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Appendix 4 From slab waveguide to optical fibre 1. By exploiting a pair of reflector to form a slab waveguide, we manage to confine the wave in one direction (x) by enforcing a resonance along that direction. As a result, the wave becomes a standing wave (in a sinusoidal form) inside the core region of the slab waveguide along x, and evanescent wave (in an exponentially decaying form) inside the top and bottom claddings. 2. The standing wave in a sinusoidal form comes from the summation of contrapropagating plane (traveling) waves, since e jkx x e jkx x cos(k x x),sin(k x x) . 3. The evanescent wave in an exponentially decaying form comes from the fact that the propagation constant of the plan (traveling) wave turns from real to imaginary, hence e jkx x e j ( j|kx |) x e|kx | x . 4. The wave propagation along z is set free. Hence we still have the free-space solution for the wave in z: e j ( kz z t ) e j ( z t ) (plane traveling wave). 5. The field must take a homogenous form as E ( x, z, t ) ( x)e j ( z t ) , with ( x) given in a form of either standing wave or evanescent wave, depending on the region where the field stays, as described in above item 2 and 3, simply because the aforementioned resonance must happen everywhere along z, otherwise the wave will be leaking at the specific location where the wave is not resonating. 6. Therefore, the field of the guided-wave by a general waveguide can always be factorized as: E ( x, y, z, t ) ( x, y ) E0 ( z, t )e j ( z t ) , with ( x, y ) representing the resonant field in the cross-sectional plane (y is restored to cover the general 3D case), e j ( z t ) the traveling wave factor along z (the direction the wave is sent), E0 a slowvarying function of z and time (t) as the wave will eventually be modulated by the base-band signal [ E0 ( z, t ) must be a slow-varying function in comparing with e j ( z t ) , namely | 2 E0 (t ) / z 2 | / | E0 (t ) || 2e j ( z t ) / z 2 | / | e j ( z t ) | 2 , | 2 E0 (t ) / t 2 | / | E0 (t ) || 2e j ( z t ) / t 2 | / | e j ( z t ) | 2 as otherwise E0 ( z, t )e j ( z t ) cannot be put into a form of f ( z t ) and the wave is no longer a traveling wave along z.] 7. The unknown field to be solved in the wave equation can then be substituted by the above factorized form to obtain: 1 2 E E E 0 E E 0 E 2 2 E 0 z E ( z , t ) 3 E0 ( z , t )e j ( z t )T2 2 j e j ( z t ) 0 ( 2 2 ) E0 ( z , t )e j ( z t ) 0 z , E ( z , t ) E0 ( z , t )[T2 ( 2 02 ) ] E0 ( z, t )( 02 2 ) 2 j 0 0 z E ( z , t ) 4 E0 ( z , t )( 02 2 ) 2 j 0 0 z 2 2 1 2 2 2 2 T where at above step 1, only a single polarization direction is put under consideration without losing generality by knowing the fact that the orthogonal polarization components are decoupled in the wave equation and the wave equation itself is linear (so that it is sufficient to consider one polarization component at a time and make a summation if multiple polarization components exist); at step 2, the 3D Laplacian operator ( 2 ) is decomposed into a 2D (in the cross-section x, y plane) Laplacian operator ( T2 ) plus a longitudinal (in z) one ( 2 / z 2 ); at step 3, the 2nd order derivative of the field slow-varying amplitude ( E0 ) is neglected due to the fact shown in above item 6; and finally at step 4, as a resonant solution in the cross-section, ( x, y ) must satisfy (the eigen-value equation) T2 ( 2 02 ) 0 , as otherwise, in the special case of E0 as a constant (i.e., without modulation) and 0 , the above equation cannot hold. 8. Hence we have the eigen-value equation: T2 ( x, y ) ( 2 02 ) ( x, y ) 0 , and the propagation equation: E0 ( z, t ) 2 02 E0 ( z, t ) 0 z 2 j held for the resonant field factor ( ) and the wave amplitude ( E0 ). By solving these two equations subject to the given waveguide structure (with a specific set of boundary conditions extracted from the waveguide for ) and the given initial condition E0 (0, t ) at the source (i.e., the base-band modulation signal at the transmitter), we will be able to obtain the general field E at any place and at any time. (We certainly will be able to find the field at the receiver end.) 9. For a slab waveguide, the eigen-value equation in above item 8 will be reduced to: 2 2 ntop cladding 2 2 2 2 2 2 d ( x) / dx ( 0 ) ( x) 0 , with 0 and 0 n ( x) 0 ncore , 2 nbottom cladding which can then be readily solved analytically as shown in the slides – the result is given as the sinusoidal (co-sine or sine) function in the core region, and as exponentially decaying functions in the top and bottom cladding. (We didn’t consider the modulation case when we firstly introduced the waveguide concept so that the field amplitude was assumed as a constant then.) 10. Unfortunately, for a general 2D dielectric waveguide, there is no closed-form analytical solution for the eigen-value problem. We have to rely on the numerical approach to find the solution. An optical fiber can be viewed as the result of winding of a slab waveguide along its centre symmetric line in its left uniform direction (y) to form a