DISPERSION & CUTOFF CHARACTERISTICS OF CIRCULAR OPTICAL WAVEGUIDE WITH HELICAL WINDING

Transcription

1 THE DISSERTATION ENTITLED DISPERSION & CUTOFF CHARACTERISTICS OF CIRCULAR OPTICAL WAVEGUIDE WITH HELICAL WINDING Submitted in partial fulfillment of the requirements For the degree of Master of Technology In Electronics Engineering With Specialization In Communication Systems Submitted by Ajay Kumar Gautam (P8EC9) Under the Guidance of Prof. B. R. Taunk & Dr. Vivekanand Mishra JULY- DEPARTMENT OF ELECTRONICS ENGINEERING SARDAR VALLABHBHAI NATIONAL INSTITUTE OF TECHNOLOGY SURAT-3957

2 SARDAR VALLABHBHAI NATIONAL INSTITUTE OF TECHNOLOGY DEPARTMENT OF ELECTRONICS ENGINEERING SURAT-3957 CERTIFICATE This is to certify that the thesis entitled Dispersion & Cutoff Characteristics of Circular Optical Waveguide with Helical Winding, submitted by Ajay Kumar Gautam (P8EC9) in the partial fulfillment of the requirements for award of Master in Technology in Electronics Engineering with specialization in Communication Systems, has satisfactorily presented during the year 9-. External Examiner Internal Examiner Internal Examiner Chairman Head of Dept. Dr.(Ms) S. Patnaik P. G. Incharge Prof. J. N. Sarvaiya SEAL OF DEPARTMENT

3 Acknowledgement This project is by far the most significant accomplishment in my life and it would be impossible without people who supported me and believed in me. I would like to extend my gratitude and my sincere thanks to my honourable, esteemed supervisor Prof. B. R. Taunk Sir and Dr. Vivekanand Mishra Sir, Department of Electronics and Communication Engineering, S.V. NIT, Surat for their immeasurable guidance and valuable time that he devoted for project. I sincerely thank for their exemplary guidance and encouragement. Their trust and support inspired me in the most important moments of making right decisions and I am glad to work with them. I want to express great thanks to Dr. Vivek Singh, Professor, of Department of Physics of Banaras Hindu University [BHU] Varanasi for providing a continuous motivation and help and as well guiding me. He has been great sources of inspiration to me and I thank them from the bottom of my heart. I would also like to thanks our Head of Department Dr. (Mrs.) S. Patnaik and Prof. B. R. Taunk (former HOD), ECED Department, S.V. NIT, Surat who provide me to all facilities and coordination. I would like to thank all my friends and especially my classmates for all the thoughtful and mind stimulating discussions we had, which prompted us to think beyond the obvious. I have enjoyed their companionship so much during my stay at S.V.NIT, Surat. I would like to thank all those who made my stay in S.V.NIT, Surat an unforgettable and rewarding experience. AJAY KUMAR GAUTAM Roll No. P8EC9

4 CONTENTS List of Figures I List of Tables. II Abstract II. Introduction. Motivation. Optical Waveguide 3. Introduction 3. The optical fiber 4.3 The numerical aperture 6.4 Types of optical fiber 7.4. Single Mode Fibers 8.4. Multimode Fibers 9.5 Mode theory for circular optical waveguide 9.5. Maxwell s Equations.5. Waveguide Equations Wave Equations for Step Index Fibers Boundary Conditions Modal Equation Modes in Step Index fibers Cutoff conditions for fiber modes 7 3. Analysis of optical waveguide with helical winding 3 3. Helix Types of Helix 3 3. Circular Optical Waveguide with conducting helical Winding Boundary Conditions Modal equation Result and Discussion Dispersion characteristics Dispersion characteristics at pitch angle ψ = Dispersion characteristics at pitch angle ψ = Dispersion characteristics at pitch angle ψ =45 4

5 4..4 Dispersion characteristics at pitch angle ψ = Dispersion characteristics at pitch angle ψ = Dependence of cutoff values V c Conclusion & Future Work Conclusion Future Work 46 Reference 48 Publication 5

6 LIST OF FIGURES Fig.. A typical optical fiber waveguide consists of thin cylindrical glass rod 3 Fig.. (a) Refractive index profile, multimode step - index fiber 4 (b) Refractive index profile, graded - index multi mode fiber 4 Fig..3 A long optical fiber carrying a light beam 5 Fig..4 Types of optical fiber 7 Fig..5 Electric field distribution for several of lower - order guided modes Fig..6 Low-order and high-order modes Fig..7 Cylindrical coordinate system used for analyzing electromagnetic wave propagation in an optical fiber 4 Fig..8 Bessel functions of first kind 8 Fig..9 Bessel functions of second kind 8 Fig.. Modified Bessel functions of first kind 9 Fig.. Modified Bessel functions of second kind Fig.. Plots of the propagation constant b as a function of V for a lower order modes 9 Fig. 3. Example of helix as coil springs 3 Fig. 3. Helix (A) Right handed, (B) Left handed 3 Fig. 3.3 Fiber with circular cross section wrapped with a sheath helix 3 Fig. 4. Dispersion Curve for pitch angle ψ = 39 Fig. 4. Dispersion Curve for pitch angle ψ = 3 4 Fig. 4.3 Dispersion Curve for pitch angle ψ = 45 4 Fig. 4.4 Dispersion Curve for pitch angle ψ = 6 4 Fig. 4.5 Dispersion Curve for pitch angle ψ = 9 43 Fig. 4.6 Dependence of cutoff values V c on the pitch angle ψ 45 I

7 LIST OF TABLES Table. Cutoff conditions for some lower order modes 8 Table 4. Cutoff V c values for some lower order modes 45 II

8 ABSTRACT The objective of this thesis is to study the properties of circular optical waveguide using Bessel function and to measure the dispersion characteristics using the helical windings at core-cladding interface. Then after, we have used helical windings to study the performance characteristics of waveguide with helical windings on dielectric material. Once the properties of helical windings have been evaluated, then we study the performance of this characteristic at 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 result has been compared. III

