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Research ArticleFlux Density through Guides with MicrostructuredTwisted Clad DB Medium

M A Baqir and P K Choudhury

Institute of Microengineering and Nanoelectronics Universiti Kebangsaan Malaysia UKM Bangi 43600 Selangor Malaysia

Correspondence should be addressed to P K Choudhury pankajukmmy

Received 21 October 2013 Accepted 2 December 2013 Published 2 January 2014

Academic Editor Razali Ismail

Copyright copy 2014 M A Baqir and P K Choudhury This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The paper deals with the study of flux density through a newly proposed twisted clad guide containing DB medium The innercore and the outer clad sections are usual dielectrics and the introduced twisted windings at the core-clad interface are treatedunder DB boundary conditions The pitch angle of twist is supposed to greatly contribute towards the control over the dispersioncharacteristics of the guideThe eigenvalue equation for the guiding structure is deduced and the analytical investigations aremadeto explore the propagation patterns of flux densities corresponding to the sustained low-order hybrid modes under the situationof varying pitch angles The emphasis has been put on the effects due to the DB twisted pitch on the propagation of energy fluxdensity through the guide

1 Introduction

Metamaterials are artificially designed mediums that oweunusual phenomena such as reversal of Snellrsquos law andDopplerrsquos effect negative reflectionrefraction and manyothersThese exotic features are basically due to the structuresof materials rather than their compositions and are used inmany interesting applications for example cloaking perfectlensing and power confinement [1ndash4] References [5 6]describe negative index chiral metamaterial based on eight-crank molecule designed structure homogeneous as well asisotropic behavior of chiral medium based on the periodicinclusion of cranks have been demonstrated

During the last couple of decades complex structuredguides have attracted the RampDcommunity primarily becauseof their varieties of potential applications that includeoptical sensing integrated optics and microwaves devicesThe electromagnetic behavior of guides can be tailored byaltering structural geometry medium composition and thenature of excited electromagnetic field References [7ndash11]describe several forms of waveguide structures and theirelectromagnetic response Reference [8] demonstrates wavepropagation through chiral nihility metamaterial a special

class of chiral medium in which the real part of permittivityand permeability simultaneously becomes zero while itschirality remains nonzero In the context of complex opticalmicro- andor nanostructures helical clad optical guidesoffer control over the dispersion behavior through suitableadjustments of the helix pitch angle [12ndash16] Such helicalforms or the twists can be written on fiber structures by theuse of current advancements in nanotechnology

The usefulness of DB boundary conditions in theconstruction of spherical and cylindrical cloaks has beenreported in the literature [3] Within the context the DBboundary conditions were introduced by Lindell et al [17ndash19] in which the normal components of both electric andmagnetic fields vanish at the interface [20ndash22] DB mediumcan be easily realized in the case of anisotropic mediumif the normal component of permittivity and permeabilitysimultaneously becomes zero that is

119899 sdot

997888rarr

119861 = 0 119899 sdot

997888rarr

119863 = 0(1)

In the case of isotropicmedium the boundary conditions canbe written as

119899 sdot

997888rarr

119867 = 0 119899 sdot

997888rarr

119864 = 0(2)

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2014 Article ID 629651 6 pageshttpdxdoiorg1011552014629651

2 Journal of Nanomaterials

In the present paper we investigate the propagationthrough twisted clad guiding medium wherein helical wind-ings are introduced as DB medium at the dielectric core-clad interface To the best of our knowledge such guidingstructures are not yet discussed in the literature and thepresent work would be the first investigation of the funda-mental electromagnetic behavior of microstructured guideswith DB medium twisted clad Rigorous analyses are madeof the associated boundary-value problem implementingMaxwellrsquos equations and the dispersion relations of the guideare deduced followed by the estimation of supported fluxdensity patterns

2 Theoretical Approach

We consider a cylindrical twisted clad guiding mediumas shown in Figure 1 the coreclad sections of which arelinear isotropic homogeneous and lossless dielectrics ADB based microstructured sheath helix is introduced atthe core-clad interface treated under the suitable boundaryconditions Although the sheath is tightly wound each ofthe turns is insulated from the neighboring ones In Figure 1Ψ represents the angle that the conducting helical windingsmakewith respect to the normal drawn on to the fiber surfacein the longitudinal direction

We assume 119899core and 119899clad as the refractive indices (RIs)of core and clad regions of the guide respectively and 119886 asthe core radius the clad section being infinitely extendedThe time 119905-harmonic and axis 119911-harmonic wave propagatesdown the guiding structure considering the guide to beelongated along the 119911-direction Such an assumption leads tothe transverse electricmagnetic field components to be of theforms as

