Chap. Introduction to optical waveguide modes PHILIPPE LALANNE (IOGS nd année) Chapter Introduction to optical waveguide modes The optical waveguide is the fundamental element that interconnects the various devices of an optical integrated circuit, just as a metallic strip does in an electrical integrated circuit. However, unlike electrical current that flows through a metal strip according to Ohm's law, optical waves travel in the waveguide in distinct optical modes. A mode, in this sense, is a spatial distribution of optical energy in one or more dimensions that remains constant in time. In this chapter, the concept of optical modes in a waveguide structure is discussed qualitatively, and key results of waveguide theory are presented with minimal proof to give the reader a general understanding of the nature of light propagation in an optical waveguide. Then, in the following Chapter, a mathematically sound development of waveguide theory is given. Excellent books exist on the topic on optical waveguides including [Yeh88], and [Vas91,Mar91], which are more advanced..1 Modes in a planar waveguide structure As shown in Fig..1, a planar waveguide is characterized by parallel planar boundaries with respect to one x direction, but is infinite in extent in the lateral directions (z and y). Of course, because it is infinite in two dimensions, it cannot be a practical waveguide for optical integrated circuits, but it forms the basis for the analysis of practical waveguides of rectangular cross section. It has therefore been treated by a number of authors [Yeh88,Vas91,Mar91]. We follow the approach of [Yeh88] to examine the possible modes in a planar waveguide, without fully solving the wave equation. Details will be given in the next chapter. 13
Chap. Introduction to optical waveguide modes PHILIPPE LALANNE (IOGS nd année) Figure -1. Diagram of the basic three-layer planar waveguide structure. Three modes are shown, representing distributions of electric field in the x direction. Theoretical description. To begin the discussion of optical modes, consider the simple three-layer planar waveguide structure of Fig..1. The layers are all assumed to be infinite in extent in the y and z directions, and layers 1 and 3 are also assumed to be semi-infinite in the x direction. Light waves are assumed to be propagating in the z direction. It has been stated previously that a mode is a spatial distribution of optical energy in one or more dimensions. An equivalent mathematical definition of a mode is that it is an electromagnetic field which is a solution of Maxwell's equations in the absence of any source. In every layer, the dielectric permittivity is uniform and it is assumed to be real (no Ohmic losses). Therefore, solving Maxwell's equations amounts to solving the wave equation in three uniform media and then to matching the continuities of the tangential electromagnetic fields at the two interfaces. For monochromatic waves with a radian frequency ω, the wave equation in a uniform medium is ( ω c) E( r) = 0 E ( r) + n ( r), (.1) where E is the electric field vector, r is the radius vector, n(r) is the index of refraction of the layer under consideration and c is the speed of light in a vacuum. The solution of Eq. (.1) is well known and takes the general form ( ik x + ik y ik z) E ( r ) = exp x y + z, (.) with E(r) being one of the Cartesian components of E(r) and k x, k y and k z being arbitrary 14
complex constants satisfying k x y z i ( ) Chap. Introduction to optical waveguide modes PHILIPPE LALANNE (IOGS nd année) + k + k = n ω c, (.3) For any i=1,, 3. The propagation constants are determined by matching the continuity of the tangential field components at the two interfaces. Since the continuity conditions are satisfied for any y and z, k y and k z are the same in every layer: they are invariants. As such, we often say that the parallel wavevector components, k y and k z in the present case, are unchanged from one medium to another. This is nothing else than Snell s law. Without loss of generality, we will take k y = 0, and we will denote by β (β = k z ) the propagation constant. Indeed what is changing as one crosses the interfaces is the perpendicular wavevector component, k x. The latter is either imaginary or real, depending on whether ( ) k = n ω c β, i = 1, or 3, is greater than or less than zero, implying that the mode solution is either sinusoidal or exponential functions of x in each of the layers. Hence the possible modes are limited to