Chapter 1
Wave Theory of Optical Waveguides
The basic concepts and equations of electromagnetic wave theory required for the comprehension of lightwave propagation in optical waveguides are presented. The light confinement and formation of modes in the waveguide are qualitatively explained, taking the case of a slab waveguide. Maxwell’s equations, boundary conditions, and the complex Poynting vector are described as they form the basis for the following chapters.
1.1. WAVEGUIDE STRUCTURE
Optical fibers and optical waveguides consist of a core, in which light is confined, and a cladding, or substrate surrounding the core, as shown in Fig. 1.1.
The refractive index of the core n1 is higher than that of the cladding n0. Therefore the light beam that is coupled to the end face of the waveguide is confined in the core by total internal reflection. The condition for total internal − reflection at the core–cladding interface is given by n1 sin /2 n0. Since = 2 − 2 the angle is related with the incident angle by sin n1 sin n1 n0, we obtain the critical condition for the total internal reflection as −1 2 − 2 ≡ sin n1 n0 max (1.1) The refractive-index difference between core and cladding is of the order of − = n1 n0 0 01. Then max in Eq. (1.1) can be approximated by 2 − 2 max n1 n0 (1.2)
1 2 Wave Theory of Optical Waveguides
Figure 1.1 Basic structure and refractive-index profile of the optical waveguide.
max denotes the maximum light acceptance angle of the waveguide and is known as the numerical aperture (NA).
The relative refractive-index difference between n1 and n0 is defined as n2 − n2 n − n = 1 0 1 0 2 (1.3) 2n1 n1 is commonly expressed as a percentage. The numerical aperture NA is related to the relative refractive-index difference by √ = NA max n1 2 (1.4)
The maximum angle√ for the propagating light within the core is given by = = max max/n1 2 . For typical optical waveguides, NA 0 21 and max = = = = 12 max 8 1 when n1 1 47 1% for n0 1 455 .
1.2. FORMATION OF GUIDED MODES
We have accounted for the mechanism of mode confinement and have indi- cated that the angle must not exceed the critical angle. Even though the angle is smaller than the critical angle, light rays with arbitrary angles are not able to propagate in the waveguide. Each mode is associated with light rays at a discrete angle of propagation, as given by electromagnetic wave analysis. Here we describe the formation of modes with the ray picture in the slab wave- guide [1], as shown in Fig. 1.2. Let us consider a plane wave propagating along the z-direction with inclination angle . The phase fronts of the plane waves are perpendicular to the light rays. The wavelength and the wavenumber of light in = the core are /n1 and kn1 k 2 / , respectively, where is the wavelength of light in vacuum. The propagation constants along z and x (lateral direction) are expressed by = kn1 cos (1.5) = kn1 sin (1.6) Formation of Guided Modes 3
Figure 1.2 Light rays and their phase fronts in the waveguide.
