Chapter 2 Matrix description of wave propagation and polarization Contents 2.1 Electromagnetic waves................................... 2–1 2.2 Matrix description of wave propagation in linear systems.............. 2–4 2.3 Matrix description of light polarization......................... 2–12 2.1 Electromagnetic waves An electromagnetic wave propagating in a dielectric material with permittivity " and permeability µ is described by Maxwell’s equations in the time domain, relating the electric field E(r,t) and magnetic field H(r,t) in the absence of sources: ( @H(r;t) r × E(r; t) = −µ @t @E(r;t) (2.1) r × H(r; t) = " @t In the frequency domain, the field components can be represented using the phasor notation F~(r)ej!t + F~∗(r)e−j!t F(r; t) = Re(F~(r)ej!t) = (2.2) 2 in which F is a field vector (either E or H) and F~(r) is a complex valued vector, representing the ! amplitude and phase of the sinusoidal time variation of F (at a frequency f = 2π ). Using the phasor notation, we can rewrite Maxwell’s equations 2.1 to get a set of differential equations relating the complex phasors representing the real field vectors. r × E~(r) = −j!µH~ (r) (2.3) r × H~ (r) = j!"E~(r) 2–1 If an electromagnetic wave propagates in a uniform medium (" and µ constants) equation 2.3 can be rewritten as (by applying the operator r × · to these equations and using r · "E~ = 0 and r · µH~ = 0) r2E~ + !2εµE~ = 0 (2.4) r2H~ + !2εµH~ = 0 called the Helmholtz equations. A general solution to these equations can be written as −jk·r jk·r F~(r) = F~+e + F~−e (2.5) in which F~(r) can be either the electric field phasor or the magnetic field phasor. The wave vector k can have an arbitrary orientation and is related to the frequency as p k = ! εµ1k (2.6) and defines the wavelength (in the material) λ as 2π jkj = (2.7) λ This is the well known plane wave solution. F~+ represents a plane wave propagating in the direction 1k, while F~− represents a plane wave propagating in the opposite direction. This can be seen by substituting the forward propagating part of equation 2.5 into equation 2.2 and assuming propagation along the z-axis (k = jkj 1z). The time variation of the field can then be written as ~ ~ F (z; t) = F+ cos(!t − kz + arg(F+)) (2.8) This means that planes of constant phase φ move along the positive z-axis with a phase velocity ! v = (2.9) ph k In what follows, we will omit the ~ and implicitly assume that we are dealing with phasors. Inserting equation 2.5 in equation 2.3 we find that −jk·r +jk·r −jk·r +jk·r −jk × E+e + jk × E−e = −j!µ(H+e + H−e ) −jk·r +jk·r −jk·r +jk·r (2.10) −jk × H+e + jk × H−e = j!"(E+e + E−e ) Identifying corresponding terms results in a relation between electric and magnetic field k × E+ = !µH+ k × E− = −!µH− (2.11) k × H+ = −!"E+ k × H− = !"E− 2–2 Figure 2.1: Orientation of field vectors for the plane wave solution This means that electric field vector, magnetic field vector and wave vector are all orthogonal as shown in figure 2.1. If we now consider the case of an isotropic z-invariant dielectric waveguide described by per- mittivity "(x; y) and permeability µ(x; y), we can suggest a forward propagating solution of the Maxwell equations of the form −jβz E(x; y; z) = [eT(x; y) + ez(x; y)1z] e −jβz (2.12) H(x; y; z) = [hT(x; y) + hz(x; y)1z] e in which eT(x; y) and hT(x; y) are vectors in the xy plane. Substituting equation 2.12 into equation 2.3 we can write βeT − jrT ez = −!µ1z × hT βh − jr h = !"1 × e T T z z T (2.13) rT · (1z × eT) = j!µhz rT · (1z × hT) = −j!"ez @ @ in which rT = 1x @x + 1y @y and β is still unknown. This set of equations forms an eigenvalue equation, for which there is only a solution for certain values of β. The corresponding field profiles feT; ez1z; hT; hz1zg are called eigenmodes or optical modes of the waveguide. It is easy to see that if −jβz jβz feT; ez1z; hT; hz1zg e is a solution of equation 2.13 then feT; −ez1z; −hT; hz1zg e is also a solution. If β is assumed to be purely real (the mode is propagating without loss) and the waveguide is lossless (" is real), then eT and hT are purely real and ez and hz are purely imaginary. One can show that all waveguide modes are orthogonal or 1 ZZ e × h 