [27]

The method investigated in this work significantly differsfrom the above described techniques Compared with [16ndash21 27] we use measurements of the propagation constant

Hindawi Publishing CorporationPhysics Research InternationalVolume 2014 Article ID 253435 9 pageshttpdxdoiorg1011552014253435

2 Physics Research International

instead of the electric field data which is typical for waveg-uides applications Our new algorithms use solution of inte-gral equation over the boundary of a waveguide and thus theyare less computationally expensive compared with methodsof [11 12] which are also based on the volume singular integralequation (VSIE) technique [8 9] The models of [11 12]require the solution of three-dimensional vector problemsand thus they aremore computationally demanding and needpowerful computer resources

We note that two mathematical models of optical waveg-uides were investigated in detail by methods of integralequations in [19 20 28ndash34] In [19 20 28ndash31] the model ofstep-indexwaveguideswas studied and in [32ndash34] themodelof waveguides without a sharp boundary was studied Areview of modern results in this field is given in [35] To solveour eigenvalue problem we are not requiring the informationabout specific values of eigenfunctions In our algorithmsit is enough only to know that the fundamental mode isexcited and then to measure its propagation constant for oneor two frequencies Main applications of our algorithm arefor example in detection of defects in metamaterials and innanoelectronics [36 37] calculation of dielectric constant ofsaline water [38 39] and in microwave imaging technologyin remote sensing [40]

An outline of this paper is a follows In Section 2 wefirst present a mathematical model of a weakly guiding step-index arbitrarily shaped optical fiber and formulate threeinverse spectral problems Then we present new numeri-cal algorithms for calculation of dielectric constants basedon an approximate solution of a nonlinear nonselfadjointeigenvalue problem for a system of weakly singular integralequations In Section 3 we present reconstruction resultswhich show efficiency and robustness of new developedalgorithms

2 Methods of Solution

21 Nonlinear Eigenvalue Problem for Transverse Wavenum-bers Let us consider an optical fiber as a regular cylindricaldielectric waveguide in a free space The cross-section ofthe waveguidersquos core is a bounded domain Ω

119894with twice

continuously differentiable boundary 120574 (see Figure 1) Theaxis of the cylinder is parallel to the 119909

3-axis Let Ω

119890=

1198772 Ω119894be the unbounded domain of the cladding Let the

permittivity be prescribed as a positive piecewise constantfunction 120576 which is equal to a constant 120576

infinin the domain Ω

119890

and to a constant 120576+gt 120576infin

in the domainΩ119894

Eigenvalue problems of optical waveguide theory [5] areformulated on the base of the set of homogeneous Maxwellequations

rotE = minus1205830

120597H

120597119905 rotH = 120576

0120576120597E

120597119905 (1)

Here E andH are electric and magnetic field vectors and 1205760

and 1205830are the free-space dielectric and magnetic constants

Nontrivial solutions of set (1) which have the form

[EH] (119909 119909

3 119905) = Re([EH] (119909) 119890

119894(1205731199093minus120596119905)) (2)

120574

x3 x1

x2

120576 = 120576+

120576 = 120576infinΩi

Ωe

Figure 1 The cross-section of a cylindrical dielectric waveguide ina free space

are called the eigenmodes of the waveguide Here positive 120596is the radian frequency 120573 is the propagation constant E andH are complex amplitudes of E andH and 119909 = (119909

1 1199092)

In forward eigenvalue problems the permittivity isknown and it is necessary to calculate longitudinal wavenum-bers 119896 = 120596radic12057601205830 and propagation constants 120573 such thatthere exist eigenmodes The eigenmodes have to satisfy to atransparency condition at the boundary 120574 and to a conditionat infinity

In inverse eigenvalue problems it is necessary to recon-struct the unknown permittivity 120576 by some information onnatural eigenmodes which exist for some eigenvalues 119896 and120573 The main question is how many observations of naturaleigenmodes are enough for unique and stable reconstructionof the permittivity

The domain Ω119890is unbounded Therefore it is necessary

to formulate a condition at infinity for complex amplitudesE and H of eigenmodes Let us confine ourselves to theinvestigation of the surface modes only The propagationconstants 120573 of surface modes are real and belong to theinterval 119866 = (119896120576

infin 119896120576+) The amplitudes of surface modes

satisfy the following condition

[EH] = 119890

minus120590119903119874(

1

radic119903) 119903 = |119909| 997888rarr infin (3)

Here 120590 = radic1205732 minus 1198962120576infingt 0 is the transverse wavenumber in

the claddingDenote by 120594 = radic1198962120576

+minus 1205732 the transverse wavenumber

in the waveguidersquos core Under the weakly guidance approx-imation [5] the original problem was reduced in [29] tothe calculation of numbers 120594 and 120590 such that there existnontrivial solutions 119906 = 119867

1= 1198672of Helmholtz equations

Δ119906 + 1205942119906 = 0 119909 isin Ω

119894

Δ119906 minus 1205902119906 = 0 119909 isin Ω

119890

(4)

which satisfy the transparency conditions

119906+= 119906minus

120597119906+

120597]=120597119906minus

120597] 119909 isin 120574 (5)

Physics Research International 3

Here ] is the normal derivative on 120574 119906minus (resp 119906+) is the limitvalue of a function 119906 from the interior (resp the exterior) of120574

Denote by 119880 the space of continuous and continuouslydifferentiable functions inΩ

119894and Ω

119890and twice continuously

differentiable functions in Ω119894and Ω

119890 satisfying to condition

(3) Let us calculate nontrivial solutions 119906 of problem (4)-(5)in the space 119880

Let 120590 gt 0 be a given number A nonzero function 119906 isin 119880is called an eigenfunction of problem (4)-(5) correspondingto a real eigenvalue 120594 if relationships (4)-(5) hold

The next theorem follows from results of [29]

Theorem 1 For any 120590 gt 0 the eigenvalues 120594 of problem(4)-(5) can be only positive isolated numbers Each number 120594depends continuously on 120590

For waveguides of circular cross-section the analogousresults about the localization of the surface modes spectrumand about the continuous dependence between the transversewavenumbers 120590 and 120594 were obtained in [5] The results of [5]are valid only for waveguides of circular cross-section by themethod of separation of variables Theorem 1 generalizes theresults of [5] for the case of an arbitrary smooth boundaryThe next theorem follows from results of [34] (see anillustration in Figure 2)

Theorem 2 The following statements hold(1) For any 120590 gt 0 there exist the denumerable set of positive

eigenvalues120594119894(120590) where 119894 = 1 2 of a finitemultiplicity with

only cumulative point at infinity(2) For any 120590 gt 0 the smallest eigenvalue 120594

1(120590) is positive

and simple (its multiplicity is equal to one) correspondingeigenfunction 119906

1can be chosen as the positive function in the

domain Ω119894

(3) 1205941(120590) rarr 0 at 120590 rarr 0

For a given 120590 gt 0 the smallest eigenvalue 1205941(120590) and

corresponding eigenfunction 1199061define the eigenmode which

is called the fundamental mode (see the bottom curve plottedby the red solid line in Figure 2) Thus Theorem 2 statesparticularly that for any 120590 gt 0 there exists exactly onefundamental mode

To compute eigenmodes we use the representation ofeigenfunctions of problem (4)-(5) in the form of single-layerpotentials [41]

119906 (119909) = int120574

Φ(120594 119909 119910) 119891 (119910) 119889119897 (119910) 119909 isin Ω119894

119906 (119909) = int120574

Ψ (120590 119909 119910) 119892 (119910) 119889119897 (119910) 119909 isin Ω119890

(6)

Here

Φ(120594 119909 119910) =119894

4119867(1)

0(1205941003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

Ψ (120590 119909 119910) =1

21205871198700(1205901003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(7)

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

1205942

1205902

Figure 2 The plot of the dispersion curves computed by thespline-collocation method for surface eigenmodes in a weaklyguiding dielectric waveguide of the circular cross-sectionThe exactsolutions are plotted by solid linesThe solution for the fundamentalmode is plotted by the lower red solid line Numerical solutionsof the residual inverse iteration method are marked by red crossesInitial approximations for SVDs are marked by blue circles

are the fundamental solutions of Helmholtz equations (4)correspondingly119867(1)