Before describing the formation of modes in detail, we must explain the phase shift of a light ray that suffers total reflection. The reflection coefficient of the totally reflected light, which is polarized perpendicular to the incident plane (plane formed by the incident and reflected rays), as shown in Fig. 1.3, is given by [2] A n sin + j n2 cos2 − n2 r = r = 1 1 0 (1.7) A − 2 2 − 2 i n1 sin j n1 cos n0
When we express the complex reflection coefficient r as r = exp −j , the amount of phase shift is obtained as n2 cos2 − n2 2 =− −1 1 0 =− −1 − 2 tan 2 tan 2 1 (1.8) n1 sin sin where Eq. (1.3) has been used. The foregoing phase shift for the totally reflected light is called the Goos–Hänchen shift [1, 3]. Let us consider the phase difference between the two light rays belonging to the same plane wave in Fig. 1.2. Light ray PQ, which propagates from point P to Q, does not suffer the influence of reflection. On the other hand, light ray RS,
Figure 1.3 Total reflection of a plane wave at a dielectric interface. 4 Wave Theory of Optical Waveguides propagating from point R to S, is reflected two times (at the upper and lower core–cladding interfaces). Since points P and R or points Q and S are on the same phase front, optical paths PQ and RS (including the Goos–Hänchen shifts caused by the two total reflections) should be equal, or their difference should be an integral multiple of 2 . Since the distance between points Q and R is 2a/tan − 2a tan , the distance between points P and Q is expressed by 2a 1 = − 2a tan cos = 2a − 2 sin (1.9) 1 tan sin
Also, the distance between points R and S is given by
2a = (1.10) 2 sin
The phase-matching condition for the optical paths PQ and RS then becomes
+ − = kn1 2 2 kn1 1 2m (1.11) where m is an integer. Substituting Eqs. (1.8)–(1.10) into Eq. (1.11) we obtain the condition for the propagation angle as m 2 tan kn a sin − = − 1 (1.12) 1 2 sin2
Equation (1.12) shows that the propagation angle of a light ray is discrete and is determined by the waveguide structure (core radius a, refractive index n1, refractive-index difference ) and the wavelength of the light source (wavenumber is k = 2 / [4]. The optical field distribution that satisfies the phase-matching condition of Eq. (1.12) is called the mode. The allowed value of propagation constant [Eq. (1.5)] is also discrete and is denoted as an eigenvalue. The mode that has the minimum angle in Eq. (1.12) m=0 is the fundamental mode; the other modes, having larger angles, are higher-order modes m 1 . Figure 1.4 schematically shows the formation of modes (standing waves) for (a) the fundamental mode and (b) a higher-order mode, respectively, through the interference of light waves. In the figure the solid line represents a positive phase front and a dotted line represents a negative phase front, respectively. The electric field amplitude becomes the maximum (minimum) at the point where two positive (negative) phase fronts interfere. In contrast, the electric field amplitude becomes almost zero near the core–cladding interface, since positive and negative phase fronts cancel out each other. Therefore the field distribution along the x-(transverse) direction becomes a standing wave and varies periodically along = = the z direction with the period p /n1 /cos 2 / . Formation of Guided Modes 5
Figure 1.4 Formation of modes: (a) Fundamental mode, (b) higher-order mode.
= 2 − 2 Since n1 sin sin n√1 n0 from Fig. 1.1, Eqs. (1.1) and (1.3) give the propagation angle as sin 2 . When we introduce the parameter sin = √ (1.13) 2 which is normalized to 1, the phase-matching Eq. (1.12) can be rewritten as
√ cos−1 + m /2 kn a 2 = (1.14) 1 The term on the left-hand side of Eq. (1.14) is known as the normalized frequency, and it is expressed by √ = v kn1a 2 (1.15)
When we use the normalized frequency v, the propagation characteristics of the waveguides can be treated generally (independent of each waveguide structure). 6 Wave Theory of Optical Waveguides
