International Journal of Advanced Engineering Technology E-ISSN

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1 International Journal of Advanced Engineering Technology E-ISSN Research Article DISPERSION & CUTTOFF CHARACTERISTICS OF CIRCULAR HELICALLY CLADDED OPTICAL FIBER Ajay Kumar Gautam, ivekanand Mishra Address for Correspondence Department of Electronics Engineering, Sardar allahhai National Institute of Technology, Surat, India ABSTRACT In this article dispersion characteristic of conventional optical waveguide with helical winding at core cladding interface has een otained. The model dispersion characteristics of optical waveguide with helical winding at core-cladding interface have een otained for five different pitch angles. This article includes dispersion characteristics of optical waveguide with helical winding, and compression of dispersion characteristics of optical waveguide with helical winding at core-cladding interface for five different pitch angles. Boundary conditions have een used to otain the dispersion characteristics and these conditions have een utilized to get the model Eigen values equation. From these Eigen value equations dispersion curve are otained and plotted for modified optical waveguide for particular values of the pitch angle of the winding and the effect of this winding has een discussed. We oserve that the effect of conducting helical winding is to reduce the cutoff values, thus increasing the numer of modes; we also oserve that for very small value of anomalous dispersion properties may occur in helically wound waveguides. We found that some curves have and gaps of discontinuities etween some value of. Thus helical pitch angle controls the modal properties of this type of optical waveguide. KEYWORDS: Dispersion Curves, Bessel Functions, Characteristics Equation, Sheath Helix, Circular Waveguide, Modal cutoff, CHCF. INTRODUCTION An optical waveguide is asically a cylindrical dielectric waveguide with a circular cross section where a high-index wave guiding core is surrounded y a low-index cladding. The index step and profile are controlled y the concentration and distriution of dopants. Silica fiers are ideal for light transmission in the visile and nearinfrared regions ecause of their low loss and low dispersion in these spectral regions. They are therefore suitale for optical communications. Even though optical fier seems quite flexile, it is made of glass, which cannot withstand sharp ending or longitudinal stress. Therefore when fier is placed inside complete cales special construction techniques are employed to allow the fier to move freely within a tue. Usually fier optic cales contain several fiers, a strong central strength memer and one or more metal sheaths for mechanical protection. Some cales also include copper pairs for auxiliary applications. Optical fiers with helical winding are known as complex optical waveguides. The use of helical winding in optical fiers makes the analysis much accurate. As the numer of propagating modes depends on the helix pitch angle, so helical winding at core cladding interface can control the dispersion characteristics of the optical waveguide [3]. The conventional optical fier having a circular core cross section which is widely used in optical communication systems []. Recently metal clad optical waveguides have een studied ecause these provide potential applications, connecting the optical components to other circuits. Metallic cladding structure on an optical waveguide is known as a TE mode pass polarizer and is commercially applied to various optical devices [4]. The propagation characteristics of optical fiers with elliptic cross section have een investigated y many researchers. Singh [5] have proposed an analytical study of dispersion characteristics of helically cladded step index optical fier with circular core. The model characteristic and dispersion curves of a hypocycloidal optical waveguide have een investigated y Ojha [6]. Present work is the study of circular optical waveguide with sheath helix [3] in etween the core and cladding region. The sheath helix is a cylindrical surface with high conductivity in a preferential direction which winds helically at constant angle around the core cladding oundary surfaces. Optical fiers with helical winding are known as complex optical waveguides. The conventional optical fier having a circular core cross section which is widely used in optical communication systems. The use of helical winding in optical fiers makes the analysis much accurate []. The propagation characteristics of optical fiers with elliptic cross section have een investigated y many researchers. Singh [3] have proposed an analytical study of dispersion characteristics of helically cladded step index optical fier with elliptical core. Present work is the study of circular optical waveguide with sheath helix in etween the core and cladding region, this work also gives the comparison of dispersion characteristic at different pitch angles. The sheath helix [] is a cylindrical surface with high conductivity in a preferential direction which winds helically at constant angle around the core cladding oundary surfaces. As the numer of propagating modes depends on the IJAET/ol.II/ Issue III/July-Septemer, /97-35

