International Journal of Advanced Engineering Technology E-ISSN

Transcription

1 Research Article ANALYTICAL STUDY OF HELICALLY CLADDED OPTICAL WAVEGUIDE WITH DIFFERENT PITCH ANGLES Mishra V.* Gautam A. K. Taunk B. R. Address for Correspondence Sr. Member IEEE, Electronics Engineering Department, Sardar Vallabhbhai National Institute of Technology Surat, India, Electronics Engineering Department, Sardar Vallabhbhai National Institute of Technology Surat, India Electronics Engineering Department Sardar Vallabhbhai National Institute of Technology Surat, India ABSTRACT 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. In this article dispersion characteristic of conventional optical waveguide with helical winding at core cladding interface has been obtained. The model dispersion characteristics of optical waveguide with helical winding at core-cladding interface have been obtained for five different pitch angles. Boundary conditions have been used to obtain the dispersion characteristics and these conditions have been utilized to get the model Eigen values equation. From these Eigen value equations dispersion curve are obtained and plotted for modified optical waveguide for particular values of the pitch angle of the winding and the effect of this winding has been discussed. The article also shows the effect in the Dispersion Curve with changing the Pitch Angle. KEYWORDS Bessel functions, dispersion curves, characteristics equation, sheath helix, circular waveguide, modal cutoff.. INTRODUCTION Some cables also include copper pairs for An optical waveguide is basically a cylindrical dielectric waveguide with a circular cross section where a high-index wave guiding core is surrounded by a low-index cladding. The index step and profile are controlled by the concentration and distribution of dopants. Silica fibers are ideal for light transmission in the visible and near-infrared regions because of their low loss and low dispersion in these spectral regions. They are therefore suitable for optical communications. Even though optical fiber seems quite flexible, it is made of glass, which cannot withstand sharp bending or longitudinal stress. Therefore when fiber is placed inside complete cables special construction techniques are employed to allow the fiber to move freely within a tube. Usually fiber optic cables contain several fibers, a strong central strength member and one or more metal sheaths for mechanical protection. auxiliary applications. Optical fibers with helical winding are known as complex optical waveguides. The use of helical winding in optical fibers makes the analysis much accurate. As the number 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 fiber having a circular core cross section which is widely used in optical communication systems []. Recently metal clad optical waveguides have been studied because 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

2 optical fibers with elliptic cross section have been investigated by many researchers. Singh [5] have proposed an analytical study of dispersion characteristics of helically cladded step index optical fiber with circular core. The model characteristic and dispersion curves of a hypocycloidal optical waveguide have been investigated by Ojha [6]. Present work is the study of circular optical waveguide with sheath helix [3] in between 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 boundary surfaces. Optical fibers with helical winding are known as complex optical waveguides. The conventional optical fiber having a circular core cross section which is widely used in optical communication systems. The use of helical winding in optical fibers makes the analysis much accurate []. The propagation characteristics of optical fibers with elliptic cross section have been investigated by many researchers. Singh [3] have proposed an analytical study of dispersion characteristics of helically cladded step index optical fiber with elliptical core. Present work is the study of circular optical waveguide with sheath helix in between 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 boundary surfaces. As the number of propagating modes depends on the helix pitch angle [], so helical winding at core-cladding interface can control the dispersion characteristics [3-7] of the optical waveguide. The winding angle of helix (ψ) can take any arbitrary value between 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 boundary 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 distribution of optical energy in one or more dimensions that remains constant in time. The mode theory, along with the ray theory, is used to describe the propagation of light along an optical fiber. The mode theory [] is used to describe the properties of light that ray theory is unable to explain. The mode theory uses electromagnetic wave behavior to describe the propagation of light along a fiber. A set of guided electromagnetic waves is called the modes [3, 6] of the fiber. For a given mode, a change in wavelength can prevent the mode from propagating along the fiber. The mode is no longer bound to the fiber. The mode is said to be cut off [3]. Modes that are

