Waveguide

12 Key excerpts on "Waveguide"

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  eBook - ePub

  Photonic Waveguides

  Theory and Applications

  • Azzedine Boudrioua(Author)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  Chapter 1

  Optical Waveguide Theory

  Optical Waveguides are structures with three layers controlling light confinement and propagation in a well defined direction inside the central layer (Figure 1.1 ).

  Figure 1.1.  Planar optical Waveguide

  Light confinement is carried out by successive total reflections on the two interface guides – substrate and guide – superstrate.

  Light propagation is governed by an interference phenomenon which occurs inside the guide between two waves; one of them undergoes two successive total reflections. For a better understanding of the guided wave propagation, we will recall the main principles of these two phenomena, total reflection and interference, inside a transparent plate with parallel faces.

  1.1. Principles of optics

  1.1.1. Total reflection phenomenon

  Let us consider an interface separating two mediums 1 and 2, which are dielectric, lossless, homogenous and isotropic with refractive indices n1 and n2 , respectively. An electromagnetic wave propagates from 1 to 2 with an angle of incidence θi related to the normal of the interface (Figure 1.2 ).

  Figure 1.2.  Reflection on an interface (medium 1/medium 2)

  The electric field of the incident wave is given by:

  [1.1] 

  [1.2] 

  is the wave vector in the vacuum (λ: wavelength in the vacuum) and Ei0 is the incident wave amplitude. The electric fields of the reflected and transmitted waves can be written:

  [1.3] 

  [1.4] 

  In addition, refraction law is given by:

  [1.5] 

  [1.6] 

  In the case of n1 > n2 , there is an incident angle

  θl

  , as:

  [1.7] 

  For

  θi

  >

  θl

  , the incident wave is totally reflected into medium 1 (total reflection) and the angle

  θt

  of the transmitted wave is complex [Bor 1999, War 1988]:

  [1.8] 

  From [1.4 ], the transmitted wave can be written as:

  [1.9] 

  This wave propagates in the Oz direction with an amplitude exponentially decreasing in the Ox direction. This is called an evanescent wave. Also, according to Fresnel’s formulae, the considered wave undergoes a phase shift compared to the incident wave, given by [Bor 1999]:

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  Principles of Optics for Engineers

  Diffraction and Modal Analysis

  • William S. C. Chang(Author)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

  They have a high index core surrounded by lower index cladding or substrate. The transverse dimensions of the core are comparable to or smaller than the optical wavelength. Guided electro- magnetic waves propagate in and around the core. A typical optical fiber and a typical channel Waveguide are illustrated in Figure 5.1. Guided-wave modes are solutions of homogeneous Maxwell’ s electromagnetic equa- tions in Waveguide structures that have a constant cross-section and infinite length in the direction of propagation. The modes of optical fibers and Waveguides are the focus of discussion in this chapter. Homogeneous solution means that these are the propagating electric and magnetic fields that satisfy the differential equations and the appropriate boundary conditions in the absence of any radiation source. What modes are excited is determined by the input radiation. Modal analyses are used in microwaves as well as optics. However, there are important differences between optical and microwave Waveguides. In microwaves, we usually have closed Waveguides inside metallic boundaries. Metals are considered to be perfect con- ductors at most microwave frequencies. Figure 5.2 illustrates a typical microwave rec- tangular Waveguide, which is surrounded by metallic walls. The boundary condition at the metal surface is that the tangential electric field is zero. Microwaves propagate within the metallic enclosure. In such closed structures, we have only a discrete set of Waveguide modes. At optical wavelengths, we avoid the use of metallic boundaries because of their strong absorption of radiation at optical frequencies. All optical Waveguides are open dielectric Waveguides. Two examples have been illustrated in Figures 5.1(a) and 5.1(b). There are at least three differences between microwave and optical Waveguide analysis. (1) The mathematics of finding the modes is more complex for open Waveguides.