9 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Chapter Introduction 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. For example, the core can be doped with Germania (GeO ) or alumina (Al O 3 ) or other oxides, such as P O 5 or TiO, for a slightly higher index than that of a silica cladding []. 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 are manufactured in three main types: multi-mode step-index, multi-mode graded-index, and single mode. Multi mode step index fiber has the largest diameter core (typically 5 to um) []. The larger distance between interfaces allows the light rays to travel the most distance when bouncing through the cable. Multi mode fibers typically carry signals with wavelengths of 85 nm or 3 nm. Optical fibers allow data signals to propagate through them by ensuring that the light signal enters the fiber at an angle greater than the critical angle of the interface between two types of glass. To use fiber optic cables for communications, electrical signals must be converted to light [], transmitted, received, and converted back from light to electrical signals. This requires optical sources and detectors that can operate at the data rates of the communications system. With the cost of optical fiber technology continuing to decrease, many of today s businesses are utilizing this technology in building distribution and/or workstation Sardar Vallabhbhai National Institute of Technology, Surat Page

10 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding applications. Optical fiber s inherent immunity to both electromagnetic interference (EMI) and radio interference (RFI), and its relatively light weight and enormous bandwidth capabilities make it ideal for voice, video and high-speed data applications. Optical fibers have a wide range of applications []. Owing to their low losses and large bandwidths, their most important applications are fiber-optic communications and interconnections. Other important applications include fiber sensors, guided optical imaging, remote monitoring, and medical 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].. Motivation 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 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, 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. Sardar Vallabhbhai National Institute of Technology, Surat Page

11 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Chapter Optical Waveguide. Introduction An optical waveguide is a structure which confines and guides the light beam by the process of total internal reflection. The most extensively used optical waveguide is the step index optical waveguide which consists of a cylindrical central core, clad by a material of slightly lower refractive index. If the refractive indices of the core and cladding are n and n respectively, then for a ray entering the fiber, if the angle of incident (at the core cladding interface) θ A is greater than the critical angle. sin, (.) n / n C Fig. A typical optical fiber waveguide consists of thin cylindrical glass rod [8] then the ray will undergo total internal reflection at that interface. Furthermore, because of the cylindrical symmetry in the fiber structure, this ray will suffer total reflection at the lower interface also and will therefore be guided through the core by repeated total internal reflections. This is the basic principle of the light guidance through the optical fiber. The simplest optical waveguide is the planner waveguide which consists of a thin dielectric film (of refractive index n ) sandwiched between materials of slightly lower refractive indices. Although all waveguides used in integrated optics are asymmetric in nature, the electromagnetic analysis of a symmetric waveguides is much easier to Sardar Vallabhbhai National Institute of Technology, Surat Page 3

12 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding understand and at same time it brings out almost all the salient points associated with the modes of a waveguide, therefore making it easier to understand the physical principles of more complicated guiding structures [7].. The optical fiber Fig. shows a glass fiber which consists of a (cylindrical) central core cladded by a material of slightly lower refractive index. The corresponding refractive index distribution is given by as shown in Fig... n; r a nr (), (.) n; r a For the ray entering the fiber, if the angle of incidence (at the core cladding interface) θ A is than the critical angle θ C, then the ray will undergo total internal reflection at that interface. (a) (b) Fig.. (a) Refractive index profile, multimode step - index fiber, (b) Refractive index profile, graded - index multi mode fiber [9] Sardar Vallabhbhai National Institute of Technology, Surat Page 4

13 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Furthermore, because of the cylindrical symmetry in the fiber structure, this ray will suffer total internal reflection at the lower interface also and will therefore be guided through the core by repeated total internal reflections. Fig..3 A long optical fiber carrying a light beam [] Fig..3 shows the actual guidance of a light beam as it propagates through a long optical fiber. It is necessary to use a cladded fiber (Fig..) rather than a bare fiber because of the fact that for transmission of light from one place to another, the fiber must be supported, and the supporting structures may considerably distort the fiber thereby affecting the guidance of the light wave. This can be avoided by choosing a sufficiently thick cladding. Further, in a fiber bundle, in the absence of the cladding, light can leak from one fiber to another. Sardar Vallabhbhai National Institute of Technology, Surat Page 5

14 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding.3 The numerical aperture Consider a ray which is incident on the entrance aperture of the fiber making an angle θ A with the axis. Let the reflected ray make an angle θ R with the axis. Assuming the outside medium to have a refractive index n (which for most practical cases is unity), we get sin A n sin (.3) n R Obviously if this ray has to suffer total internal reflection at the core cladding interface, n sin θ R (= cos θ R ) > n Thus sin R n n And we must have n n n n sin A n n n If (n n ) n then for all values of θ A, total internal reflection will occur. Assuming n =, the maximum value of sin θ A for a ray to be guided is given by sin A,max ( n n ) ; when, n n ; when, n n (.4) Thus, if a cone of light is incident on one end of fiber, it will be guided through it provided the semi angle of the cone is less than A,max. This angle is a measure of the light gathering power of the fiber and as such, one defines the numerical aperture (NA) of the fiber the following equation NA ( n n ) n ( ) (.5) Sardar Vallabhbhai National Institute of Technology, Surat Page 6

15 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding where, n n n n (.6) n n Also n nthis is indeed the case for all practical optical fibers..4 Types of optical fiber Optical fibers are characterized by their structure and by their properties of transmission. Basically, optical fibers are classified into two types. The first type is single mode fibers. The second type is multimode fibers. As each name implies, optical fibers are classified by the number of modes that propagate along the fiber. As previously explained, the structure of the fiber can permit or restrict modes from propagating in a fiber. The basic structural difference is the core size. Single mode fibers are manufactured with the same materials as multimode fibers. Single mode fibers are also manufactured by following the same fabrication process as multimode fibers. Fig..4 Types of optical fiber [] Sardar Vallabhbhai National Institute of Technology, Surat Page 7