119864

1199031= minus

119895

119906

2[120573119906119860119869

1015840

V (119906119903) +119895120596120583V119903

119861119869V (119906119903)] 119890119895(120596119905minus120573119911+V120593)

(3a)

119867

1199031=

119895

119906

2[

119895120596120583V119903

119860119869V (119906119903) minus 1205731199061198611198691015840

V (119906119903)] 119890119895(120596119905minus120573119911+V120593)

(3b)

119864

1205931= minus

119895

119906

2[

119895120573V119903

119860119869V (119906119903) minus 1205961205831199061198611198691015840

V (119906119903)] 119890119895(120596119905minus120573119911+V120593)

(4a)

119867

1205931= minus

119895

119906

2[120596120583119906119860119869

1015840

V (119906119903) +119895120573V119903

119861119869V (119906119903)] 119890119895(120596119905minus120573119911+V120593)

(4b)wherein 119906 =

radic(119899core1198960)

2

minus 120573

2 Similarly the transversecomponents of fields in the clad section can be written as

119864

1199032= minus

119895

119908

2[120573119908119862119870

1015840

V (119908119903) +119895120596120583V119908

119863119870V (119908119903)] 119890119895(120596119905minus120573119911+V120593)

(5a)

119867

1199032=

119895

119908

2[

119895120596120583V119903

119862119870V (119908119903) minus 1205731199081198631198701015840

V (119908119903)] 119890119895(120596119905minus120573119911+V120593)

(5b)

Sheath helix turnsy

z

Core

Clad

a

x

120595

Figure 1 Longitudinal view of the guiding structure

119864

1205932= minus

119895

119908

2[

119895120573V119903

119862119870V (119908119903) minus 1205961205831199081198631198701015840

V (119908119903)] 119890119895(120596119905minus120573119911+V120593)

(6a)

119867

1205932= minus

119895

119908

2[120596120583119908119862119870

1015840

V (119908119903) +119895120573V119903

119863119870V (119908119903)] 119890119895(120596119905minus120573119911+V120593)

(6b)

with 119908 =radic120573

2minus (119899clad1198960)

2 In the above set of equations 120583and 120576 are respectively the permeability and the permittivityof the medium 120596 is the angular frequency in the unboundedmedium 120573 is the propagation constant in the medium 119896

0is

the free-space propagation constant V is the azimuthal modeindex and primes represent differentiation with respect tothe arguments of field functions Also 119860 119861 119862 and 119863 areunknown constants to be estimated by the use of boundaryconditions

Now considering the existence of DB twisted cladboundary conditions will essentially include vanishing nor-mal components of electricmagnetic fields and continuoustangential components of the same at the medium interfaceImplementation of this ultimately leads to the evolutionof a set of four homogeneous equations followed by thedispersion relation (as obtained by collecting the coefficientsof unknown constants) given as

119891 (120573) =

1

4

120596

2

120583120576119908

2

119870Vminus1 (119886119908) + 119870V+1 (119886119908)2

sin2120595

times [

V21205962120583120576119886

2119906

4119869

2

V (119886119906) (cos120595 minus sin120595)2

minus

120573

2

4

119869Vminus1 (119886119906) minus 119869V+1 (119886119906)2

(sin120595 minus cos120595)2]

+ 119870V (119886119908) cos120595 +V120573119886119908

2119869V (119886119906) sin120595

times 119870V (119886119908) cos120595 minusV120573119886119908

2119870V (119886119908) cos120595

times

V21205831205761205962

119886

2119906

4119869

2

V (119886119906) minus1

4

120573

2

(119869Vminus1 (119886119906) minus 119869V+1 (119886119906))2

times (sin120595 minus cos120595)2 = 0(7)

Journal of Nanomaterials 3

The unknown coefficients could be determined solvingthe aforesaid four equations obtained after implementingthe boundary conditions After some rigorous mathematicalsteps the coefficients119860 119861 and119862 can be determined in termsof the coefficient119863 as follows

119860 = minus119895

119906

2

119886119869

1015840

V (119906119886)Θ (1199062

119886120574Δ minus Ξ120585) + 119908

2

119886 (Θ120585 minus 120596120576119870V (119908119886))

120596120576119886119908

2120574 (Θ120585 minus 120596120576119870V (119908119886) 120575)