those shown in Fig... Consider how the mode shape changes as a function of β, for the case of constant frequency ω and n > n3 > n1. This relative ordering of the indices is quite a common case, corresponding, for example, to a waveguiding layer of index n formed on a substrate with smaller index n3, surrounded by air of index n1. As we will see in the next Chapter, it is a necessary condition for waveguiding in Layer that n is greater than both n1 and n3. When > n ( ω c) β, the function E(x) must be exponential in all three regions and only the mode shape shown as (a) in Fig.. could satisfy the boundary conditions at the interfaces. This mode is not physically realizable because the field increases unboundedly in Layers 1 and 3, implying infinite energy. Modes (b) and (c) are well confined guided modes, generally referred to as the zeroth order and first order transverse electric modes, TE 0 and TE 1 [Yeh88]. For values of β between n ( ω c) and n3 ( ω c) such modes can be supported. If β is greater than n ( ω c) x i 1 but less than n3 ( ω c), a mode like that in (d) will result. This type of mode, which is confined at the air interface but sinusoidally varying at the substrate, is often called a substrate radiation mode. It can be supported by the waveguide structure, but because it is continually losing energy from the waveguiding region to the substrate Region 3 as it propagates, it tends to be damped out over at short distance. Hence it is not very useful in signal transmission, but, in fact, it may be very useful in coupler applications such as the tapered coupler. This type of coupler will be discussed in Chapter 6. If β is less than ( ) n ω c 1 the solution for E(x) is oscillatory in all three 15
Chap. Introduction to optical waveguide modes PHILIPPE LALANNE (IOGS nd année) regions of the waveguide structure. These modes are not guided modes because the energy is free to spread out of the waveguiding region. They are generally referred to as the air radiation modes of the waveguide structure. Of course, radiation is also occurring at the substrate interface. Figure -. Diagram of the possible modes in a planar waveguide [Yeh88]. Dispersion diagram. A powerful representation of the mode diagram is the ω(β) dispersion diagram where we plot the energy (proportional to ω) as a function of the momentum β. This representation that is directly inspired from the band diagram of electrons in solid state is shown in Fig..3 for a symmetric slab waveguide (n 1 = n 3 ) or for a fiber with a core refractive index n surrounded by a clad material with a refractive index n 1. We start by plotting the material light curves of the materials ω/c=β/n i, i = 1,, which are straight lines if one assumes that the materials are non-dispersive. They are shown with the dashed lines. The red curve represents the fundamental guided mode that has no cut-off for symmetric planar waveguides or fibers. Even if material dispersion is neglected (n 1 and n are independent of ω), n eff varies from n 1 at small ω s to n for large ω s. It is important to realize that the dispersion dn eff /dω is in general much larger than any material dispersion, especially close to cut-off. What is explaining the large dispersion is the mode profile variation with the frequency. For small ω s, the wavelength is much larger than the core size and the mode essentially 16
Chap. Introduction to optical waveguide modes PHILIPPE LALANNE (IOGS nd année) spreads into the cladding material so that the electromagnetic field essentially feels the cladding. As w increases, the waveguide becomes multimode (only the fundamental mode without cutoff is shown) and the fundamental mode becomes more and more confined into the core. One should remember that the transverse size required to confine a wave into a material of refractive index n is typically λ/n, so that for large ω s, the core size is much larger than λ/n and the electromagnetic field is almost entirely confined into the core (the electromagnetic energy contained in the evanescent tail in the clad becomes negligible in comparison with that contained in the large core. Figure -3. Dispersion diagram of a photonic fiber. The dashed lines are the dispersion curves of plane waves propagating in bulk (uniform) materials with refractive indices n 1 and n. The red curve is the fundamental guided mode which is bounded between the two light lines, because n 1 < n eff < n.. Cutoff conditions We shall see in Chapter 3, when Maxwell s equations is formally solved, subject to appropriate boundary conditions at the interface, that β can have any value when it is less than kn3, but only discrete values of β are allowed in the range between n ( ω c) and n ( ω c) 17 3. These discrete values of β correspond to the various modes TEj, j = 0, 1,,... (or TMk, k = 0, 1,,...). The number of modes that can be supported depends on the thickness t of the waveguiding layer and on ω, n 1, n and n 3. For given t, n 1, n and n 3, there is a cutoff frequency ω c below which