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  Advances in Microwaves

  Volume 1

  • Leo Young(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  Guided waves occur only in the presence of inhomogeneities such as clearly definable boundaries, for example, highly reflecting metal surfaces or changes in refractive index. Unlike free waves, guided waves are made to follow the path of the Waveguide, and for this reason possess distinct advantages, though the waves are subjected to an attenuation caused by the Waveguide and con-sequently the components of the electromagnetic field propagate according to F = f(u,v)exp(-*z) (3) where u and ν are the coordinates in the transverse plane of the Waveguide f(u,v) describes the pattern of the mode, ζ is the axial coordinate, and α is the attenuation coefficient. In many cases the radiation field and guided waves cannot be properly separated [4] and then the usefulness of the above concepts is rather limited. Nevertheless we assume for the purposes of the present discussion that substantially pure guided waves characterized by exponential attenuation can be realized at optical frequencies, even though a free wave in the optical band would also be characterized by an exponential attenuation owing to the absorption in the medium, the atmosphere. The important point of distinction is that a guided wave will follow substantially the Waveguide axis despite any changes of direction, whereas a free wave spreads out in many directions and its course cannot be changed except by the introduction of special devices (such as mirrors) in the path of the rays. B . TYPES OF GUIDED PROPAGATION At the outset it is helpful to recognize that electromagnetic waves can be guided only by one of two possible processes: (1) by reflection between highly conducting/reflecting walls, and (2) refraction at gradients of refractive index. The two processes are clearly distinct. Moreover, the principles of the design of guiding systems are the same whether we deal with radio frequencies or frequencies corresponding to the visible part of the spectrum.

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  Microwave Engineering

  Concepts and Fundamentals

  • Ahmad Shahid Khan(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  This allows a wave-guide to handle only a small range of frequencies both above and below the frequency of operation. Dimension ‘ b ’ commonly varies between 0.2 λ and 0.5 λ and is governed by the breakdown voltage of the dielectric, which is usually air. In centimetric and millimetric ranges, Waveguide is the key player in a majority of applications wherein the wave is guided by the four walls. 3.3 Rectangular Waveguide To explore the guiding principles of wave propagation through a Waveguide, Figure 3.4 shows a rectangular Waveguide along with its dimensions and coordinate system. The wave is assumed to be propagating along its z -axis. As before, the time variation is accounted by e j ω t and the space variation along the z -axis by e − γ z = e − ( α + j β ) z , where parameters α , β and γ have the same mean-ing as discussed in Chapter 2. Since the Waveguide configuration contains four conducting walls, boundary conditions along the x - and y -axes include the continuity of E tan and H norm at x = 0 and a , for all values of y and at y = 0 and b for all values of x . In case of Waveguides the behaviour of waves is also governed by Maxwell’s equations, which are written as below. a b (a) (b) (c) m n n m m λ /4 λ /4 λ /4 λ /4 λ /4 λ /4 n and p q p q p q b a a b FIGURE 3.3 Impact of change of frequency. (a) Normal operating frequency, (b) more than operating fre-quency and (c) less than operating frequency.

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  Microwave and RF Vacuum Electronic Power Sources

  • Richard G. Carter(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  4 1 2 Waveguides 2.1 Introduction Modern vacuum tubes are power amplifiers and oscillators which require the use of Waveguides or coaxial lines to convey RF power into and out of them. The pur- pose of this chapter is to provide a summary of those topics which are important for the design of vacuum tubes. It also provides a foundation for the discussions of resonators in Chapter 3, and of slow-wave structures in Chapter 4. Section 2.2 sum- marises the theory of hollow metal Waveguides, and two-conductor transmission lines, having uniform cross-sections. This leads to a discussion of practical coaxial lines, and rectangular, ridged and circular Waveguides in Section 2.3. The properties of simple discontinuities in rectangular Waveguides are considered in Section 2.4 followed by a discussion of matching techniques in Section 2.5. Sections 2.6 and 2.7 examine methods of coupling between Waveguides of different cross-sections without, and with, changes in the mode of propagation. The final section reviews the different kinds of vacuum windows which are used in coaxial lines and rect- angular Waveguides. The theory and practice of Waveguides and Waveguide com- ponents is covered by many books and the reader is referred to them for detailed information [1–5]. 2.2 Waveguide Theory The propagation of electromagnetic waves in a source-free region, filled with a uniform material of permittivity ε and permeability μ, is governed by the wave equations ∇ − ∂ ∂ = 2 2 2 0 E E εμ t (2.1) and ∇ − ∂ ∂ = 2 2 2 0 H H εμ t , (2.2) where E is the electric field and H is the magnetic field [1]. We shall assume that the waves are guided in the z direction by conducting boundaries whose shape Waveguides 42 4 2 does not vary with z. The vector operator can be decomposed into transverse and longitudinal parts ∇ = ∇ + ∂ ∂ T z ˆ , z (2.3) where ˆ z is the unit vector in the z direction.

