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Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts

Victor Bârsan

Published Mar 23, 2012• 4 citations
Publisher

Abstract

In this chapter, we shall expose several analogies between oscillatory phenomena in mecahanics and optics.The main subject will be the analogy between propagation of electromagnetic waves in dielectrics and of electrons in various time-independent potentials.The basis of this analogy is the fact that both wave equations for electromagnetic monoenergetic waves (i.e. with well-defined frequency), obtained directly from the Maxwell equations, and the time-independent Schrodinger equation are Helmholtz equations; when specific restrictions -like behaviour at infinity and boundary conditions -are imposed, they generate similar eigenvalues problems, with similar solutions.The benefit of such analogies is twofold.First, it could help a researcher, specialized in a specific field, to better understand a new one.For instance, they might efficiently explain the fiber-optics properties to people already familiar with quantum mechanics.Also, even if such researchers work frequently with the quantum mechanical wave function, the electromagnetic modal field may provide an interesting vizualization of quantum probability density field [1].Second, it provides the opportunity of cross-fertilization between (for instance) electromagnetism and optoelectronics, through the development of ballistic electron optics in two dimensional (2D) electron systems (2DESs); transferring concepts, models of devices and experiments from one field to another stimulate the progress in both domains.Even if in modern times the analogies are not credited as most creative approaches in physics, in the early days of developement of science the perception was quite different."Men's labour ... should be turned to the investigation and observation of the resemblances and analogies of things... for these it is which detect the unity of nature, and lay the fundation for the constitution of the sciences.",considers Francis Bacon, quoted by [1].Some two centuries later, Goethe was looking for the "ultimate fact" -the Urphänomenon -specific to every scientific discipline, from botanics to optics [2], and, in this investigation, attributed to analogies a central role.However, if analogies cannot be considered anymore as central for the scientific investigation, thay could still be pedagogically useful and also inspiring for active scientific research.Let us describe now shortly the structure of this chapter.The next two sections are devoted to a general description of the two fields to be hereafter investigated: metallic and dielectric waveguides (Section 2) and 2DESs (Section 3).The importance of these topics for the development of optical fibers, integrated optics, optoelectronics, transport Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts

DOI
10.5772/34589

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In this chapter, we shall expose several analogies between oscillatory phenomena in mecahanics and optics.The main subject will be the analogy between propagation of electromagnetic waves in dielectrics and of electrons in various time-independent potentials.The basis of this analogy is the fact that both wave equations for electromagnetic monoenergetic waves (i.e. with well-defined frequency), obtained directly from the Maxwell equations, and the time-independent Schrodinger equation are Helmholtz equations; when specific restrictions -like behaviour at infinity and boundary conditions -are imposed, they generate similar eigenvalues problems, with similar solutions.The benefit of such analogies is twofold.First, it could help a researcher, specialized in a specific field, to better understand a new one.For instance, they might efficiently explain the fiber-optics properties to people already familiar with quantum mechanics.Also, even if such researchers work frequently with the quantum mechanical wave function, the electromagnetic modal field may provide an interesting vizualization of quantum probability density field [1].Second, it provides the opportunity of cross-fertilization between (for instance) electromagnetism and optoelectronics, through the development of ballistic electron optics in two dimensional (2D) electron systems (2DESs); transferring concepts, models of devices and experiments from one field to another stimulate the progress in both domains.Even if in modern times the analogies are not credited as most creative approaches in physics, in the early days of developement of science the perception was quite different."Men's labour ... should be turned to the investigation and observation of the resemblances and analogies of things... for these it is which detect the unity of nature, and lay the fundation for the constitution of the sciences.",considers Francis Bacon, quoted by [1].Some two centuries later, Goethe was looking for the "ultimate fact" -the Urphänomenon -specific to every scientific discipline, from botanics to optics [2], and, in this investigation, attributed to analogies a central role.However, if analogies cannot be considered anymore as central for the scientific investigation, thay could still be pedagogically useful and also inspiring for active scientific research.Let us describe now shortly the structure of this chapter.The next two sections are devoted to a general description of the two fields to be hereafter investigated: metallic and dielectric waveguides (Section 2) and 2DESs (Section 3).The importance of these topics for the development of optical fibers, integrated optics, optoelectronics, transport Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts
Let us return from optics to mechanics and explore the analogy to its fullest extent. In optics the old system of mechanics corresponds to intellectually operating with isolated mutually independent light rays. The new undulatory mechanics corresponds to the wave theory of light . Erwin Schrödinger, Nobel lecture , 1933 In this chapter we shall see that electromagnetic waves and electrons feature a number of common properties under conditions of spatial confinement. Simple and familiar problems from introductory quantum mechanics and textbook wave optics are recalled in this chapter to emphasize the basic features of waves in spatially inhomogeneous media. Herewith we make a first step towards understanding the properties of electrons and electromagnetic waves in nanostructures and notice that these properties in many instances are counterparts. Different formulas and statements of this chapter can be found in handbooks on quantum mechanics and wave optics. A few textbooks on quantum mechanics do consider analogies of propagation and reflection phenomena in wave optics with those in wave mechanics. Isomorphism of the Schrödinger and Helmholtz equations In Chapter 2 we discussed that an electron in quantum mechanics is described by the wave function, the square of its absolute value giving the probability of finding an electron at a specific point in space. This function satisfies the Schrödinger equation (2.56) which is the second-order differential equation with respect to space and the first-order differential equation with respect to time.
During these three years we focused on several topics in quantum otpics and quantum electrodynamics. A central theme in our investigations is that of the photon wave function. Can quantum optics and quantum electrodynamics experiments be described simply, in position space, with the help of a wave function describing the photon(s) featured in the experiment ? The answer to that question is not quite obvious: the usual description of photons takes place in the reciprocal space of wave vectors. But these experiments call for a wave mechanical description in the position representation, as is done in quantum mechanics textbooks in situations featuring massive particles. Moreover, in a recent experiment [1], single photon trajectories through a Young two-slit setup have been observed. In order to try and describe these trajectories formally, it is natural to build a wave mechanical formalism for photons. We therefore studied in detail the formal construction of the photon wave function, an object which was little studied until the 1990s. We also studied the properties of the photon wave function in the presence of sources.To do that, we considered several open (interacting) quantum systems. We saw that there exists in principle an infinite number of possibilities when defining the photon wave function. We emphasised several criteria on the basis of which it appears that only three choices for the wave function are interesting. One of them coincides with an object introduced and used by Glauber [2] to study light detection andthe correlations of the electromagnetic field in the quantum regime. We also saw that, in the absence of sources, the propagation equation for a single photon is formally equivalent to Maxwell’s equations. At low photon numbers, the wave function formalism can be very useful. We adapted it to interacting systems,first, to cavity quantum electrodynamics (QED) [3], in particular to the experiments carried out by Serge Haroche’s group [4]. We proposed a simple model to describe photons in QED cavities. With this model, and with the helpof the photon wave function, we studied the propagation of photons escaping a cavity. We also constructed the Lindblad master equation without introducing nonunitary quantum jumps (also see [5]). We finally investigated the spacetime evolution of a photon which is emitted during the decay of an atomic electron. After having carefully studied the dynamics of the electronic decay, especially at very short times [6, 7], we set out to describe the emitted electromagnetic field as rigorously as possible. This emitted field, surprisingly, does not evolve causally. Though this is not entirely unexpected in view of Hegerfeldt’s theorem, which states [8] that causality is impossible for quantum systems which are described by a Hamiltonian with a spectrum which is bounded by below, we identified [9] two other sources of non causality. One of them was predicted qualitatively by Shirokov [10], while the other one, which is completely new as far as we can tell, is still to be better understood