3 bound at one wavelength may not exist at longer wavelengths. The wavelength at which a mode ceases to be bound is called the cutoff wavelength [] for that mode. However, an optical fiber is always able to propagate at least one mode. This mode is referred to as the fundamental mode [6] of the fiber. The fundamental mode can never be cut off. We can take a case of a fiber with circular crosssection wound with a sheath helix at the coreclad interface (Figure ). A sheath helix can be assumed by winding a very thin conducting wire around the cylindrical surface so that the spacing between 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 <<. 3. WAVEGUIDE WITH CONDUCTING HELICAL WINDING We consider the case of a fiber with circular cross section wrapped with a sheath helix at core clad boundary 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 control the propagation behavior of such fibers [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 arbitrary value between to π/. This type of fibers is referred to as circular helically cladded fiber (CHCF). This analysis requires the use of cylindrical coordinate system ( r, φ, z) [8] with the z axis being the direction of propagation. 4. BOUNDARY CONDITIONS Tangential component of the electric field in the direction of the conducting winding should be zero, and in the direction perpendicular to the helical winding, the tangential component of both the electric field and magnetic field must be continuous, so we have following boundary condition [7] with helix. E sin z ψ Eφ cosψ E sin + = () z ψ Eφ cosψ + = () ( Ez Ez ) cosψ ( Eφ Eφ ) sin ( z z ) ψ ( φ φ ) = (3) ψ H H sin + H H cos = (4) ψ preferential direction. The pitch angle can Figure : Fiber with circular cross section wrapped with a sheath helix

4 5. MODEL EQUATION The guided mode along this type of fiber can be analyzed in a standard way, with the cylindrical coordinates system( r, φ, z). In order to have a guided field the following conditions must be 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 be written as, The expressions for E z and H z inside the core are, when (r < a) j j z j t E z A J ( ua ) e φ β + = ω (5) z ( ) H B J 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) j j z j t E Z C K ( u a ) e φ β + = ω (7) z ( ) H DK ua e φ β ω j j z+ j t = (8) where, A, B, C, D are arbitrary constants which are to be evaluated from the boundary conditions. Also J ( ua) Bessel functions. and K ( wa) are the For a guided mode, the propagation constant lies between two limitsβ andβ. n k= k β k = n k then a field distribution is generated which will has an oscillatory behavior in the core and a decaying behavior in the cladding. The energy then is propagated along fiber without any loss. Where π k = is free space propagation λ constant. The transverse field components can be obtained by using Maxwell s standard relations. So the electric and magnetic field components E ϕ and H ϕ can be 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 The expressions for E ϕ and H ϕ inside the core are, when (r > a) j β Eφ = j CK ( wa) wdk '( wa) e µω w a j β Hφ = j DK ( wa) ωε wck '( wa) e w + a jφ jβ z+ jωt jφ jβ z+ jωt jφ jβ z+ jωt jφ jβ z+ jωt Now put these transverse field components equations into boundary conditions, we get following four unknown equations involving four unknown arbitrary constants β jµω AJ ( ua) + cos ψ + BJ '( ua) cos ψ = u a β jµω CK ( wa) + cos ψ + DK '( wa) cos ψ = w β jµω AJ ( ua) cosψ sin ψ BJ '( ua) sin ψ u a β jµω CK ( wa) cosψ sin ψ + DK '( wa) sin ψ = w If (9) () () () (3) (4) (5)

5 jωε β AJ '( ua) cos ψ + BJ ( ua) + cosψ u a jωε β + CK wa DK wa + = w '( ) cos ψ ( ) cosψ 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 B B B3B4 CC C3C4 = (7) D D D3D4 where, β A = J ( ua) + cosψ u a jµω A = J '( ua) cosψ (8) A3= A4= B= B= β B3 = K ( wa) + cosψ (9) jµω B4 = K '( wa) cosψ w β C = J ( ua) cosψ u a jµω C = J '( ua) β C3 = K ( wa) cosψ () jµω C4 = K '( wa) w jωε D = J '( ua) cosψ β D = J ( ua) + cosψ u a jωε D3 = K '( wa) cosψ w () β D4 = K ( wa) + cosψ (6)

6 J ( ) k ua β J '( ua) u + cosψ cos ψ. J '( ua ) u a u J ( ua ) K ( wa) k K '( wa) w + + = K '( wa ) w a w K ( wa ) After eliminating unknown constants from equations (7), (8), (9), () & (), we get the characteristic equation (). Equation () is standard characteristic equation, and is used for model dispersion properties and model cutoff conditions. 6. SIMULATION RESULTS AND DISCUSSION It is now possible to interpret the characteristic equation (Equation ) in numerical terms. This will give us an insight into model properties of our wave guide. where, b & V are known as normalization propagation constant & normalized frequency parameter respectively. We make some simple calculations based on Equations (4) and (5). We choose n =.5, n =.46 and λ =.55µm. We take = for simplicity, but the result is valid for any value of. In order to plot the dispersion relations, we plot the normalized frequency parameter V against the normalization propagation constant b. we considered five special cases corresponding to the values of pitch angle ψ as, 3, 45, 6 and 9. From the above figures we observe β cosψ cos ψ () that, they all have standard expected shape, but 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 observe that another effect of the conducting helical winding is to reduce the cutoff values, thus increasing the number of modes. This effect is undesirable for the possible use of these waveguide for long distance communication. An anomalous feature in the dispersion curves is observable 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 resemblance with standard dispersion curves, and have no cutoff values. This means that for very small value of V anomalous dispersion properties may occur in helically wound waveguides. 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 cosψ cos ψ ( β / k) n aw b= = V n n (3) (4) π a = + = V ( u w ) a ( n n ) λ (5)