16 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding.4. Single Mode Fibers The core size of single mode fibers is small. The core size (diameter) is typically around 6 to m. A fiber core of this size allows only the fundamental or lowest order mode to propagate around a 3 nanometer (nm) wavelength. Single mode fibers propagate only one mode, because the core size approaches the operational wavelength. The value of the normalized frequency parameter (V) relates core size with mode propagation. In single mode fibers, V is less than or equal to.45. When V is.45, single mode fibers propagate the fundamental mode down the fiber core, while high-order modes are lost in the cladding. For low V values, most of the power is propagated in the cladding material. Power transmitted by the cladding is easily lost at fiber bends. The value of V should remain near the.45 level. V a n n an (.7) ( ) ( ) where V is known as waveguide parameter, V number or V parameter. Practical single mode fibers have varying from.% to.5% and typical core diameters in the range 6 m. Single mode fibers have a lower signal loss and a higher information capacity (bandwidth) than multimode fibers. Single mode fibers are capable of transferring higher amounts of data due to low fiber dispersion. Basically, dispersion is the spreading of light as light propagates along a fiber. Dispersion mechanisms in single mode fibers are discussed in more detail later in this chapter. Signal loss depends on the operational wavelength. In single mode fibers, the wavelength can increase or decrease the losses caused by fiber bending. Single mode fibers operating at wavelengths larger than the cutoff wavelength lose more power at fiber bends. They lose power because light radiates into the cladding, which is lost at fiber bends. In general, single mode fibers are considered to be low-loss fibers, which increase system bandwidth and length. Sardar Vallabhbhai National Institute of Technology, Surat Page 8

17 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding.4. Multimode Fibers As their name implies, multimode fibers propagate more than one mode. Multimode fibers can propagate over modes. The number of modes propagated depends on the core size and numerical aperture (NA). As the core size and NA increase, the number of modes increases. Typical values of fiber core size and NA are 5 to m and. to.9, respectively. A large core size and a higher NA have several advantages. Light is launched into a multimode fiber with more ease. The higher NA and the larger core size make it easier to make fiber connections. During fiber splicing, core-to-core alignment becomes less critical. Another advantage is that multimode fibers permit the use of light-emitting diodes (LEDs). Single mode fibers typically must use laser diodes. LEDs are cheaper, less complex, and last longer. LEDs are preferred for most applications. Multimode fibers also have some disadvantages. As the number of modes increases, the effect of modal dispersion increases. Modal dispersion (intermodal dispersion) means that modes arrive at the fiber end at slightly different times. This time difference causes the light pulse to spread. Modal dispersion affects system bandwidth. Fiber manufacturers adjust the core diameter, NA, and index profile properties of multimode fibers to maximize system bandwidth..5 Mode theory for circular optical waveguide 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 Sardar Vallabhbhai National Institute of Technology, Surat Page 9

18 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding describe the propagation of light along a fiber. A set of guided electromagnetic waves is called the modes of the fiber [3, 4, 5]. 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. 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 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 of the fiber. The fundamental mode can never be cut off. The wavelength that prevents the next higher mode from propagating is called the cutoff wavelength of the fiber. An optical fiber that operates above the cutoff wavelength (at a longer wavelength) is called a single mode fiber. An optical fiber that operates below the cutoff wavelength is called a multimode fiber. In a fiber, the propagation constant of a plane wave is a function of the wave's wavelength and mode. The change in the propagation constant for different waves is called dispersion. The change in the propagation constant for different wavelengths is called chromatic dispersion. The change in propagation constant for different modes is called modal dispersion. Maxwell's equations describe electromagnetic waves or modes as having two components. The two components are the electric field, E(x, y, z), and the magnetic field, H(x, y, z). The electric field, E, and the magnetic field, H, are at right angles to each other. Modes traveling in an optical fiber are said to be transverse. The transverse modes, shown in Fig..5, propagate along the axis of the fiber. The mode field patterns shown in Fig..5 are said to be transverse electric (TE). In TE modes, the electric field is perpendicular to the direction of propagation. The magnetic field is in the direction of propagation. Another type of transverse mode is the transverse magnetic (TM) mode. TM modes are opposite to TE modes. In TM modes, the magnetic field is perpendicular to the direction of propagation. The electric field is in the direction of propagation. Fig..5 shows only TE modes. Sardar Vallabhbhai National Institute of Technology, Surat Page

19 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Fig..5 Electric field distribution for several of lower - order guided modes [3] The TE mode field patterns shown in Fig..5 indicate the order of each mode. The order of each mode is indicated by the number of field maxima within the core of the fiber. For example, TE has one field maxima. The electric field is a maximum at the center of the waveguide and decays toward the core-cladding boundary. TE is considered the fundamental mode or the lowest order standing wave. As the number of field maxima increases, the order of the mode is higher. Generally, modes with more than a few (5-) field maxima are referred to as high-order modes [3]. The order of the mode is also determined by the angle the wavefront makes with the axis of the fiber. Fig..6 illustrates light rays as they travel down the fiber. These light rays indicate the direction of the wavefronts. High-order modes cross the axis of the fiber at steeper angles. Low-order and high-order modes are shown in Fig..6. Fig..6 Low-order and high-order modes [3] Sardar Vallabhbhai National Institute of Technology, Surat Page

20 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Notice that the modes are not confined to the core of the fiber. The modes extend partially into the cladding material. Low-order modes penetrate the cladding only slightly. In loworder modes, the electric and magnetic fields are concentrated near the center of the fiber. However, high-order modes penetrate further into the cladding material. In high-order modes, the electrical and magnetic fields are distributed more toward the outer edges of the fiber. This penetration of low-order and high-order modes into the cladding region indicates that some portion is refracted out of the core. The refracted modes may become trapped in the cladding due to the dimension of the cladding region. The modes trapped in the cladding region are called cladding modes. As the core and the cladding modes travel along the fiber, mode coupling occurs. Mode coupling is the exchange of power between two modes. Mode coupling to the cladding results in the loss of power from the core modes..5. Maxwell s Equations To analyze the optical waveguide we need to consider Maxwell s equations that give the relationships between the electric and magnetic fields. Assuming a linear dielectric material having no currents and free charges, these equations take the form B XE (.8a) t D XH (.8b) t. D (.8c). B (.8d) Where D = E and B = µh. The parameter is the permittivity (or dielectric constant) and µ is the permeability of the medium. A relationship defining the wave phenomena of the electromagnetic fields can be derived from Maxwell s equations [4]. Taking the curl of Eq. (.8a) and making use of Eq. (.8b) yields Sardar Vallabhbhai National Institute of Technology, Surat Page

