To describe the behaviour of electromagnetic radiation in absorbing/gain media in a general manner, we derive a more general wave equation by defining a complex refractive index. [17] L.N. This, as well as the small size of the optical components, is the key factor for the success of integrated photonic systems. 38 2 REVIEW OF THE ELECTROMAGNETIC THEORY OF LIGHT Let us consider an EM monochromatic plane wave, characterised by its angular frequency i and wavevector ki , incident from a homogeneous medium (1) to a planar frontier separating a different homogeneous medium (2). Gathers together and summarizes the numerous integrated optical components and devices emerging from the research laboratories. In this situation, the electric and magnetic fields are given by: || E i E i [E ix , 0, E iz ] (2.111) Hi H i [0, H iy , 0] (2.112) where the symbols || and denote vectors parallel and perpendicular to the incident plane, respectively. The most common way of preventing boundary reflection using FFT-BPM is the insertion of artificial absorption regions adjacent to the pertinent boundaries [9]. For the description of the electromagnetic field in a material medium it is necessary to define two additional vectorial magnitudes: the electric displacement vector D (r, t) and the magnetic flux density vector B (r, t). The search for the propagation constant of a guided mode is carried out by establishing the condition of an exponentially decreasing behaviour of the electric field in the cover and substrate regions. In order to calculate the reflectivity in a case of hard incidence, it is necessary to evaluate cos t included in the formulae for the reflection and transmission coefficients. 53 In this case the field pattern corresponds to a radiation mode, where the light cannot be confined in the film but leaks to the cover and substrate regions, as can be seen in Figure 3.23. Copyright 2022, The Research Foundation for the State University of New York, AIM Photonics257 Fuller RoadAlbany, New York12203 USA. The modulus of the atten- uation vector at defined in Section 2.1.6 is now given by c in the cover region and by s in the substrate. The three media have refractive indices nc (cover), nf (film) and ns (substrate), and are separated by planar boundaries perpen- dicular to the x-axis, the light propagation being along the z-axis. 116 4 COUPLED MODE THEORY: WAVEGUIDE GRATINGS Lateral distance (m) 5 0 5 (a) (b) Lateral distance (mm) 5 0 5 0 0.5 1.0 1.5 0 0.5 1.0 1.5 (c) Propagation distance (mm) (d) Propagation distance (mm) Figure 4.14 Light propagation in a waveguide system which consists of two parallel step-index waveguides, separated by a fixed distance. Low voltage control. [2] R. Stoffer, H.J.W.M. The quantities c and t denote the phase changes at the surface x = 0 and at the turning point xt beneath the surface. The surface peak index of the refractive index distribution n0 is calculated by evaluating the index function at q = 0.75 (arrows), 91 The optical elements present in integrated photonic devices should include basic components for the generation, focusing, splitting, junction, coupling, isolation, polar- isation control, switching, modulation, filtering and light detection, ideally all of them Opto-electronics Electro-optics Waveguide technology Non-linear optics Acousto-optics Integrated photonics Figure 1.2 Confluence of various disciplines into integrated photonics, 13 In particular, we will see that for confined radiation the wave equation admits a discrete number of solutions, called guided modes. Please try again. Perhaps, second to LiNbO3 the IIIV semiconductor compounds (mainly GaAs and InP) are the substrates with greatest impact on integrated optics technology, and are probably the materials with the most promising future in this field [10, 11]. There is an intermediate approximation, which starts from the wave equation but ignores coupling terms between the transversal components of the fields, and for that reason this method is usually referred to as semi-vectorial (FD-SVBPM). Vetterling and B.P. This course discusses the fundamentals and design principles of emerging silicon photonic integrated circuits and systems integrating such technology. Fundamental of Photonics - AbeBooks This is valid provided that xi and xi+1 are sufficiently close to each other. You do not have subscription access to this journal. This method is of interest, 131 This type of waveguide is the best choice for fabricating integrated photonic devices. 5.5 SPATIAL FREQUENCIES FILTERING 151 and the low frequencies components where the higher amount of optical field power is concentrated are retained. 94 3 THEORY OF INTEGRATED OPTIC WAVEGUIDES p is the effective refractive index NI calculated previously. Following the path taken in electronics of reducing devices to their ultimately fundamental forms, for instance single-electron transistors, now optical devices have also been scaled down, creating the increasingly active research fields of integrated and coupled photonic systems. [8] J.E. Come to MIT July 22 - 26 for a week of intensive short courses on integrated photonics: materials, devices, EPDA software training and integrated photonic circuit design, chip fabrication, packaging, testing, and system applications. Radiation is allowed to freely escape the problem region without appreciable reflection, whereas radiation flux back into the region is prevented. Try Numerade free. In fact, although the ray optics treatment can give some interesting results, for a complete understanding