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  Photonics Modelling and Design

  • Slawomir Sujecki(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  55 3 Optical Waveguides Guided wave optics, realised both in optical fibre and planar technologies, has numerous applications. Optical fibres are the backbone of modern high-speed tele-communication systems. They are also indispensable for the flexible optical power delivery in medical and manufacturing laser applications, for high-speed computer interconnects, and for decreasing the weight and improving immunity to external electromagnetic fields in modern sensor and measurement systems. Planar optical Waveguides, on the other hand, are used for beam formation in edge emitting laser diodes, as sensors in modern sensor and measurement systems, to provide amplifica-tion in optical telecommunication systems, and in high-speed external laser diode modulators. The optical fibre technology currently allows the fabrication of low loss optical Waveguides providing both single mode and multimode wave guiding at a number of wavelengths and for various maximum optical power levels. The fibre optic tech-nology is fairly advanced and provides many high-quality optical components. The weakness of fibre optic technology lays in the large dimensions of the components and in its limited suitability for low-cost mass production. Planar technology, on the other hand, has the potential to provide the low-cost large-scale production of compact optical components, which can meet the demands of the modern telecom systems and other applications. INTRODUCTION TO OPTICAL Waveguide THEORY Optical Waveguides are characterised by very large longitudinal dimensions that eas-ily extend over 100,000s of wavelengths and lateral dimensions ranging from nano-metre-scale structural details to 100 or more wavelengths. The optical Waveguides typically vary slowly along the direction of the wave propagation and are terminated by an abrupt dielectric discontinuity (Figure 3.1).

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  Optical Physics

  • Ariel Lipson, Stephen G. Lipson, Henry Lipson(Authors)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  10 Optical Waveguides and modulated media In this chapter we shall meet examples of electromagnetic wave propagation in systems containing fine dielectric structure on a scale of the order of the wavelength, where the scalar-wave approximation is inadequate. Clearly, in these cases we have to solve Maxwell’s equations directly. On writing the equations, we shall discover that they bear a close similarity to those of quantum mechanics, where the dielectric constant in Maxwell’s equations is analogous to the potential in Schrödinger’s equation. This opens up a vast arsenal of methods, both analytical and numerical, which have been developed for their solution. We first discuss the optical Waveguide , already familiar in everyday life as the optical fibre, which has caused a revolution in the communications industry (Agrawal (2002)). The second topic is the dielectric multilayer system which, in its simplest form (the quarter-wave anti-reflection coating) has been with us for more than a century, but can today be used to make optical filters of any degree of complexity (MacLeod (2001)). Following these examples, we shall briefly discuss their application to photonic crystals , structures with periodic refractive index leading to optical band gaps, whose behaviour can immediately be understood in terms of the quantum anal-ogy ( Joannopoulos et al . (2008)). Photonic crystals have always existed. First, many of the bright colours of insects, butterflies and bird feathers are the result of diffraction by microscopic periodic structures (Fig. 10.1), and these can usually be recognized by the tendency of the colours to change with direction of obser-vation (as opposed to flowers, which have colours of chemical origin) (Zi et al . (2003); Biró et al . (2007)). Second, opals are periodically ordered collections of microscopic silica spheres, and their colours result from their periodicity.