21 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding E X ( X E) ( X H) t t (.9a) Using the vector identity X ( X E) (. E) E And using Eq. (.8c), Eq. (.9a) becomes E t E (.9b) Similarly, by taking the curl of Eq. (.8b), it can be shown that H t H (.9c) Equations (.9b) and (.9c) are the standard wave equations [4, 5]..5. Waveguide Equations Consider electromagnetic waves propagating along the cylindrical fiber shown in Fig..7. For this fiber, a cylindrical coordinates system ( r,, z) is defined with the z axis lying of the waveguide. If the electromagnetic waves are to propagate along the z axis, they will have a functional dependence of the form E E (, ) j t z r e ( ) (.a) H H (, ) j t z r e ( ) (.b) Which are harmonic in time t and coordinate z. The parameter β is the z component of the propagation vector and will be determined by the boundary conditions on the electromagnetic fields at the core cladding interface. An optical mode refers to a specific solution of the wave equation that satisfies the appropriate boundary conditions and has the property that its spatial distribution does not change with propagation [4, 5]. Sardar Vallabhbhai National Institute of Technology, Surat Page 3

22 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding When Eq. (.a) and (.b) are substituted into Maxwell s curl equations, we have, from Eq. (.8a) Ez jr E j H r r r E r z j Er jh r E re jh r z (.a) (.b) (.c) And, from Eq. (.8b), H z jr H j E r H r z j Hr j E r r r H rh j E r z (.a) (.b) (.c) Fig..7 Cylindrical coordinate system used for analyzing electromagnetic wave propagationin an optical fiber [4] By eliminating variables these equations can be rewritten such that, when E z and H z are known, the remaining transverse components E r, E ϕ, H r and H ϕ can be determined. For Sardar Vallabhbhai National Institute of Technology, Surat Page 4

23 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding example E ϕ or H r can be eliminated from Eq. (.a) and Eq. (.b) so that the component H ϕ or E r, respectively, can be found in terms of E z or H z. These yields E r j E z q r r Hz (.3a) E j E H z z q r r (.3b) H r j H z q r r Ez (.3c) H j H E z z q r r (.3d) where q = - β = k - β. Substitution of Eq. (.3c) and Eq. (.3d) into Eq. (.c) gives the wave equation [4-6] in cylindrical coordinates, Ez Ez Ez Ez qe z r r r r z (.4) And substitution of Eq. (.3a) and Eq. (.3b) into Eq. (.c) gives, H z H z H z H z qhz r r r r z (.5) It is interesting to note that Eq. (.4) and Eq. (.5) each contain either only E z or only H z. This implies that the longitudinal components of E and H are uncoupled and can be chosen arbitrary provided that they satisfy Eq. (.4) and Eq. (.5). However the coupling of E z and H z is required by the boundary conditions of the electromagnetic field components. If the boundary conditions do not lead to coupling between the field components, mode solutions can be obtained in which either E z = or H z =. When E z = the modes are called transverse electric or TE modes, and when H z = they are called transverse magnetic or TM modes. Hybrid mode exists if both H z and E z are nonzero. These are designed as HE or EH a mode, depending on whether Hz and Ez, respectively, can makes a larger contribution to the transverse field. The fact that the hybrid modes are Sardar Vallabhbhai National Institute of Technology, Surat Page 5

24 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding present in optical waveguides makes their analysis more complex than in simpler case of hollow metallic waveguides where only TE and TM modes are found..5.3 Wave Equations for Step Index Fibers We can use Eq. (.4) and Eq. (.5) to find the guided modes in a step index fiber. A standard mathematical procedure for solving equations such as Eq. (.4) is to use the separation of variables method, which assumes a solution of the form E z = AF (r) F (ϕ) F 3 (z) F 4 (t) (.6) Assume the time- and z- dependent factors are given by F 3 (z)f 4 (t) =e j(ωt βz) (.7) Since the wave is sinusoidal in time and propagates in the z direction. Also because of the symmetry of the waveguide, each field component must not change when the coordinate ϕ is increased by π. We thus assume a periodic function of the form ( ) j F e (.8) Thus constant can be positive or negative, but it must be an integer since the field must be periodic in ϕ with a period of π. Now substituting Eq. (.7) and Eq. (.8) into Eq. (.6), the wave equation for E z [Eq. (.4)] becomes F F q F r r r r (.9) This is well - known differential equation for Bessel functions [5]. An exactly identical equation can be derived for H z. Consider a homogeneous core of refractive index n and radius a, which is surrounded by an infinite cladding of index n. The reason for assuming an infinitely thick cladding is that the guided modes in the core have exponentially decaying fields outside the core and these must have insignificant values at the outer boundary of the cladding. In practice, Sardar Vallabhbhai National Institute of Technology, Surat Page 6

25 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding optical fiber designed with claddings that are sufficiently thick so that the guided mode field does not reach the outer boundary of the cladding. To get an idea of field patterns, the electric field distributions for several of the lower guided modes in a symmetrical slab were shown in Fig..6. The fields vary harmonically in the guiding region of refractive index n and decay exponentially outside of this region. Eq. (.9) must now be solved for the regions inside and outside the core. For the inside region the solutions for the guided modes must remain finite as r, whereas on the outside the solutions must decay to zero as r. As Eq. (.9) is standard differential equation for Bessel function, so we must take solution in the form of Bessel function. But first we have to choose appropriate Bessel function for solution of Eq. (.9) We have a variety of solutions to the Bessel s equation depending upon the parameters and q. is an integer and a positive quantity. Depending upon the choice of q i.e., a) real, b) imaginary, c) complex, we get different solutions to the Bessel s equation. So to choose the proper solution, let us now look at the plot of the Bessel functions for various possibilities of q (argument). There are three different types of Bessel functions depending upon the nature of q. Let us now look at the plot of the Bessel functions for various possibilities of q (argument). There are three different types of Bessel functions depending upon the nature of q. If q is real then the solutions are J qr Bessel functions of first kind Y ( qr) Bessel functions of second kind The quantity is called the order of the function and ( qr ) is called the argument of the function. Plots of the two functions as a function of their arguments are shown in the Fig..8 and Fig..9. Sardar Vallabhbhai National Institute of Technology, Surat Page 7

26 Y v (qr) J v (qr) Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding J (qr) J (qr) J (qr) qr Fig..8 Bessel functions of first kind [5].8 Y (qr) Y (qr) Y (qr) qr Fig..9 Bessel functions of second kind [5] We can see from Fig..8 that except J, all the other Bessel functions of first kind go to zero as the argument goes to zero. Only J approaches as its argument approaches Sardar Vallabhbhai National Institute of Technology, Surat Page 8