2 International Journal of Advanced Engineering Technology E-ISSN helix pitch angle [], so helical winding at corecladding interface can control the dispersion characteristics [3-7] of the optical waveguide. The winding angle of helix (ψ) can take any aritrary value etween to π/. In case of sheath helix winding [], cylindrical surface with high conductivity in the direction of winding which winds helically at constant pitch angle (ψ) around the core cladding oundary surface. We assume that the waveguide have real constant refractive index of core and cladding is n and n respectively (n > n ). In this type of optical wave guide which we get after winding, the pitch angle controls the model characteristics of optical waveguide. THEORETICAL BACKGROUND The optical waveguide is the fundamental element that interconnects the various devices of an optical integrated circuit, just as a metallic strip does in an electrical integrated circuit. However, unlike electrical current that flows through a metal strip according to Ohm s law, optical waves travel in the waveguide in distinct optical modes. A mode, in this sense, is a spatial distriution of optical energy in one or more dimensions that remains constant in time. The mode theory, along with the ray theory, is used to descrie the propagation of light along an optical fier. The mode theory [] is used to descrie the properties of light that ray theory is unale to explain. The mode theory uses electromagnetic wave ehavior to descrie the propagation of light along a fier. A set of guided electromagnetic waves is called the modes [3, 6] of the fier. For a given mode, a change in wavelength can prevent the mode from propagating along the fier. The mode is no longer ound to the fier. The mode is said to e cut off [3]. Modes that are ound at one wavelength may not exist at longer wavelengths. The wavelength at which a mode ceases to e ound is called the cutoff wavelength [] for that mode. However, an optical fier is always ale to propagate at least one mode. This mode is referred to as the fundamental mode [6] of the fier. The fundamental mode can never e cut off. We can take a case of a fier with circular cross-section wound with a sheath helix at the core-clad interface (Figure ). A sheath helix can e assumed y winding a very thin conducting wire around the cylindrical surface so that the spacing etween the nearest two windings is very small and yet they are insulated from each another. In our structure, the helical windings are made at a constant helix pitch angle (ψ). We assume that (n - n ) / n <<. WAEGUIDE WITH CONDUCTING HELICAL WINDING We consider the case of a fier with circular cross section wrapped with a sheath helix at core clad oundary as shown in Figure. In our structure, the helical windings are made at a constant angle ψ the helix pitch angle. The structure has high conductivity in a preferential direction. The pitch angle can control the propagation ehavior of such fiers [3]. We assume that the core and cladding regions have the real refractive indices n and n (n > n ), and (n -n ) / n <<. The winding is right handed and the direction of propagation is positive z direction. The winding angle of the helix (pitch angle - ψ) can take any aritrary value etween to π/. This type of fiers is referred to as circular helically cladded fier (CHCF). This analysis requires the use of cylindrical coordinate system φ [8] with the z axis eing the direction ( r,, z) of propagation. Figure : Fier with circular cross section wrapped with a sheath helix IJAET/ol.II/ Issue III/July-Septemer, /97-35