0(119911) is Hankel function of the first kind

1198700(119911) is Macdonald function and unknown densities 119891 and

119892 belong to the Holder space 1198620120572(120574) Then 119891 and 119892 satisfyto the following system of boundary integral equations for119909 isin 120574

A11(120594) 119891 +A

12 (120590) 119892 = 0

A21(120594) 119891 +A

22 (120590) 119892 = 0

(8)

where

(A11(120594) 119891) (119909) = int

120574

Φ(120594 119909 119910) 119891 (119910) 119889119897 (119910)

(A12 (120590) 119892) (119909) = minus int

120574

Ψ (120590 119909 119910) 119892 (119910) 119889119897 (119910)

(A21(120594) 119891) (119909) =

1

2119891 (119909) + int

120574

120597Φ (120594 119909 119910)

120597] (119909)119891 (119910) 119889119897 (119910)

(A22 (120590) 119892) (119909) =

1

2119892 (119909) minus int

120574

120597Ψ (120590 119909 119910)

120597] (119909)119892 (119910) 119889119897 (119910)

(9)

Suppose that the boundary 120574 is parametrically defined bya function 119903 = 119903(119905) of 119905 isin [0 2120587] and this parametrizationis regular We consider functions from 119862

0120572(120574) and 1198621120572(120574)

as Holder-continues and Holder-continuously differentiable2120587-periodic functions of parameter 119905 Then for 119909 = 119909(119905) isin 120574


where 119905 isin [0 2120587] system (8) is transformed to the followingsystem of equations

1198711199011+ 11986111(120594 120590) 119901

1+ 11986112(120594 120590) 119901

2= 0

1199012+ 11986121(120594 120590) 119901

1+ 11986122(120594 120590) 119901

2= 0

(10)

Here new unknown functions are defined as follows

1199011 (120591) = (119891 (119910) minus 119892 (119910))

100381610038161003816100381610038161199031015840(120591)10038161003816100381610038161003816 119901

2 (120591) = 119891 (119910) + 119892 (119910)

119910 = 119910 (120591) 120591 isin [0 2120587]

(11)

Integral operators are defined by the following relationshipsfor 119905 isin [0 2120587]

(1198711199011) (119905) = minus

1

2120587int

2120587

0

ln1003816100381610038161003816100381610038161003816sin 119905 minus 120591

2

10038161003816100381610038161003816100381610038161199011 (120591) 119889120591

(119861119894119895(120594 120590) 119901

(119895)) (119905) =

1

2120587int

2120587

0

ℎ119894119895(120594 120590 119905 120591) 119901

(119895)(120591) 119889120591

(12)

where

ℎ11(120594 120590 119905 120591) = 2120587 (119866

11(120594 119905 120591) + 119866

12 (120590 119905 120591))

ℎ12(120594 120590 119905 120591) = 2120587 (119866

11(120594 119905 120591) minus 119866

12 (120590 119905 120591))100381610038161003816100381610038161199031015840(120591)10038161003816100381610038161003816

ℎ21(120594 120590 119905 120591) = 4120587 (119866

21(120594 119905 120591) + 119866

22 (120590 119905 120591))

ℎ22(120594 120590 119905 120591) = 4120587 (119866

21(120594 119905 120591) minus 119866

22 (120590 119905 120591))100381610038161003816100381610038161199031015840(120591)10038161003816100381610038161003816

11986611(120594 119905 120591) = Φ (120594 119909 119910) +

1

2120587ln1003816100381610038161003816100381610038161003816sin 119905 minus 120591

2

1003816100381610038161003816100381610038161003816

11986612 (120590 119905 120591) = Ψ (120590 119909 119910) +

1

2120587ln1003816100381610038161003816100381610038161003816sin 119905 minus 120591

2

1003816100381610038161003816100381610038161003816

11986621(120594 119905 120591) =

120597Φ (120594 119909 119910)

120597] (119909)

11986622 (120590 119905 120591) =

120597Ψ (120590 119909 119910)

120597] (119909)

(13)

The linear operator 119871 1198620120572(120574) rarr 119862

1120572(120574) is continuous

and continuously invertible (for details see eg [29]) It wasproved in [29] that for each 120594 gt 0 120590 gt 0 the linear operators

11986121(120594 120590) 119861

22(120594 120590) 119862

0120572(120574) 997888rarr 119862

0120572(120574)

11986111(120594 120590) 119861

12(120594 120590) 119862

0120572(120574) 997888rarr 119862

1120572(120574)

(14)

are compact Therefore system (8) is equivalent to thefollowing Fredholm operator equation of the second kind

119860 (120594 120590)119908 = (119868 + 119861 (120594 120590))119908 = 0 (15)

where 119908 = (1199081 1199082)119879

1199081= 1198711199011isin 1198621120572(120574) 119908

2= 1199012isin 1198620120572(120574) (16)

and the compact operator 119861 acting in the Banach space119882 =

1198621120572(120574) times 119862

0120572(120574) is defined as follows

119861119908 = [11986111119871minus1

11986112

11986121119871minus1

11986122

] [1199081

1199082

] (17)

by 119868 the identical operator is denotedLet120590 gt 0 be a given number A nonzero element119908 isin 119882 is

called an eigenfunction of the operator-valued function119860(120594)corresponding to a characteristic value 120594 gt 0 if (15) holds

It follows from results of [29] that original problem (4)-(5)is spectrally equivalent to (15) Namely the following theoremis true

Theorem 3 Let 120590 gt 0 be a given number The followingstatements hold

(1) If a function 119908 is the eigenfunction of the operator-valued function 119860(120594) corresponding to a characteristic value1205940gt 0 then the function 119906 defined by equalities (6) where

120594 = 1205940

119891 =1199082

2+119871minus11199081

2100381610038161003816100381611990310158401003816100381610038161003816

119892 =1199082

2minus119871minus11199081

2100381610038161003816100381611990310158401003816100381610038161003816

(18)

is the eigenfunction of problem (4)-(5) corresponding to theeigenvalue 120594

0

(2) Each eigenfunction of problem (4)-(5) corresponding toan eigenvalue 120594

0gt 0 can be represented in the form of single-

layer potentials (6) with some Holder-continuous densities 119891and 119892 respectively At the same time the function

119908 = (119871 ((119891 minus 119892)10038161003816100381610038161003816119903101584010038161003816100381610038161003816) 119891 + 119892) (19)

is the eigenfunction of the operator-valued function 119860(120594)

corresponding to the characteristic value 1205940

Let us formulate the nonlinear spectral problem for trans-verse wavenumbers as follows Suppose that the boundary 120574of the waveguidersquos cross-section and the number 120590 gt 0 aregiven It is necessary to calculate all characteristic values 120594 ofthe operator-valued function 119860(120594) in some given interval

A spline-collocation method was proposed in [31] fornumerical calculations of characteristic values 120594 of theoperator-valued function 119860(120594) for given 120590 and problem (15)for each fixed120590was reduced to a nonlinear finite-dimensionaleigenvalue problem of the form

119860119899(120594)119908119899= 0 (20)

where 119899 is the number of collocation points For numericalsolution of obtained nonlinear finite-dimensional eigenvalueproblem we use the residual inverse iteration method [42]

Any iterative numerical method for computations of thenonlinear eigenvalues 120594 needs good initial approximationsfor each given 120590 Usually initial approximations are chosenby a physical intuition using prior information on solutionsIf we model fundamentally new types of waveguides solve


inverse problems or investigate defects in fibers thenwemaynot have accurate prior information on solutions

In this case we can investigate spectral properties of thematrix 119860

119899(120594) as a function of variable 120594 for each fixed 120590 For

each given point in an investigated domain of parameters 120594and 120590 we calculate the condition number of matrix 119860