and description of light propagation within guiding structures, it is nec- essary to consider light as electromagnetic waves, and use the formalism developed in the previous chapter. Indeed, the total disappearance of the beam as it crosses the limit of the computational region is the expected result from a physical point of view. From equations (2.9) and (2.10), one obtains respectively that the normal components of the fields D and B should be kept across the boundary, that is: (D Normal )Medium 1 = (D Normal )Medium 2 at the interface (2.90) (B Normal )Medium 1 = (B Normal )Medium 2 at the interface (2.91) On the other hand, by using Maxwells equations (2.11) and (2.12) respectively, the conditions of continuity across the interface of the tangential components of the E and H fields are obtained: (E Tangential )Medium 1 = (E Tangential )Medium 2 at the interface (2.92) (H Tangential )Medium 1 = (H Tangential )Medium 2 at the interface (2.93), 47 5.3 METHOD BASED ON FINITE DIFFERENCES (FD-BPM) 143 z x (a) (b) (c) Figure 5.5 Diagram showing the three schemes used in the simulation of light propagation based on finite differences. It is necessary to remember that filtering windows must be used with caution, because some essential frequency components of the optical field which could be necessary for the correct description of the optical propagation can be eliminated, and thus the simulation would lose accuracy. While the simple finite difference method (implicit standard scheme) allows a first-order approximation in the propagation step O( z + x 2 ), the Crank-Nicolson method establishes a second-order approximation in the propaga- tion O( z2 + x 2 ). The reflection coefficient of the grating depends on the length of the grating region and on the modulation refractive index depth. This kind of interaction can occurs also when the a and b modes are the same mode (in the same waveguide), but with opposite propagation directions. It is worth noting that here the speed of light has been obtained only using values of electric and magnetic constants. Lambeck, Remote Opto-Chemical Sensing with Extreme Sensitiv- ity: Design, Fabrication and Performance of a Pigtailed Integrated Optical Phase-Modulated Mach-Zehnder Interferometer System, Sensors and Actuators B 61, 100127 (1999). First, we sample N (q) in descending order N0 > N1 > N2 > . This book presents the basic concepts of waveguides necessary to understand and describe integrated photonic devices, from Maxwells equations to the modelling of light propagation in arbitrary guiding structures. On the other hand, the phase shift of the reflected waves, TM and TE in (2.144) and (2.145), are calculated from the following expressions: tan(TM /2) = n1 B/n2 cos i (2.148) tan(TE /2) = n2 B/n1 cos i (2.149) The magnitudes of the phase shifts TE and TM are very important when establishing the condition of light propagation in planar optical waveguides, which gives rise to the calculation of the allowed propagating modes, as we will see in the next chapter. 2.1.5 Polarisation of electromagnetic waves As we have seen, the electric and magnetic fields of a plane EM wave oscillate in a direction perpendicular to the direction of propagation defined by the wavevector k, and therefore the plane EM waves are transversal. In this case (F = 1, = ), the efficiency of power transfer simplifies to: |B(0)|2 = = tanh2 (L) (4.93) |A(0)|2 Following this formula, even in the most favourable case of F = 1, the efficiency can never reach the unity. The theory that describes the interaction between modes, or the new modes that arise from the modified structure, is known as the coupled mode theory (CMT). First, as the mismatching between the two guides increases, the fraction of power transfer decreases. Substituting the optical field in the Helmholtz equation, it follows that: 2u u 2 2 2 + 2iK = + u + (k 2 K 2 )u (5.4) z z x 2 y 2, 147 Therefore, as long as the real part of kx is positive, the contribution to the overall change in energy from this boundary will always be negative, i.e., radiative energy can only flow out of the problem region. The transversal components of the electric and magnetic fields are calculated by differentiation of the equations (3.83)(3.86). After a masters degree completed with a thesis on Luminescent Solar Concentrators, he received his PhD under the direction of Professor F. Cusso, with a thesis on the topic of Materials for Colour Centre Lasers. The main goal pursued by integrated photonics is therefore the miniaturisation of optical systems, similar to the way in which inte- grated electronic circuits have miniaturised electronic devices, and this is possible thanks to the small wavelength of the light, which permits the fabrication of circuits and compact photonic devices with sizes of the order of microns. of each connecting element. We design and build new semiconductors, transistors that are faster, communication channels . [19] Y.H. Please use a different way to share. [20] H.V. ratio of electrons and light wave photons. There are several methods that can be used to calculate the effective refractive index of the modes in waveguide structures with graded index profiles. In fact, this mode has an effective index of 2.1679, which is very close to that of the substrate value (2.1675), indicating that the mode is very close to the cut-off. The emergence of novel photonic devices, as well as resulting in the important connection between optics and electronics, has given rise to other sub-disciplines within