The relationship between normalized frequency v and (propagation constant ), Eq. (1.14), is called the dispersion equation. Figure 1.5 shows the dispersion curves of a slab waveguide. The crossing point between = cos−1 + m /2 / = and v gives m for each mode number m, and the propagation constant m is obtained from Eqs. (1.5) and (1.13). It is known from Fig. 1.5 that only the fundamental mode with m = 0 can = exist when v Figure 1.5 Dispersion curves of a slab waveguide. Maxwell’s Equations 7 c is called the cutoff (free-space) wavelength. The waveguide operates in a single = mode for wavelengths longer than c. For example, c 0 8 m when the core = = = = width 2a 3 54 m for the slab waveguide of n1 1 46 0 3% n0 1 455 . 1.3. MAXWELL’S EQUATIONS Maxwell’s equations in a homogeneous and lossless dielectric medium are written in terms of the electric field e and magnetic field h as [5] h × e =− (1.17) t e × h = (1.18) t where and denote the permittivity and permeability of the medium, respec- = tively. and are related to their respective values in a vacuum of 0 × −12 = × −7 8 854 10 F/m and 0 4 10 H/m by = 2 0n (1.19a) = 0 (1.19b) where n is the refractive index. The wavenumber of light in the medium is then expressed as [5] √ √ = = = n 0 0 kn (1.20) In Eq. (1.20), is an angular frequency of the sinusoidally varying electromag- netic fields with respect to time; k is the wavenumber in a vacuum, which is related to the angular frequency by √ k = = (1.21) 0 0 c In Eq. (1.21), c is the light velocity in a vacuum, given by 1 c = √ = 2 998 × 108 m/s (1.22) 0 0 The fact that the units for light velocity c are m/s is confirmed from the units of the permittivity 0 F/m and permeability 0 [H/m] as 1 m m m = √ = = F/m H/m F · H A · s/V V · s/A s 8 Wave Theory of Optical Waveguides When the frequency of the electromagnetic wave is f Hz , it propagates c/f m in one period of sinusoidal variation. Then the wavelength of electro- magnetic wave is obtained by c /k 2 = = = (1.23) f f k where = 2 f. When the electromagnetic fields e and h are sinusoidal functions of time, they are usually represented by complex amplitudes, i.e., the so-called phasors. As an example consider the electric field vector e t =E cos t + (1.24) where E is the amplitude and is the phase. Defining the complex amplitude of e(t)by E =Eej (1.25) Eq. (1.24) can be written as e t = Re Eej t (1.26) We will often represent e(t)by e t = Eej t (1.27) instead of by Eq. (1.24) or (1.26). This expression is not strictly correct, so when we use this phasor expression we should keep in mind that what is meant by Eq. (1.27) is the real part of Eej t. In most mathematical manipulations, such as addition, subtraction, differentiation and integration, the replacement of Eq. (1.26) by the complex form (1.27) poses no problems. However, we should be careful in the manipulations that involve the product of sinusoidal functions. In these cases we must use the real form of the function (1.24) or complex conjugates [see Eqs. (1.42)]. When we consider an electromagnetic wave having angular frequency and propagating in the z direction with propagation constant , the electric and magnetic fields can be expressed as e = E r ej t− z (1.28) h = H r ej t− z (1.29) Maxwell’s Equations 9 where r denotes the position in the plane transverse to the z-axis. Substituting Eqs. (1.28) and (1.29) into Eqs. (1.17) and (1.18), the following set of equations are obtained in Cartesian coordinates: ⎧ E ⎪ z + =− ⎪ j Ey j 0Hx ⎪ y ⎪ ⎪ E ⎪ − − z =− ⎪ j Ex j 0Hy ⎪ x ⎪ ⎪ Ey Ex ⎨⎪ − =−j H x y 0 z (1.30) ⎪ H ⎪ z + = 2 ⎪ j Hy j 0n Ex ⎪ y ⎪ ⎪ H ⎪ − − z = 2 ⎪ j Hx j 0n Ey ⎪ x ⎪ ⎪ H H ⎩ y − x = j n2E x y 0 z The foregoing equations are the bases for the analysis of slab and rectangular waveguides. For the analysis of wave propagation in optical fibers, which are axially symmetric, Maxwell’s equations are written in terms of cylindrical coordinates: ⎧ ⎪ E ⎪ 1 z + =− ⎪ j E j 0Hr ⎪ r ⎪ ⎪ E ⎪ − − z =− ⎪ j Er j 0H ⎪ r ⎪ ⎪ 1 1 E ⎨⎪ rE − r =−j H r r r 0 z (1.31) ⎪ 1 H ⎪ z + = 2 ⎪ j H j 0n Er ⎪ r ⎪ ⎪ H ⎪ − − z = 2 ⎪ j Hr j 0n E ⎪ r ⎪ ⎪ 1 1 H ⎩ rH − r = j n2E r r r 0 z Maxwell’s Eqs. (1.30) or (1.31) do not determine the electromagnetic field completely. Out of the infinite possibilities of solutions of Maxwell’s equations, we must select those that also satisfy the boundary conditions of the respective problem. The most common type of boundary condition occurs when there are discontinuities in the dielectric constant (refractive index), as shown in Fig. 1.1. 