cylinder. By doing so, the eigen-value equation in above item 8 will take the form of: T2 ( x, y ) ( 2 02 ) ( x, y ) 0 , 2 (r , ) 1 (r , ) 1 2 (r , ) 2 [( ) 2 n 2 (r ) 02 ] (r , ) 0 2 2 r r r r c where we have transformed the equation from the Cartesian system to the polar system in order to accommodate the boundary condition in cylindrical symmetry. With the rotational symmetry, the solution stands as a combined azimuthal resonance given in the form of a standing wave e jm [i.e., cos(m ) and sin( m ) ] and a radial resonance given by a family of the Bessel functions. The latter is obviously not in a closed-form. 11. The wave propagation equation as given in above item 8 takes the same form as the one shown in Lecture Note #3, although they are derived by different approaches. For further discussions of optical fibers, please refer to Lecture Note #3. 12. A summary of the knowledge we have learnt so far: Coulomb’s law that describes the static electronic interaction → implicit expression of the static electric field by specifying its divergence (given by the source charge distribution) and curl (free) → Biot-Savert’s law that describes the static magnetic interaction in a symmetric fashion to the Coulomb’s law → implicit expression of the static magnetic field by specifying its divergence (free) and curl (given by the current density, also known as the Ampere’s law) → Faraday’s law showing the fact that the total electric and magnetic field is conserved in a combined space and time domain, hence the time-varying magnetic field supports the curl of the electric field (be careful with the negative sign!) → modified Ampere’s law to remove the inconsistency lies in between the original Ampere’s law and the charge conservation law, by adding on a displacement 3 current term, hence the time-varying electric field supports the curl of the magnetic field → Maxwell’s equations are obtained as a general and full description of the electromagnetic field → Faraday’s law + modified Ampere’s law leads to a sustainable oscillation between the electric and magnetic fields in both space and time domain, the fact that the oscillation walks away from the source as time goes (a nature related to the curl operation) implies the existence of the traveling wave (E and M field oscillation) that can propagate away from its excitation source → to quantify this effect, we extract the electric and magnetic wave equations from Maxwell’s equations by eliminating the magnetic field and electric field in Faraday’s and modified Ampere’s equations, respectively → the solution to the wave equation in free space (sourceless and homogenous) is a plane traveling wave having its wave vector with the value of ( / c)n (2 / )n and with its direction pointing right at the direction of wave propagation, having the electric and magnetic field both polarized in a plane perpendicular to the wave propagation direction, having the electric and magnetic field polarized orthogonally [i.e., the wave vector or the wave propagation direction, the electric field, and the magnetic field must stay all orthogonally, from which we immediately come to the conclusion that the traveling electromagnetic wave can only exist in a 3D or higher-dimensioned space], and having the ratio between the electric and magnetic field amplitude given as the free-space impedance characterized by / → reflection and refraction (transmission) of the plane wave at the dielectric interface (boundary between two different dielectric media) [please remember the Snell’s law and the formula for reflectivity under normal incidence!] → the idea to build a slab waveguide → extraction of the general waveguide concept, i.e., the guided wave must take a factorized form with one factor given as a stationary distribution (resonant/standing wave inside the core region and evanescent/decaying wave in claddings) in the cross-sectional (x, y) plane (conventionally termed as “mode”) and the other as a traveling wave term f ( z t ) ~ A( z, t )e j ( z t ) ; its amplitude (A) can either be a constant (for DC wave propagation in applications such as power transmission) or a slow-varying function in terms of z and t compared to the traveling wave factor e j ( z t ) (when the wave is modulated by base-band signals) → the field in the wave equation for a waveguide problem can then be substituted by its factorized form, which lead to an eigen-value equation and an wave propagation equation; the former governs the crosssectional stationary field (mode) distribution, whereas the latter governs the wave amplitude → solve the eigen-value problem subject to the boundary condition defined by the waveguide cross-sectional structure to obtain the mode; solve the wave propagation equation subject to the initial condition to obtain the wave amplitude; their product then gives the total field and the entire waveguide problem is solved Please prepare your cheat sheet for the midterm by following this summary and record equations/expressions/formulas along with the path shown above once you feel it is necessary. 4