27 I v (qr/j) Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding zero. All Bessel functions of first kind have oscillatory behavior and their amplitude slowly decreases as the argument increases. Fig..9 shows the behavior of the Neumann function as a function of its argument, qr. The important thing to note is, the Bessel functions of first kind J are finite for all values of the argument, whereas the Bessel functions of second kind are finite for all values of argument except zero. When the argument tends to zero, the Bessel functions of second kind tend to. If q is imaginary, we get solutions of the Bessel s equation as I qr / j Modified Bessel functions of first kind K ( qr / j) Modified Bessel functions of second kind Since q is imaginary, ( qr / j ) is a real quantity. So the argument of the modified Bessel functions is real. 3 5 I (qr) I (qr) I (qr) qr/j Fig.. Modified Bessel functions of first kind [5] Sardar Vallabhbhai National Institute of Technology, Surat Page 9

28 K v (qr/j) Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding.8 K (qr) K (qr) K (qr) qr/j Fig.. Modified Bessel functions of second kind [5] The modified Bessel functions are shown in the Fig.. and Fig... The I functions are monotonically increasing functions of qr / j, and K functions are monotonically decreasing functions of ( qr / j ). If q is complex, then the solutions are H () qr Hankel functions of first kind H () qr Hankel functions of second kind But as our medium is lossless, in this case q can either be real or imaginary, so no need to study the case when the q is complex. Now q, where is the propagation constant of the wave along the z direction. If we assume the situation is lossless i.e. when the wave travels in the z direction, its amplitude does not change as a function of z, then should be a real quantity. If become imaginary, the function j z e becomes an exponentially decaying function, and there is no wave propagation. For wave propagation inside an optical fiber Sardar Vallabhbhai National Institute of Technology, Surat Page

29 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding we assume that the material is lossless. Then the dielectric constant is a real quantity. This makes a real quantity. Also for a propagating mode is a real quantity. Hence, q is also a real quantity albeit it can be positive or negative. In other words, q can be real or imaginary depending upon whether is greater or lesser than. For a lossless case, we have a solution which is a linear combination of Bessel functions of first kind and modified Bessel functions of second kind. As far as guided wave propagation is concerned, the fields should have oscillatory behavior inside the core, and in cladding the field must decay monotonically. Therefore it is obvious that inside the core the Modified Bessel function is not the proper solution. Only Bessel function of first kind could be solutions inside the core. Let us now re-look at the two functions, Bessel functions of first kind (Fig..8) and Bessel functions of second kind (Fig..9), and make following observations. Bessel functions of first kind: The functions J ur are finite for all values of r. Bessel functions of second kind: The functions Y ( ur) start from at r and have finite value for all other values of r. For the core r represents the axis of fiber. Therefore if a Bessel function of second kind is chosen as a solution, the field strength would be at the axis of the fiber which is inconsistent with the physical conditions. The fields must be finite all over the cross section of the core. So the Bessel function of second kind cannot be the solution if r point is included in the region under consideration. Therefore we can conclude that only J ur modal fields inside the core of an optical fiber. is the appropriate solution for the Let us now look at the modified Bessel s functions, as shown in figures. For modified Bessel s functions of the st kind (Fig..), as r increases, that is, as we move away from the axis of the fiber the field monotonically increases and when r field goes to infinity. Since the energy source is inside the core, the fields cannot grow indefinitely Sardar Vallabhbhai National Institute of Technology, Surat Page

30 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding away from the core. The only acceptable situation is that the field decays away from the core i.e., for larger values of r. This behavior is correctly given by the Modified Bessel function of second kind, K ( wr) (Fig..). So we can conclude that the modified Bessel function of st kind I ( wr) is not appropriate solution in the cladding. The correct solution would be only Modified Bessel function of nd kind, K ( wr). In all then, the fields inside the core are given by J ( ur) and in the cladding are given by K ( wr). Thus for r < a, the solutions are Bessel function of first kind of order. For these functions we use the common designation J ur. Here, n u k with k. The expressions for E z and H z inside the core are, when (r < a) Ez AJ ( ua) e j j z jt (.) z H BJ ua e j j zj t (.) Outside of the core the solutions of Eq..9 are given by modified Bessel functions of the n second kind, K ( wa), where w k with k. The expressions for E z and H z outside the core are, when (r > a) z E CK wa e z j j zj t (.) H DK wa e j j zj t (.3) where A, B, C, D are arbitrary constants which are to be evaluated from the boundary conditions. Also J ua and K ( wa) are the Bessel functions. For a guided mode, the propagation constant lies between two limits and. If n k k k n k then a field distribution is generated which will has an Sardar Vallabhbhai National Institute of Technology, Surat Page

31 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding oscillatory behavior in the core and a decaying behavior in the cladding. The energy then is propagated along fiber without any loss. Where constant. k is free space propagation.5.4 Boundary Conditions The solutions for β must be determined from the boundary conditions [4, 7]. The boundary conditions requires that the tangential components E and Ez of E inside and outside of the dielectric interface at r = a must be the same, and similarly for the tangential components H and H. z The boundary conditions are then given as: At r a, E E H H E (.4a) E (.4b) z z H (.4c) H (.4d) z z The boundary conditions give four equations in terms of arbitrary constants, A, B, C, D and the modal phase constant..5.5 Modal Equation Consider the first tangential components of E, for the z component we have, from Eq. (.) at inner core cladding boundary (E = E z ) and from Eq. (.) at the outside of the boundary (E = E z ), that Ez Ez AJ( ua) CK( wa) (.5) The ϕ component is found from Eq. (.3b) inside the core the factor q is given by q u k (.6) Sardar Vallabhbhai National Institute of Technology, Surat Page 3

32 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding n where k, while outside the core q w k (.7) n with k. Substituting Eq. (.) and Eq. (.) into Eq. (.3b) to find E ϕ. Similarly, using Eq. (.) and Eq. (.3) to determine E ϕ, yields, at r = a, j j E E A J ( ua) B uj '( ua) u a j j C K ( wa) D wk '( wa) w a (.8) where the prime indicates differentiation with respect to the argument. Similarly, for tangential components of H it is readily shown that, at r = a, Hz Hz BJ( ua) DK( wa) (.9) j j H H B J ( ua) A uj '( ua) u a j j D K ( wa) C wk '( wa) w a (.3) Eq. (.5), Eq. (.8), Eq. (.9) and Eq. (.3) are set of four equations with four unknown coefficients, A, B, C and D. A solution to these equations exists only if the determinant of these coefficients is zero, that is, A B C D A B C D A3 B3 C3 D3 A4 B4 C4 D4 (.3) where, Sardar Vallabhbhai National Institute of Technology, Surat Page 4