119863

119861 =

119906119869V (119906119886)Θ (1199062

119886120574Δ minus Ξ120585) + 119908

2

119886 (Θ120585 minus 120596120576119870V (119908119886) 120575)

119908

2119886120574 (Θ120585 minus 120596120576119870V (119886119908) 120575)

119863

119862 = 119895

119906

2

119886120574Δ minus 120596

2

120583120576119906119908

3

119870

1015840

V (119908119886) 120585

120596120576119870V (119886119908) (1199082119886 cos120595 minus 120573V sin120595) 120585 minus 120575

119863

(8)

where

120574 = 120596

2

120583120576119906

2

119869V (119906119886) 1198691015840

V (119906119886) sin120595

minus 120573119869

1015840

V (119906119886) (1199062

119886119869V (119906119886) cos120595 + 120573V119869V (119906119886) sin120595)

120585 = 119869

2

V (119906119886) (1199062

119886 cos120595 + 120573V sin120595) minus 120573119906211988621198691015840V (119906119886) sin120595

Θ = 120596120598119870V (119908119886) (1199082

119886 cos120595 minus 120573V sin120595)

Δ = 119908

2

119870V (119908119886) cos120595 + 120573V119870V (119908119886) sin120595

Ξ = 120596

2

120583120576119908

3

119870V (119908119886) sin120595

120575 = 120596120576119906119908119886

2

119870V (119908119886) sin120595(9)

Now the flux density in the guide along the direction ofpropagation can be estimated and following the procedurein [11] it will finally assume the forms corresponding to thecoreclad sections as follows

(119878

119911)Core =

120573

2119906

4(120576|119860|

2

+ 120583|119861|

2

) (

V119903

)

2

119869

2

V (119906119903) + (119906

2

)

2

120590

(119878

119911)Clad =

120573

2119908

4(120576|119862|

2

+ 120583|119863|

2

) (

V119903

)

2

119870

2

V (119908119903) + (119908

2

)

2

Λ

(10)

with

120590 = 119869

2

Vminus1 (119906119903) + 1198692

V+1 (119906119903) minus 2119869Vminus1 (119906119903) 119869V+1 (119906119903)

Λ = 119870

2

Vminus1 (119908119903) + 1198702

V+1 (119908119903) minus 2119870Vminus1 (119908119903)119870V+1 (119908119903) (11)

Equations (10) can be written in terms of only one unknownconstant for example in terms of119863 as determined by the useof (8)

3 Results and Discussion

We now investigate the propagation features of flux densitythrough DB twisted clad microstructured guide Howeverfor this purpose it remains essential to evaluate the allowedvalues of propagation constants corresponding to differentsustained modes under the conditions when the DB twistsare typically oriented perpendicular as well as parallel to

Table 1 Hybrid mode propagation constants

Pitch angle120595

Coreradius a Mode Propagation constant

120573 (mminus1)

0∘10120583m

EH01 5970 times 106

EH11 5975 times 106

EH21 5968 times 106

20120583mEH01 5960 times 106

EH11 5965 times 106

EH21 5970 times 106

90∘10120583m

EH01 5850 times 106

EH11 5890 times 106

EH21 5870 times 106

20120583mEH01 5950 times 106

EH11 5980 times 106

EH21 5965 times 106

the direction of wave propagation For the computationalpurpose we consider the RI values of the core and the cladsections as 1485 and 1470 respectively and the operatingwavelength is taken to be 155 120583m The chosen parametersare indeed the widely accepted ones in the case of modelingand simulations related to the electromagnetic properties ofoptical fibers Also we consider two typical values of coreradius as 10 120583m and 20 120583m the clad radius being fixed at100 120583m The behavior of normalized flux density throughcoreclad regions is studied considering three low-orderhybrid EH

01 EH11 and EH

21modes having propagation

constants as obtained by the use of eigenvalue (7) as shownin Table 1

Figures 2(a) and 2(b) respectively demonstrate the nor-malized flux density patterns through core and clad regionsof guide having core radius 10120583m under the situation whenthe DB twist is oriented perpendicular to the direction ofpropagation that is Ψ = 0

∘ In all the figures the resultscorresponding to the EH

01 EH11 and EH

21modes are

represented by black dash-dot solid red and dashed bluelines We observe from Figure 2(a) that the flux generallyexhibits an oscillatory trend in the core region of the guidingmedium Also in the central region of the core EH