0 · 1 dS = Cδ 0 (2.14) 2 i i z ii The fields ei and hi are defined as e = e + e 1 i T;i z;i z (2.15) hi = hT;i + hz;i1z 2–3 Figure 2.2: Schematics of an (optical) circuit and correspond with an eigenvalue βi. In the case of a lossless waveguide equation 2.14 can be rewritten, due to the above mentioned properties of the fields, as ZZ 1 ∗ e × h 0 · 1 dS =Cδ 0 (2.16) 2 i i z ii 2.2 Matrix description of wave propagation in linear systems 2.2.1 Introduction In this section we will gain insight into the global transmission properties of electromagnetic cir- cuits. The results are applicable to microwave circuits as well as to optical circuits. We will look at the circuit as a black box, which exchanges energy with the outside through several physical out- lets that can be optical waveguides or mere free space electromagnetic beams as shown in figure 2.2. In figure 2.2 ai and bi represent the complex amplitude of the ingoing and outgoing normal- ized electromagnetic mode (carrying unit power) using the phasor notation. This means that the transversal electric and magnetic field at position z=0 (with position z=0 chosen to be at the intersection of the outlet and the surface S of the black box) of port i can be written as j!t ET;i(x; y; t) = Re (ai + bi)eT;i(x; y)e j!t (2.17) HT;i(x; y; t) = Re (ai − bi)hT;i(x; y)e 2–4 with eT;i(x; y) and hT;i(x; y) being the electric and magnetic transversal field profile of the normal- ized electromagnetic mode respectively. The ports are considered to be lossless waveguides and the electrical field is assumed to be zero everywhere on S outside the ports. Every outlet has its own propagation axis and it is essential to assume that electromagnetic modes are well confined around these axes, so that effective outlets can be defined outside which the fields are negligible. In figure 2.2, the outlet ports are assumed to carry only one mode. This is not a restriction as dif- ferent outlet ports may physically coincide to describe multi mode ports (due to the orthogonality of the waveguide modes). In the following matrix formalisms no attention is paid to the internal details of the circuit. The only requirements are that the system has to be passive and linear. A circuit is passive if there are no outgoing waves generated, when there are no incident waves. 2.2.2 Scattering matrix description As the ports are considered to be lossless waveguides, the orthonormality relation for the fields can be written as ZZ 1 ∗ e × h 0 · 1 dS =δ 0 (2.18) 2 i i z ii integrated over the surface S and the unit vector 1z directed inwards. The power absorbed in the circuit is obtained by adding the net powers entering the various ports X 2 2 y y P = (jaij − jbij ) = A A − B B (2.19) with A being the column vector of incident field amplitudes and B the column vector with the amplitudes of the outgoing fields. Here y represents the Hermitian conjugate of the vector A, being the complex conjugate of the transpose. The output wave vector B is completely determined by A otherwise there could be two output vectors B and B’ for the same input vector A and the difference B - B’ would correspond to a zero input field. This is forbidden by the assumption that the circuit is passive. Since we assumed the N-port circuit to be linear, the relation between A and B can be described by an NxN scattering matrix S: B = SA (2.20) T T with A = (a1; a2; :::; aN ) and B = (b1; b2; :::; bN ) . If all the terms of A are zero except ai = 1, the output waves are given by the i-th column of S. The diagonal term Sii is the reflection coefficient of mode i, the non-diagonal terms Ski are the transmission coefficients from the mode i towards the mode k. 2–5 Figure 2.3: Moving the port planes General properties of S matrices • Moving the port planes The ports are changed when the surface S enclosing the circuit is changed. Let lµ be the displacement of port µ counted positively if the port is moved towards the inside. The new circuit is then described by the scattering matrix S’ related to the original scattering matrix S by S0 = XSX (2.21) where X is the diagonal matrix formed with the elements exp(jkµlµ) with kµ the propagation constant of mode µ.