photonics. scalability of programmable integrated photonics and provide an This parameter can be determined after imposing the continuity of the longitudinal component of the magnetic field Hz at the boundaries between region I and regions IV and V, that is, at the positions y = b and y = 0. Indeed, a planar wave must have an infinite spatial extension, and this is not the case when dealing with optical waveguides modes that have transversal dimensions of the order of the wavelength of the light, typically of a few microns. Light Propagation in Waveguides: The Beam Propagation Method. These two propagation constants are therefore the propagation constants of the coupled waveguide system above described, whose wave descriptions are given by: e = Ae fa + Be fb (4.64) o = Ao fa + Bo fb (4.65) where Ae and Ao are constants determined by the initial condition of light injection. Opt. 3.1 Optical Waveguides: Basic Geometries The basic element in integrated photonic technology is the optical waveguide. Appendix 7: Implementation of the Crank-Nicolson Propagation Scheme. tuning resolution step in BUL units (the lower the Equations are available to subscribers only. The structure consists initially of a symmetric planar waveguide in step, with a length of 200 m, which splits into two branches of 550 m length; from a distance of 750 m the two branches become parallel. The Fourier terms describing this type of periodic structure are given by: sin qa q (x) = q = 0 (n2f n2c ) (4.140) q with q = 0, 0 < a < 1, in the region corresponding to h < x < h, and q (x) = 0 elsewhere. As we can seen, after launching light into the left waveguide, a total power transfer occurs after a propagation distance of 500 m. This distance can be directly calculated by computing the coupling coefficient using the CMT, and the results are given in Figure 4.13. Appendix 1: Complex Notation of the Electric and Magnetic Fields. If the modes are collinear, two cases can be distinguished in the coupled field behaviour: when the two waves have the same prop- agation direction (unidirectional coupling), and when the coupled waves have opposite directions (bi-directional coupling). Figure 5.13 shows the evolution of the transversal light distribution in a one- dimensional straight waveguide when a Gaussian beam with constant phase is launched 10 Lateral distance (m) 0 0 10 0 0.5 1 1.5 2 2.5 Propagation distance (mm) Figure 5.13 Light propagation, using FD-BPM with TBC, in a multi-mode straight waveguide. By substituting the solution for planar structures given by equations (3.12) and (3.13) in Maxwells equations (3.6) and (3.7), we obtain the following coupled equations: Hx = (/0 )Ey (3.14) Hz = (i/0 )Ey /x (3.15) iHx + Hz /x = i0 n2 (x)Ey (3.16) that relate the field amplitude components Ey , Hx and Hz . In the FD-BPM method, the Helmholtz scalar wave equation (equation (5.6)) in partial derivatives is approximated by a finite difference scheme, which can be expressed as: uj (z + z) uj (z) uj 1 (z) 2uj (z) + uj +1 (z) 2iK = + k02 (n2 n20 )uj (z) (5.20) z x2 where uj (z) is the optical filed at the position (j x, z) with j = 1, 2, . For waveguides fabricated by deposition techniques the lateral definition of the optical circuits is usually carried out by means of etching after the deposition of the material onto the whole substrate surface. In addition, a particular function can be executed using different elements, whose design may differ substantially. If the propagation constant of the mode is far from the cut-off ( k0 n1 ), the electromagnetic field is confined mainly in the core (region I), and only a small fraction of the energy carried by the optical mode spreads out to the surrounding regions (regions II, III, IV and V). Heideman and P.V. Thus, the vectors kr and kt must lie in the plane defined by the ki vector, 48 Electro-optics deals with the study of optical devices in which the electrical interaction plays a relevant role in controlling the flow of light, such as electro-optic modulators, or certain types of lasers. To calculate the intensity associated with the wave we use the expressions given in equations (2.38) and (2.39), as well as. In order to visualise these effects, let us analyse the behaviour of a 3 tilted Gaus- sian beam propagating in a one-dimensional homogeneous medium. Choosing the same Gaussian excitation, but now shifted 6 m respect to the waveguide centre, we obtain the propagation depicted in Figure 5.16. Waveguide I 1.0 Normalised power 0.8 0.6 F=1 Waveguide II F = 0.3 0.4 0.2 0.0 p/2g p/g Length of the coupling region Figure 4.11 Variation of the relative optical power in waveguides I and II as a function of the propagation distance, in the case of synchronous waveguides (F = 1, continuous line) and asynchronous waveguides with = 0 (F = 0.3, dashed line). The relation between the position change in depth of the ray and its advance in the propagation direction is x x nc N nf n(x ) z xt ns Figure 3.28 Left: graded refractive index profile, showing the position of the effective index position N of the mode and the corresponding turning point xt . On the contrary, this type of periodic structures is effective in producing strong interactions between modes of equal order ( = ), in particular for contradirectional coupling between a mode propagating in one direction and the same mode propagating in oppo- site direction. Therefore, in order to properly describe light propagation in a medium, whether vacuum or a material, it is necessary in general to know six scalar functions, with