11hybrid

mode has higher energy than the other two EH01

and EH21

modes Further flux densities due to the EH01

and EH11

modes decrease with the increase in radial distance whereasthat due to the EH

21mode exhibits a minor increase only

In the clad section as Figure 2(b) shows the aforesaidthree hybrid modes exhibit similar flux characteristics andit remains extremely small as compared to that in thecore region At the core-clad interface it shows relativelylarge value and then step-falls with a little increase in cladradius and become almost vanishing as the radius is furtherincreased

Figures 3(a) and 3(b) correspond to the situations whenthe DB helical windings are put parallel to the optical axisthat is the helix pitch Ψ = 90

∘ We observe in this casethat the confinements of flux in the core section (Figure 3(a))exhibit almost similar patterns as observed in the case when


02

04

06

08

10Sz

c (au

)

20 40 60 80 10

r (m) times10minus6

(a)

10

20

30

40

50

60

70

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus8

(b)

Figure 2 Flux density through the core (a) and clad (b) corresponding to 10120583m core radius and 0∘ helix pitch

Szc (

au)

20

05

10

15

20

25

30

35

40 60 80 10r (m)

times10minus8

times10minus6

(a)

100

80

60

40

20

000004 000006 000008 00001r (m)

Szcl

(au

)times10minus12

(b)


05

50 times 10minus6

10

15

20

Szc (

au)

000001 0000015 000002

r (m)

times10minus6

(a)

10

12

14

80

60

40

20

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus17

(b)



50 times 10minus6

05

10

15

20

000001 0000015 000002

Szc (

au)

r (m)

times10minus16

(a)

20

40

60

80

10

12

14

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus17

(b)


DB twists are oriented in the direction perpendicular to thewave propagation (Figure 2(a)) but its values are drasticallyreduced corresponding to all the sustained modes Other-wise the EH

01and EH

11modes correspond to higher power

than the EH21mode in the central region of core section In

the clad section too the flux is further reduced (Figure 3(b))as compared to the case of Figure 2(b) exhibiting similardensity patterns as observed before for the situation of Ψ =0

∘ Thus comparing the results of Figures 2 and 3 we noticethe effect of introducing DB sheath helix structure at thedielectric core-clad interface We find that making the angleof pitch (ofDBhelix) as 90∘ greatly reduces the propagation ofpower and therefore a change in pitch wouldmake the guideto amplify or attenuate the signal depending upon lowering orincreasing its valueThis clearly demonstrates the importanceof such guiding structures

Figures 4 and 5 correspond to the situation when the coreradius is doubled that is increased to 20 120583mWe observe thatthe trends of flux density patterns are almost similar to thesituation as observed before for 10 120583m but the oscillations offlux are increased in this case This is very much obvious dueto more room for the fields to be sustained thereby causingdispersion and so forth However in this as well we observethat the increase of pitch of DB twist essentially reduces theflow of flux density as evidenced by Figures 4 and 5

4 Conclusion

The eigenvalue equation is deduced for the aforesaid struc-ture and the propagation features of the low-order hybridmodes are estimated Investigations are made of the energyflux density patterns corresponding to the sustained modesand the effects are observed due to the alterations of pitchangles of the microstructured DB twists introduced at thedielectric core-clad interface The flux patterns reveal pro-found effect of helix pitch and making the DB windings par-allel to the axis of guide greatly reduces power confinementsin the guiding region As such these guides may be usedfor the purpose of amplifying andor attenuating signals inoptical systems

Conflict of Interests

Theauthors declare that there is no conflict of interest regard-ing the publication of this manuscript

Acknowledgments

The authors are grateful to Professors S Shaari and BYMajlis for constant encouragement and help Also they arethankful for the Ministry of Higher Education (Malaysia)for providing financial support to the work The authors aregrateful to the anonymous reviewer for making constructivecriticisms

References

[1] V G Veselago ldquoThe electrodynamics of substances with simult-aneously negative values of 120576 and 120583rdquo Soviet Physics Uspekhi vol10 no 4 pp 509ndash514 1968

[2] J B Pendry ldquoNegative refraction makes a perfect lensrdquo PhysicalReview Letters vol 85 no 18 pp 3966ndash3969 2000

[3] A D Yaghjian ldquoExtreme electromagnetic boundary conditionsand their manifestation at the inner surfaces of spherical andcylindrical cloaksrdquo Metamaterials vol 4 no 2-3 pp 70ndash762010

[4] N Landy and D R Smith ldquoA full-parameter unidirectionalmetamaterial cloak formicrowaverdquoNatureMaterials vol 12 pp25ndash28 2013