119899

cond (119860119899(120594)) =

1205881

120588119899

(21)

where 1205881and 120588119899are maximal and minimal singular values of

matrix If for given120590 a number120594 is equal to a nonlinear eigen-value of119860

119899(120594) then the condition number is equal to infinity

Therefore the numbers 120594 for which the condition number isbig enough are good approximations for eigenvalues Singularvalues are calculated by singular value decomposition (SVD)method

119860119899(120594) = 119880119878119881 119878 = diag (120588

1 1205882 120588

119899) (22)

where 119880 119881 are unitary matrices and 119878 is a diagonal matrixthe singular numbers form 119878 The calculations are based onunitary transformations of the matrix 119860 and therefore arestable

In the next step for each 120590 in the investigated intervalwe use obtained initial approximations for numerical cal-culations of nonlinear eigenvalues 120594 by the residual inverseiteration method

On the base of solution of nonlinear eigenvalue problem(15) for transverse wavenumbers we solve the forward andinverse spectral problems for weakly guiding step-indexwaveguides

22 Forward Spectral Problem Clearly if for given120590 the char-acteristic values 120594 of the operator-valued function119860(120594) werecalculated and also if the permittivities 120576

+ 120576infinare known then

the longitudinal wavenumber 119896 and the propagation constant120573 are calculated by the following explicit formulas

1198962=1205902+ 1205942

120576+minus 120576infin

1205732=120576+1205902+ 120576infin1205942


(23)

For each given 120576+ 120576infin and 119894 the function 120594

119894(120590) generates

a function 1205732 of variable 1198962 As an example in Figure 3 wepresent the plot of the computed function 120573

2= 1205732(1198962)

corresponding to the fundamental mode (see the bottomcurve plotted by the red solid line in Figure 2) of the circularwaveguide Here 120576

+= 2 120576

infin= 1 and 119894 = 1 We

will use two points marked by circle and by square in thenext sections as test points for numerical solution of inversespectral problems

23 Inverse Spectral Problems In this subsection we presentthree algorithms for approximate solution of three inversespectral problems The algorithms are based on previousnumerical solution of nonlinear eigenvalue problem (15) fortransverse wavenumbers namely on numerical calculationsof characteristic values 120594

1(120590) of the operator-valued function

119860(120594) for fundamental mode and for 120590 in an appropriateinterval

0 1 2 3 4 50

1

2

3

4

5

6

7

8

1205732

k2

120576+ = 2 120576infin = 1

k2 = 1 1205732 = 10409

k2 = 3 1205732 = 39652

Figure 3 The red solid line presents the plot of the computedfunction 1205732 = 120573

2(1198962) corresponding to the fundamental mode of

the circular waveguide

231 Algorithm for Reconstruction of the Permittivity of theCore The first inverse problem is formulated as followsSuppose that the boundary 120574 of the waveguidersquos cross-sectionand the permittivity 120576

infinof the cladding are given Suppose

that the propagation constant 120573 of the fundamental modeis measured for a given wavenumber 119896 The measuring canbe done by experimental methods described for examplein [3] It is necessary to calculate the permittivity 120576

+of the

waveguidersquos coreThe solution of this inverse spectral problem is calculated

in the following way First we compute the number

120590 = radic1205732 minus 1198962120576infin (24)

which is calculated for given values of 120573 119896 and 120576infin Then

the transverse wavenumber 120594(120590) is calculated by the spline-collocation method for the obtained 120590 of the fundamentalmode This number is calculated by an interpolation of thefunction 120594

1(120590) with respect to the points obtained when the

nonlinear spectral problem for transverse wavenumbers wasnumerically solved Finally the permittivity of the waveg-uidersquos core is calculated by the following explicit formula

120576+=1205942+ 1205732

1198962 (25)

232 Algorithm for Reconstruction of the Permittivity of theCladding Suppose that the permittivity of the core is givenand that the propagation constant120573 of the fundamentalmodeis measured for a given wavenumber 119896 It is necessary tocalculate the permittivity 120576

infinof the waveguidersquos cladding


0 05 1 151

15

2

25

3

35

Cut-off points

120576 +

120576infin

k2 = 1 1205732 = 10409

k2 = 3 1205732 = 39652 120576infin = 1 120576+ = 2

Figure 4 The blue and the green solid lines are plots of function120576+= 120576+(120576infin) for given pairs of 119896 and 120573 of the fundamental mode

The unique exact solution of the inverse spectral problem is markedby the red circle The approximate solution obtained by the spline-collocation method coincides with the red circle for the exact 120573For randomly noised 120573 approximate solutions are intersections ofdashed lines They belong to the red rhomb

The solution of this problem is calculated in the followingway First we compute the number

120594 = radic1198962120576+minus 1205732 (26)

which is calculated for given values of 120573 119896 and 120576+ Second

the transverse wavenumber 120590 is calculated by the spline-collocation method for the obtained 120594 for the fundamentalmode Third the permittivity of the waveguidersquos cladding iscalculated by the following explicit formula

120576infin=1205732minus 1205902

1198962 (27)

233 Algorithm for Reconstruction of Two Permittivities of theCore and of the Cladding The full variant of our problem isthe reconstruction of both permittivities of the core and ofthe claddingThe solution for the fundamental mode of non-linear eigenvalue problem (15) for transverse wavenumbersgives us an implicit function 120576

+of the variable 120576

infinfor each

fixed longitudinal wavenumber 119896 and propagation constant120573 For example in Figure 4 the blue and the green solid linesare plots of this function for given pairs of 119896 and 120573

The intersection of these lines marked by the red circleis the unique exact solution of our problem Therefore we

calculate the permittivities 120576+and 120576infin

as the solution of thefollowing nonlinear system of two equations

120594(radic1205732

1minus 1198962

1120576infin) = radic119896

2

1120576+minus 1205732

1

120594 (radic1205732

2minus 1198962

2120576infin) = radic119896

2

2120576+minus 1205732

2

(28)

Here 1198961

and 1198962

are some given distinct longitudinalwavenumbers 120573

1and 120573

2are correspondingmeasured propa-

gation constants and 120594 is the function of variable 120576infinfor fixed

119896119895and 120573

119895 119895 = 1 2

3 Results and Discussion

Let us describe numerical results obtained for a nonlineareigenvalue problem (15) for transverse wavenumbers Wecompare our numerical results with the well-known exactsolution for the circular waveguide obtained by themethod ofseparation of variables (see eg [5]) In Figure 2 we presentsome dispersion curves for surface eigenwaves of the circularwaveguide Here the exact solution is plotted by solid lines

We started our numerical calculations with computationsof initial approximations for nonlinear eigenvalues 120594 To dothat we have used SVD as was described in Section 21 Cal-culated initial approximations are marked in Figure 2 by bluecircles The blue circles are only the initial approximationsfor eigenvalues 120594 We apply these initial approximations asstart points in the residual inverse iteration method Usingthis iterationmethod for each given 120590 we solved numericallythe finite-dimensional nonlinear eigenvalue problem (20)depending on eigenvalues 120594 The numerical solution whichresulted in this method is marked in Figure 2 by red crosses

Note here that applying previously calculated initialapproximations for nonlinear eigenvalues 120594 we numericallysolve this problem directly and without any prior physicalinformation Figure 2 illustrates that the initial approxima-tions are good enough and we can use them in the majorityof cases without iterative refinement of solutions

Our next numerical experiment shows reconstruction ofcorersquos permittivity using the algorithm in Section 231 Themathematical analysis of existence of the solution of theoriginal spectral problem (4)-(5) for transverse wavenumbersis presented in Theorems 1 and 2 An illustration of thetheoretical results is shown in Figure 2 Analyzing Figure 2we observe that the fundamental mode (see the red solidcurve in Figure 2) exists for each wavenumber 119896 gt 0 Thefundamental mode is unique its dispersion curve does notintersect with any other curves and well separated fromthemTherefore the inverse spectral problemrsquos solution existsand is unique for each wavenumber 119896 gt 0 this solutiondepends continuously on given data In other words theinverse spectral problem is well posed by Hadamard We cansee an example of this continuity dependence in Figure 5The red solid line is the plot of the function 120576