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  Lightwave Communications

  • George C. Papen, Richard E. Blahut(Authors)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  3.2.2 Guided Waves Wave optics must be used whenever the wavelength of the light is comparable to or larger than the dimensions of the guiding structure. Wave propagation for the complex field E ( r , ω) is described by the vector Helmholtz equation as given in ( 2.3.15 ): ∇ 2 E ( r , ω) + n 2 ( r , ω) k 2 0 E ( r , ω) = 0 , (3.2.16) where k 0 = 2 π/λ 0 is the free-space wavenumber. Applying the boundary conditions to a Waveguide geometry that does not vary in the z direction produces the characteristic spatial solutions for each ω called eigenmodes or simply modes . Expressions for these modes will be developed in Section 3.3 . A guided mode is a mode for which the power is confined to a local cross-sectional region as defined by the Waveguide geometry. An unguided mode is a solution for the same structure whose power is not confined to the waveguiding structure and is lost into the cladding. To determine the lightwave field in a Waveguide at a distance L , the incident monochromatic field at the input to the Waveguide is first decomposed into a linear combination of modes. The spatial characteristics of the guided modes are determined in this chapter. The field in each mode, indexed by j , is propagated using a mode-dependent propa-gation constant β j , which depends on the geometry of the Waveguide. The temporal and propagation characteristics of the individual modes, which depend on β j as a function of ω , are determined in Chapter 4 . The spatial period of the wave in the propagation direction is given by 2 π/β j . The field in each guided mode at distance L is summed to produce the total field at that distance. 122 3 The Guided Lightwave Channel 3.2.3 Guided Photon Streams Modeling lightwave signal propagation using a discrete photon-optics signal model based on photon 5 streams is complementary to modeling the signal using a continu-ous signal model based on rays or waves.

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  The Theory of Electromagnetism

  • D. S. Jones, I. N. Sneddon, S. Ulam, M. Stark(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  in radar and television. As the frequency of the signal increases the dimensions of the line have to be reduced to prevent the propagation of the T E n mode. There is therefore an increased risk of breakdown if the same power is being carried. Also the losses, as indicated by (24), steadily increase with frequency and there will be additional losses in the dielectric supports holding the inner conductor. These shortcomings are absent in the circular and rectangular Waveguides becauses, as can easily be confirmed, the attenuation is less than that of a coaxial line when the frequency is about 1.2 times the cut-off frequency. Thus at wavelengths of a few centimetres it is more efficient to use a rectangular or circular Waveguide than a coaxial line. J U N C T I O N S To serve a useful purpose a Waveguide must be connected to a transmitter, receiver or antenna. Such a connection will materially affect the field in the guide. The next few sections will be concerned with the theory necessary for estimating this effect. 5.6 General Waveguide junction A typical junction is shown in Fig. 5.6. It consists of T and a number L of Waveguide regions. The boundary S of T is closed by L terminal planes Τ λ , . . ., T L each of which is perpendicular to the axis of the guide in which it is placed. The positions of the terminal planes are, to a large extent, arbitrary but it will be assumed that they FIG. 5.6. A general Waveguide'junction. THE THEORY OF WaveguideS 255 are far enough from the junction for any non-propagating modes which may be present at the junction to be essentially zero. We shall also assume that only the fundamental mode can propagate in each guide; the theory can be extended in a straightforward manner to the more general case. The field on te terminal planes may then be expressed completely in terms of the fundamental modes. The region T may contain dielectrics, conducting substances and the like. It will be assumed that they are linear and passive.