[5] A J Garcia-Collado G J Molina-Cuberos J Margineda M JNunez and EMartın ldquoIsotropic and homogeneous behavior ofchiral media based on periodical inclusions of cranksrdquo IEEEMicrowave and Wireless Components Letters vol 20 no 3 pp175ndash177 2010

[6] G JMolina-Cuberos A J Garcıa-Collado I Barba and JMar-gineda ldquoChiralmetamaterials with negative refractive index co-mposed by an eight-cranksmoleculerdquo IEEEAntennas andWire-less Propagation Letters vol 10 pp 1488ndash1490 2011

[7] P K Choudhury andRA Lessard ldquoAn estimation of power tra-nsmission through a doubly clad optical fiber with an annularcorerdquo Microwave and Optical Technology Letters vol 29 no 6pp 402ndash405 2001


[8] S Tretyakov I Nefedov A Sihvola S Maslovski and C Sim-ovski ldquoWaves and energy in chiral nihilityrdquo Journal of Electro-magnetic Waves and Applications vol 17 no 5 pp 695ndash7062003

[9] S C YeowMH Lim andPKChoudhury ldquoA rigorous analysisof the distribution of power in plastic clad linear tapered fibersrdquoOptik vol 117 no 9 pp 405ndash410 2006

[10] PKChoudhury andWK Soon ldquoOn the transmission by liquidcrystal tapered optical fibersrdquo Optik vol 122 no 12 pp 1061ndash1068 2011

[11] M A Baqir and P K Choudhury ldquoOn the energy flux through auniaxial chiral metamaterial made circular waveguide underPMC boundaryrdquo Journal of Electromagnetic Waves and Appli-cations vol 26 no 16 pp 2165ndash2175 2012

[12] U N Singh O N Singh II P Khastgir and K K Dey ldquoDis-persion characteristics of a helically cladded step-index opticalfibermdashanalytical studyrdquo Journal of the Optical Society of AmericaB vol 12 no 7 pp 1273ndash1278 1995

[13] D Kumar and O N Singh II ldquoModal characteristic equationand dispersion curves for an elliptical step-index fiber with aconducting helical winding on the core-cladding boundarymdashananalytical studyrdquo Journal of Lightwave Technology vol 20 no 8pp 1416ndash1424 2002

[14] C C Siong and P K Choudhury ldquoPropagation characteristicsof tapered core helical cald dielectric optical fibersrdquo Journal ofElectromagnetic Waves and Applications vol 23 no 5 pp 663ndash674 2009

[15] K Y Lim P K Choudhury and Z Yusoff ldquoChirofibers with hel-ical windingsmdashan analytical investigationrdquo Optik vol 121 no11 pp 980ndash987 2010

[16] M Ghasemi and P K Choudhury ldquoOn the sustainment of opt-ical power in twisted clad dielectric cylindrical fiberrdquo Journal ofElectromagneticWaves and Applications vol 27 no 11 pp 1382ndash1391 2013

[17] I V Lindell and A H Sihvola ldquoElectromagnetic boundary andits realization with anisotropic metamaterialrdquo Physical ReviewE vol 79 no 2 Article ID 026604 2009

[18] I V Lindell HWallen andA Sihvola ldquoGeneral electromagnet-ic boundary conditions involving normal field componentsrdquoIEEE Antennas andWireless Propagation Letters vol 8 pp 877ndash880 2009

[19] I V Lindell and A H Sihvola ldquoElectromagnetic boundary con-ditions defined in terms of normal field componentsrdquo IEEETransactions on Antennas and Propagation vol 58 no 4 pp1128ndash1135 2010

[20] I V Lindell A Sihvola L Bergamin and A Favaro ldquoRealiza-tion of the D1015840B1015840 boundary conditionrdquo IEEE Antennas andWire-less Propagation Letters vol 10 pp 643ndash646 2011

[21] I V Lindell andAH Sihvola ldquoSHDB boundary condition real-ized by pseudochiral mediardquo IEEE Antennas andWireless Prop-agation Letters vol 12 pp 591ndash594 2013

[22] M A Baqir and P K Choudhury ldquoPropagation through uni-axial anisotropic chiral waveguide under DB boundary condi-tionsrdquo Journal of Electromagnetic Waves and Applications vol27 no 6 pp 783ndash793 2013