+of 1205732 for the

fundamental mode


3 35 4 45 51

12

14

16

18

2

22

24

26

1205732

120576 +

Randomly noised

Exact 1205732 = 39652

k2 = 3 120576infin = 1

Figure 5 The red solid line is the plot of function 120576+= 120576+(1205732) for

the fundamental mode The approximate solution obtained by thespline-collocation method is marked by the blue circle for the exact120573 and by black points for the randomly noised 120573

In our computations by analogy with [13 14] we haveintroduced random noise in the propagation constant as

120573 = 120573 (1 + 119901120572) (29)

where 120573 = 398518 is the exact measured propagationconstant 120572 isin (minus1 1) are randomly distributed numbersand 119901 is the noise level In our computations we have used119901 = 005 and thus the noise level was 5

Some numerical results of reconstruction of 120576+are pre-

sented in Figure 5 The approximated value of 120576+for the

noise-free data is marked in Figure 5 by the blue circleApproximated values of 120576

+for randomly distributed noise 120573

with the 5 noise level are marked in Figure 5 by the blackpoints Using this figure we observe that the approximatesolutions even for the randomly noised 120573were stable We canconclude that for the unique and stable reconstruction of theconstant waveguide permittivity 120576

+ it is enough to measure

the propagation constant 120573 of the fundamental mode for onlyone wavenumber 119896

Absolutely analogous results were obtained for recon-struction of claddingrsquos permittivity using the algorithm inSection 232 As in Figure 5 the red solid line in Figure 6is the plot of continuous function 120576

infinof squared 120573 for the

fundamental mode The approximate solution obtained bythe spline-collocationmethod is marked by the blue circle forthe exact 120573 and by black points for the randomly noised 120573Using this figure we observe that the approximate solutionsfor the randomly noised 120573 were stable in this case too For

35 4 45 50

02

04

06

08

1

12

14

16

18

1205732

120576 infinRandomly noised

Exact 1205732 = 39652

k2 = 3 120576+ = 2

Figure 6 The red solid line is the plot of function 120576infin= 120576infin(1205732) for


this test we can also conclude that for the unique and stablereconstruction of the constant permittivity 120576

infin it is enough

to measure the propagation constant 120573 of the fundamentalmode for only one wavenumber 119896

Finally we show simultaneous reconstruction of bothpermittivities of core and of cladding using the algorithmin Section 233 The approximate solution obtained by thespline-collocation method is marked in Figure 4 by the redcircle for the exact propagation constantWe have introducedthe random noise in the propagation constant similarlywith above described tests The approximate solutions forthe noised 120573 are presented in Figure 4 as intersections ofdashed lines Using Figure 4 we observe that the approximatesolutions for the randomly noised 120573 belong to the red rhomband are stableTherefore we can conclude that for the uniqueand stable reconstruction of the dielectric constants 120576

infinand

120576+ it is enough to measure the propagation constant 120573 of the

fundamental mode for two distinct wavenumbers 119896

4 Conclusion

In this work we have formulated three inverse spectralproblems and proved that they are well posed It is importantto note that in our algorithms any information on specificvalues of eigenfunctions is not required For solution of theseinverse problems it is enough to know that the fundamentalmode is excited and then to measure its propagation con-stant for one or for two frequencies This approach satisfies


the practice of physical experiments because usually the fun-damental mode is excited in waveguides in real applications[3] Moreover the fundamental mode can be excited only forenough wide range of frequencies

For the approximate solution of the inverse problemswe propose to solve the nonlinear spectral problem fortransverse wavenumbers in order to compute the dispersioncurve for the fundamental mode These calculations aredone accurately by the spline-collocationmethod Finally wehave demonstrated unique and stable reconstruction of thepermittivities using new inverse algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thankMichael Havrilla and GeorgeHanson for helpful discussions concerning the topics coveredin this work This work was funded by the subsidy allocatedto Kazan Federal University for the state assignment in thesphere of scientific activities the work was supported alsoby RFBR and by Government of Republic Tatarstan Grant12-01-97012-r povolzhrsquoe a The research of Larisa Beilina wassupported by the Swedish Research Council the SwedishFoundation for Strategic Research (SSF) through theGothen-burg Mathematical Modelling Centre (GMMC) and by theSwedish Institute

References

[1] L Solymar and E ShamoninaWaves in Metamaterials OxfordUniversity Press Oxford UK

[2] M Havrilla A Bogle M Hyde IV and E Rothwell ldquoEMmaterial characterization of conductor backed media using aNDEmicrostrip proberdquo Studies in Applied Electromagnetics andMechanics vol 38 no 2 pp 210ndash218 2014

[3] M D Janezic and J A Jargon ldquoComplex permittivity deter-mination from propagation constant measurementsrdquo IEEEMicrowave and Wireless Components Letters vol 9 no 2 pp76ndash78 1999

[4] M T Chu andGHGolub Inverse Eigenvalue ProblemsTheoryAlgorithms and Applications Oxford University Press 2005

[5] A W Snyder and J D Love Optical Waveguide TheoryChapman amp Hall London UK 1983

[6] G Nakamura and M Sini ldquoOn the near field measurementfor the inverse scattering problem for ocean acousticsrdquo InverseProblems vol 20 no 5 pp 1387ndash1392 2004

[7] A RammMultidimensional Inverse Scattering Problems WileyNew York NY USA 1992

[8] K Kobayashi Y Shestopalov and Y Smirnov ldquoInvestigation ofelectromagnetic diffraction by a dielectric body in a waveguideusing the method of volume singular integral equationrdquo SIAMJournal on Applied Mathematics vol 70 no 3 pp 969ndash9832009

[9] Y V Shestopalov and V V Lozhechko ldquoDirect and inverseproblems of the wave diffraction by screens with arbitrary finiteinhomogeneitiesrdquo Journal of Inverse and Ill-Posed Problems vol11 no 6 pp 643ndash654 2003

[10] E Eves K Murphy and V Yakovlev ldquoReconstruction ofcomplex permittivity with neural network-controlled FDTDmodelingrdquoThe Journal ofMicrowave Power and ElectromagneticEnergy vol 41 p 1317 2007

[11] Y Shestopalov and Y Smirnov ldquoExistence and uniqueness ofa solution to the inverse problem of the complex permittivityreconstruction of a dielectric body in a waveguiderdquo InverseProblems vol 26 no 10 Article ID 105002 2010

[12] Y Shestopalov and Y Smirnov ldquoDetermination of permittivityof an inhomogeneous dielectric body in a waveguiderdquo InverseProblems vol 27 no 9 Article ID 095010 2011

[13] L Beilina andM V Klibanov Approximate Global Convergenceand Adaptivity for Coefficient Inverse Problems Springer NewYork NY USA 2012

[14] L Beilina and M V Klibanov ldquoA new approximate mathe-matical model for global convergence for a coefficient inverseproblem with backscattering datardquo Journal of Inverse and Ill-Posed Problems vol 20 no 4 pp 513ndash565 2012

[15] L Beilina and M V Klibanov ldquoThe philosophy of the approx-imate global convergence for multi-dimensional coefficientinverse problemsrdquo Complex Variables and Elliptic Equationsvol 57 no 2ndash4 pp 277ndash299 2012

[16] L Beilina N T Thanh M V Klibanov and M A FiddyldquoReconstruction from blind experimental data for an inverseproblem for a hyperbolic equationrdquo Inverse Problems vol 30no 2 Article ID 025002 2014

[17] L Beilina N T Thanh M V Klibanov and J B MalmbergldquoReconstruction of shapes and refractive indices from backscat-tering experimental data using the adaptivityrdquo Inverse Problemshttparxivorgabs14045862

[18] L Beilina and M V Klibanov ldquoReconstruction of dielec-trics from experimental data via a hybrid globally conver-gentadaptive inverse algorithmrdquo Inverse Problems vol 26 no12 Article ID 125009 2010

