Guided Waves in Layered Media Guided Waves in
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Guided Waves in Layered Media
Guided Waves in Layered Media Layered media can support confined electromagnetic propagation. These modes of propagation are the so-called guided waves, and the structures that support guided waves are called waveguides.
Symmetric Slab Waveguides • Dielectric slabs are the simplest optical waveguides. • It consists of a thin dielectric layer sandwiched between two semi-infinite bounding media. • The index of refraction of the guiding layer must be greater than those of the surround media.
Symmetric Slab Waveguides • The following equation describes the index profile of a symmetric dielectric waveguide:
Symmetric Slab Waveguides • The propagation of monochromatic radiation along the z axis. Maxwell’s equation can be written in the form
Symmetric Slab Waveguides • For layered dielectric structures that consist of homogeneous and isotropic materials, the wave equation is • The subscript m is the mode number
Symmetric Slab Waveguides • For confined modes, the field amplitude must fall off exponentially outside the waveguide. • Consequently, the quantity (nω/c)^2 -β^2 must be negative for |x| > d/2.
Symmetric Slab Waveguides • For confined modes, the field amplitude must fall off sinusoidally inside the waveguide. • Consequently, the quantity (nω/c)^2 -β^2 must be positive for |x| < d/2.
Guided TE Modes • The electric field amplitude of the guided TE modes can be written in the form
Guided TE Modes • The mode function is taken as
Guided TE Modes • The solutions of TE modes may be divided into two classes: for the first class and for the second class • The solution in the first class have symmetric wavefunctions, whereas those of the second class have antisymmetric wavefunctions.
Guided TE Modes • The propagation constants of the TE modes can be found from a numerical or graphical solution.
Guided TE Modes
Guided TM Modes • The field amplitudes are written
Guided TM Modes • The wavefunction H(x) is
Guided TM Modes • The continuity of Hy and Ez at the two (x=±(1/2)d) interface leads to the eigenvalue equation
Asymmetric Slab Waveguides • The index profile of a asymmetric slab waveguides is as follows • n 2 is greater than n 1 and n 3, assuming n 1<n 3<n 2
Asymmetric Slab Waveguides Typical field distributions corresponding to different values of β
Guided TE Modes • The field component Ey of the TE mode can be written as • The function Em(x) assumes the following forms in each of the three regions
Guided TE Modes • By imposing the continuity requirements , we get • or
Guided TE Modes • The normalization condition is given by • Or equivalently,
Guided TM Modes • The field components are
Guided TM Modes • The wavefunction is
Guided TM Modes
Guided TM Modes • We define the parameter • At long wavelengths, such that No confined mode exists in the waveguide.
Guided TM Modes • As the wavelength decreases such that One solution exists to the mode condition
Guided TM Modes • As the wavelength decreases further such that Two solutions exist to the mode condition
Guided TM Modes • The mth satifies
Surface Plasmons • Confined propagation of electromagnetic radiation can also exist at the interface between two semiinfinite dielectric homogeneous media. • Such electromagnetic surface waves can exist at the interface between two media, provided the dielectric constants of the media are opposite in sign. • Only a single TM mode exist at a given frequency.
Surface Plasmons • A typical example is the interface between air and silver where n 1^2 = 1 and n 3^2 = -16. 40 i 0. 54 at λ = 6328(艾).
Surface Plasmons • For TE modes, by putting t=0 in Eq. (11. 2 -5), we obtain the mode condition for the TE surface waves, p + q =0 Where p and q are the exponential attenuation constants in media 3 and 1. • It can never be satisfied since a confined mode requires p>0 and q>0.
Surface Plasmons • For TM waves, the mode funcion Hy(x) can be written as • The mode condition can be obtained by insisting on the continuity of Ez at the interface x = 0 of from Eq. (11. 2 -11)
Surface Plasmons • The propagation constant β is given by • A confined propagation mode must have a real propagation constant, since <0,
Surface Plasmons • The attenuation constants p and q are given
Surface Plasmons • The electric field components are given
Surface Plasmons • Surface wave propagation at the interface between a metal and a dielectric medium suffers ohmic losses. The propagation therefore attenuates in the z direction. This corresponding to a complex propagation constants β • Where α is the power attenuation coefficient.
Surface Plasmons • In the case of a dielectric-metal interface, is a real positive number and is a complex number (n – iκ)^2, and the propagation constant of the surface wave is given
Surface Plasmons • In terms of the dielectric constants • The propagation and attenuation constants can be written as
Surface Plasmons • The attenuation coefficient can also be obtained from the ohmic loss calculation and can be written as • σ is the conductivity
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