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Helical Windings on Circular Optical Waveguide and Induced Effects on its Dispersion Curve Ajay Kumar Gautam Amit Kumar Katariya Izhar Ahmed Departmant of Electronics & Communication Engineering Dev Bhoomi Institute of Technology Dehradun, India Departmant of Electronics & Communication Engineering Dev Bhoomi Institute of Technology Dehradun, India Departmant of Electronics & Communication Engineering Dev Bhoomi Institute of Technology Dehradun, India [email protected] [email protected] [email protected] 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. The conventional optical fiber having a circular core cross – section which is widely used in optical communication systems [1]. 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 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. Keywords Bessel Functions, Dispersion Curves, Characteristics Equation, Sheath Helix, Circular Waveguide, Modal Cutoff INTRODUCTION We 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. Some cables also include copper pairs for 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]. 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 [1]. The propagation characteristics of optical fibers with elliptic cross – section have been investigated by many researchers. Singh [13] 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 [12] 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 [2], 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 0 to π/2. In case of sheath helix winding [1], 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 n1 and n2 respectively (n1 > n2). In this type of optical wave guide which we get after winding, the pitch angle controls the model characteristics of optical waveguide.. Theoritical 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 [10] 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 [13, 16] 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 [13]. Modes that are 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 [11] 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 [16] 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 core-clad interface (Figure 1). 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 (n1-n2) / n1 << 1. 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 1. Figure 1: Fiber with circular cross – section wrapped with a sheath helix 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 behavior of such fibers [23]. We assume that the core and cladding regions have the real refractive indices n1 and n2 (n1 > n2), and (n1-n2) / n1 << 1. 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 0 to π/2. This type of fibers is referred to as circular helically cladded fiber (CHCF). This analysis requires the use of cylindrical coordinate system ( r , , z ) [18] with the z – axis being the direction of propagation. 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 [17] with helix. Ez1sin E1cos 0 (1) Ez 2 sin E 2cos 0 (2) Ez1 Ez 2 cos E1 E 2 sin 0 (3) H z1 H z 2 sin H1 H 2 cos 0 (4) conditions must be satisfied n2 k k2 k1 n1k , where n1 and n2 are refractive indices or core and cladding regions respectively. The solution of the axial field components can be written as, Modal Equation The guided mode along this type of fiber can be analyzed in a standard way, with the cylindrical coordinates system ( r , , z ) . In order to have a guided field the following The expressions for Ez and Hz inside the core are, when (r < a) Ez1 AJ (ua)e j j z jt H z1 BJ ua e (5) j j z jt (6) The expressions for Ez and Hz outside the core are, when (r > a) EZ 2 CK (ua)e j j z jt H z 2 DK ua e where, A, B, C , D are arbitrary constants which are to be evaluated from the boundary conditions. Also K ( wa) J ua and are the Bessel functions. For a guided mode, the propagation constant lies between two limits 2 and 1 . If n2 k k2 k1 n1k then a (7) j j z jt (8) behavior in the core and a decaying behavior in the cladding. The energy then is propagated along fiber without any loss. Where k 2 is free – space propagation constant. The transverse field components can be obtained by using Maxwell’s standard relations. field distribution is generated which will has an oscillatory The expressions for Eϕ and Hϕ inside the core are, when (r < a) j j AJ (ua) uBJ '(ua) e j j z jt 2 u a j H1 2 j BJ (ua ) 1uAJ '(ua ) e j j z jt u a E1 (9) (10) The expressions for Eϕ and Hϕ inside the core are, when (r > a) j j CK ( wa ) wDK '( wa ) e j j z jt w2 a j 2 j DK ( wa ) 2 wCK '( wa ) e j j z jt w a E 2 H 2 (11) (12) Now put these transverse field components equations into boundary conditions, we get following four unknown equations involving four unknown arbitrary constants j AJ (ua) sin 2 cos BJ '(ua) cos 0 u a u j CK ( wa ) sin 2 cos DK '( wa ) cos 0 wa w (13) (14) j AJ (ua ) cos 2 sin BJ '(ua) sin u a u j CK ( wa) cos 2 sin DK '( wa) sin 0 wa w j1 AJ '(ua ) cos BJ (ua ) sin 2 cos u a u j 2 CK '( wa ) cos DK ( wa ) sin 2 cos 0 wa w (15) (16) Equations (13), (14), (15) and (16) 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 A1 A2 A3 A4 B1 B 2 B3 B 4 0 C1C 2 C 3 C 4 (17) D1 D 2 D3 D 4 where, A1 J (ua) sin 2 cos ua j A2 J '(ua) cos u A3 0 A4 0 B1 0 B2 0 B3 K ( wa) sin 2 cos wa j B 4 K '( wa) cos w C1 J (ua) cos 2 sin u a j C 2 J '(ua) sin u C 3 K ( wa) cos 2 sin wa j C 4 K '( wa) sin w (18) (19) (20) j1 D1 J '(ua) cos u D 2 J (ua) sin 2 cos u a j 2 D3 K '( wa ) cos w (21) D 4 K ( wa) sin 2 cos wa After eliminating unknown constants from equations (17), (18), (19), (20) & (21), we get the following characteristic equation. 2 2 J (ua) k1 J '(ua ) u cos 2 sin 2 cos J '(ua ) u a u J (ua ) 2 k 2 2 K '( wa ) K ( wa ) w cos 2 0 sin 2 cos K '( wa) wa w K ( wa ) (22) Equation (22) is standard characteristic equation, and is used for model dispersion properties and model cutoff conditions. Simulation Results and Discussion It is now possible to interpret the characteristic equation (Equation 22) in numerical terms. This will give us an insight into model properties of our waveguide. 2 2 J (ua) k1 J '(ua ) u cos 2 sin 2 cos J '(ua ) u a u J (ua ) 2 k 2 2 K '( wa ) K ( wa ) w cos 2 0 sin 2 cos K '( wa) wa w K ( wa ) (23) 2 2 aw ( / k ) n2 b 2 2 V n1 n2 (24) 2 a 2 2 V 2 (u 2 w2 )a 2 (n1 n2 ) (25) 2 2 where, b & V are known as normalization propagation constant & normalized frequency parameter respectively. We make some simple calculations based on Equations (24) and (25). We choose n1=1.50, n2=1.46 and λ =1.55µm. We take 1 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 00, 300, 450, 600 and 900. 1 0.9 0.8 0.7 b 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 V Figure 2: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 00 1 0.9 0.8 0.7 b 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 V Figure 3: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 300 1 0.9 0.8 0.7 b 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 V Figure 4: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 450 1 0.9 0.8 0.7 b 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 V Figure 5: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 600 1 0.9 0.8 0.7 b 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 V Figure 6: Dispersion Curve of normalized propagation constant b as a function of V for a lower – order modes for pitch angle ψ = 900 From the above figures we observe 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 = 300, 450 and 600 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. 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 modes 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. An anomalous feature in the dispersion curves is observable for ψ = 300, 450 and 600 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. REFERENCES [1] Ajay Kumar Gautam, "Analytical Study of Helically Cladded Optical Waveguide with Different Pitch Angles", International Journal of Advanced Engineering Technology, Vol. II, Issue I, pp. - 144-153, January - March, 2011. [2] Ajay Kumar Gautam, "Dispersion & Cut-off Characteristics of Circular Helically Cladded Optical Fiber", International Journal of Advanced Engineering Technology, Vol. II, Issue III, pp. - 297-305, July-September, 2011. [3] V.N. Mishra, Vivek Singh, B. Prasad, S. P. Ojha (2000). 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