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Wave Theory of Optical Waveguides 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 001. 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 021 and max = = = = 12 max 81 when n1 147 1% for n0 1455. 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 kn1k 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 + − = kn12 2 kn11 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<vc /2. vc determines the single-mode condition of the slab waveguide—in other words, the condition in which higher-order modes are cut off. Therefore it is called the cutoff v-value. When we rewrite the cutoff condition in terms of the wavelength we obtain √ = 2 c an1 2 (1.16) vc 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 08 m when the core = = = = width 2a 354 m for the slab waveguide of n1 146 03%n0 1455. 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 8854 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 00 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.
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