[19] A V Kuzhuget L Beilina M V Klibanov A Sullivan LNguyen and M A Fiddy ldquoBlind backscattering experimentaldata collected in the field and an approximately globallyconvergent inverse algorithmrdquo Inverse Problems vol 28 no 9Article ID 095007 2012

[20] M V Klibanov M A Fiddy L Beilina N Pantong andJ Schenk ldquoPicosecond scale experimental verification of aglobally convergent algorithm for a coefficient inverse problemrdquoInverse Problems vol 26 no 4 Article ID 045003 30 pages2010

[21] N T Thanh L Beilina M V Klibanov and M A FiddyldquoReconstruction of the refractive index from experimen-tal backscattering data using a globally convergent inversemethodrdquo SIAM Journal on Scientific Computing vol 36 no 3pp B273ndashB293 2014

[22] N TThanh L BeilinaM V Klibanov andM A Fiddy ldquoImag-ing of buried objects from experimental backscattering radarmeasurements using a globally convergent inverse algorithmrdquohttparxivorgabs14063500v1

[23] M Asadzadeh and L Beilina ldquoA posteriori error analysisin a globally convergent numerical method for a hyperbolic


coefficient inverse problemrdquo Inverse Problems vol 26 no 11Article ID 115007 115007 pages 2010

[24] L Beilina and M V Klibanov ldquoSynthesis of global convergenceand adaptivity for a hyperbolic coefficient inverse problem in3Drdquo Journal of Inverse and Ill-Posed Problems vol 18 no 1 pp85ndash132 2010

[25] L Beilina ldquoAdaptive finite element method for a coefficientinverse problem for Maxwellrsquos systemrdquo Applicable Analysis vol90 no 10 pp 1461ndash1479 2011

[26] L Beilina and C Johnson ldquoA posteriori error estimationin computational inverse scatteringrdquo Mathematical Models ampMethods in Applied Sciences vol 15 no 1 pp 23ndash35 2005

[27] L Beilina and E Karchevskii ldquoThe layer-stripping algorithm forreconstruction of dielectrics in an optical fiberrdquo in Inverse Prob-lems and Applications Springer Proceedings in Mathematics ampStatistics 2014

[28] E M Karchevskii ldquoStudy of spectrum of guided waves ofdielectric fibersrdquo in Proceedings of the Mathematical Methods inElectromagnetic Theory (MMET rsquo98) vol 2 pp 787ndash788 1998

[29] E M Karchevskii ldquoAnalysis of the eigenmode spectra ofdielectric waveguidesrdquo Computational Mathematics and Math-ematical Physics vol 39 no 9 pp 1493ndash1498 1999

[30] E M Karchevskii ldquoThe fundamental wave problem for cylin-drical dielectric waveguidesrdquoDifferential Equations vol 36 no7 pp 1109ndash1111 2000

[31] A O Spiridonov and E M Karchevskiy ldquoProjection methodsfor computation of spectral characteristics of weakly guidingoptical waveguidesrdquo in Proceedings of the International Confer-ence Days on Diffraction (DD rsquo13) pp 131ndash135 May 2013

[32] E M Karchevskii and S I Solovrsquoev ldquoInvestigation of a spectralproblem for the Helmholtz operator in the planerdquo DifferentialEquations vol 36 no 4 pp 631ndash634 2000

[33] E M Kartchevski A I Nosich and GW Hanson ldquoMathemat-ical analysis of the generalized natural modes of an inhomoge-neous optical fiberrdquo SIAM Journal on Applied Mathematics vol65 no 6 pp 2033ndash2048 2005

[34] A Frolov and E Kartchevskiy ldquoIntegral equation methods inoptical waveguide theoryrdquo in Inverse Problems and Large-ScaleComputations vol 52 of Springer Proceedings in Mathematics ampStatistics pp 119ndash133 2013

[35] E Karchevskiy and Y Shestopalov ldquoMathematical and numer-ical analysis of dielectric waveguides by the integral equationmethodrdquo in Proceedings of the Progress in ElectromagneticsResearch Symposium (PIERS rsquo13) pp 388ndash393 StockholmSweden August 2013

[36] M P H Andresen H E Krogstad and J Skaar ldquoInversescattering of two-dimensional photonic structures by layerstrippingrdquo Journal of the Optical Society of America B OpticalPhysics vol 28 no 4 pp 689ndash696 2011

[37] J D Joannopoulos R DMeade and J NWinn Photonic Crys-tals Molding the Flow of Light Princeton University PrincetonNJ USA 1995

[38] L A Klein and C T Swift ldquoAn improved model for the dielectric constant of sea water at microwave frequenciesrdquo IEEETransactions on Antennas and Propagation vol 25 no 1 pp104ndash111 1977

[39] A Stogryn ldquoEquations for calculating the dielectric constantof saline waterrdquo IEEE Transactions on Microwave Theory andTechniques vol 19 no 8 pp 733ndash736 1971

[40] M Pastorino Microwave Imaging John Wiley amp Sons Hobo-ken NJ USA 2010

[41] D L Colton and R Kress Integral Equation Methods inScattering Theory Pure and Applied Mathematics (New York)John Wiley amp Sons New York NY USA 1983

[42] A Neumaier ldquoResidual inverse iteration for the nonlineareigenvalue problemrdquo SIAM Journal on Numerical Analysis vol22 no 5 pp 914ndash923 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


instead of the electric field data which is typical for waveg-uides applications Our new algorithms use solution of inte-gral equation over the boundary of a waveguide and thus theyare less computationally expensive compared with methodsof [11 12] which are also based on the volume singular integralequation (VSIE) technique [8 9] The models of [11 12]require the solution of three-dimensional vector problemsand thus they aremore computationally demanding and needpowerful computer resources

We note that two mathematical models of optical waveg-uides were investigated in detail by methods of integralequations in [19 20 28ndash34] In [19 20 28ndash31] the model ofstep-indexwaveguideswas studied and in [32ndash34] themodelof waveguides without a sharp boundary was studied Areview of modern results in this field is given in [35] To solveour eigenvalue problem we are not requiring the informationabout specific values of eigenfunctions In our algorithmsit is enough only to know that the fundamental mode isexcited and then to measure its propagation constant for oneor two frequencies Main applications of our algorithm arefor example in detection of defects in metamaterials and innanoelectronics [36 37] calculation of dielectric constant ofsaline water [38 39] and in microwave imaging technologyin remote sensing [40]

An outline of this paper is a follows In Section 2 wefirst present a mathematical model of a weakly guiding step-index arbitrarily shaped optical fiber and formulate threeinverse spectral problems Then we present new numeri-cal algorithms for calculation of dielectric constants basedon an approximate solution of a nonlinear nonselfadjointeigenvalue problem for a system of weakly singular integralequations In Section 3 we present reconstruction resultswhich show efficiency and robustness of new developedalgorithms

2 Methods of Solution

21 Nonlinear Eigenvalue Problem for Transverse Wavenum-bers Let us consider an optical fiber as a regular cylindricaldielectric waveguide in a free space The cross-section ofthe waveguidersquos core is a bounded domain Ω

119894with twice

continuously differentiable boundary 120574 (see Figure 1) Theaxis of the cylinder is parallel to the 119909

3-axis Let Ω

119890=

1198772 Ω119894be the unbounded domain of the cladding Let the

permittivity be prescribed as a positive piecewise constantfunction 120576 which is equal to a constant 120576

infinin the domain Ω

119890

and to a constant 120576+gt 120576infin

in the domainΩ119894

Eigenvalue problems of optical waveguide theory [5] areformulated on the base of the set of homogeneous Maxwellequations

rotE = minus1205830

120597H

120597119905 rotH = 120576

0120576120597E

120597119905 (1)

Here E andH are electric and magnetic field vectors and 1205760

and 1205830are the free-space dielectric and magnetic constants

Nontrivial solutions of set (1) which have the form

[EH] (119909 119909

3 119905) = Re([EH] (119909) 119890

119894(1205731199093minus120596119905)) (2)