33 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding A to A4 are coefficients of A in Eq. (.5), Eq. (.8), Eq. (.9) and Eq. (.3). B to B4 are coefficients of B in Eq. (.5), Eq. (.8), Eq. (.9) and Eq. (.3). C to C4 are coefficients of C in Eq. (.5), Eq. (.8), Eq. (.9) and Eq. (.3). D to D4 are coefficients of D in Eq. (.5), Eq. (.8), Eq. (.9) and Eq. (.3). Also, A J ( ua) A J ( ua) au A3 j A J ua u 4 '( ) B j B J '( ua) u B3 J ( ua) B4 J ( ua) au C K ( wa) C K ( wa) aw C3 j C K wa w 4 '( ) D j D K '( wa) w D3 K ( wa) D4 K ( wa) aw (.3a) (.3b) (.3c) (.3d) Evaluation of the above determinant yields the following eigenvalue equation for β. (.33) Sardar Vallabhbhai National Institute of Technology, Surat Page 5

34 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding J ua K wa J ua K wa uj ua wk wa uj ua wk wa a u w ' ' ' ' k k Eq. (.33) is called characteristic equation [4, 5]. The characteristic equation contains three unknowns namely uw,,. So using the Eq. (.6), Eq. (.7) & Eq. (.33) we can find the modal propagation constant..5.6 Modes in Step Index fibers Eq. (.33) is called the characteristic equation. Its contain J - type Bessel functions. The J - type Bessel functions are similar to harmonic functions since they exhibit oscillatory behavior for real k, as is the case for sinusoidal functions. Because of the oscillatory behavior of J, there will be m roots of Eq. (.33) for a given value. These roots are designated by, and the corresponding modes [4, 5] are either TE, TM, EH or HE m m m. If we take z m m H, all field components are expressed in terms of Ez and whatever fields we get, they do not have any magnetic field component in the direction of propagation. We call this mode the Transverse Magnetic mode (TM mode). Similarly if Ez, the mode is called the Transverse Electric mode (TE mode). If both the longitudinal components of the fields ( E and z H z ) are non-zero then we call the mode the Hybrid mode. This mode is a combination of TE and TM modes. For a hybrid mode, if we calculate the contribution by or Ez and H to the transverse fields, one of them i.e. E z H z would dominate. Depending upon which of them contributes more, we can subclassify the Hybrid modes. If Ez Dominates EH mode If H z DominatesHE mode. Each of the above modes is characterized by two indices, and m (solution number). The mode are therefore designated as TEm, TM m & EH, HE m m. z For the dielectric fiber waveguide, all modes are hybrid modes except those which. When the right hand side of Eq. (.33) vanishes and two different equations result. These are J ' ua K' wa uj ua wk wa (.34a) Sardar Vallabhbhai National Institute of Technology, Surat Page 6

35 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding k J ' ua K ' wa k uj ua wk wa (.34b) Using recurrence relation for the Bessel function, we have J '( x) J ( x) and K '( x) K ( x). Put these recurrence relation for the Bessel function in Eq. (.34a) and Eq. (.34b), we get J ua K wa uj ua wk wa (.35a) Which corresponds to TE m modes (E z = ), and k J ua K wa k uj ua wk wa (.35b) Which corresponds to TM m modes (H z = ). When the situation is more complex and numerical methods are needed to solve Eq. (.33) exactly..5.7 Cutoff conditions for fiber modes The cutoff condition [4] is the point at which a mode is no longer bound to the core region. So its field no longer decays on the outside of the core. The cutoffs for the various modes can be found by solving Eq. (.33) in the limit w. This is, in general, fairly complex, so that only the results, which are listed in Table.. Sardar Vallabhbhai National Institute of Technology, Surat Page 7

36 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Table. Cutoff conditions for some lower order modes [4] Mode Cutoff condition TE m, TM J ( ua) m HE m, EH J ( ua) m J ( ua) EH m HE m n n ua J( ua) J( ua) The permissible range of β for bound solutions is therefore n k k k n k (.36) Where k / is the free space propagation constant. An important parameter connected with the cutoff condition is the normalized frequency V (also called the V number or V parameter) [4] defined by a a ( ) ( ) V u w a n n NA (.37) which is dimensionless number that determines how many modes a fiber can support. The number of modes that can exist in a wave guide as a function of V may be conveniently represented in terms of a normalized propagation constant b [4] defined by ( / k) n aw b V n n (.38) A plot of b as function of V is shown in Fig..3 for few of the lower order modes. Sardar Vallabhbhai National Institute of Technology, Surat Page 8

37 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Fig.. Plots of the propagation constant b as a function of V for a lower order modes [5] This figure shows that each mode can exist only for values of V that exceed a certain limiting value. The modes are cutoff when β/k = n. The HE mode has no cutoff and ceases to exist only when the core diameter is zero. This is the principle on which the single mode fiber is based. By appropriately choosing a, n and n so that a / V ( n n ).45 (.39) Which is the value at which the lowest order Bessel function J =, all modes except the HE mode are cutoff. Sardar Vallabhbhai National Institute of Technology, Surat Page 9

38 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Chapter 3 Analysis of optical waveguide with helical winding 3. Helix A helix [8] is a type of space curve, i.e. a smooth curve in three - dimensional space. It is characterized by the fact that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases (Fig. 3.). Fig. 3. Example of helix as coil springs [9] 3.. Types of Helix Helices can be either right-handed or left-handed. With the line of sight along the helix's axis, if a clockwise screwing motion moves the helix away from the observer, then it is called a right-handed helix (Fig. 3.); if towards the observer then it is a left-handed helix. Handedness (or chirality) is a property of the helix, not of the perspective: a right-handed helix cannot be turned or flipped to look like a left-handed one unless it is viewed in a Sardar Vallabhbhai National Institute of Technology, Surat Page 3