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Journal ofNanomaterials


In the present paper we investigate the propagationthrough twisted clad guiding medium wherein helical wind-ings are introduced as DB medium at the dielectric core-clad interface To the best of our knowledge such guidingstructures are not yet discussed in the literature and thepresent work would be the first investigation of the funda-mental electromagnetic behavior of microstructured guideswith DB medium twisted clad Rigorous analyses are madeof the associated boundary-value problem implementingMaxwellrsquos equations and the dispersion relations of the guideare deduced followed by the estimation of supported fluxdensity patterns

2 Theoretical Approach

We consider a cylindrical twisted clad guiding mediumas shown in Figure 1 the coreclad sections of which arelinear isotropic homogeneous and lossless dielectrics ADB based microstructured sheath helix is introduced atthe core-clad interface treated under the suitable boundaryconditions Although the sheath is tightly wound each ofthe turns is insulated from the neighboring ones In Figure 1Ψ represents the angle that the conducting helical windingsmakewith respect to the normal drawn on to the fiber surfacein the longitudinal direction

We assume 119899core and 119899clad as the refractive indices (RIs)of core and clad regions of the guide respectively and 119886 asthe core radius the clad section being infinitely extendedThe time 119905-harmonic and axis 119911-harmonic wave propagatesdown the guiding structure considering the guide to beelongated along the 119911-direction Such an assumption leads tothe transverse electricmagnetic field components to be of theforms as

119864

1199031= minus

119895

119906

2[120573119906119860119869

1015840

V (119906119903) +119895120596120583V119903

119861119869V (119906119903)] 119890119895(120596119905minus120573119911+V120593)

(3a)

119867

1199031=

119895

119906

2[

119895120596120583V119903

119860119869V (119906119903) minus 1205731199061198611198691015840

V (119906119903)] 119890119895(120596119905minus120573119911+V120593)

(3b)

119864

1205931= minus

119895

119906

2[

119895120573V119903

119860119869V (119906119903) minus 1205961205831199061198611198691015840

V (119906119903)] 119890119895(120596119905minus120573119911+V120593)

(4a)

119867

1205931= minus

119895

119906

2[120596120583119906119860119869

1015840

V (119906119903) +119895120573V119903

119861119869V (119906119903)] 119890119895(120596119905minus120573119911+V120593)

(4b)wherein 119906 =

radic(119899core1198960)

2

minus 120573

2 Similarly the transversecomponents of fields in the clad section can be written as

119864

1199032= minus

119895

119908

2[120573119908119862119870

1015840

V (119908119903) +119895120596120583V119908

119863119870V (119908119903)] 119890119895(120596119905minus120573119911+V120593)

(5a)

119867

1199032=

119895

119908

2[

119895120596120583V119903

119862119870V (119908119903) minus 1205731199081198631198701015840

V (119908119903)] 119890119895(120596119905minus120573119911+V120593)

(5b)

Sheath helix turnsy

z

Core

Clad

a

x

120595

Figure 1 Longitudinal view of the guiding structure

119864

1205932= minus

119895

119908

2[

119895120573V119903

119862119870V (119908119903) minus 1205961205831199081198631198701015840

V (119908119903)] 119890119895(120596119905minus120573119911+V120593)

(6a)

119867

1205932= minus

119895

119908

2[120596120583119908119862119870

1015840

V (119908119903) +119895120573V119903

119863119870V (119908119903)] 119890119895(120596119905minus120573119911+V120593)

(6b)

with 119908 =radic120573

2minus (119899clad1198960)

2 In the above set of equations 120583and 120576 are respectively the permeability and the permittivityof the medium 120596 is the angular frequency in the unboundedmedium 120573 is the propagation constant in the medium 119896

0is

the free-space propagation constant V is the azimuthal modeindex and primes represent differentiation with respect tothe arguments of field functions Also 119860 119861 119862 and 119863 areunknown constants to be estimated by the use of boundaryconditions

Now considering the existence of DB twisted cladboundary conditions will essentially include vanishing nor-mal components of electricmagnetic fields and continuoustangential components of the same at the medium interfaceImplementation of this ultimately leads to the evolutionof a set of four homogeneous equations followed by thedispersion relation (as obtained by collecting the coefficientsof unknown constants) given as

119891 (120573) =

1

4

120596

2

120583120576119908

2

119870Vminus1 (119886119908) + 119870V+1 (119886119908)2

sin2120595

times [

V21205962120583120576119886

2119906

4119869

2

V (119886119906) (cos120595 minus sin120595)2

minus

120573

2

4

119869Vminus1 (119886119906) minus 119869V+1 (119886119906)2

(sin120595 minus cos120595)2]