120574

x3 x1

x2

120576 = 120576+

120576 = 120576infinΩi

Ωe

Figure 1 The cross-section of a cylindrical dielectric waveguide ina free space

are called the eigenmodes of the waveguide Here positive 120596is the radian frequency 120573 is the propagation constant E andH are complex amplitudes of E andH and 119909 = (119909

1 1199092)

In forward eigenvalue problems the permittivity isknown and it is necessary to calculate longitudinal wavenum-bers 119896 = 120596radic12057601205830 and propagation constants 120573 such thatthere exist eigenmodes The eigenmodes have to satisfy to atransparency condition at the boundary 120574 and to a conditionat infinity

In inverse eigenvalue problems it is necessary to recon-struct the unknown permittivity 120576 by some information onnatural eigenmodes which exist for some eigenvalues 119896 and120573 The main question is how many observations of naturaleigenmodes are enough for unique and stable reconstructionof the permittivity

The domain Ω119890is unbounded Therefore it is necessary

to formulate a condition at infinity for complex amplitudesE and H of eigenmodes Let us confine ourselves to theinvestigation of the surface modes only The propagationconstants 120573 of surface modes are real and belong to theinterval 119866 = (119896120576

infin 119896120576+) The amplitudes of surface modes

satisfy the following condition

[EH] = 119890

minus120590119903119874(

1

radic119903) 119903 = |119909| 997888rarr infin (3)

Here 120590 = radic1205732 minus 1198962120576infingt 0 is the transverse wavenumber in

the claddingDenote by 120594 = radic1198962120576

+minus 1205732 the transverse wavenumber

in the waveguidersquos core Under the weakly guidance approx-imation [5] the original problem was reduced in [29] tothe calculation of numbers 120594 and 120590 such that there existnontrivial solutions 119906 = 119867

1= 1198672of Helmholtz equations

Δ119906 + 1205942119906 = 0 119909 isin Ω

119894

Δ119906 minus 1205902119906 = 0 119909 isin Ω

119890

(4)

which satisfy the transparency conditions

119906+= 119906minus

120597119906+

120597]=120597119906minus

120597] 119909 isin 120574 (5)




119894and Ω















domain Ω119894

(3) 1205941(120590) rarr 0 at 120590 rarr 0





119906 (119909) = int120574

Φ(120594 119909 119910) 119891 (119910) 119889119897 (119910) 119909 isin Ω119894

119906 (119909) = int120574

Ψ (120590 119909 119910) 119892 (119910) 119889119897 (119910) 119909 isin Ω119890

(6)

Here

Φ(120594 119909 119910) =119894

4119867(1)

0(1205941003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

Ψ (120590 119909 119910) =1

21205871198700(1205901003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(7)

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

1205942

1205902






A11(120594) 119891 +A

12 (120590) 119892 = 0

A21(120594) 119891 +A

22 (120590) 119892 = 0

(8)

where

(A11(120594) 119891) (119909) = int

120574

Φ(120594 119909 119910) 119891 (119910) 119889119897 (119910)

(A12 (120590) 119892) (119909) = minus int

120574

Ψ (120590 119909 119910) 119892 (119910) 119889119897 (119910)

(A21(120594) 119891) (119909) =

1

2119891 (119909) + int

120574

120597Φ (120594 119909 119910)

120597] (119909)119891 (119910) 119889119897 (119910)

(A22 (120590) 119892) (119909) =

1

2119892 (119909) minus int

120574

120597Ψ (120590 119909 119910)

120597] (119909)119892 (119910) 119889119897 (119910)

(9)


0120572(120574) and 1198621120572(120574)




1198711199011+ 11986111(120594 120590) 119901

1+ 11986112(120594 120590) 119901

2= 0

1199012+ 11986121(120594 120590) 119901

1+ 11986122(120594 120590) 119901

2= 0

(10)


1199011 (120591) = (119891 (119910) minus 119892 (119910))

100381610038161003816100381610038161199031015840(120591)10038161003816100381610038161003816 119901

2 (120591) = 119891 (119910) + 119892 (119910)

119910 = 119910 (120591) 120591 isin [0 2120587]

(11)


(1198711199011) (119905) = minus

1

2120587int

2120587

0

ln1003816100381610038161003816100381610038161003816sin 119905 minus 120591

2

10038161003816100381610038161003816100381610038161199011 (120591) 119889120591

(119861119894119895(120594 120590) 119901

(119895)) (119905) =

1

2120587int

2120587

0

ℎ119894119895(120594 120590 119905 120591) 119901

(119895)(120591) 119889120591

(12)

where

ℎ11(120594 120590 119905 120591) = 2120587 (119866

11(120594 119905 120591) + 119866

12 (120590 119905 120591))

ℎ12(120594 120590 119905 120591) = 2120587 (119866

11(120594 119905 120591) minus 119866

12 (120590 119905 120591))100381610038161003816100381610038161199031015840(120591)10038161003816100381610038161003816

ℎ21(120594 120590 119905 120591) = 4120587 (119866

21(120594 119905 120591) + 119866

22 (120590 119905 120591))

ℎ22(120594 120590 119905 120591) = 4120587 (119866

21(120594 119905 120591) minus 119866

22 (120590 119905 120591))100381610038161003816100381610038161199031015840(120591)10038161003816100381610038161003816

11986611(120594 119905 120591) = Φ (120594 119909 119910) +

1

2120587ln1003816100381610038161003816100381610038161003816sin 119905 minus 120591

2

1003816100381610038161003816100381610038161003816

11986612 (120590 119905 120591) = Ψ (120590 119909 119910) +

1

2120587ln1003816100381610038161003816100381610038161003816sin 119905 minus 120591

2

1003816100381610038161003816100381610038161003816

11986621(120594 119905 120591) =

120597Φ (120594 119909 119910)

120597] (119909)

11986622 (120590 119905 120591) =

120597Ψ (120590 119909 119910)

120597] (119909)

(13)




11986121(120594 120590) 119861

22(120594 120590) 119862

0120572(120574) 997888rarr 119862

0120572(120574)

11986111(120594 120590) 119861

12(120594 120590) 119862

0120572(120574) 997888rarr 119862

1120572(120574)

(14)


119860 (120594 120590)119908 = (119868 + 119861 (120594 120590))119908 = 0 (15)

where 119908 = (1199081 1199082)119879

1199081= 1198711199011isin 1198621120572(120574) 119908

2= 1199012isin 1198620120572(120574) (16)


1198621120572(120574) times 119862


119861119908 = [11986111119871minus1

11986112

11986121119871minus1

11986122

] [1199081

1199082

] (17)






120594 = 1205940

119891 =1199082

2+119871minus11199081

2100381610038161003816100381611990310158401003816100381610038161003816

119892 =1199082


2100381610038161003816100381611990310158401003816100381610038161003816

(18)


0




119908 = (119871 ((119891 minus 119892)10038161003816100381610038161003816119903101584010038161003816100381610038161003816) 119891 + 119892) (19)





119860119899(120594)119908119899= 0 (20)








119899

cond (119860119899(120594)) =

1205881

120588119899

(21)





119860119899(120594) = 119880119878119881 119878 = diag (120588

1 1205882 120588

119899) (22)







1198962=1205902+ 1205942


1205732=120576+1205902+ 120576infin1205942


(23)


119894(120590) generates


2= 1205732(1198962)


+= 2 120576

infin= 1 and 119894 = 1 We





0 1 2 3 4 50

1

2

3

4

5

6

7

8

1205732

k2

120576+ = 2 120576infin = 1

k2 = 1 1205732 = 10409

k2 = 3 1205732 = 39652







+of the








120576+=1205942+ 1205732

1198962 (25)




0 05 1 151

15

2

25

3

35

Cut-off points

120576 +

120576infin

k2 = 1 1205732 = 10409

k2 = 3 1205732 = 39652 120576infin = 1 120576+ = 2




120594 = radic1198962120576+minus 1205732 (26)



120576infin=1205732minus 1205902

1198962 (27)



infinfor each





120594(radic1205732

1minus 1198962

1120576infin) = radic119896

2

1120576+minus 1205732

1

120594 (radic1205732

2minus 1198962

2120576infin) = radic119896

2

2120576+minus 1205732

2

(28)