Analogy is a fundamental tool for understanding Mother Nature, since it associates various phenomena related by common properties or comparable behavior. In particular, the analogies between classical physics and quantum world show the fact that similar scientiffic formalisms apply to phenomena that are completely different. The role of mathematics is essential, because the analogy principle exist in the fact that totally different systems can be modeled by similar mathematical equations. Speciffically, analogies between quantum mechanics and wave optics have been emphasized since the beginning of quantum mechanics: wave effects like interference and diffraction were taken from optics and applied to exhibit the wavy idea of quantum particles, as electrons, neutrons and atoms. After the complete development of quantum theory the exchange of ideas in the opposite way started to happen. In the recent decade quantum-classical analogies have seen a great revival thanks to quantum control schemes for transfer of populations, such as excitation with rapid adiabatic passage, stimulated Raman adiabatic passage and composite pulses. Now in this dissertation we will use the concepts of composite pulses and adiabatic evolution, from the realm of coherent quantum control, to demonstrate: (a) novel robust polarization manipulation devices; (b) efficient broadband and scalable frequency conversion schemes as well as optical parametric ampliffication schemes;(c) several new optical ...
This chapter shows that the laws of conventional optics are a consequence of the linearity of Maxwell's equations, which permit the superposition of different waves. Experiments on electron diffraction show that the interference phenomena associated with these "matter waves" obey the familiar laws of physical optics. The chapter provides the linearly independent solutions of the Schrodinger equation form a complete set of functions. The form of the Schrodinger equation is intimately connected with the requirement of gauge invariance . In classical transformation theory, the Hamiltonian H is the generator of displacements in time. In classical mechanics the total linear and angular momenta are constants of the motion if the Hamiltonian is invariant under translations and rotations, respectively. The chapter focuses on eigenfunctions, and in particular, on stationary states. It presents several illustrations of the fact that there frequently are more than one linearly independent energy eigenfunction belonging to an energy eigenvalue.
It is critical to understand the nature of the propagation medium in terms of the available resonances which couple to the electromagnetic field. The detailed specification of the complex index of refraction as a function of frequency is the subject of spectroscopy. Since propagation media cover all phases of matter, the spectroscopy of gases, solids, and liquids is introduced in this chapter. Topics are not exhaustively covered, but rather with sufficient depth to support the practical applications that will come later. The chapter begins with a formal review of topics covered in stationary-state quantum mechanics necessary for the development of spectroscopy. The next section develops models to calculate spectral line positions, which are necessary to characterize a spectral line. The full development of spectral line parameters is not only the goal of this chapter but the next two as well. The remaining sections address the essential elements of the spectroscopy of gases, solids and liquids. Both classical and quantum models are used. Time-independent quantum mechanics allows the description of the quantized energy level structure of matter. This is the central topic of this chapter. Time-dependent quantum mechanics is necessary for the development of a theory covering transitions between stationary-state energy levels and is the topic of Chapter 5. The classical concept of light is that of an oscillating continuous wave field, as presented in the first two chapters. This is consistent with the classical electrodynamics based on Maxwell’s equations. However, the work by Planck on blackbody radiation and Einstein on the photoelectric effect showed that optical fields also have a quantized or particle-like nature. Planck, in 1900, suggested that the energy of light, E, be quantized according to . . . E = h f [J] (3.1) . . . where f is frequency (sec−1) and h is Planck’s constant (h = 6.6260755(40) × 10−34 J-sec). This allowed a theoretical description of blackbody radiation for the first time. However, it was Einstein, who, a few years later, made Eq. 3.1 more credible by applying it to explain the photoelectric effect.
Optical fibers are the backbone of the world's communication systems. They also represent the simplest example of a photonic device, where photons in waveguides replace electrons in wires. More sophisticated devices than this can be made, in which all the optical components – including multiple waveguided modes – are integrated on a chip. It is important to understand the quantum noise properties of these systems. While in some respects these systems can be treated as one-dimensional, there are additional features. Dispersion is due not just to material properties, but also to the waveguide geometry. The nonlinearity comes from a combination of material and geometric properties as well. There is, of course, a quantum aspect to these photonic devices, in which quantum noise occurs due to random effects caused by nonlinearity and gain. Optical devices based on waveguides are, of course, not just simple fibers. One can have multiple waveguides with various types of linear coupling. The intrinsic nonlinearity of the dielectric leads to four-wave mixing, which can be an important source of correlated photons, both for fundamental tests of quantum mechanics, and for applications in quantum information. It is often essential to include effects due to anisotropies of various types, as well as material dispersion. In dielectrics, another effect that should be included is the coupling of vibrational modes of the (solid) dielectric to the propagating field, which gives rise to Raman and Brillouin scattering effects. This allows us to obtain a more detailed and correct theory of quantum noise than the simplified case treated in Chapter 3.