Chap. Introduction to optical waveguide modes PHILIPPE LALANNE (IOGS nd année) waveguiding cannot occur. This ω c corresponds to a long wavelength cutoff λc. Since wavelength is often a fixed parameter in a given application, the cutoff problem is frequently stated by asking the question, for a given wavelength, what indices of refraction must be chosen in the three layers to permit waveguiding of a given mode? For the special case of the so-called asymmetric waveguide, in which n1 is very much less than n3, it can be shown (Chapter 3) that the required indices of refraction are related by ( m+ 1) ( λ ) ( 3 n ) n = n, (.4) n3 = 0 t where the mode number m = 0, 1,,, and λ 0 = πc/ ω is the vacuum wavelength. The change in index of refraction required for waveguiding of the lower-order modes is surprisingly small. For example, in a gallium arsenide (GaAs) waveguide with n equal to 3.5 and with t on the order of λ0, Eq. (.4) predicts that a Δn on the order of only 0.01 is sufficient to support waveguiding of the TE0 mode. Because only a small change in index is needed, a great many different methods of waveguide fabrication, including doping of the core medium, have proven effective in a variety of materials, such as glass, solgel, semiconductors, and dielectrics, the specific case of metals deserving specific attention..3 Experimental observation of waveguide modes Since the waveguides in optical integrated circuits are typically only a few micrometers thick, observation of the optical mode profile across a given dimension cannot be accomplished without a relatively elaborate experimental set-up, featuring at least 1000 magnification. One such system, which works particularly well for semiconductor waveguides operating at telecom wavelengths, is depicted in Fig..4. The sample, with its waveguide at the top surface, is fixed atop an x-y-z micropositioner. TE-polarized light from a high-resolution tuneable external laser source (150-160 nm) is launched into the sample waveguides using a polarization-maintaining fibber connected to a microlensed fibber. The coupling is critical in the x-direction. For semiconductor waveguides, it is in general weakly efficient because of the large transverse-mode profile mismatch between the waveguide mode and the focused beam from the fibber lenslet (beam waist of µm). Microscope objective lenses with a high numerical aperture, used for output image magnification, are also mounted on a translational stage to facilitate the critical alignment that is required. For visual observation of the waveguide mode, 18
Chap. Introduction to optical waveguide modes PHILIPPE LALANNE (IOGS nd année) the output facet of the waveguide can be imaged onto an InGaAs camera, or onto an InGaAs photodiode for quantitative transmission measurements. The lowest order mode (m = 0) appears as a single band of light, while higher order modes have a correspondingly increased number of bands, as shown in Fig..5. It is also possible to visualize the coupling between the fiber lenslet and the waveguide by adding a microscope objective above the sample. The light that is imaged onto the camera can also be used to see the impact of the inevitable waveguide loss due to surface roughness. Figure -4. Diagram of an experimental setup than can be used to measure optical mode shapes and out-of-plane leakage.. Figure -5. Optical mode patterns in a planar waveguide, a TE 0, b TE 1, c TE. In the planar guide, light is unconfined in the y direction, and is limited, as shown in the photos, only by the extent of spreading of the input laser beam. For the corresponding TE nm patterns of a rectangular waveguide, see [Mar91]. 19
Chap. Introduction to optical waveguide modes PHILIPPE LALANNE (IOGS nd année) References [Mar91] D. Marcuse, Theory of dielectric optical waveguides, nd Ed. (Academic, 1991). [Vas91] C. Vassallo, Optical waveguide concepts (Elsevier, Amsterdam, 1991). [Yeh88] P. Yeh, Optical waves in layered media, J. Wiley and Sons eds., New York, 1988. 0