Here 1198961

and 1198962


1and 120573



119896119895and 120573

119895 119895 = 1 2






+of 1205732 for the

fundamental mode


3 35 4 45 51

12

14

16

18

2

22

24

26

1205732

120576 +

Randomly noised

Exact 1205732 = 39652

k2 = 3 120576infin = 1




120573 = 120573 (1 + 119901120572) (29)












35 4 45 50

02

04

06

08

1

12

14

16

18

1205732


Exact 1205732 = 39652

k2 = 3 120576+ = 2




infin it is enough



infinand



4 Conclusion







Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119894and Ω















domain Ω119894

(3) 1205941(120590) rarr 0 at 120590 rarr 0





119906 (119909) = int120574

Φ(120594 119909 119910) 119891 (119910) 119889119897 (119910) 119909 isin Ω119894

119906 (119909) = int120574

Ψ (120590 119909 119910) 119892 (119910) 119889119897 (119910) 119909 isin Ω119890

(6)

Here

Φ(120594 119909 119910) =119894

4119867(1)

0(1205941003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

Ψ (120590 119909 119910) =1

21205871198700(1205901003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(7)

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

1205942

1205902






A11(120594) 119891 +A

12 (120590) 119892 = 0

A21(120594) 119891 +A

22 (120590) 119892 = 0

(8)

where

(A11(120594) 119891) (119909) = int

120574

Φ(120594 119909 119910) 119891 (119910) 119889119897 (119910)

(A12 (120590) 119892) (119909) = minus int

120574

Ψ (120590 119909 119910) 119892 (119910) 119889119897 (119910)

(A21(120594) 119891) (119909) =

1

2119891 (119909) + int

120574

120597Φ (120594 119909 119910)

120597] (119909)119891 (119910) 119889119897 (119910)

(A22 (120590) 119892) (119909) =

1

2119892 (119909) minus int

120574

120597Ψ (120590 119909 119910)

120597] (119909)119892 (119910) 119889119897 (119910)

(9)


0120572(120574) and 1198621120572(120574)




1198711199011+ 11986111(120594 120590) 119901

1+ 11986112(120594 120590) 119901

2= 0

1199012+ 11986121(120594 120590) 119901

1+ 11986122(120594 120590) 119901

2= 0

(10)


1199011 (120591) = (119891 (119910) minus 119892 (119910))

100381610038161003816100381610038161199031015840(120591)10038161003816100381610038161003816 119901

2 (120591) = 119891 (119910) + 119892 (119910)

119910 = 119910 (120591) 120591 isin [0 2120587]

(11)


(1198711199011) (119905) = minus

1

2120587int

2120587

0

ln1003816100381610038161003816100381610038161003816sin 119905 minus 120591

2

10038161003816100381610038161003816100381610038161199011 (120591) 119889120591

(119861119894119895(120594 120590) 119901

(119895)) (119905) =

1

2120587int

2120587

0

ℎ119894119895(120594 120590 119905 120591) 119901

(119895)(120591) 119889120591

(12)

where

ℎ11(120594 120590 119905 120591) = 2120587 (119866

11(120594 119905 120591) + 119866

12 (120590 119905 120591))

ℎ12(120594 120590 119905 120591) = 2120587 (119866

11(120594 119905 120591) minus 119866

12 (120590 119905 120591))100381610038161003816100381610038161199031015840(120591)10038161003816100381610038161003816

ℎ21(120594 120590 119905 120591) = 4120587 (119866

21(120594 119905 120591) + 119866

22 (120590 119905 120591))

ℎ22(120594 120590 119905 120591) = 4120587 (119866

21(120594 119905 120591) minus 119866

22 (120590 119905 120591))100381610038161003816100381610038161199031015840(120591)10038161003816100381610038161003816

11986611(120594 119905 120591) = Φ (120594 119909 119910) +

1

2120587ln1003816100381610038161003816100381610038161003816sin 119905 minus 120591

2

1003816100381610038161003816100381610038161003816

11986612 (120590 119905 120591) = Ψ (120590 119909 119910) +

1

2120587ln1003816100381610038161003816100381610038161003816sin 119905 minus 120591

2

1003816100381610038161003816100381610038161003816

11986621(120594 119905 120591) =

120597Φ (120594 119909 119910)

120597] (119909)

11986622 (120590 119905 120591) =

120597Ψ (120590 119909 119910)

120597] (119909)

(13)




11986121(120594 120590) 119861

22(120594 120590) 119862

0120572(120574) 997888rarr 119862

0120572(120574)

11986111(120594 120590) 119861

12(120594 120590) 119862

0120572(120574) 997888rarr 119862

1120572(120574)

(14)


119860 (120594 120590)119908 = (119868 + 119861 (120594 120590))119908 = 0 (15)

where 119908 = (1199081 1199082)119879

1199081= 1198711199011isin 1198621120572(120574) 119908

2= 1199012isin 1198620120572(120574) (16)


1198621120572(120574) times 119862


119861119908 = [11986111119871minus1

11986112

11986121119871minus1

11986122

] [1199081

1199082

] (17)






120594 = 1205940

119891 =1199082

2+119871minus11199081

2100381610038161003816100381611990310158401003816100381610038161003816

119892 =1199082


2100381610038161003816100381611990310158401003816100381610038161003816

(18)


0




119908 = (119871 ((119891 minus 119892)10038161003816100381610038161003816119903101584010038161003816100381610038161003816) 119891 + 119892) (19)





119860119899(120594)119908119899= 0 (20)








119899

cond (119860119899(120594)) =

1205881

120588119899

(21)





119860119899(120594) = 119880119878119881 119878 = diag (120588

1 1205882 120588

119899) (22)







1198962=1205902+ 1205942


1205732=120576+1205902+ 120576infin1205942


(23)


119894(120590) generates


2= 1205732(1198962)


+= 2 120576

infin= 1 and 119894 = 1 We





0 1 2 3 4 50

1

2

3

4

5

6

7

8

1205732

k2

120576+ = 2 120576infin = 1

k2 = 1 1205732 = 10409

k2 = 3 1205732 = 39652







+of the








120576+=1205942+ 1205732

1198962 (25)




0 05 1 151

15

2

25

3

35

Cut-off points

120576 +

120576infin

k2 = 1 1205732 = 10409

k2 = 3 1205732 = 39652 120576infin = 1 120576+ = 2




120594 = radic1198962120576+minus 1205732 (26)



120576infin=1205732minus 1205902

1198962 (27)



infinfor each





120594(radic1205732

1minus 1198962

1120576infin) = radic119896

2

1120576+minus 1205732

1

120594 (radic1205732

2minus 1198962

2120576infin) = radic119896

2

2120576+minus 1205732

2

(28)

Here 1198961

and 1198962


1and 120573



119896119895and 120573

119895 119895 = 1 2






+of 1205732 for the

fundamental mode


3 35 4 45 51

12

14

16

18

2

22

24

26

1205732

120576 +

Randomly noised

Exact 1205732 = 39652

k2 = 3 120576infin = 1




120573 = 120573 (1 + 119901120572) (29)












35 4 45 50

02

04

06

08

1

12

14

16

18

1205732


Exact 1205732 = 39652

k2 = 3 120576+ = 2




infin it is enough



infinand



4 Conclusion







Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1198711199011+ 11986111(120594 120590) 119901

1+ 11986112(120594 120590) 119901

2= 0

1199012+ 11986121(120594 120590) 119901

1+ 11986122(120594 120590) 119901

2= 0

(10)


1199011 (120591) = (119891 (119910) minus 119892 (119910))

100381610038161003816100381610038161199031015840(120591)10038161003816100381610038161003816 119901

2 (120591) = 119891 (119910) + 119892 (119910)

119910 = 119910 (120591) 120591 isin [0 2120587]

(11)