3 International Journal of Advanced Engineering Technology E-ISSN BOUNDARY CONDITIONS Tangential component of the electric field in the direction of the conducting winding should e zero, and in the direction perpendicular to the helical winding, the tangential component of oth the electric field and magnetic field must e continuous, so we have following oundary condition [7] with helix. E + E = () z φ E sin z ψ Eφ ( z z ) ψ ( φ φ ) ( z z ) ψ ( φ φ ) + = () E E cos E E sin = (3) ψ H H sin + H H cos = (4) ψ MODEL EQUATION The guided mode along this type of fier can e analyzed in a standard way, with the cylindrical coordinates system ( r, φ, z). In order to have a guided field the following conditions must e satisfied n k= k β k = n k, where n and n are refractive indices or core and cladding regions respectively. The solution of the axial field components can e written as, The expressions for E z and H z inside the core are, when (r < a) E AJ ua e φ β + = ω (5) z ( ) j j z j t z ( ) H BJ ua e φ β ω j j z+ j t = (6) The expressions for E z and H z outside the core are, when (r > a) E CK ua e φ β + = ω (7) Z ( ) j j z j t z ( ) H DK ua e φ β ω j j z+ j t = (8) where, A, B, C, D are aritrary constants which are to e evaluated from the oundary conditions. Also ( ua) J and K ( wa) are the Bessel functions. For a guided mode, the propagation constant lies etween two limitsβ andβ. If nk = k β k = nk then a field distriution is generated which will has an oscillatory ehavior in the core and a decaying ehavior in the cladding. The energy then is propagated along fier without any loss. Where k π λ = is free space propagation constant. The transverse field components can e otained y using Maxwell s standard relations. So the electric and magnetic field components E ϕ and H ϕ can e written as, The expressions for E ϕ and H ϕ inside the core are, when (r < a) j β Eφ = j AJ ( ua) ubj '( ua) e µω u a j β Hφ= j BJ ( ua) ωεuaj '( ua) e u + a jφ jβ z+ jωt jφ jβ z+ jωt The expressions for E ϕ and H ϕ inside the core are, when (r > a) j β jφ jβ z+ jωt Eφ = j CK ( wa) wdk '( wa) e µω w a () j β jφ jβ z+ jωt Hφ = j DK ( wa) ωε wck '( wa) e w + a () Now put these transverse field components equations into oundary conditions, we get following four unknown equations involving four unknown aritrary constants β jµω AJ + cos ψ + BJ '( ua) cos ψ = (3) β jµω CK ( wa) + cos ψ + DK '( wa) cos ψ = (4) (9) () IJAET/ol.II/ Issue III/July-Septemer, /97-35

4 International Journal of Advanced Engineering Technology E-ISSN β jµω AJ sin ψ BJ '( ua) sin ψ β jµω CK ( wa) sin ψ + DK '( wa) sin ψ = jωε β AJ ' cos ψ + BJ ( ua) + u a (6) jωε β + CK '( wa) cos ψ DK ( wa) + = w Equations (3), (4), (5) and (6) will yield a non trivial solution if the determinant whose elements are the coefficient of these unknown constants is set equal to zero. Thus we have A A A3 A4 where, B B B3 B4 CCC3C4 D D D3D4 β A = J + jµω A = J ' A3= A4= B= B= β B3 = K ( wa) + jµω B4 = K '( wa) β C = J jµω C = J ' β C3 = K ( wa) jµω C4 = K '( wa) jωε D = J ' β D = J + jωε D3 = K '( wa) β D4 = K ( wa) + (5) = (7) (8) (9) () () IJAET/ol.II/ Issue III/July-Septemer, /97-35

5 International Journal of Advanced Engineering Technology E-ISSN After eliminating unknown constants from equations (7), (8), (9), () & (), we get the following characteristic equation. J ( ) k ua J '( ua) u sin cos cos J '( ua ) u a u J ( ua ) β ψ + ψ ψ K ( ) k wa β K '( wa) w + cos K + ψ = '( wa ) w a w K ( wa ) Equation () is standard characteristic equation, and is used for model dispersion properties and model cutoff conditions. SIMULATION RESULTS AND DISCUSSION It is now possile to interpret the characteristic equation (Equation ) in numerical terms. We now make some simple calculations ased on equation (3), equation (4) and equation (5). This will give us an insight into model properties of our waveguide. J ( ) k ua J '( ua) u sin cos cos J '( ua ) u a u J ( ua ) β ψ + ψ ψ K ( ) k wa K '( wa) w + + = K '( wa ) w a w K ( wa ) β cos ψ ( β / k) n aw = = n n () (3) (4) π a = + = ( u w ) a ( n n ) λ (5) where, & are known as normalization propagation constant & normalized frequency parameter respectively. We make some simple calculations ased on Equations (4) and (5). We choose n =.5, n =.46 and λ =.55µm. We take = for simplicity, ut the result is valid for any value of. In order to plot the dispersion relations, we plot the normalized frequency parameter against the normalization propagation constant. we considered five special cases corresponding to the values of pitch angle ψ as, 3, 45, 6 and Figure : Dispersion Curve of normalized propagation constant as a function of for a lower order modes for pitch angle ψ = IJAET/ol.II/ Issue III/July-Septemer, /97-35