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  Handbook of Nanophysics

  Nanoelectronics and Nanophotonics

  • Klaus D. Sattler(Author)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  37 -1 37.1 Introduction The term nanophotonics may be understood as abbreviation from “photonics of nanostructures.” This rapidly evolving research area deals with light interaction in nanostructured materials and their applications in photonic devices like light sources, modulators, detectors, etc. 37.1.1 Basic Types of Waveguides for Nanophotonics Optical Waveguides are structures that are able to confine and guide optical electromagnetic field. Classical Waveguides (optical fibers, Figure 37.1a) use refractive index difference between the core and the cladding layer to guide light by total internal reflection. When the size of Waveguide approaches the wavelength of light, the confinement decreases and losses increase. Therefore, the size of practical classical Waveguides is limited to several hundreds of nanometers. Somewhat bet-ter confinement can be achieved in more complex Waveguides with guiding properties determined by the formation of pho-tonic bands due to their regular spatial structure (Figure 37.1b and c): (1) photonic crystal Waveguides using “defects” in peri-odic structures of photonic crystals to confine light and (2) plasmonic Waveguides based on metallic nanostructures with plasmon resonance (see, e.g., Pavesi and Guillot 2006). Due to limited space, we are going to describe only a special type of the “classical” Waveguides formed by luminescing silicon nanocrystals (Si-nc). 37.1.2 Silicon Nanophotonics The recent years can be characterized by the association of micro-electronics with optoelectronics or photonics. While the micro-electronics is based almost exclusively on silicon (indirect band-gap semiconductor), photonic light sources are currently made out of direct band-gap III–V compounds (family of GaAs materials). In order to reduce the material diversity, an effort to develop an effi-cient, electrically pumped silicon-compatible light-emitting device is becoming very strong (Pavesi and Lockwood 2004).

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  Semiconductor Laser Theory

  • Prasanta Kumar Basu, Bratati Mukhopadhyay, Rikmantra Basu(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  In this way, the ray propagates in the core layer, being confined within it. This is light guidance by the core layer. This simplified approach might tempt anyone to conclude that the wave-guide will guide light whenever the incidence angle exceeds the critical angle. This is not, however, true as will be shown next. 5.3 Reflection Coefficients The complex amplitudes of the electric fields associated with the incident and reflected rays, denoted by E i and E r , respectively, in Figure 5.1, are related to each other by E rE r i = (5.2) (a) (b) n 2 n 1 n 2 n 2 n 1 θ 1 θ 1 θ θ 2 E i E r E t FIGURE 5.1 (a) Reflection and refraction in two dielectric media; (b) ray path showing total internal reflec-tion and guidance of mode in a symmetric slab Waveguide. 139 Waveguides and Resonators where r is the complex reflection coefficient that depends on both the angle of incidence and the polarization of the light ray. The polarization of a light ray is related to the direction of the electric field vector associated with the EM wave. The transverse electric (TE) condition refers to the situation when the electric fields of the waves are perpendicular to the plane of incidence, that is, the plane containing the wave normal and normal to the interface. Similarly, the transverse magnetic (TM) case occurs when the magnetic field of the wave is normal to the plane of incidence.

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  Emerging Waveguide Technology

  • Kok Yeow You(Author)
  • 2018(Publication Date)
  • IntechOpen

    (Publisher)

  Introduction Conventional optical Waveguides work on the principle to guide waves in a material sur -rounded by other material media, the refractive index of the material should be slightly higher than that of surrounded media such that light can bounce along the Waveguide by means of total internal reflections at the boundaries between different media. The indefinite guiding progress the waves from successive boundaries which must interfere constructively to gener -ate a continuous and stable interference pattern along the Waveguide. If the interference pat -tern in not fully constructive, the waves cancel, owing to the self-destruction. The conventional © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. optical Waveguides are primarily the most common type of thin film optical filters used widely as narrowband filters in laser cavities, optical telecommunications, and light modulators [ 1]. However, the realization of sub-nanometer narrowband filters with thin film technique require hundreds of optical thin-films stack with stringent tolerances over thicknesses and refractive index variations [ 2 ]. Resonant Waveguide Gratings (RWGs) are a new class of narrowband fil -ters and are widely used in applications such as polarizers [ 3 ], laser cavity reflectors [ 3, 4], light modulators [ 5 ], biosensors [ 6 ], and wavelength division multiplexing [ 7 ]. Such narrowband reflectance/transmittance spectral characteristics can be observed by sub-wavelength grating structures in terms of resonance anomalies [ 8, 9] with numerous potential applications. RWG operates with resonance effects with relatively simpler structure of few layers.

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