(1198711199011) (119905) = minus

1

2120587int

2120587

0

ln1003816100381610038161003816100381610038161003816sin 119905 minus 120591

2

10038161003816100381610038161003816100381610038161199011 (120591) 119889120591

(119861119894119895(120594 120590) 119901

(119895)) (119905) =

1

2120587int

2120587

0

ℎ119894119895(120594 120590 119905 120591) 119901

(119895)(120591) 119889120591

(12)

where

ℎ11(120594 120590 119905 120591) = 2120587 (119866

11(120594 119905 120591) + 119866

12 (120590 119905 120591))

ℎ12(120594 120590 119905 120591) = 2120587 (119866

11(120594 119905 120591) minus 119866

12 (120590 119905 120591))100381610038161003816100381610038161199031015840(120591)10038161003816100381610038161003816

ℎ21(120594 120590 119905 120591) = 4120587 (119866

21(120594 119905 120591) + 119866

22 (120590 119905 120591))

ℎ22(120594 120590 119905 120591) = 4120587 (119866

21(120594 119905 120591) minus 119866

22 (120590 119905 120591))100381610038161003816100381610038161199031015840(120591)10038161003816100381610038161003816

11986611(120594 119905 120591) = Φ (120594 119909 119910) +

1

2120587ln1003816100381610038161003816100381610038161003816sin 119905 minus 120591

2

1003816100381610038161003816100381610038161003816

11986612 (120590 119905 120591) = Ψ (120590 119909 119910) +

1

2120587ln1003816100381610038161003816100381610038161003816sin 119905 minus 120591

2

1003816100381610038161003816100381610038161003816

11986621(120594 119905 120591) =

120597Φ (120594 119909 119910)

120597] (119909)

11986622 (120590 119905 120591) =

120597Ψ (120590 119909 119910)

120597] (119909)

(13)




11986121(120594 120590) 119861

22(120594 120590) 119862

0120572(120574) 997888rarr 119862

0120572(120574)

11986111(120594 120590) 119861

12(120594 120590) 119862

0120572(120574) 997888rarr 119862

1120572(120574)

(14)


119860 (120594 120590)119908 = (119868 + 119861 (120594 120590))119908 = 0 (15)

where 119908 = (1199081 1199082)119879

1199081= 1198711199011isin 1198621120572(120574) 119908

2= 1199012isin 1198620120572(120574) (16)


1198621120572(120574) times 119862


119861119908 = [11986111119871minus1

11986112

11986121119871minus1

11986122

] [1199081

1199082

] (17)






120594 = 1205940

119891 =1199082

2+119871minus11199081

2100381610038161003816100381611990310158401003816100381610038161003816

119892 =1199082


2100381610038161003816100381611990310158401003816100381610038161003816

(18)


0




119908 = (119871 ((119891 minus 119892)10038161003816100381610038161003816119903101584010038161003816100381610038161003816) 119891 + 119892) (19)





119860119899(120594)119908119899= 0 (20)








119899

cond (119860119899(120594)) =

1205881

120588119899

(21)





119860119899(120594) = 119880119878119881 119878 = diag (120588

1 1205882 120588

119899) (22)







1198962=1205902+ 1205942


1205732=120576+1205902+ 120576infin1205942


(23)


119894(120590) generates


2= 1205732(1198962)


+= 2 120576

infin= 1 and 119894 = 1 We





0 1 2 3 4 50

1

2

3

4

5

6

7

8

1205732

k2

120576+ = 2 120576infin = 1

k2 = 1 1205732 = 10409

k2 = 3 1205732 = 39652







+of the








120576+=1205942+ 1205732

1198962 (25)




0 05 1 151

15

2

25

3

35

Cut-off points

120576 +

120576infin

k2 = 1 1205732 = 10409

k2 = 3 1205732 = 39652 120576infin = 1 120576+ = 2




120594 = radic1198962120576+minus 1205732 (26)



120576infin=1205732minus 1205902

1198962 (27)



infinfor each





120594(radic1205732

1minus 1198962

1120576infin) = radic119896

2

1120576+minus 1205732

1

120594 (radic1205732

2minus 1198962

2120576infin) = radic119896

2

2120576+minus 1205732

2

(28)

Here 1198961

and 1198962


1and 120573



119896119895and 120573

119895 119895 = 1 2






+of 1205732 for the

fundamental mode


3 35 4 45 51

12

14

16

18

2

22

24

26

1205732

120576 +

Randomly noised

Exact 1205732 = 39652

k2 = 3 120576infin = 1




120573 = 120573 (1 + 119901120572) (29)












35 4 45 50

02

04

06

08

1

12

14

16

18

1205732


Exact 1205732 = 39652

k2 = 3 120576+ = 2




infin it is enough



infinand



4 Conclusion







Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








119899

cond (119860119899(120594)) =

1205881

120588119899

(21)





119860119899(120594) = 119880119878119881 119878 = diag (120588

1 1205882 120588

119899) (22)







1198962=1205902+ 1205942


1205732=120576+1205902+ 120576infin1205942


(23)


119894(120590) generates


2= 1205732(1198962)


+= 2 120576

infin= 1 and 119894 = 1 We





0 1 2 3 4 50

1

2

3

4

5

6

7

8

1205732

k2

120576+ = 2 120576infin = 1

k2 = 1 1205732 = 10409

k2 = 3 1205732 = 39652







+of the








120576+=1205942+ 1205732

1198962 (25)




0 05 1 151

15

2

25

3

35

Cut-off points

120576 +

120576infin

k2 = 1 1205732 = 10409

k2 = 3 1205732 = 39652 120576infin = 1 120576+ = 2




120594 = radic1198962120576+minus 1205732 (26)



120576infin=1205732minus 1205902

1198962 (27)



infinfor each





120594(radic1205732

1minus 1198962

1120576infin) = radic119896

2

1120576+minus 1205732

1

120594 (radic1205732

2minus 1198962

2120576infin) = radic119896

2

2120576+minus 1205732

2

(28)

Here 1198961

and 1198962


1and 120573



119896119895and 120573

119895 119895 = 1 2






+of 1205732 for the

fundamental mode


3 35 4 45 51

12

14

16

18

2

22

24

26

1205732

120576 +

Randomly noised

Exact 1205732 = 39652

k2 = 3 120576infin = 1




120573 = 120573 (1 + 119901120572) (29)












35 4 45 50

02

04

06

08

1

12

14

16

18

1205732


Exact 1205732 = 39652

k2 = 3 120576+ = 2




infin it is enough



infinand



4 Conclusion







Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 05 1 151

15

2

25

3

35

Cut-off points

120576 +

120576infin

k2 = 1 1205732 = 10409

k2 = 3 1205732 = 39652 120576infin = 1 120576+ = 2




120594 = radic1198962120576+minus 1205732 (26)



120576infin=1205732minus 1205902

1198962 (27)



infinfor each





120594(radic1205732

1minus 1198962

1120576infin) = radic119896

2

1120576+minus 1205732

1

120594 (radic1205732

2minus 1198962

2120576infin) = radic119896

2

2120576+minus 1205732

2

(28)

Here 1198961

and 1198962


1and 120573



119896119895and 120573

119895 119895 = 1 2






+of 1205732 for the

fundamental mode


3 35 4 45 51

12

14

16

18

2

22

24

26

1205732

120576 +

Randomly noised

Exact 1205732 = 39652

k2 = 3 120576infin = 1




120573 = 120573 (1 + 119901120572) (29)












35 4 45 50

02

04

06

08

1

12

14

16

18

1205732


Exact 1205732 = 39652

k2 = 3 120576+ = 2




infin it is enough



infinand



4 Conclusion







Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




3 35 4 45 51

12

14

16

18

2

22

24

26

1205732

120576 +

Randomly noised

Exact 1205732 = 39652

k2 = 3 120576infin = 1




120573 = 120573 (1 + 119901120572) (29)












35 4 45 50

02

04

06

08

1

12

14

16

18

1205732


Exact 1205732 = 39652

k2 = 3 120576+ = 2




infin it is enough



infinand



4 Conclusion







Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Reconstruction of Dielectric Constants of Core and Cladding of...

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