6 International Journal of Advanced Engineering Technology E-ISSN Figure 3: Dispersion Curve of normalized propagation constant as a function of for a lower order modes for pitch angle ψ = Figure 4: Dispersion Curve of normalized propagation constant as a function of for a lower order modes for pitch angle ψ = 45 IJAET/ol.II/ Issue III/July-Septemer, /97-35

7 International Journal of Advanced Engineering Technology E-ISSN Figure 5: Dispersion Curve of normalized propagation constant as a function of for a lower order modes for pitch angle ψ = Figure 6: Dispersion Curve of normalized propagation constant as a function of for a lower order modes for pitch angle ψ = 9 From the aove figures we oserve that, they all means that one effect of conducting helical winding have standard expected shape, ut except for lower is to split the modes and remove a degeneracy order modes they comes in pairs, that is cutoff which is hidden in conventional waveguide without values for two adjacent mode converge. This windings. We also oserve that another effect of IJAET/ol.II/ Issue III/July-Septemer, /97-35

8 International Journal of Advanced Engineering Technology E-ISSN the conducting helical winding is to reduce the cutoff values, thus increasing the numer of modes. This effect is undesirale for the possile use of these waveguide for long distance communication. An anomalous feature in the dispersion curves is oservale for ψ = 3, 45 and 6 for this type of waveguide near the lowest order mode. It is found that on the left of the lowest cutoff values, portions of curves appear which have no resemlance with standard dispersion curves, and have no cutoff values. This means that for very small value of anomalous dispersion properties may occur in helically wound waveguides. We found that some curves have and gaps of discontinuities etween some value of. These represent the and gaps or foridden ands of the structure. These are induced y the periodicity of the helical windings. We now come to tale. we note particularly that the dependence of the cutoff value ( c ) on ψ is such that as ψ is increased there is a drastic fall in c at ψ =3 and then a small increase as ψ goes from 3 to 6 ; then is a quick rise as ψ changes from 6 to 9 (Figure 7). Thus the two most sensitive regions in respect of the influence of helical pitch angle ψ on the cutoff values and the model properties of waveguides are ranges from ψ = to ψ = 3 and ψ = 6 to ψ = 9 and these ranges of pitch angle expected to have potential applications with ψ as a means for controlling the model properties. Tale : Cutoff c values for some lower order modes ψ c c c c c c c c c c Angle in Degree Figure 7: Dependence of cutoff values c on the pitch angle ψ CONCLUSION From the aove results we oserve that, they all have standard expected shape, ut except for lower order modes they comes in pairs, that is cutoff values for two adjacent mode converge. This means that one effect of conducting helical winding is to split the modes and remove a degeneracy which is hidden in conventional waveguide without windings. We also oserve that another effect of the conducting helical winding is to reduce the cutoff values, thus increasing the numer of modes. This effect is undesirale for the possile use of these waveguide for long distance communication. An anomalous feature in the dispersion curves is oservale for ψ = 3, 45 and 6 for this type of waveguide near the lowest order mode. It is found IJAET/ol.II/ Issue III/July-Septemer, /97-35