39 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding mirror, and vice versa. Most hardware screws are right-handed helices. Fig. 3. Helix (A) Right handed, (B) Left handed [] The pitch of a helix is the width of one complete helix turn, measured parallel to the axis of the helix. A double helix consists of two (typically congruent) helices with the same axis, differing by a translation along the axis, which may or may not measure half the pitch. A conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of the angle indicating direction from the axis. An example is the Corkscrew [] roller coaster at Cedar Point amusement park. A circular helix has constant band curvature and constant torsion. A curve is called a general helix or cylindrical helix if its tangent makes a constant angle with a fixed line in space. A curve is a general helix if and only if the ratio of curvature to torsion [] is constant. A sheath helix [4] can be approximated by winding a very thin conducting wire around the cylindrical surface so that the spacing between the adjacent windings is very small and yet they are insulated from each other as shown in Fig Sardar Vallabhbhai National Institute of Technology, Surat Page 3

40 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding 3. Circular Optical 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 Fig Fig 3.3 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 [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,, z) [4] with the z axis being the direction of propagation. 3.3 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. Sardar Vallabhbhai National Institute of Technology, Surat Page 3

41 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding E sin z E cos E sin (3.a) z E cos z z (3.b) E E cos E E sin (3.c) z z H H sin H H cos (3.d) 3.4 Model 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 conditions must be satisfied n k k 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) Ez AJ ( ua) e j j z jt (3.a) z H BJ ua e j j zj t (3.b) The expressions for E z and H z outside the core are, when (r > a) z E CK wa e z j j zj t (3.c) H DK wa e j j zj t (3.d) where A, B, C, D are arbitrary constants which are to be evaluated from the boundary conditions. Also J ua and K ( wa) are the Bessel functions. For a guided mode, the propagation constant lies between two limits and. If n k k 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 constant. k is free space propagation Sardar Vallabhbhai National Institute of Technology, Surat Page 33

42 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding 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 E j AJ ( ua) ubj '( ua) e u a j H j BJ ( ua) uaj '( ua) e u a j j zjt j j zjt (3.3a) (3.3b) The expressions for E ϕ and H ϕ outside the core are, when (r > a) j E j CK ( wa) wdk '( wa) e w a j H j DK ( wa) wck '( wa) e w a j j zjt j j zjt (3.4a) (3.4b) 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 cos BJ '( ua) cos u a u (3.4a) j CK( wa) sin cos DK '( wa) cos w a w (3.4b) j AJ( ua) cos sin BJ '( ua) sin u a u j CK( wa) cos sin DK '( wa) sin w a w (3.4c) j AJ '( ua) cos BJ( ua) sin cos u u a j CK wa DK wa w w a '( ) cos ( ) sin cos (3.4d) Sardar Vallabhbhai National Institute of Technology, Surat Page 34

43 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Eq. (3.4a), Eq. (3.4b), Eq. (3.4c) and Eq. (3.4d) 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 CCC3C4 D D D3 D4 (3.5) where, A J ( ua) sin cos ua j A J '( ua) cos u A3 A4 B B B3 K ( wa) sin cos wa j B4 K '( wa) cos w C J ( ua) cos sin ua j C J '( ua) sin u C3 K ( wa) cos sin wa j C4 K '( wa) sin w (3.6a) (3.6b) (3.6c) Sardar Vallabhbhai National Institute of Technology, Surat Page 35

44 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding j D J '( ua) cos u D J ( ua) sin cos ua j D3 K '( wa) cos w D4 K ( wa) sin cos wa (3.6d) Evaluation of the above determinant yields the following eigenvalue equation for β. The determinant can be solve as A A A3 A4 B B B3B4 CCC3C4 D D D3D4 B B3B4 B B3B4 B B B4 B B B3 A CC3C4 A CC3C4 A3 CCC4 A4 CC3C3 (3.7) D D3D4 D D3D4 D D D4 D D D3 Using Eq. 3.6, we get A A A3 A4 A A B B B3B4 B3B4 CCC3C4 CC C3C4 D D D3D4 D D D3D4 A A A3 A4 B3B4 B3B4 B B B3B4 A CC3C4 A CC3C4 CCC3C4 D D3D4 D D3D4 D D D3D4 A A A3 A4 B B B3B4 A B3CD4 C4D B4( CD3 C3D) CCC3C4 D D D3D4 A B3 CD4 C4D B4( CD3 C3D) Sardar Vallabhbhai National Institute of Technology, Surat Page 36

45 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding A A A3 A4 B B B3B4 AB 3C D4 AB 3C 4D AB 4CD3 AB 4C3D CCC3C4 D D D3D4 AB3CD 4 AB3C 4D AB4CD3 AB4C3D (3.8) After eliminating unknown constants from Eq. (3.8) and Eq. (3.6), we get the following characteristic equation. J ( ) k ua J '( ua) u sin cos cos J '( ua ) u a u J ( ua ) K ( ) k wa K '( wa) w K '( wa ) w a w K ( wa ) sin cos cos (3.8) Eq. (3.9) is standard characteristic equation, and is used for model dispersion properties and model cutoff conditions. Sardar Vallabhbhai National Institute of Technology, Surat Page 37

46 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Chapter 4 Results & Discussion It is now possible to interpret the characteristic equation (Eq. 4.) in numerical terms. 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 ) K ( ) k wa K '( wa) w K '( wa ) w a w K ( wa ) sin cos cos (4.) ( / k) n aw b V n n (4.) a a ( ) ( ) V u w a n n NA (4.3) where b & V are known as normalization propagation constant & normalized frequency parameter respectively. We make some simple calculations based on Eq. 4. and Eq We choose n =.5, n =.46 and λ =.55µm. We take for simplicity, but the result is valid for any value of. 4. Dispersion characteristics 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. Sardar Vallabhbhai National Institute of Technology, Surat Page 38

47 b Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding 4.. Dispersion characteristics at pitch angle ψ = To obtain the dispersion curve for this case put ψ = in Eq. 4.. We now get J ( ua) k J '( ua) K ( wa) k K '( wa) u J '( ua) u a u J ( ua) K '( wa) w a w K ( wa) w (4.4) Dispersion curve corresponding to Eq. 4.4 is shown in Fig V Fig. 4. Dispersion Curve for pitch angle ψ = Sardar Vallabhbhai National Institute of Technology, Surat Page 39

48 b Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding 4.. Dispersion characteristics at pitch angle ψ = 3 To obtain the dispersion curve for this case put ψ = 3 in Eq. 4.. We now get 3 3 k J ( ua) J '( ua) u J '( ua ) u a 4 u J ( ua ) 3 3 k K ( wa) K '( wa) w K '( wa ) w a 4 w K ( wa ) (4.5) Dispersion curve corresponding to Eq. 4.5 is shown in Fig V Fig. 4. Dispersion Curve for pitch angle ψ = 3 Sardar Vallabhbhai National Institute of Technology, Surat Page 4