+ 119870V (119886119908) cos120595 +V120573119886119908

2119869V (119886119906) sin120595

times 119870V (119886119908) cos120595 minusV120573119886119908

2119870V (119886119908) cos120595

times

V21205831205761205962

119886

2119906

4119869

2

V (119886119906) minus1

4

120573

2

(119869Vminus1 (119886119906) minus 119869V+1 (119886119906))2

times (sin120595 minus cos120595)2 = 0(7)



119860 = minus119895

119906

2

119886119869

1015840

V (119906119886)Θ (1199062

119886120574Δ minus Ξ120585) + 119908

2

119886 (Θ120585 minus 120596120576119870V (119908119886))

120596120576119886119908

2120574 (Θ120585 minus 120596120576119870V (119908119886) 120575)

119863

119861 =

119906119869V (119906119886)Θ (1199062

119886120574Δ minus Ξ120585) + 119908

2

119886 (Θ120585 minus 120596120576119870V (119908119886) 120575)

119908

2119886120574 (Θ120585 minus 120596120576119870V (119886119908) 120575)

119863

119862 = 119895

119906

2

119886120574Δ minus 120596

2

120583120576119906119908

3

119870

1015840

V (119908119886) 120585


119863

(8)

where

120574 = 120596

2

120583120576119906

2

119869V (119906119886) 1198691015840

V (119906119886) sin120595

minus 120573119869

1015840

V (119906119886) (1199062

119886119869V (119906119886) cos120595 + 120573V119869V (119906119886) sin120595)

120585 = 119869

2

V (119906119886) (1199062


Θ = 120596120598119870V (119908119886) (1199082

119886 cos120595 minus 120573V sin120595)

Δ = 119908

2

119870V (119908119886) cos120595 + 120573V119870V (119908119886) sin120595

Ξ = 120596

2

120583120576119908

3

119870V (119908119886) sin120595

120575 = 120596120576119906119908119886

2

119870V (119908119886) sin120595(9)


(119878

119911)Core =

120573

2119906

4(120576|119860|

2

+ 120583|119861|

2

) (

V119903

)

2

119869

2

V (119906119903) + (119906

2

)

2

120590

(119878

119911)Clad =

120573

2119908

4(120576|119862|

2

+ 120583|119863|

2

) (

V119903

)

2

119870

2

V (119908119903) + (119908

2

)

2

Λ

(10)

with

120590 = 119869

2

Vminus1 (119906119903) + 1198692

V+1 (119906119903) minus 2119869Vminus1 (119906119903) 119869V+1 (119906119903)

Λ = 119870

2

Vminus1 (119908119903) + 1198702

V+1 (119908119903) minus 2119870Vminus1 (119908119903)119870V+1 (119908119903) (11)





Pitch angle120595


120573 (mminus1)

0∘10120583m

EH01 5970 times 106

EH11 5975 times 106

EH21 5968 times 106

20120583mEH01 5960 times 106

EH11 5965 times 106

EH21 5970 times 106

90∘10120583m

EH01 5850 times 106

EH11 5890 times 106

EH21 5870 times 106

20120583mEH01 5950 times 106

EH11 5980 times 106

EH21 5965 times 106


01 EH11 and EH





01 EH11 and EH

21modes are


11hybrid


and EH21


and EH11







02

04

06

08

10Sz

c (au

)

20 40 60 80 10

r (m) times10minus6

(a)

10

20

30

40

50

60

70

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus8

(b)


Szc (

au)

20

05

10

15

20

25

30

35

40 60 80 10r (m)

times10minus8

times10minus6

(a)

100

80

60

40

20

000004 000006 000008 00001r (m)

Szcl

(au

)times10minus12

(b)


05

50 times 10minus6

10

15

20

Szc (

au)

000001 0000015 000002

r (m)

times10minus6

(a)

10

12

14

80

60

40

20

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus17

(b)



50 times 10minus6

05

10

15

20

000001 0000015 000002

Szc (

au)

r (m)

times10minus16

(a)

20

40

60

80

10

12

14

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus17

(b)



01and EH






4 Conclusion




Acknowledgments


References































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





119860 = minus119895

119906

2

119886119869

1015840

V (119906119886)Θ (1199062

119886120574Δ minus Ξ120585) + 119908

2

119886 (Θ120585 minus 120596120576119870V (119908119886))

120596120576119886119908

2120574 (Θ120585 minus 120596120576119870V (119908119886) 120575)