9 International Journal of Advanced Engineering Technology E-ISSN that on the left of the lowest cutoff values, portions of curves appear which have no resemlance with standard dispersion curves, and have no cutoff values. This means that for very small value of anomalous dispersion properties may occur in helically wound waveguides. We found that some curves have and gaps of discontinuities etween some value of. These represent the and gaps or foridden ands of the structure. These are induced y the periodicity of the helical windings. Thus helical pitch angle controls the modal properties of this type of optical waveguide. REFRENCES. Kumar, D. and O. N. Singh II, Modal characteristics equation and dispersion curves for an elliptical step-index fier with a conducting helical winding on the corecladding oundary - An analytical study, IEEE, Journal of Light Wave Technology, ol., No. 8, 46 44, USA, August.. Watkins, D. A., Topics in Electromagnetic Theory, John Wiley and Sons Inc., NY, N. Mishra, ivek Singh, B. Prasad, S. P. Ojha, Optical Dispersion curves of two metal - clad lightguides having doule convex lens core cross sections, Wiley, Microwave and Optical Technology Letters, ol. 4, No. 4, 9-3, New York, Fe,. 4..N. Mishra,. Singh, B. Prasad, S. P. Ojha, An Analytical investigation of dispersion characteristic of a lightguide with an annular core cross section ounded y two cardioids, Wiley, Microwave and Optical Technology Letters, ol. 4, No. 4, 9-3, New York, Fe,. 5.. Singh, S. P. Ojha, B. Prasad, and L. K. Singh, Optical and microwave Dispersion curves of an optical waveguide with a guiding region having a core cross section with a lunar shape, Optik, 67-7, Singh, S. P. Ojha, and L. K. Singh, Model Behaviour, cutoff condition, and dispersion characteristics of an optical waveguide with a core cross section ounded y two spirals, microwave Optical Technology Letter, ol., -4, Singh, S. P. Ojha, and B. Prasad, weak guidance modal dispersion characteristics of an optical waveguide having core with sinusoidally varying gear shaped cross section, microwave Optical Technology Letter, ol., 9-33, Gloge D., Dispersion in weakly guiding fiers, Appl. Optics, ol., , P. K. Choudhury, D. Kumar, and Z. Yusoff, F. A. Rahman, An analytical investigation of four-layer dielectric optical fiers with au nano-coating - A comparison with three-layer optical fiers, PIER 9, 69-86, 9.. Keiser G., Optical Fier Communications, Chap., 3 rd edition McGraw-Hill, Singapore,.. Jia Ming-Liu, Photonic Devices, Camridge University Press, UK, 5.. Kumar, D. and O. N. Singh II, Towards the dispersion relations for dielectric optical fiers with helical windings under slow and fast wave considerations a comparative analysis, PIER, ol. 8, 49 4, Kumar, D. and O. N. Singh II, An analytical study of the modal characteristics of annular step index fier of elliptical cross section with two conducting helical windings on the two oundary surfaces etween the guiding and non guiding regions Optik, ol. 3, No. 5, 93-96,. 4. Singh, U. N., O. N. Singh II, P. Khastgir and K. K. Dey Dispersion characteristics of helically cladded step index optical fier analytical study J. Opt. Soc. Am. B, 73-78, M. P. S. Rao, ivek Singh, B. Presad and S. P. Ojha Model characteristic and dispersion curves of hypocycloidal optical waveguide Optik,, No., 8-85, Ajoy Ghatak and K. Thyagarajan, Optical Electronics Camridge University Press, India, Kumar, D. and O. N. Singh II, Some special cases of propagation characteristics of an elliptical step index fier with a conducting helical winding on the core cladding oundary An analytical treatment, Optik ol., No., ,. 8. Govind P. Agrawal, Fier Optic Communication Systems, 3 rd edition, A John Wiley & Sons, Inc., Pulication, New York,. IJAET/ol.II/ Issue III/July-Septemer, /97-35