49 b Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding 4..3 Dispersion characteristics at pitch angle ψ = 45 To obtain the dispersion curve for this case put ψ = 45 in Eq. 4.. We now get J ( ua) k J '( ua) u J '( ua ) u a u J ( ua ) K ( wa) k K '( wa) w K '( wa ) w a w K ( wa ) (4.6) Dispersion curve corresponding to Eq. 4.6 is shown in Fig V Fig. 4.3 Dispersion Curve for pitch angle ψ = 45 Sardar Vallabhbhai National Institute of Technology, Surat Page 4

50 b Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding 4..4 Dispersion characteristics at pitch angle ψ = 6 To obtain the dispersion curve for this case put ψ = 6 in Eq. 4.. We now get 3 k J ( ua) J '( ua) u J '( ua ) u a 4 u J ( ua ) 3 k K ( wa) K '( wa) w K '( wa ) w a 4 w K ( wa ) (4.7) Dispersion curve corresponding to Eq. 4.7 is shown in Fig V Fig. 4.4 Dispersion Curve for pitch angle ψ = 6 Sardar Vallabhbhai National Institute of Technology, Surat Page 4

51 b Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding 4..5 Dispersion characteristics at pitch angle ψ = 9 To obtain the dispersion curve for this case put ψ = 9 in Eq. 4.. We now get J ( ua) k J '( ua) K( wa) k K '( wa) u w (4.8) J '( ua) u J ( ua) K '( wa) w K ( wa) Dispersion curve corresponding to Eq. 4.8 is shown in Fig V Fig. 4.5 Dispersion Curve for pitch angle ψ = 9 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. Sardar Vallabhbhai National Institute of Technology, Surat Page 43

52 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding An anomalous feature in the dispersion curves is observable for ψ = 3, 45 and 6 for this type of waveguide (Fig. 4., Fig. 4.3 and Fig. 4.4) 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. 4. Dependence of cutoff values V c We now come to table 4.. we note particularly that the dependence of the cutoff V value (V c ) on ψ is such that as ψ is increased there is a drastic fall in V 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 (Fig. 4.6). Table 4. Cutoff V c values for some lower order modes ψ V c V c V c V c V c V c V c V c V c Sardar Vallabhbhai National Institute of Technology, Surat Page 44

53 Vc Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Angle in Degree Fig. 4.6 Dependence of cutoff values V c on the pitch angle ψ 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. Sardar Vallabhbhai National Institute of Technology, Surat Page 45

54 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Chapter 5 Conclusion & Future Work 5. Conclusion From the above results (Fig. 4., Fig. 4., Fig. 4.3, Fig. 4.4 and Fig. 4.5) 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 = 3, 45 and 6 for this type of waveguide (Fig. 4., Fig. 4.3 and Fig. 4.4) 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. 5. Future Work In present work right handed helical winding is applied, left handed helical winding can be applied and the effects on dispersion characteristics can be studied. Also in addition left handed and right handed helical winding can be applied simultaneously to the fiber, Sardar Vallabhbhai National Institute of Technology, Surat Page 46

55 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding and result can be compared with present work. We have not studied polarization in this work; this implies that the helical winding possible introduces important changes in polarization properties. This can be considered for future work. Present work consists the analysis and description of model characteristics, considering, for simplicity, although the result can be analyzed for any value of, so in future work more values of can be consider for more results. We can also consider other type of fiber waveguides like, elliptical, triangular and square and study the model characteristics for the mentioned waveguides and results can be compare. Optical waveguides have their importance in versatile applications, viz. communication purposes, sensing technology as well as integrated optical devices, so this type of waveguides can be used for the above applications, this will surely improve the efficiency and operation of the applied area. Sardar Vallabhbhai National Institute of Technology, Surat Page 47

56 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding References [] Jia Ming-Liu, Photonic Devices, Cambridge University Press, UK, 5. [] [3] 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, 8. [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,. [5] 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, 995. [6] 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, 999. [7] Ajoy Ghatak and K. Thyagarajan, Optical Electronics Cambridge University Press, India, 8. [8] vg [9] [] [] [] [3] [4] Keiser G., Optical Fiber Communications, Chap., 3 rd edition McGraw- Hill, Singapore,. [5] [6] ical%communication/course_home-m3.html Sardar Vallabhbhai National Institute of Technology, Surat Page 48

57 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding [7] Kumar, D. and O. N. Singh II, Modal characteristic equation and dispersion curves for an elliptical step index fiber with a conducting helical winding on the core cladding boundary An analytical study IEEE, Journal of Light Wave Technology, Vol., No.8, 46-44, USA, August. [8] [9] [] [] [] [3] 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., ,. [4] [5] Govind P. Agrawal, Fiber Optic Communication Systems, 3 rd edition A John Wiley & Sons, Inc., Publication, New York,. Sardar Vallabhbhai National Institute of Technology, Surat Page 49

58 Dispersion and Cutoff Characteristics of Circular Optical Waveguide with Helical Winding Publication [] Ajay Kumar Gautam, Dr. Vivekanand Mishra and Prof. B.R. Taunk, Dispersion Characteristic of Optical Waveguide with Helical Winding for Different Pitch Angle, National Conference on Electronics, Communication & Instrumentation, CSE Jhansi, e Manthan,, 7 73, nd - 3 rd April. Publication (Under Communication) [] Ajay Kumar Gautam, Dr. Vivekanand Mishra and Prof. B.R. Taunk, Modal Dispersion Characteristics of Circular Optical Waveguide with Helical Winding - A Comparison for Different Pitch Angles, International Conference on Advances in Computing and Communication, NIT Hamirpur,. [3] V. Mishra, A. K. Gautam, B. R. Taunk, Effect of Helical Pitch Angles on Dispersion Characteristics of Circular Optical Waveguide having Helical Windings on Core - Cladding Interface, International Journal for Laser Physics,. [4] V. Mishra, A. K. Gautam, B. R. Taunk, Dispersion & Cuttoff Characteristics of Circular Helically Cladded Optical Fiber International Journal on Engineering & Technology,. Sardar Vallabhbhai National Institute of Technology, Surat Page 5

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