their dependence of the position and the time. In this equation we have defined, for the sake of simplicity, the parameters 1 (ns /nf )2 and 2 (nc /nf )2 = 1 a(1 1 ). ., the major and minor axis of the ellipse will lie along the x and y axis. By using several RF signals simultaneously, it is even possible to drop several wavelengths. If the medium is not linear, it should be necessary to include additional terms involving power expansion of the electric and magnetic fields. If we assume propagation 0 ba bb b qK 2 Figure 4.20 Relation between the propagation constants of the modes (a and b ), the wave- vector of the grating (K) and the mismatching parameter (2 ) in the coupling induced by a waveguide grating structure. Bring your club to Amazon Book Clubs, start a new book club and invite your friends to join, or find a club thats right for you for free. 62 3 THEORY OF INTEGRATED OPTIC WAVEGUIDES being the angular frequency and the propagation constant of the wave. The result is that for coupling orders |q| = 1 the coupling coefficients between TETE and TMTM modes are very small, thus the energy transfer via high order coupling do not take place. Anisotropic Media: Crystal Optics and Polarization (PDF - 2.5 MB) 13 Quantum Nature of Light and Matter 14-16 Schrdinger Equation and Stationary States. In particular, the sum of plane waves described in (2.55) and (2.56), and those described by (2.58) and (2.59) will give us a different, but valid, solution of the wave equation: E = E0 [cos(t kz)ux sin(t kz)uy ] (2.60) and H = H0 [cos(t kz)uy + sin(t kz)ux ] (2.61) In order to examine the polarisation character of this new wave, let us study the curve described by the tip of the electric field vector at a fixed plane, for instance, the plane defined by z = 0. In addition, as we are dealing with propagating modes polarised mainly along the x direction, we find that Ez Ex . This new waveguide can easily be solved by conventional methods applied to pq planar structures. 4.2.2 Mathematical description of waveguide gratings Regardless of the type, a waveguide diffraction grating is a periodic structure that can be described by the spatial change on the dielectric permittivity caused by the grating in the original waveguide structure, that is, in the canonical structure. The step-index planar waveguide is the simplest structure for light confinement, and is formed by a uniform planar film with a constant refractive index (homogeneous film, nf = constant), surrounded by two dielectric media of lower refractive indices [3]. The rapid boom of monomode optical fibre systems which started in the 1980s was the perfect niche market for these advanced integrated pho- tonic devices that were waiting in the research laboratories. These material properties make possible a wide range of integrated electronic and photonic circuits. The grating is inherently a wavelength selective element, and thus the grating period must be calculated for the specific working wavelength. These two relations can be E H k Figure 2.1 Triad defined by the electric field, magnetic field and wavevector, for a plane wave, 41 The relations between the incident, reflected and transmitted waves are obtained by setting the adequate boundary conditions for the fields at the planar interface, which are derived directly from Maxwells equations. Vetterling and B.P. This is a dummy description. The modal spectrum corresponding to this propagation is represented in Figure 5.17, where now it can be observed that the six propagating modes supported by the waveg- uide have been excited, although with different amount of energy. Appendix 6 FAST FOURIER ALGORITHM Replaces Dat( ) by its discrete Fourier transform, if isign% is input as 1; or replaces Dat( ) by nn% times its inverse discrete Fourier transform, if isign% is input as 1. In order to obtain the peak refractive index n0 at the surface we first need to estimate the value q = q0 , which is done by substituting xt = 0 into equation (3.73), where the guided mode indices m are now denoted by the artificial set of modes of indices q. considerable number of new applications in the fields of First, as An integrated optical phase modulator performs a controlled shift on the phase of a light beam (Figure 1.6p), and consists of a channel waveguide fabricated on a substrate with the possibility of changing its refractive index by means of an externally applied field (thermal, acoustic, electric, etc.). This coupled system corresponds to the case where the two interacting modes are synchronised. Here we will explain two widely used methods: Marcatilis method and the effective index method. Click here to see what's new. 4.1 MODAL COUPLING 109 d A(z) = iab B(z)e i(b a )z (4.48) dz d B(z) = iba A(z)e +i(b a )z (4.49) dz where the coefficients ab and ba are the coupling coefficients between the modes a and b and vice versa, respectively. The major and minor axis of the electric and magnetic fields step in BUL (. Q ) in descending order N0 > N1 > N2 > as as... The medium is not linear, it is even possible to drop several wavelengths function be. Photonic integrated circuits and systems integrating such technology applied to pq planar structures on modulation! Denote the phase changes at the turning point xt beneath the surface x = 0 at! Grating is inherently a wavelength selective element, and thus the grating period must be for. Emerging silicon photonic integrated circuits and systems integrating such technology appendix 1: Notation. 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