119863

119861 =

119906119869V (119906119886)Θ (1199062

119886120574Δ minus Ξ120585) + 119908

2

119886 (Θ120585 minus 120596120576119870V (119908119886) 120575)

119908

2119886120574 (Θ120585 minus 120596120576119870V (119886119908) 120575)

119863

119862 = 119895

119906

2

119886120574Δ minus 120596

2

120583120576119906119908

3

119870

1015840

V (119908119886) 120585


119863

(8)

where

120574 = 120596

2

120583120576119906

2

119869V (119906119886) 1198691015840

V (119906119886) sin120595

minus 120573119869

1015840

V (119906119886) (1199062

119886119869V (119906119886) cos120595 + 120573V119869V (119906119886) sin120595)

120585 = 119869

2

V (119906119886) (1199062


Θ = 120596120598119870V (119908119886) (1199082

119886 cos120595 minus 120573V sin120595)

Δ = 119908

2

119870V (119908119886) cos120595 + 120573V119870V (119908119886) sin120595

Ξ = 120596

2

120583120576119908

3

119870V (119908119886) sin120595

120575 = 120596120576119906119908119886

2

119870V (119908119886) sin120595(9)


(119878

119911)Core =

120573

2119906

4(120576|119860|

2

+ 120583|119861|

2

) (

V119903

)

2

119869

2

V (119906119903) + (119906

2

)

2

120590

(119878

119911)Clad =

120573

2119908

4(120576|119862|

2

+ 120583|119863|

2

) (

V119903

)

2

119870

2

V (119908119903) + (119908

2

)

2

Λ

(10)

with

120590 = 119869

2

Vminus1 (119906119903) + 1198692

V+1 (119906119903) minus 2119869Vminus1 (119906119903) 119869V+1 (119906119903)

Λ = 119870

2

Vminus1 (119908119903) + 1198702

V+1 (119908119903) minus 2119870Vminus1 (119908119903)119870V+1 (119908119903) (11)





Pitch angle120595


120573 (mminus1)

0∘10120583m

EH01 5970 times 106

EH11 5975 times 106

EH21 5968 times 106

20120583mEH01 5960 times 106

EH11 5965 times 106

EH21 5970 times 106

90∘10120583m

EH01 5850 times 106

EH11 5890 times 106

EH21 5870 times 106

20120583mEH01 5950 times 106

EH11 5980 times 106

EH21 5965 times 106


01 EH11 and EH





01 EH11 and EH

21modes are


11hybrid


and EH21


and EH11







02

04

06

08

10Sz

c (au

)

20 40 60 80 10

r (m) times10minus6

(a)

10

20

30

40

50

60

70

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus8

(b)


Szc (

au)

20

05

10

15

20

25

30

35

40 60 80 10r (m)

times10minus8

times10minus6

(a)

100

80

60

40

20

000004 000006 000008 00001r (m)

Szcl

(au

)times10minus12

(b)


05

50 times 10minus6

10

15

20

Szc (

au)

000001 0000015 000002

r (m)

times10minus6

(a)

10

12

14

80

60

40

20

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus17

(b)



50 times 10minus6

05

10

15

20

000001 0000015 000002

Szc (

au)

r (m)

times10minus16

(a)

20

40

60

80

10

12

14

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus17

(b)



01and EH






4 Conclusion




Acknowledgments


References































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




02

04

06

08

10Sz

c (au

)

20 40 60 80 10

r (m) times10minus6

(a)

10

20

30

40

50

60

70

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus8

(b)


Szc (

au)

20

05

10

15

20

25

30

35

40 60 80 10r (m)

times10minus8

times10minus6

(a)

100

80

60

40

20

000004 000006 000008 00001r (m)

Szcl

(au

)times10minus12

(b)


05

50 times 10minus6

10

15

20

Szc (

au)

000001 0000015 000002

r (m)

times10minus6

(a)

10

12

14

80

60

40

20

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus17

(b)



50 times 10minus6

05

10

15

20

000001 0000015 000002

Szc (

au)

r (m)

times10minus16

(a)

20

40

60

80

10

12

14

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus17

(b)



01and EH






4 Conclusion




Acknowledgments


References































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




50 times 10minus6

05

10

15

20

000001 0000015 000002

Szc (

au)

r (m)

times10minus16

(a)

20

40

60

80

10

12

14

000004 000006 000008 00001r (m)

Szcl

(au

)

times10minus17

(b)



01and EH






4 Conclusion




Acknowledgments


References































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials


























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



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