10 Wave Theory of Optical Waveguides At the boundary the tangential components of the electric field and magnetic field should satisfy the conditions 1 2 Et = Et (1.32) 1 2 Ht = Ht (1.33) where the subscript t denotes the tangential components to the boundary and the superscripts (1) and (2) indicate the medium, respectively. Equations (1.32) and (1.33) mean that the tangential components of the electromagnetic fields must be continuous at the boundary. There are also natural boundary conditions that require the electromagnetic fields to be zero at infinity. 1.4. PROPAGATING POWER Consider Gauss’s theorem (see Section 10.1) for vector A in an arbitrary volume V · A dv = A · n ds (1.34) V S where n is the outward-directed unit vector normal to the surface S enclosing V and dv and ds are the differential volume and surface elements, respectively. When we set A = e × h in Eq. (1.34) and use the vector identity · e × h = h · × e − e · × h (1.35) we obtain the following equation for electromagnetic fields: h · × e − e · × h dv = e × h · n ds (1.36) V S Substituting Eqs. (1.17) and (1.18) into Eq. (1.36) results in e h e · + h · dv =− e × h · n ds (1.37) t t V S The first term in Eq. (1.37) e W e · = e · e ≡ e (1.38) t t 2 t Propagating Power 11 represents the rate of increase of the electric stored energy We and the second term h W h · = h · h ≡ h (1.39) t t 2 t represents the rate of increase of the magnetic stored energy Wh, respectively. Therefore, the left-hand side of Eq. (1.37) gives the rate of increase of the electromagnetic stored energy in the whole volume V ; in other words, it repre- sents the total power flow into the volume bounded by S. When we replace the =− outward-directed unit vector n by the inward-directed unit vector uz n , the total power flowing into the volume through surface S is expressed by = − × · = × · P e h n ds e h uz ds (1.40) S S Equation (1.40) means that e × h is the vector representing the power flow, × · and its normal component to the surface e h uz gives the amount of power flowing through unit surface area. Therefore, vector e × h represents the power- flow density, and S = e × h W/m2 (1.41) is called the Poynting vector. In this equation, e and h denote instantaneous fields as functions of time t. Let us obtain the average power-flow density in an alternating field. The complex electric and magnetic fields can be expressed by 1 e t = Re Eej t = Eej t + E∗e−j t (1.42a) 2 1 h t = Re Hej t = Hej t + H∗e−j t (1.42b) 2 where ∗ denotes the complex conjugate. The time average of the normal com- ponent of the Poynting vector is then obtained as · = × · S uz e h uz 1 = Eej t + E∗e−j t × Hej t + H∗e−j t · u 4 z 1 1 = E × H∗ + E∗ × H · u = Re E × H∗ · u (1.43) 4 z 2 z where denotes a time average. Then the time average of the power flow is given by 1 P = Re E × H∗ · u ds (1.44) 2 z S 12 Wave Theory of Optical Waveguides Since E × H∗ often becomes real in the analysis of optical waveguides, the time average propagation power in Eq. (1.44) is expressed by 1 P = E × H∗ · u ds (1.45) 2 z S REFERENCES [1] Marcuse, D. 1974. Theory of Dielectric Optical Waveguides. New York: Academic Press. [2] Born, M. and E. Wolf. 1970. Principles of Optics. Oxford: Pergamon Press. [3] Tamir, T. 1975. Integrated Optics. Berlin: Springer-Verlag. [4] Marcuse, D. 1972. Light Transmission Optics. New York: Van Nostrand Rein-hold. [5] Stratton, J. A. 1941. Electromagnetic Theory. New York: McGraw-Hill.