7 b V Figure : Dispersion Curve of normalized propagation constant b as a function of V for a lower order modes for pitch angle ψ = b Figure 3: Dispersion Curve of normalized propagation constant b as a function of V for a lower order modes for pitch angle ψ = 3 V

8 b V Figure 4: Dispersion Curve of normalized propagation constant b as a function of V for a lower order modes for pitch angle ψ = b Figure 5: Dispersion Curve of normalized propagation constant b as a function of V for a lower order modes for pitch angle ψ = 6 V

9 b Figure 6: Dispersion Curve of normalized propagation constant b as a function of V We found that some curves have band gaps of discontinuities between some value of V. These represent the band gaps or forbidden bands of the structure. These are induced by the periodicity of the helical windings. 7. CONCLUSION From the above results we observe that, the effect of the conducting helical winding is to reduce the cutoff values, thus increasing the number of modes. This effect is undesirable for the possible use of these waveguide for long distance communication. We also observe that, all curves have standard expected shape, but 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. for a lower order modes for pitch angle ψ = 9 V An anomalous feature in the dispersion curves is observable 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 resemblance with standard dispersion curves, and have no cutoff values. This means that for very small value of V anomalous dispersion properties may occur in helically wound waveguides. We found that some curves have band gaps of discontinuities between some value of V. These represent the band gaps or forbidden bands of the structure. These are induced by 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 fiber with a conducting helical winding on

10 the core-cladding boundary - An analytical study, IEEE, Journal of Light Wave Technology, Vol., No. 8, 46 44, USA, August.. Watkins, D. A., Topics in Electromagnetic Theory, John Wiley and Sons Inc., NY, V.N. Mishra, Vivek Singh, B. Prasad, S. P. Ojha, Optical Dispersion curves of two metal - clad lightguides having double convex lens core cross sections, Wiley, Microwave and Optical Technology Letters, Vol. 4, No. 4, 9-3, New York, Feb,. 4. V.N. Mishra, V. Singh, B. Prasad, S. P. Ojha, An Analytical investigation of dispersion characteristic of a lightguide with an annular core cross section bounded by two cardioids, Wiley, Microwave and Optical Technology Letters, Vol. 4, No. 4, 9-3, New York, Feb,. 5. V. 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, V. Singh, S. P. Ojha, and L. K. Singh, Model Behaviour, cutoff condition, and dispersion characteristics of an optical waveguide with a core cross section bounded by two spirals, microwave Optical Technology Letter, Vol., - 4, V. 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, Vol., 9-33, Gloge D., Dispersion in weakly guiding fibers, Appl. Optics, Vol., , P. K. Choudhury, D. Kumar, and Z. Yusoff, F. A. Rahman, An analytical investigation of four-layer dielectric optical fibers with au nano-coating - A comparison with three-layer optical fibers, PIER 9, 69-86, 9.. Keiser G., Optical Fiber Communications, Chap., 3 rd edition McGraw-Hill,Singapore,.. Jia Ming-Liu, Photonic Devices, Cambridge University Press, UK, 5.. Kumar, D. and O. N. Singh II, Towards the dispersion relations for dielectric optical fibers with helical windings under slow and fast wave considerations a comparative analysis, PIER, Vol. 8, 49 4, Kumar, D. and O. N. Singh II, An analytical study of the modal characteristics of annular step index fiber of elliptical cross section with two conducting helical windings on the two boundary surfaces between the guiding and non guiding regions Optik, Vol. 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 fiber analytical study J. Opt. Soc. Am. B, 73-78, M. P. S. Rao, Vivek 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 Cambridge University Press, India, Kumar, D. and O. N. Singh II, Some special cases of propagation characteristics of an elliptical step index fiber with a conducting helical winding on the core cladding boundary An analytical treatment, Optik Vol., No., ,. 8. Govind P. Agrawal, Fiber Optic Communication Systems, 3 rd edition, A John Wiley & Sons, Inc., Publication, New York,.